* Universite des Sciences et Techniques de Lille

Transcription

1 XXII. INTERPRETATION OF QUADRUPOLAR POWDER SPECTRA: STATIC AND MAS EXPERIMENTS. (TUTORIAL SESSION) J.P. AMOUREUX*, C. FERNANDEZ* and P. GRANGER** * Universite des Sciences et Techniques de Lille Villeneuve d'ascq Cedex, France ** Universite de Strasbourg, Institut de Chimie BP 296R Strasbourg Cedex, France ABSTRACT. The basic equations for the solid state NMR line shapes of quadrupolar nuclei are presented and are used in microcomputer programs for the simulation of spectra with or without magic angle spinning. 1. Introduction The majority of the elements (71 %) in the periodic table have at least one isotope with a quadrupole moment. Contrary to nuclei with spin 1/2, those quadrupolar nuclei have several transitions which depend not only on magnetic interactions but also on interaction between their nuclear electric quadrupole moment Q and the electric field gradient q created by the surroundings. Then the chemical information contained in an NMR spectrum of such a nucleus is not only limited to chemical shifts but can also provide parameters (with some chemical significance) which describe the quadrupolar interactions. Those parameters depend on the local symmetry around the nucleus. In the following, the effect of the quadrupolar interaction is always assumed to be small with respect to the Zeeman intemction and is. therefore deduced from a perturbation calculation. First order perturbation shows that the transition (1/2 ~ -1/2), when it exists, is not shifted, but that other transitions (m ~ m-l) are shifted to a frequency v'. Only second order perturbation leads to a small shift v" of the (1/2 ~ -1/2) transition. As the magic angle experiment has no "magic" properties on the second order quadrupolar interaction, MAS will not average this intemction and the sample will display a powder pattern for this central transition. All the theoretical spectra shown below have been computed with the QUADVAS-SIM program (see below). In this program, rotation sidebands, broadenings, left-shifted points, RF pulse dumtions and instrumental bandwidths (emitter, receiver) are taken into account for all transitions and all angles of the spinner for a given nucleus. Moreover, phase and base-line corrections can be applied, and both chemical shift anisotropy (CSA) and quadrupolar interactions may be taken into account. 2. Basic theory 2.1. DEFINITIONS As usual, the quadrupolar interaction is described by the quadrupole coupling constant X = e2qzz Q/h and the asymmetry parameter" = (qxx-qyy)/qzz with 0 ~" ~ 1 [1] P 287. P. Granger and R. K. Harris (eds.) Multinuclear Magnetic Resonance in Liquids and Solids - Chemical Applications, KlllWer Academic Publishers. 409

2 410 We will call VI the Lannorfrequency including the chemical shift and vq = 3xJ21(21-1) the quadrupolar frequency, and we will use the parameter: a = (VQ2/v1 )[1(1+1)-3/4] (1) For a powder sample the frequency VG of the centre of gravity of the band corresponding to a given transition is given by [2] vg(m, m-1)-v/= - (vq2/30v/ )[1(1+1) - 3-9m(m-I)](1+112/3) (2) for m '# 1/2 and vg(1/2,-1/2)-v/ = - a (1+11 2/3)/30 (3) for the (1/2 ~ -1/2) transition. This induced quadrupole shift, typical of the solid state, is independent of the techniques used: STATIC, MAS, V AS. 2.2 STATIC SAMPLE If we call a and ~ the angles expressing the orientation of Bo in the principal axes of the electric field gradient tensor, then we have for the first order shift [1]. v'(m, m-1) = (vql4)(1-2m)[3 cos2~-1+11 sin2~ cos2a] (4) For the second order shift of the (1/2 ~ -1/2) transition we obtain: V"(1/2,-l/2)= a [A cos4~+b cos2~+c]/144 = a [A sin4~ -(2A+B)sin2p+4112]/144 where (5) A (a) = 9 (3-11 cos 2a)2 B (a) = cos 2a cos2 2a C (a) = cos 2a + 9rt2 cos2 2a 2A(a) + B(a) = 12( cos 2a) This second order shift becomes simpler for the case, which is often met experimentaly, where 11 = 0: v" (1/2, -1/2) = (a /16)(1 - cos2 P)(l - 9 cos2 P) (6) The total width of the two symmetrical transitions (m, mol) + (l-m, -m) is then given by [3] AV'TS = II-2m l.vq (7) and the interval between the two peaks of these transitions is equal to : Av'PS = AV'TS (1-11)/2 (8)

3 411 o.,z 0,5 0,8 1 o 2 4y (1-2m)vQ Figure 1. Static powder samples; a) sum of a pair of satellite lines (m~m-l)+(l-m-+m) transitions as a function of 11; b) Schematic aspect of one satellite.

4 412 JfTl = 0 then the interval between two consecutive peaks is equal to vq. For the transition (1/2 ~ -1/2) the total width is :.1v"TS = ( Tl + Tl2)a /144 (9) Figure 1 presents the aspect of the satellite lines according to Tl MAS WITH INFINITE SPINNER SPEED [3] Again if we call yand 0 the polar angles expressing the orientation of the spinner axis in the principal axes of the electric field gradient tensor and if we observe the transition (1/2~-1/2) we have: v"(i/2,-1/2)= -a [D cos4y+e cos2y+f]/288 = - a [D sin4y-(2d+e)sin2y+4(6+tl2)]/288 (10) where: D(O) = 7(3-Tl cos 20)2 E(o) = Tl2 + 48Tl cos Tl2 cos2 20 F(o) = 15-6Tl cos Tl2 cos2 20 2D(0)+E(0) = 4(18 + Tl2-9rl cos 20) and the associated total width of this central transition is : Table 1 summarises some typical values of.1v"t for the static:s and MAS :R cases for different spin and Tl values, and figure 2 presents the aspect of the central band according to Tl with or without magic angle spinning at infinite speed. (11) I 3/2 5/2 7/2 9/2 Tl S R S R S R S R l a Table 1. Values of.1v"t and a in vq2/v[ units MAS WITH FINITE SPINNER SPEED vr [4] For this experimental case, a modulation of the transition frequencies occurs and the result can be related to the previous one 2.3 [4].

5 413 =0 0,1 0,2 0,3 w 0,4 0,5 0,6 0,7 0,8 O,g 1 Figure 2. Central transition (1/2 ~ -1/2) of half integer spin in a powder sample: 0 static, II MAS at infinite speed.

6 414 The spectrum then presents spinning sidebands equally spaced with an interval equal to VR which is the rotational frequency of the spinner. The total intensity of each transition remains constant and is scattered into the central line and its sidebands. When VR becomes very large with respect to the static bandwidth of the transitions, only the centreband is observed and it corresponds to the case where vr = 00 previously described in 2.3. This "infinite" speed is the optimal condition for the observation of the (1/2 4-1/2) transition but is nearly never obtained for the other transitions (m 4 m-l), which leads to numerous sidebands. For all values of I, vq and VI, the spectrum of the (1/2 4-1/2) transition depends only on 11 and a/vr when the frequency scale is divided by a, and the integrated intensity of the centre band depends only on [7]: x = a /3)/4 VR 2.5. THEORETICAL EXAMPLES (2) Some theoretical results obtained from the previous formulae and calculated on a microcomputer are presented on figures 1 and 2. The first one depicts the evolution of a (m 4 m-i) transition according to 11 for a static sample, and the second one presents the evolution according to 11 of the (1/2 4-1/2) transition for the static case and the magic angle experiment with VR = 00. These shapes may help the spectroscopist to estimate the order of magnitude of 11 and especially to recognize if one can assume to be in the simpler case corresponding to 11 = O. The use of these pictures will be exemplified in the next section. 3. Applications The spectra used in this section have been simulated with the programs of the authors starting from parameters obtained on experimental samples. This part will introduce the spectroscopist to the field of the interpretation of NMR spectra of quadrupolar nuclei observed to the solid state. The parameters deduced from this analysis can be used to obtain the quadrupolar coupling constants, the asymmetry factors and so on to be used as starting parameters in the simulation of spectra or as rough values of the parameters to interpret chemical structures Na SPECTRA FOR NaN03 (SPIN 3/2) The whole spectrum simulated at 26 MHz is presented in figure 3a for the static case and the expansion of the central 0/2 4-1/2) transition in figure 3b. The result of MAS experiment on the central band is shown in figure 3c. We must say that the experimental spectra corresponding to figure 3a cannot be obtained using a single pulse experiment since the dead time will distort the spectrum; the use of an echo is then recommended. Comparison of figure 3b with figure 2 shows that T] = O. The interval between the two external lines on the static spectrum, figure 3a, is equal to vq = 168 khz. The shoulders of the (3/2 4 1/2) or (-1/2 4-3/2) transitions are generally too weak to be observed. Using expression (1) giving a and with VI = 26 MHz we obtain from (3) :

7 415 23Na (1=3/2) ±1i2 STAT.IC Echo-Method a) STATIC t~ b) v MAS ±1 fz c) \ SW=8kHz I V Figure 3. 23Na NMR of NaN03, a) the complete spectrum, b) the central band, c) MAS spectrum of the central band.

8 416 va (1/2, -1/2)- vi = -109Hz = -4ppm, from (9) : f1v"ts = 564 Hz and from (11) f1v"tr = 233 Hz for the MAS experiment. As we have used a value of vr»564 Hz, this last parameter is of no use in the interpretation of the MAS spectrum and we can use the case vr = 00 presented in Al SPECfRA FOR KAI(S04h 12H20 (SPIN 5/2) The spectra simulated at 26 MHz are presented in figure 4 ; the static case in a) with its expansion b) and the MAS experiment c). Here also the comparison of figure 4b with figure 2 allows us to deduce 1'\ = O. Therefore, from the interval between the small peaks of figure 4a we deduce vq = 60 khz. Then as previously va (1/2, -1/2)-vl = -37Hz = -1,4 ppm, f1v"ts = 192 Hz and f1v"tr = 79 Hz. The sidebands arising from the transitions (m ~ m-l) with m -:t= 1/2 are now visible on the MAS spectrum. Practically they disappear as soon as the spinner axis is not perfectly adjusted to the magic angle. From the sharp sidebands we can deduce the speed of rotation of the sample: vr = 4 khz Al SPECfRA OF TRANSITION ALUMINA This experiment, depicted in figure 5, is performed at a higher frequency of 156 MHz. The original sample contains two different types of nuclei with the same proportion. It is clear from the static measurement that one of the species gives 1'\ = 0 but that this is not the case for the second. For the first one, from the total width of the central band f1v"ts = 80 khz measured on the spectrum, we can deduce: a = and then we obtain vq = 3 MHz which leads to va(l/2, -1/2) - VI = -99 ppm. It is then obvious from the values ofvq and SW that no transitions (m ~ m-l) (m -:t= 1/2) can be observed, and from the width of the central band no instrument can reach a spinning speed greater than 80 khz, then spinning sidebands will occur on all MAS spectra as appears on figure 5b. For the second species we must then use more information since we have now to determine 1'\. For this purpose we draw the theoretical curve presented in figure 6 which gives the ratio f1v"p/f1v"t of the distance between the two sharp maxima of the (1/2 ~ - 1/2) transition to f1v"ts versus 1'\. The measured value is 0.4 on the spectrum which gives 1'\ A better value may be obtained on an expanded spectrum. Formula 9 then allows the determination of a which then gives vq = 1.2 MHz, and as previously we then have va (1/2, -1/2)- VI = 17 ppm. Notice that these parameters are obtained with some error, but they are good starting values for fitting the experimental spectra with the theoretical ones. We remark that the MAS spectrum is not of a high interest since it is more complicated than the static one even with a high spinning speed: VR = 10 khz Al SPECfRA IN NO-DISTRIBUTED ZEOLITES Only the central band can be observed and is presented in figure 6 at two different Larmor frequencies. The two spectra obtained on the static sample do not look like those of figure 2. It is then necessary to suppose that at least two types of aluminium atoms are present. This is confmned by the magic angle experiment at high field. We then have, as previously, aluminium atoms in sites with 1'\ = 0 (which is the most abundant) and a second type in sites with 1'\ > O. It is then easy from the spectra at 26 and 156 MHz to deduce, as for the sample presented in section 3.3, the different parameters for the first

9 Al (1=5/2) ± 1/2 ~ ± 3/2, ± 1/2 ±5/2,±3/2~ ~ a) STATIC SW=240kHz Echo-Method ± 1/2 STATIC SW=lOkHz b)..--- ± 1/2 MAS Sideband (± 3/2, ± 1/2) / Sideband ~± 5/2, ± 3/2) c) SW=lOkHz Figure 4.27 Al spectra of KAI(S04h, 12H20, a) the entire spectrum, b) the central band, c) MAS spectra.

10 418 STATIC I SW:700 khz \ Figure Al spectra of transition alumina.

11 O.O+-~_...-_~-.- ~_...-_~~---.:>.,-_ tlv"p/tlv"t Figure 6. Graphical relationslup between 11 and tlv"p/av"t (see text). species. We then obtain for the lower Lannor frequency: 11 = 0, V"TS = Hz, vq = 450 khz, vg(1/2, -12) -VI = -80 ppm and tlv"tr = 4.4 khz. Similar results are deduced from the higher Lannor frequency spectrum except that vg(l/2, -1/2) -VI = -2.2 ppm and tlv"tr = 0.7 khz. The parameters of the second species are more difficult to measure since overlapping prevents precise measurement. Nevertheless we can use the same strategy as for the previous case 3.3. We notice that the static spectra observed at two frequencies are homothetical for each species but they do not have the same shift. We are then able to find the left limit of the spectrum of the second species observed at low frequency using the high field experiment. We then find tlv"ts = 15.7 khz from which we can deduce using figure 6 the value: 11 = 0.5 and then as usual vq = 450 khz and VG (1/2, -1/2) -VI = -86 ppm. The MAS experiment shows that sidebands have almost disappeared on the high field spectra since the tlv"tr values are less than the spinning rate. It is then possible to measure the relative abundance of the two types of aluminium atom using careful integration or simulation. On the contrary, the magic angle experiment at low field where tlv"tr is of the order of VR leads to spinning sidebands giving a less interesting spectrum.

12 420 v/=26mhz kHI -tl STATIC v/=156mhz 10kHI I, MAS MAS v/=156 MHz -1'! 1 10kHI, Figure 7.27 Al spectra of no-distributed zeolite at two different Larmor frequencies.

13 INTENSITIES OF THE CENTRAL BAND WITH OR WITHOUT MAS [7] General aspecr Figure 8 gives the evolution of the total intensity of the central band as a function of X2 where X is the parameter defined in (12). At an infinite speed, all the signal is displayed in the centre band, but as the speed decreases, the intensity of this band decreases and for 27 Al in zeolite with VI = 26 MHz, vq ::: 450 khz, T\ ::: 0 and VR ::: 2.85 khz (which gives X ::: 5.47 ), 46% of the total intensity is observed. Figure 9 represents the theoretical spectra obtained for different experimental situations for T\ ::: 0 and T\ ::: 1. This relationship leads to the conclusion that at VR ::: 0, X ~oo the intensity of the central band becomes zero. But as the speed decreases, the intervals between the spinning sidebands decrease and they begin to overlap the central resonance. As the intensities of the central band and of the spinning sidebands represent the total intensity of the (1/2 ~ -1/2) transition, the intensity of the central band with its overlapping spinning sidebands always represents the total intensity and at VR ::: 0 the static spectrum represents the total intensity. Problems may arise for intermediate values of X where the total intensity is scattered among the different spinning sidebands. In those cases, the correct intensity must include all those lines, and care must be taken that all lines fall into the spectral width MAS N=O Fifure 8. Proportion of the total intensity of the central band observed as a function of X (see text).

14 422 x-. x.. O VIt"="., ",.1 MAS 4.. V-V.t a x ts x.... STATIC T1=O 4. v-vt a MAS x= Figure 9. Evolution of the spectrum of a central band according to X for two cases: T1 = 0 and 11 = 1.

15 423 STATIC kh z o 12.5kHz I o MAS _ skH~ o 12.5kHz Figure 10. Simulated spectra of Na2S04 and Na2Mo04 with and without magic angle spinning Some examples Figure 10 represents the simulation of the static and the MAS 23Na spectra of Na2S04 and Na2M004 observed previously [5, 6]. The parameters deduced from those experimental results are respectively: vq = 1.3 MHz, ll.v"ts = 14.3 khz and ll.v"tr = 4.6kHz for the flrst one and vq = 1.3 MHz, ll.v"ts = 9.2 khz and ll.v"tr = 3.8 khz for the second one, which leads to the simulated spectra of figure 10. The published MAS spectra use a spinning speed VR = 4.9 khz, this gives X = 3.0 for Na2S04 and X = 2.7 for Na2Mo04. According to the relationship given in flgure 8, only 75% of the total intensity is observed for the flrst one and 80% for the second. This must be taken into account when intensity problems are discussed. 4. Conclusion Solid state NMR spectra of quadrupolar nuclei in powder samples appear usually as complicated broad bands even for MAS experiments. The easiest way to improve the results and to identify a nucleus with a half integer spin in different sites is to increase the static magnetic fleld to its maximum available value. However, if chemical shift

16 424 anisotropy has a significant contribution, it is recommended to perform experiments in two different fields. At low field the quadrupolar interaction is dominant whereas in high field the reverse is observed. If magic angle spinning experiments are of a high interest in the presence of chemical shift anisotropy, when quadrupolar interactions are dominant such experiment are of interest only when the spinning frequency is large compared to the static spectrum width. In that case the centre band integrated intensity is easily evaluated. 5. Programs As previously said all the spectra presented in this chapter were simulated using programs for microcomputers. The following programs will be distributed by BRUKER- SPECTROSPIN: QUADVAS-SIM : QUADV AS-FIT : QUADMAS-FIT : 2D-QUADSTAT-SIM : 2D-QUADV AS-SIM : QUADECHO: Simulation of all possible spectra in VAS experiments. Refinement of the experimentallineshape corresponding to the (±l/2) central transition of half-integer spins in V AS experiments. Refinement of the experimental integrated intensities when numerous isolated sidebands are observed in MAS experiments. Simulation of the two dimensional nutation spectra of the central transition with static samples. Simulation of the two dimensional nutation spectra of the central transition in V AS experiments. Simulation of all possible spectra observed with a quadrupole echo. References [1] CPo Slichter, Principles of magnetic resonance. Springer Verlag Berlin, second Ed. (1978) [2] F. Lefebvre, J.P. Amoureux, C. Fernandez and E.G. Derouane, 1. Chern. Phys., 86, 6070 (1987). [3] A. Samoson, E. Kundla and E. Lippmaa, 1. Magn. Res., 49, 350 (1982). [4] J. Herzfeld and A.E. Berger, 1. Chern. Phys.,73, 6021 (1980). [5] W. Gauss, S. Gunther and A.R. Haase, Naturforsch., A33, 934(1978). [6] G.F. Lynch and S.L. Segel, Can. 1. Phys., 50, 567 (1972). [7] J.P. Amoureux, C. Fernandez and F.Lefebvre, Magn. Reson. Chern., 28, 5 (1990).