- Email: [email protected]
A 2H NMR spin-lattice relaxation study of -NH3poststaggered+ group dynamics in polycrystalline l -alanine
SOLID STATE Nuclear Magnetic Resonance
ELSEVIER
Solid State Nuclear
Magnetic Resonance 6 (1996) 47-53
A 2H NMR spin-lattice relaxation study of -NH; dynamics in polycrystalline L-alanine
group
M.A.K. Williams, R.D. Keenan, T.K. Halstead * of Chemistry,
Department
The Unil,ersity
of York, Heslington,
York,
YOI 5DD,
UK
Received 17 June 1995; accepted 19 July 1995
Abstract The *H spin-lattice relaxation times T,, and T,o of the -ND; group in polycrystalline L-alanine were measured over the temperature ranges 250-425 K and 300-425 K, respectively. The relaxation data were fitted to analytical expressions for three-fold reorientation, giving a correlation time T, = 1.5 X lo-l4 s exp(40.2 kJ mol-‘/RT), which is in good agreement with the results of a limited lineshape study. T,, values derived from numerical simulations, taking into consideration geometrical distortion of the -ND: group and experimental imperfections, are compared with values given by an analytical expression for three-fold jumps and with the experimental results. The extraction of average site quadrupolar coupling constants is discussed. Keywords:
NMR;
Deuterium;
Spin-lattice
relaxation;
Three-fold
1. Introduction There has been much written on the determination of molecular dynamics from the simulation of one-dimensional 2H spectra [1,2]. Studies of this type involving the reorientation of C-D and N-D bonds are plentiful, both where jump sites are related by molecular symmetry and where they are not [3-51. A complementary NMR approach to the experimental determination of molecular dynamics, often used in cases where the dipolar interaction is dominant, uses the temperature dependence of the spin-lattice relaxation time to map temperature onto correlation
* Corresponding author. 0926.2040/96/$15.00 0 1996 Elsevier SSDI 0926.2040(95)01204-4
Science
B.V.
All rights
reorientation;
L-Alanine
time, and hence to extract dynamic parameters. While deuterium NMR inversion-recovery experiments involving series of partially relaxed spectra are fashionable, and do allow diffusional and discrete dynamic models to be distinguished by their different T,, anisotropy, relatively few studies have adopted the purely time-domain approach and measured (T,,), directly by monitoring the recovery of the quadrupolar echo. This is the simplest experimental approach to relaxationtime measurement, having the advantages of gaining reasonable signal-to-noise ratios with far fewer transients than are needed for the accurate determination of the relaxation times of selected orientations, and dispensing with the additional data processing necessitated by subsequent Fourier transformation. In a polycrystalline powreserved
48
M.A.K.
Williams
et al. /Solid
State Nuclear
der sample, the relaxation time for each molecular group is determined by the particular orientation in the laboratory (B,,) frame of its motionally averaged electric field gradient (EFG) principal axis system (PAS) and, therefore, the measured spin-lattice relaxation time (T,,), is the average of many angular dependent relaxation times. Analytical expressions exist which relate T,, to the correlation time for various jump and diffusive dynamical models, and it should be noted that the derivation of theoretical expressions for powder-average relaxation times can be greatly simplified by exploiting the orthogonality relations of the Wigner rotation matrices [2]. Whereas these expressions are valid only for symmeteric reorientation motions involving sites with the same quadrupolar coupling constant e*qQ/h, and the same asymmetry parameter 7, programs such as MXQET and its descendants [6-91 enable the simulation of spectral lineshapes for jump models involving sites with arbitrary orientations and different e2qQ/h and n values. Later versions also allow the simulation of spinlattice relaxation behaviour for dynamic models involving asymmetric jumps. Furthermore, these simulations consider the effects of finite pulse widths and distortions of the quadrupolar echo. We have sought to test the success of these two approaches in predicting the relaxation behaviour
Table 1 Arrhenius
parameters
for the reorientation
of the -ND:
group
Magnetic
Resonance
6 (1996)
47-53
of the 2H NMR signal from an asymmetric -ND: group, by comparing the results of calculations using the expression [lo]:
x [J&P 4 + 4J2Po,
41
(1)
for the powder-average Zeeman spin-lattice relaxation rate for three-fold reorientation, where J(w, 7,) = r,/(l + w2r:), T, is the correlation time and 0 is the angle between the axis of reorientation and eq, and the results of computer simulations of an inversion-recovery experiment using the program MXETl, with the experimental results of a study on deuterated L-alanine. A comparison of these models is of general interest. The amino acid L-alanine crystallizes in zwitterionic form in the orthorhombic space group P2,2,2, with a = 0.6032 nm, b = 1.2343 nm, c = 0.5784 nm, and 2 = 4 [ll-131. The molecular dynamics of the methyl and -NH: groups have been studied using various techniques, including proton NMR relaxation [14,15], Raman and infra-red spectroscopy [ 16,171, phonon dispersion [18], deuterium NMR lineshape studies [19-211 and molecular dynamics simulations and normal mode calculations [22]. The methyl group dynam-
in L-alanine
Technique
E (kJ mol-‘)
‘H 7-1, ‘H T,p ” Raman spectroscopy ‘H Lineshape b Molecular dynamics
38.6 33.6 37.0 40.5 31.2
1.6 5.0 1.3 -
Ref. 14 15 16 21 22
‘H Lineshape 2H TIZ 2H T,z/TIQ
41.5 * 1.7 40.2 k 0.5 39.8 f 0.8
1.0 f 0.3 1.5 + 0.3 1.6 k 0.3
This work This work This work
a This study was carried out on L-[ lSN]alanine. ’ There is an error in this paper. which relates the infinite temperature exchange rate A, correlation time. In fact, for a three-site jump model A, = l/37,,. The 7O value extracted that obtained from the proton study than stated therein.
to the reciprocal of infinite temperature from this study is considerably closer to
M.A.K. Williams et al. /Solid State Nuclear Magnetic Resonance 6 (1996) 47-53
its are of special interest because of the unusually high activation energy of the reorientation, which is ascribed to steric hindrance effects. The Arrhenius parameters for the reorientation of the -NH: group obtained from these and other investigations are shown in Table 1.
49
to ensure full magnetization recovery. Selected 2H lineshapes were obtained from the Fourier transformation of solid echoes with delay periods 40 and 80 ks. Signal processing included apodization with an exponential window function, equivalent to a Lorentzian broadening of 1 kHz. 2.1. Relaxation simulations
2. Experimental
MXETl requires both system-specific parameters (e*qQ/h, 7, Euler angles of the PAS EFG at different sites, exchange matrix and rate constant) and experimental parameters (such as echo delays and pulse widths). The deuterated -NH: group in L-alanine is asymmetric and, in addition, each site has a unique quadrupolar coupling tensor. The tensorial components e’qQ/h and 71 for each site have previously been measured in the course of a deuterium nuclear quadrupole resonance (NQR) study [23]. Whereas it is crucial for the generation of reliable lineshape simulations to take the site-specific values of e2qQ/h into account, this is irrelevant for the calculation of spin-lattice relaxation behaviour. Many reorientational jumps of the -ND: group occur during the time T,, and, therefore, on this time-scale the three deuterons, which are constrained to move as unit, experience only an average quadrupolar coupling interaction. Indeed, as the T,, mechanism is inherently dependent on the motion of the spin between sites, although an individual site can be assigned a unique value of e2qQ/h it cannot be labelled by a T,, value. In the relaxation simulations carried out here, the
Polycrystalline N-deuterated L-alanine, (2amino-d,-propanoic acid-d) was crystallized by slow evaporation from a solution of L-alanine (Sigma > 98%) in D,O at 303 K. Slow evaporation was crucial to avoid the formation of another polymorph [20]. The powder X-ray diffraction patterns of the deuterated crystals and the original protonated sample were indistinguishable. All measurements were carried out with a Bruker MSL-300 spectrometer operating at 46.073 MHz. The temperature was controlled by a conventional gas flow system, to within * 1 K. 2H Zeeman spin-lattice relaxation times, T,, were measured over the temperature range 250-425 K using a saturation-recovery pulse sequence. The magnetization recovery was monitored using a solid echo with a delay 7 of 40 ps. The 90” pulse lengths were typically between 6.5 and 7.5 ps duration. The 2H spin-lattice relaxation times of quadrupolar alignment T,o were measured using a Jeener-Broekaert pulse sequence (90”,-r45”,,-t-45”,-r-echo, with T = 40 PSI. For Tie measurements, recycle delays were at least 5T,,
Table 2 Assignment Label
of
e’qQ/h
and 77 to the deuteron 0
sites in the -ND;
d
group N-D
(nm)
HI H3
94.7 22.0 98.2
Average
70.03 b
HZ
a The sites are labelled in the crystal b Value in the molecular frame.
108.9 339.7 220.6
frame
by the polar
in L-alanine
. ‘0
0.1861 0.1780 0.1828
angles
0 and C#J,as determined
e’qQ/h
?)
174.5 145.2 166.9
0.072 0.027 0.043
162.7
0.047
&Hz)
from
neutron
diffraction
data [ll].
50
M.A.K.
Williams
et al. /Solid
State Nuclear
root-mean-square average value of e’qQ/h was assigned to each site. Table 2 shows the assignments of e2qQ/h and n to the crystallographically unique deuterons in the -ND: group. The assignment was carried out assuming a correlation between the e*qQ/h values and hydrogen-bond lengths [23]. Assuming that the principal component of the electric field gradient tensor eq at a particular deuteron site is directed along the N-D bond, then the polar angles 8 and 4 given in Table 2 are equivalent to two of the Euler angles describing the orientation of the PAS of the EFG tensor in the crystal frame. These Euler angles are needed in order to perform the numerical calculations. The third Euler angle, relating to the orientation of the tensorial x\: and yy components around the principal axis, can only be obtained from a singlecrystal study carried out at low temperature (i.e. in the rigid lattice) and, to our knowledge, this has not been done. The small magnitudes of the asymmetry parameters in L-alanine, however, warrant the often made approximation n = 0, and this renders the third Euler angle obsolete. (The validity of this assumption was investigated using MXETl.) It should be noted that a rigid lattice single-crystal study would also allow the unequivocal assignment of the tensorial components. There has previously been a single-crystal study of the -ND: group in L-alanine [24] but, as this was carried out at elevated temperatures, only the motionally averaged components of the quadrupolar interaction tensor could be determined. It is interesting to note that, although the principal axis of the average tensor was found to lie nearly 4” away from the C-N bond, this is likely to be a result of the asymmetry of the -ND: group rather than any deviation of eq from the direction of the N-D bond. Support for this approximation is provided by a low-temperature single-crystal NMR study on deuterated glycine [25], which showed that the principal components of the -ND: group deuterium EFG tensors deviated from the N-D bonds by less than 1”. Therefore, in carrying out the numerical calculations, it has been assumed that eq lies along the N-D bond. Further details of MXETl are given elsewhere [7].
Magnetic
Resonance
6 (1996)
47-53
3. Results and discussion 3.1. Relaxation measurements
Fig. 1 shows the results of a 2H spin-lattice relaxation study of the -ND: group in polycrystalline L-alanine. Powder-average Zeeman spinlattice relaxation times (T,,), were measured over the temperature range 250-425 K. Spinlattice relaxation times of quadrupolar alignment ( TIQ)P measured at temperatures around the T,, minimum are also shown on Fig. 1. The analytical expression, analogous to Eq. (11, for (Tro),, is given by [lo]: (--&--~=~(~)‘~sin4
O+sinZ
28)
(2) The initial analysis was carried out by performing an appropriately weighted non-linear leastsquares fit of Eq. (1) to the ( TIZ)P data, assuming an Arrhenius dependence of correlation time on temperature, and allowing the infinite temperature correlation time rO, activation energy E and e’qQ/h all to be free-fitting parameters. The Arrhenius parameters obtained from this analysis are given in Table 1. The ( TIZIP and (T,, jI, data were also simultaneously fitted to Eqs. (1) and (21, respectively, and the single set of extracted
IO41
2.5
3.0 lOOOK/
3.5
4.0
T
Fig. 1. Experimental ‘H spin-lattice relaxation times (7’,Z)P (0) and (T,o)P (0) of the -ND; group in polycrystalline L-alanine. The solid line is the best fit to Eqs. (1) and (2).
M.A.K.
i
10
Williams
100
et al. /Solid
State Nuclear
1000
7, / 1 o-g s Fig. 2. Theoretical plots of ‘H (TIZjP versus 7C for the three-fold reorientation of the -ND: group in L-alanine. The solid line was calculated using the analytical expression given in Eq. Cl), with e2&/h = 162.7 kHz and 0 = 70.03”. The points (a) were determined from an inversion-recovery experiment simulated using the program MXETl.
parameters is also shown Table 1. The lines shown in Fig. 1 were calculated from Eqs. (1) and (2) using this set of parameters. Both fits had a correlation coefficient of 0.998, and the extracted values of TV and E are in good agreement with those reported previously. In terms of the analytical expressions describing three-fold reorientation, the (T,,), and (TIQ)P data are self-consistent. Although the fits are in excellent agreement, the e’qQ/h values extracted in this simplistic manner (157.8 * 0.4 and 157.7 -t 0.4 kHz for the (T,z& and the simultaneous (T,,), and (Tlo&, fits, respectively) are slightly less than the average (162.7 kHz) of the three unique e2qQ/h values, falling outside the estimated experimental 95% confidence limits. Attention is now turned to the results of the numerical simulation. Fig. 2 shows a plot of ( Tlz)p versus correlation time for three-fold reorientation of the -ND: group in L-alanine. One set of data, shown as a solid line, was calculated using the analytical expression given in Eq. cl), using e*qQ/h = 162.7 kHz and 8 = 70.03”. The second set of data (0) was obtained, using MXETl, from the simulation of an inversion-recovery experiment and subsequent exponential fitting of the recovery of the solid-echo maximum. For these simulations e2qQ/h = 162.7 kHz and 77 = 0 for all sites, as
Magnetic
Resonance
6 (1996)
47-53
51
discussed previously. It should be noted that the reorientation rate in the numerical simulation carried out using MXETl is defined by an exchange rate k. For a three-site jump model this is related to the correlation time by 7, = 1/3k [25l. The results of the numerical calculations shown here were obtained using 100 powder increments. Selected simulations carried out using 500 powder increments showed that there was no advantage in increasing the fineness of the angular tiling. As would be expected, there is broad agreement between the two approaches. In the intermediate region, however, there is a slight discrepency between the predictions of the analytical expression and the numerical calculation. (It is worth commenting that the observed degree of discrepency between the methods of calculation is more sensitive to changes in the echo delay than it is to the pulse widths involved.) The maximum deviation in this region (in this simulation about 10%) occurs where the well-known attenuation of echo intensity [1,2] is close to its maximum. Consequently, experimental (T,, jp values determined in this region are likely to have larger uncertainties associated with them and, therefore, any deviations from the form of the analytical expression are unlikely to be detectable. In other words, in terms of the extraction of Arrhenius parameters it is unlikely to be better to generate ( TLZ& numerically and use the comparison with experimental values to obtain k(T), rather than to fit the experimental data to the analytical expression. On closer examination of the (T,, jP minimum (see inset in Fig. 21, a small but significant discrepency can be seen to exist. If we compare the predictions of the two approaches with the experimental results, we see that the value of (TIZjP at the minimum (2.92 ms), obtained from the analytical expression with e2qQ/h = 162.7 kHz (solid line), is less than that obtained experimentally (3.05 ms, Fig. l), which is consistent with the value of e*qQ/h of 157.8 kHz obtained by fitting. In contrast, the value of the (T,,), minimum (3.04 ms) determined from the simulation using e*qQ/h = 162.7 kHz (Fig. 2) is in excellent agreement with the experimental minimum. In conclu-
52
M.A.K.
Williams
et al. /Solid
State Nuclear
sion, whereas analytical expressions allow relatively straightforward fitting of experimental data and the extraction of reliable motional parameters, values of e2qQ/h obtained from the depth of the minimum in this manner may not relate directly to the average e*qQ/h value of the sites involved in the motional pathway, although they are likely to be close (within 4% in this work). The results indicate that better agreement with experiment can be obtained by performing numerical simulations; however, this approach is more time consuming. 3.2. Lineshape measurements
For completeness, the temperature dependence of the spectral lineshapes obtained from the -ND: group in L-alanine was briefly examined. As mentioned previously, the program MXETl can be used to simulate lineshapes, accounting for finite pulse widths, echo distortions, site-unique values of e*qQ/h and 7, and correcting for the Lorentzian broadening introduced by signal processing and for Gaussian broadening
Magnetic
Resonance
T = 80 ps
k/s-’
T/K
IA.-5*4x 293
-150
frequency/kHz Fig. 3. Experimental height and obtained
47-53
caused by dipole-dipole interactions. Taking detailed account of dipole-dipole interactions is not straightforward owing to, among other things, the different effects of intra- and inter-molecular homo- and hetero-nuclear interactions on the formation of the solid echo. It is significantly easier to place an upper limit on the value of dipolar broadening. (The latest lineshape simulation descendant of MXQET, namely NDMXEX [9], considers the effects of N-D dipolar coupling explictly for ND, groups.) Typically, the values of l-3 kHz employed for dipolar broadening are determined empirically from low-temperature lineshapes. To ensure that the comparisons of the experimental lineshapes with those calculated were realistic, the maximum possible value of Gaussian broadening was obtained from a simple calculation of the total rigid lattice second moment. In this case, the contributions from all other deuteron, proton and nitrogen spins within a sphere of 0.15 nm radius, centered on each non-equivalent reference nucleus, were calculated. The total rigid lattice second moment for the three -ND: group deuterons was calculated
* = 40 ps T/K
6 (1996)
k/s-’
6.0 x 10’
280 n
n
1.3 x 10’
150
-150
150
frequencylkl-fz
and simulated ‘H spectra of the -ND; group in polycrystalline L-alanine. Lineshapes from solid ethos with delays of 40 and 80 KS. The exchange rate is k = l/37,.
are shown
at the same
M.A.K.
Williams
et al. /Solid
State Nuclear
to be 1.08 X lo8 rad* s-*, which is equivalent to a Gaussian broadening of 3.9 kHz. This was taken as the maximum feasible value for the purpose of the simulations. Spectral lineshapes obtained in the intermediate regime with echo delays of 40 and 80 ps are shown in Fig. 3 together with simulations carried out using MXETl. There is excellent agreement between the experimental and theoretical lineshapes, and also good agreement with a previous more extensive lineshape study [21]. The values of E and 70 consistent with the spectra shown in Fig. 3 are given in Table 1. These values are in good agreement with the relaxation study conducted here, and with previous studies using various techniques. It is worth commenting that in this study it was found that k(H)/k(D) = 1.07 + 0.35, compared to a value of 2l/* predicted by the simple harmonic oscillator model. In other words, the dynamics of the -ND: group are very similar to those of the -NH: group [141. In the case of glycine, the deuteration of the -NH: group has no significant effect on E, but the value of TV is reduced by a factor of 8 [25], whereas there is no evidence of a similar change in the case of Lalanine. Acknowledgements
This work was supported by a MAFF/DTI LINK programme. R.D.K. wishes to thank EPSRC and Nest16 for a CASE award. References [l]
R.J. Schadt, E.J. Cain and A.D. 97 (1993)8387.
English,
J. Phys. Chem.,
Magnetic
Resonance
6 (1996)
47-53
53
[2] T.M. Alam and G.P. Drobny, Chem. Rev., 91 (1991) 1545. [3] J.H. Simpson, W. Laing, D.M. Rice and F.E. Karasz, Macromolecules, 25 (1992) 3068. [4] D. Hentschel, H. Sillescu and H.W. Spiess, Polymer, 25 (1984) 1078. [S] N.J. Heaton, R.L. Vold and R.R. Void, J. Am. Chem. Sot., 111 (1989) 3211. [6] MS. Greenfield, A.D. Ronemus, R.L. Void, R.R. Void, P.D. Ellis and T.R. Raidy, J. Magn. Reson., 72 (1987) 89. [7] R.R. Void and R.L. Void, Adv. Magn. Opt. Reson., 16 (1991) 85. [8] T.H. Lin, R.R. Vold and R.L. Vold, J. Magn. Reson., 95 (1991) 71. [9] T.H. Lin and R.R. Vold, J. Magn. Resonan. Ser. A., 113 (1995) 271. 1101D.A. Torchia and A. Szabo, J. Magn. Reson., 49 (1982) 107. 1111M.S. Lehmann, T.F. Koetzle and W.C. Hamilton, J. Am. Chem. Sot., 94 (1972) 2657. 20 [=I H.J. Simpson, Jr., and R.E. Marsh, Acta Crystallogr., (1966) 550. R.E. Marsh and R. Bianchi, J. Phys. Chem., t131 R. Destro, 92 (1988) 966. W.S. Hinshaw, M.G. Hutchins, R.O.I. [141 E.R. Andrew, Sjoblom and P.C. Canepa, Mol. Phys., 32 (1975) 795. [I51 P. Jackson and R.K. Harris, J. Chem. Sot., Faraday Trans., 91 (1995) 9805. [161 S. Forss, J. Raman Spectrosc., 12 (1982) 266. L. Genzel, F. Kremer and L. Santa, Spec[171 J. Bandekar, trochim. Acta, Part A, 39 (1983) 357. [181 D. Durand, M.J. Field, M. Quilichini and J.C. Smith, Biopolymers., 33 (1993) 725. C.H. Niu and D.A. Torchia, J. Am. [I91 LX Batchelder, Chem. Sot., 105(6) (1983) 2228. and R.G. Griffin, J. Chem. La K. Beshah, E.T. Olejniczak Phys., 86(9) (1987) 4730. ml J.R. Long, B.Q. Sun, A. Bowen and R.G. Griffin, J. Am. Chem. Sot., 116(26) (1994) 11950. Kc J. Garen, M.J. Field, G. Kneller, M. Karplus and J. Smith, J. Chim. Phys., 88 (1991) 2587. [23] M.J. Hunt and A.L. MacKay, J. Magn. Reson., 15 (1974) 402. [24] D. Warner, T. Gorvin and W. Derbyshire, Mol. Phys., 14(3) (1968) 281. [25] C. Miiller, W. Schajor, H. Zimmermann and U. Haeberlen, J. Magn. Reson., 56 (1984) 235.
ELSEVIER
Solid State Nuclear
Magnetic Resonance 6 (1996) 47-53
A 2H NMR spin-lattice relaxation study of -NH; dynamics in polycrystalline L-alanine
group
M.A.K. Williams, R.D. Keenan, T.K. Halstead * of Chemistry,
Department
The Unil,ersity
of York, Heslington,
York,
YOI 5DD,
UK
Received 17 June 1995; accepted 19 July 1995
Abstract The *H spin-lattice relaxation times T,, and T,o of the -ND; group in polycrystalline L-alanine were measured over the temperature ranges 250-425 K and 300-425 K, respectively. The relaxation data were fitted to analytical expressions for three-fold reorientation, giving a correlation time T, = 1.5 X lo-l4 s exp(40.2 kJ mol-‘/RT), which is in good agreement with the results of a limited lineshape study. T,, values derived from numerical simulations, taking into consideration geometrical distortion of the -ND: group and experimental imperfections, are compared with values given by an analytical expression for three-fold jumps and with the experimental results. The extraction of average site quadrupolar coupling constants is discussed. Keywords:
NMR;
Deuterium;
Spin-lattice
relaxation;
Three-fold
1. Introduction There has been much written on the determination of molecular dynamics from the simulation of one-dimensional 2H spectra [1,2]. Studies of this type involving the reorientation of C-D and N-D bonds are plentiful, both where jump sites are related by molecular symmetry and where they are not [3-51. A complementary NMR approach to the experimental determination of molecular dynamics, often used in cases where the dipolar interaction is dominant, uses the temperature dependence of the spin-lattice relaxation time to map temperature onto correlation
* Corresponding author. 0926.2040/96/$15.00 0 1996 Elsevier SSDI 0926.2040(95)01204-4
Science
B.V.
All rights
reorientation;
L-Alanine
time, and hence to extract dynamic parameters. While deuterium NMR inversion-recovery experiments involving series of partially relaxed spectra are fashionable, and do allow diffusional and discrete dynamic models to be distinguished by their different T,, anisotropy, relatively few studies have adopted the purely time-domain approach and measured (T,,), directly by monitoring the recovery of the quadrupolar echo. This is the simplest experimental approach to relaxationtime measurement, having the advantages of gaining reasonable signal-to-noise ratios with far fewer transients than are needed for the accurate determination of the relaxation times of selected orientations, and dispensing with the additional data processing necessitated by subsequent Fourier transformation. In a polycrystalline powreserved
48
M.A.K.
Williams
et al. /Solid
State Nuclear
der sample, the relaxation time for each molecular group is determined by the particular orientation in the laboratory (B,,) frame of its motionally averaged electric field gradient (EFG) principal axis system (PAS) and, therefore, the measured spin-lattice relaxation time (T,,), is the average of many angular dependent relaxation times. Analytical expressions exist which relate T,, to the correlation time for various jump and diffusive dynamical models, and it should be noted that the derivation of theoretical expressions for powder-average relaxation times can be greatly simplified by exploiting the orthogonality relations of the Wigner rotation matrices [2]. Whereas these expressions are valid only for symmeteric reorientation motions involving sites with the same quadrupolar coupling constant e*qQ/h, and the same asymmetry parameter 7, programs such as MXQET and its descendants [6-91 enable the simulation of spectral lineshapes for jump models involving sites with arbitrary orientations and different e2qQ/h and n values. Later versions also allow the simulation of spinlattice relaxation behaviour for dynamic models involving asymmetric jumps. Furthermore, these simulations consider the effects of finite pulse widths and distortions of the quadrupolar echo. We have sought to test the success of these two approaches in predicting the relaxation behaviour
Table 1 Arrhenius
parameters
for the reorientation
of the -ND:
group
Magnetic
Resonance
6 (1996)
47-53
of the 2H NMR signal from an asymmetric -ND: group, by comparing the results of calculations using the expression [lo]:
x [J&P 4 + 4J2Po,
41
(1)
for the powder-average Zeeman spin-lattice relaxation rate for three-fold reorientation, where J(w, 7,) = r,/(l + w2r:), T, is the correlation time and 0 is the angle between the axis of reorientation and eq, and the results of computer simulations of an inversion-recovery experiment using the program MXETl, with the experimental results of a study on deuterated L-alanine. A comparison of these models is of general interest. The amino acid L-alanine crystallizes in zwitterionic form in the orthorhombic space group P2,2,2, with a = 0.6032 nm, b = 1.2343 nm, c = 0.5784 nm, and 2 = 4 [ll-131. The molecular dynamics of the methyl and -NH: groups have been studied using various techniques, including proton NMR relaxation [14,15], Raman and infra-red spectroscopy [ 16,171, phonon dispersion [18], deuterium NMR lineshape studies [19-211 and molecular dynamics simulations and normal mode calculations [22]. The methyl group dynam-
in L-alanine
Technique
E (kJ mol-‘)
‘H 7-1, ‘H T,p ” Raman spectroscopy ‘H Lineshape b Molecular dynamics
38.6 33.6 37.0 40.5 31.2
1.6 5.0 1.3 -
Ref. 14 15 16 21 22
‘H Lineshape 2H TIZ 2H T,z/TIQ
41.5 * 1.7 40.2 k 0.5 39.8 f 0.8
1.0 f 0.3 1.5 + 0.3 1.6 k 0.3
This work This work This work
a This study was carried out on L-[ lSN]alanine. ’ There is an error in this paper. which relates the infinite temperature exchange rate A, correlation time. In fact, for a three-site jump model A, = l/37,,. The 7O value extracted that obtained from the proton study than stated therein.
to the reciprocal of infinite temperature from this study is considerably closer to
M.A.K. Williams et al. /Solid State Nuclear Magnetic Resonance 6 (1996) 47-53
its are of special interest because of the unusually high activation energy of the reorientation, which is ascribed to steric hindrance effects. The Arrhenius parameters for the reorientation of the -NH: group obtained from these and other investigations are shown in Table 1.
49
to ensure full magnetization recovery. Selected 2H lineshapes were obtained from the Fourier transformation of solid echoes with delay periods 40 and 80 ks. Signal processing included apodization with an exponential window function, equivalent to a Lorentzian broadening of 1 kHz. 2.1. Relaxation simulations
2. Experimental
MXETl requires both system-specific parameters (e*qQ/h, 7, Euler angles of the PAS EFG at different sites, exchange matrix and rate constant) and experimental parameters (such as echo delays and pulse widths). The deuterated -NH: group in L-alanine is asymmetric and, in addition, each site has a unique quadrupolar coupling tensor. The tensorial components e’qQ/h and 71 for each site have previously been measured in the course of a deuterium nuclear quadrupole resonance (NQR) study [23]. Whereas it is crucial for the generation of reliable lineshape simulations to take the site-specific values of e2qQ/h into account, this is irrelevant for the calculation of spin-lattice relaxation behaviour. Many reorientational jumps of the -ND: group occur during the time T,, and, therefore, on this time-scale the three deuterons, which are constrained to move as unit, experience only an average quadrupolar coupling interaction. Indeed, as the T,, mechanism is inherently dependent on the motion of the spin between sites, although an individual site can be assigned a unique value of e2qQ/h it cannot be labelled by a T,, value. In the relaxation simulations carried out here, the
Polycrystalline N-deuterated L-alanine, (2amino-d,-propanoic acid-d) was crystallized by slow evaporation from a solution of L-alanine (Sigma > 98%) in D,O at 303 K. Slow evaporation was crucial to avoid the formation of another polymorph [20]. The powder X-ray diffraction patterns of the deuterated crystals and the original protonated sample were indistinguishable. All measurements were carried out with a Bruker MSL-300 spectrometer operating at 46.073 MHz. The temperature was controlled by a conventional gas flow system, to within * 1 K. 2H Zeeman spin-lattice relaxation times, T,, were measured over the temperature range 250-425 K using a saturation-recovery pulse sequence. The magnetization recovery was monitored using a solid echo with a delay 7 of 40 ps. The 90” pulse lengths were typically between 6.5 and 7.5 ps duration. The 2H spin-lattice relaxation times of quadrupolar alignment T,o were measured using a Jeener-Broekaert pulse sequence (90”,-r45”,,-t-45”,-r-echo, with T = 40 PSI. For Tie measurements, recycle delays were at least 5T,,
Table 2 Assignment Label
of
e’qQ/h
and 77 to the deuteron 0
sites in the -ND;
d
group N-D
(nm)
HI H3
94.7 22.0 98.2
Average
70.03 b
HZ
a The sites are labelled in the crystal b Value in the molecular frame.
108.9 339.7 220.6
frame
by the polar
in L-alanine
. ‘0
0.1861 0.1780 0.1828
angles
0 and C#J,as determined
e’qQ/h
?)
174.5 145.2 166.9
0.072 0.027 0.043
162.7
0.047
&Hz)
from
neutron
diffraction
data [ll].
50
M.A.K.
Williams
et al. /Solid
State Nuclear
root-mean-square average value of e’qQ/h was assigned to each site. Table 2 shows the assignments of e2qQ/h and n to the crystallographically unique deuterons in the -ND: group. The assignment was carried out assuming a correlation between the e*qQ/h values and hydrogen-bond lengths [23]. Assuming that the principal component of the electric field gradient tensor eq at a particular deuteron site is directed along the N-D bond, then the polar angles 8 and 4 given in Table 2 are equivalent to two of the Euler angles describing the orientation of the PAS of the EFG tensor in the crystal frame. These Euler angles are needed in order to perform the numerical calculations. The third Euler angle, relating to the orientation of the tensorial x\: and yy components around the principal axis, can only be obtained from a singlecrystal study carried out at low temperature (i.e. in the rigid lattice) and, to our knowledge, this has not been done. The small magnitudes of the asymmetry parameters in L-alanine, however, warrant the often made approximation n = 0, and this renders the third Euler angle obsolete. (The validity of this assumption was investigated using MXETl.) It should be noted that a rigid lattice single-crystal study would also allow the unequivocal assignment of the tensorial components. There has previously been a single-crystal study of the -ND: group in L-alanine [24] but, as this was carried out at elevated temperatures, only the motionally averaged components of the quadrupolar interaction tensor could be determined. It is interesting to note that, although the principal axis of the average tensor was found to lie nearly 4” away from the C-N bond, this is likely to be a result of the asymmetry of the -ND: group rather than any deviation of eq from the direction of the N-D bond. Support for this approximation is provided by a low-temperature single-crystal NMR study on deuterated glycine [25], which showed that the principal components of the -ND: group deuterium EFG tensors deviated from the N-D bonds by less than 1”. Therefore, in carrying out the numerical calculations, it has been assumed that eq lies along the N-D bond. Further details of MXETl are given elsewhere [7].
Magnetic
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6 (1996)
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3. Results and discussion 3.1. Relaxation measurements
Fig. 1 shows the results of a 2H spin-lattice relaxation study of the -ND: group in polycrystalline L-alanine. Powder-average Zeeman spinlattice relaxation times (T,,), were measured over the temperature range 250-425 K. Spinlattice relaxation times of quadrupolar alignment ( TIQ)P measured at temperatures around the T,, minimum are also shown on Fig. 1. The analytical expression, analogous to Eq. (11, for (Tro),, is given by [lo]: (--&--~=~(~)‘~sin4
O+sinZ
28)
(2) The initial analysis was carried out by performing an appropriately weighted non-linear leastsquares fit of Eq. (1) to the ( TIZ)P data, assuming an Arrhenius dependence of correlation time on temperature, and allowing the infinite temperature correlation time rO, activation energy E and e’qQ/h all to be free-fitting parameters. The Arrhenius parameters obtained from this analysis are given in Table 1. The ( TIZIP and (T,, jI, data were also simultaneously fitted to Eqs. (1) and (21, respectively, and the single set of extracted
IO41
2.5
3.0 lOOOK/
3.5
4.0
T
Fig. 1. Experimental ‘H spin-lattice relaxation times (7’,Z)P (0) and (T,o)P (0) of the -ND; group in polycrystalline L-alanine. The solid line is the best fit to Eqs. (1) and (2).
M.A.K.
i
10
Williams
100
et al. /Solid
State Nuclear
1000
7, / 1 o-g s Fig. 2. Theoretical plots of ‘H (TIZjP versus 7C for the three-fold reorientation of the -ND: group in L-alanine. The solid line was calculated using the analytical expression given in Eq. Cl), with e2&/h = 162.7 kHz and 0 = 70.03”. The points (a) were determined from an inversion-recovery experiment simulated using the program MXETl.
parameters is also shown Table 1. The lines shown in Fig. 1 were calculated from Eqs. (1) and (2) using this set of parameters. Both fits had a correlation coefficient of 0.998, and the extracted values of TV and E are in good agreement with those reported previously. In terms of the analytical expressions describing three-fold reorientation, the (T,,), and (TIQ)P data are self-consistent. Although the fits are in excellent agreement, the e’qQ/h values extracted in this simplistic manner (157.8 * 0.4 and 157.7 -t 0.4 kHz for the (T,z& and the simultaneous (T,,), and (Tlo&, fits, respectively) are slightly less than the average (162.7 kHz) of the three unique e2qQ/h values, falling outside the estimated experimental 95% confidence limits. Attention is now turned to the results of the numerical simulation. Fig. 2 shows a plot of ( Tlz)p versus correlation time for three-fold reorientation of the -ND: group in L-alanine. One set of data, shown as a solid line, was calculated using the analytical expression given in Eq. cl), using e*qQ/h = 162.7 kHz and 8 = 70.03”. The second set of data (0) was obtained, using MXETl, from the simulation of an inversion-recovery experiment and subsequent exponential fitting of the recovery of the solid-echo maximum. For these simulations e2qQ/h = 162.7 kHz and 77 = 0 for all sites, as
Magnetic
Resonance
6 (1996)
47-53
51
discussed previously. It should be noted that the reorientation rate in the numerical simulation carried out using MXETl is defined by an exchange rate k. For a three-site jump model this is related to the correlation time by 7, = 1/3k [25l. The results of the numerical calculations shown here were obtained using 100 powder increments. Selected simulations carried out using 500 powder increments showed that there was no advantage in increasing the fineness of the angular tiling. As would be expected, there is broad agreement between the two approaches. In the intermediate region, however, there is a slight discrepency between the predictions of the analytical expression and the numerical calculation. (It is worth commenting that the observed degree of discrepency between the methods of calculation is more sensitive to changes in the echo delay than it is to the pulse widths involved.) The maximum deviation in this region (in this simulation about 10%) occurs where the well-known attenuation of echo intensity [1,2] is close to its maximum. Consequently, experimental (T,, jp values determined in this region are likely to have larger uncertainties associated with them and, therefore, any deviations from the form of the analytical expression are unlikely to be detectable. In other words, in terms of the extraction of Arrhenius parameters it is unlikely to be better to generate ( TLZ& numerically and use the comparison with experimental values to obtain k(T), rather than to fit the experimental data to the analytical expression. On closer examination of the (T,, jP minimum (see inset in Fig. 21, a small but significant discrepency can be seen to exist. If we compare the predictions of the two approaches with the experimental results, we see that the value of (TIZjP at the minimum (2.92 ms), obtained from the analytical expression with e2qQ/h = 162.7 kHz (solid line), is less than that obtained experimentally (3.05 ms, Fig. l), which is consistent with the value of e*qQ/h of 157.8 kHz obtained by fitting. In contrast, the value of the (T,,), minimum (3.04 ms) determined from the simulation using e*qQ/h = 162.7 kHz (Fig. 2) is in excellent agreement with the experimental minimum. In conclu-
52
M.A.K.
Williams
et al. /Solid
State Nuclear
sion, whereas analytical expressions allow relatively straightforward fitting of experimental data and the extraction of reliable motional parameters, values of e2qQ/h obtained from the depth of the minimum in this manner may not relate directly to the average e*qQ/h value of the sites involved in the motional pathway, although they are likely to be close (within 4% in this work). The results indicate that better agreement with experiment can be obtained by performing numerical simulations; however, this approach is more time consuming. 3.2. Lineshape measurements
For completeness, the temperature dependence of the spectral lineshapes obtained from the -ND: group in L-alanine was briefly examined. As mentioned previously, the program MXETl can be used to simulate lineshapes, accounting for finite pulse widths, echo distortions, site-unique values of e*qQ/h and 7, and correcting for the Lorentzian broadening introduced by signal processing and for Gaussian broadening
Magnetic
Resonance
T = 80 ps
k/s-’
T/K
IA.-5*4x 293
-150
frequency/kHz Fig. 3. Experimental height and obtained
47-53
caused by dipole-dipole interactions. Taking detailed account of dipole-dipole interactions is not straightforward owing to, among other things, the different effects of intra- and inter-molecular homo- and hetero-nuclear interactions on the formation of the solid echo. It is significantly easier to place an upper limit on the value of dipolar broadening. (The latest lineshape simulation descendant of MXQET, namely NDMXEX [9], considers the effects of N-D dipolar coupling explictly for ND, groups.) Typically, the values of l-3 kHz employed for dipolar broadening are determined empirically from low-temperature lineshapes. To ensure that the comparisons of the experimental lineshapes with those calculated were realistic, the maximum possible value of Gaussian broadening was obtained from a simple calculation of the total rigid lattice second moment. In this case, the contributions from all other deuteron, proton and nitrogen spins within a sphere of 0.15 nm radius, centered on each non-equivalent reference nucleus, were calculated. The total rigid lattice second moment for the three -ND: group deuterons was calculated
* = 40 ps T/K
6 (1996)
k/s-’
6.0 x 10’
280 n
n
1.3 x 10’
150
-150
150
frequencylkl-fz
and simulated ‘H spectra of the -ND; group in polycrystalline L-alanine. Lineshapes from solid ethos with delays of 40 and 80 KS. The exchange rate is k = l/37,.
are shown
at the same
M.A.K.
Williams
et al. /Solid
State Nuclear
to be 1.08 X lo8 rad* s-*, which is equivalent to a Gaussian broadening of 3.9 kHz. This was taken as the maximum feasible value for the purpose of the simulations. Spectral lineshapes obtained in the intermediate regime with echo delays of 40 and 80 ps are shown in Fig. 3 together with simulations carried out using MXETl. There is excellent agreement between the experimental and theoretical lineshapes, and also good agreement with a previous more extensive lineshape study [21]. The values of E and 70 consistent with the spectra shown in Fig. 3 are given in Table 1. These values are in good agreement with the relaxation study conducted here, and with previous studies using various techniques. It is worth commenting that in this study it was found that k(H)/k(D) = 1.07 + 0.35, compared to a value of 2l/* predicted by the simple harmonic oscillator model. In other words, the dynamics of the -ND: group are very similar to those of the -NH: group [141. In the case of glycine, the deuteration of the -NH: group has no significant effect on E, but the value of TV is reduced by a factor of 8 [25], whereas there is no evidence of a similar change in the case of Lalanine. Acknowledgements
This work was supported by a MAFF/DTI LINK programme. R.D.K. wishes to thank EPSRC and Nest16 for a CASE award. References [l]
R.J. Schadt, E.J. Cain and A.D. 97 (1993)8387.
English,
J. Phys. Chem.,
Magnetic
Resonance
6 (1996)
47-53
53
[2] T.M. Alam and G.P. Drobny, Chem. Rev., 91 (1991) 1545. [3] J.H. Simpson, W. Laing, D.M. Rice and F.E. Karasz, Macromolecules, 25 (1992) 3068. [4] D. Hentschel, H. Sillescu and H.W. Spiess, Polymer, 25 (1984) 1078. [S] N.J. Heaton, R.L. Vold and R.R. Void, J. Am. Chem. Sot., 111 (1989) 3211. [6] MS. Greenfield, A.D. Ronemus, R.L. Void, R.R. Void, P.D. Ellis and T.R. Raidy, J. Magn. Reson., 72 (1987) 89. [7] R.R. Void and R.L. Void, Adv. Magn. Opt. Reson., 16 (1991) 85. [8] T.H. Lin, R.R. Vold and R.L. Vold, J. Magn. Reson., 95 (1991) 71. [9] T.H. Lin and R.R. Vold, J. Magn. Resonan. Ser. A., 113 (1995) 271. 1101D.A. Torchia and A. Szabo, J. Magn. Reson., 49 (1982) 107. 1111M.S. Lehmann, T.F. Koetzle and W.C. Hamilton, J. Am. Chem. Sot., 94 (1972) 2657. 20 [=I H.J. Simpson, Jr., and R.E. Marsh, Acta Crystallogr., (1966) 550. R.E. Marsh and R. Bianchi, J. Phys. Chem., t131 R. Destro, 92 (1988) 966. W.S. Hinshaw, M.G. Hutchins, R.O.I. [141 E.R. Andrew, Sjoblom and P.C. Canepa, Mol. Phys., 32 (1975) 795. [I51 P. Jackson and R.K. Harris, J. Chem. Sot., Faraday Trans., 91 (1995) 9805. [161 S. Forss, J. Raman Spectrosc., 12 (1982) 266. L. Genzel, F. Kremer and L. Santa, Spec[171 J. Bandekar, trochim. Acta, Part A, 39 (1983) 357. [181 D. Durand, M.J. Field, M. Quilichini and J.C. Smith, Biopolymers., 33 (1993) 725. C.H. Niu and D.A. Torchia, J. Am. [I91 LX Batchelder, Chem. Sot., 105(6) (1983) 2228. and R.G. Griffin, J. Chem. La K. Beshah, E.T. Olejniczak Phys., 86(9) (1987) 4730. ml J.R. Long, B.Q. Sun, A. Bowen and R.G. Griffin, J. Am. Chem. Sot., 116(26) (1994) 11950. Kc J. Garen, M.J. Field, G. Kneller, M. Karplus and J. Smith, J. Chim. Phys., 88 (1991) 2587. [23] M.J. Hunt and A.L. MacKay, J. Magn. Reson., 15 (1974) 402. [24] D. Warner, T. Gorvin and W. Derbyshire, Mol. Phys., 14(3) (1968) 281. [25] C. Miiller, W. Schajor, H. Zimmermann and U. Haeberlen, J. Magn. Reson., 56 (1984) 235.