Molecular motion studied by NMR powder spectra. I. Lineshape calculation for axially symmetric sheilding tensors

CHEMICALPHYSICS6 (1974) 217-225.8 NORTH.HOLLA.ND PUBLISHINGCOMPANY

MOLECULAR 1. LINESHAPE

MOTION

CALCULATION

STUDIED

BY NMR POWDER SPECTRA.

FOR AXIALLY

SYMMETRIC

SHIELDING

TENSORS

H.W. SPIES Max-PIP&-lnrlirule, Depnrrmenl

of Molecuhr

lyrics,

6900 Heidebag,

FRC

Received9 August 1974

The effect of molecularreorientationon the NhlR powder spectraof nuclei with spin I = l/2. governedby mag nelic shieldingtensors,has been investigated.Numericalcalculationswere perfotmed for the case of axially symmetric shieldingtensors. Examplesdiscussedexplicitly are the nuclei at the comers of a regularoctahedron, tecrahedron, and trigonalbipyramid. In all cases it was atimed that the molecules can jump between their equdiirium positions, thereby interchangingthe nuclei. The calculationderaibed is completely general and unbe used (0 obtain the lineshapes for molecules of lower symmetry as well. The spectradisplay chafacleristicfeatureseasily rccog tible 51experiments,especially for valuesof the jump frequency 7-l. such that Awr = I (Aw = wi -wL), But even for much smallerjump frequencies,where Awr m 30, the deviationsfrom the Powder patternwithout motion are signilicant.Comparisonwith lineshape calculations for liquids shows that the spectrafor the same valuesof the jump frequency differ markedly for reorienlational models applicable to liquids or solids. ExPerimentalexamples

are presented in an accompanyingpaper.

1. Introduction The frequency shifts due to the anisotropy of the nuclear magnetic shielding can now be determined by a number of techniques, e.g., multiple pulse [ 11, proton enhanced nuclear induction spectroscopy [2], and h&h field NMR. These frequency shifts are less than about 50 kHz for typical nuclei with spin I = l/2 like ‘H, lgF, 31P, 13C, 15N even for the highest magnetic fields currentiy employed in NMR. This frequency range is also quite often encountered for slow motions of molecules in solids [3]. Therefore one can expect that the NMR specnu are influenced strongly by molecular motion. In fact, it hasbeen noticed - e.g., in hexamethylbenzene - that the 13C shielding tensors for the ring carbons are axially symmetric at room temperature due to the rapid rotation of the molecules about their sixfold axes, whereas they clearly deviate from axial symmetry at liquid nitrogen temperature [4], where the rotation is frozenin. The limitof rapidmotion about an arbitrary axis, where the rotation is fast compared with the relevant NMR frequency shifts, has

been discussed before [S 1. Such a motion will always have the result that the effective shielding tensor is axially symmetric [S]. The region where the time scale of the motion is comparable with the NMR frequency splittings is especially interesting and one can hope to be able to distinguish between different types of models for molecular reorientation. Clearly it is immaterial whether we consider NMR lineshapes governed by shielding tensors s or ESR lineshapes dominated by the g tensor. For the case of an axially symmetric g tensor, lineshape calculations for liquids have been performed [6,7], with which we can compare our results for solids. In this paper we consider the effect of reorientation of simplemolecules of high symmetry on the NMR powder spectra. This will enable us to aunalyzespectra in order to get information about the type of motion as well as values for jump frequencies. The following paper presents experimental results on two simpIe

systems,namelysolidwhite phosphorusconsistingof P4 tetrahedra, and solid iron pentacarbonyl consisting of Fe(CO)5 trigoual bipyramids.

218

H.W.Spies, Molecubr motion from NMR pm&~ ~ectra I

2. Lineshapc calculation

2.1. Theoreticctldescription

The NMRpowder spectrum is a superposition of spectra for the different nuclei in the sample. For simplicity we shall consider explicitly only the case where the nuclei are positioned on a 3-, 4. or 6.fold symmetry axis, for which the nuclear magnetic shielding tensor is axially symmetric. Typical examples are: the nuclei at the comers of a regularoctahedron or tetrahedron. For a rigid solid, where all types of reorientation have ceased to exist, the NMR frequency for each nucleus depends only on the orientation of ff,, relative to the corresponding principal axes system of e and is independent of the frequencies of all other nuclei. The lineshape function of the powder spectrum I(w) is then simply given by I(o)=SSl’(w,B,~)sinBdBd9.

0)

I’(w, 8,#) is the lineshape function for all the nuclei, for which the orientation of the magnetic field Ho relative to the principal axes system of o is givenby the polar angles8 and 4. If we neglect dipolar coupling, magnetic field inhomogeneities,etc.,I’(w, 0, 9) is givenby a h-function at o = w1 where w, is the resonance frequency of the nuclei in that orientation and is a function of 0 and I#LFor an axially symmetric u tensor a @dependence does not exist and the integration over d is trivial. In order to be able to introduce ihe effect of mom tion, we shall rewrite eq. (I) so that the signalis a superposition of spectra due to individualmolecules rather than individualnuclei:

the Oi are determined by 9 and IQand are not treated ;1sindependent variables. Clearly in the limiting case, where the molecules are fued in space, the result is the same in both cases. JZq.(2) is particularly suited, however, to introduce the motion of the spins since only the spectra of the individualmolecules are affected by the motion and the integration over all possibleorientations can be performed as before. If we introduce the frequency Q, which may, but need not be a singleparameter, to describe the motion of the spins, the nuclear signalwill now depend on R in the followingway: Jw,W =JJGJ,

kJ , , .... wn], S&9, ~1 sin 9d9dg, (3)

where f’(w, [w,, .... wn], !2,19,q) describes the spectrum of a molecule under the influence of the motion of the spins.This lineshape for a jumping molecule can now be calculated in the usual way one treats spin exchange [8,9]: I’(@, [a, I ..-t w,],~,S,~)=Re{W*A-‘*1),

(4)

where the components Wiof the ndimensional vector Ware the probabilities of the system having the frequency wi, 1 is an n-dimensionalunit vector and the ndimensional matrix A is defined by A=i(o-wE)trr.

(5)

where now 9, IPare the polar anglesdescribingthe direction of HOrelative to the molecules. &J, I-, , yf’ ---,CA+],19,lp) is the lineshapefunction

Here o and WE are diagonal matrices with elements wi and the constant w, respectively. The transitions between the n frequencies Widue to the motion are described by the matrix I, the elements of which are linear combinations of the frequencies R. In the most simple case of an octahedron or tetrahedron one will assumethat the motion can be described by a singlefrequency, the inverse mean lifetime of a given configuration 7-l. But it is clear that the treatment is completely general,so that more complicated types of motion. where rotations about different axes have different frequencies, can be treated in exactly the same way. The same holds for the case

for the specttun of a set of molecules with the same

where the shie!ding tensors are not axially symmetric.

(2)

orientation with respect to Ho. This spectrum consists of up to n equal peaks at frequencieswl, w2, .... w, for the n magneticallydifferent sites in the molecule. In order to indicate this, the frequencies Wi are put as parameters in extra parentheses. Note, however, that

2.2. Computational aspects

The numerical calculation of the lineshape function f(w) according to eq. (3) can be divided into two parts.

H. W. Spiess, Moledar

morionfromMHR powderspecrm.

First, the calculation of the spectra for molecules under the influence of jumps after eqs. (4) and (5) and second, the integration over all possible orientations of molecules in the powder. Our aim was to keep the calculation as general as possible. so that the generalizations mentioned at the end of the last section can easily be incorporated. Therefore we do the integration directly without applying expansion techniques [7]. In other words, we calculate the spectra for a number of selected orientations of the molecules relative to Ho and ask for the statistical weight of these spectra. As a consequence, the first part of the numerical computation, the calculation of spectra for the molecules, is straightforward and need not be discussed here. For the discussion of the integration we have to consider the spectra of the molecules in fixed orientations only, since the effect of motion of the molecules is introduced independently from the integration. A “brute force” integration is possible because the part of the sphere over which the integration must be done is greatly reduced by the symmetry of the problem. Therefore we shall now first discuss some symmetry arguments, then describe the “brute force” integration, and fiially discuss an approximate method which was actually used in the calculations. The symmetry of the problem may be considerably higher than the symmetry of the molecule. The reason is that we need only consider symmetric second rank tensors because only the symmetric part of ct enters into the NMR frequencies in first order (see also ref. [lo]). The increase in symmetry is especially pronounced if we treat a shielding tensor in an approximation as being axially symmetric, although it is not required to be so by molecular symmetry. We will assume in the following that the system has at least one twofold symmetry axis. Without loss of generality we can take the axis of highest symmetry as the z axis (9 = 0) of a spherical coordinate system. Because of the inversion symmetry of symmetric second rank tensors even fold rotation axes are always connected

with the corresponding

mirror planes. As a

I

x9

hexagonal symmetry and is limited to 180” in the

cases we are considering. In order to perform the numerical calculation of I(w) given in eq. (3), we have to sample the frequency. We will only consider axially symmetric cr tensors explicitly. The unique axis will be called z axis or [Iaxis. Then the x and y elements of o are equal and wiUbe labelled L The frequency interval that we have to sample for a non-moving molecule is then Aw = o,, al. There are obviously two ways to sample the frequency. The more appeating one at first sight is to sample ihe frequency in equal intervals. Then the angles 9 and up.for which we calculate spectra of mofecules, have to be chosen accordingly. AU pairs of 9, cp for which the frequency for a given nucleus of the molecule is the same, define a curve of constant frequency, which could be observed in a singlecrystal Ill]. If we are defmg with axially symmetric shield-

ing tensors, these curvesare circles (tlor great circles, however) about the unique axes of the shielding tensors. Each sampling point in frequency then corresponds to a set of curves of constant frequency, one for each nucleus in the molecule. These curves cross each other and divide the surface of the sphere into polygons. As an example fig. la shows such curves of constant frequency for the nuclei at the corner of an octahedron. In this case we have three sets of curves of constant frequency and by symmetry three curves - all belongingto the same division of the frequency interval Aw - always cross each other in rare point. Thus the sphere is divided in triangles the sides of which are in general not geodesic lines, however, and each triangle corresponds to a certain spectrum. Therefore by calculating the areas of the corresponding triangles the integration over 9 and 9 can be performed. But already for a molecule with as high a symmetry as a tetrahedron the pattern of the curves of constant frequency is almost prohibitively difficult as shown in fig. lb. Now in general, curves of constant frequency belonging to the same division of Au but belooing to

consequence in most practical cases the sphere will be divided by planes at 9 = 90” and at 9 = const. Thus if one asks for aU possible orientations of Ho relative to

more than two different positions in the molecule no longer cross in one point. Therefore we have to dea1 with general polygons the sides of which are not

the molecule it is sufficient to consider a segment of the upper half of the sphere defied by two adjacent mitror planes. Typically 9 has to be varied from 0'to 90”, but the range for cpcan be as small as 30” for

geodesic lines. In principle, however, one can proceed as before. A possible way to do this is described in the appendix. It should be mentioned that the calculation of the areas of the spherical polygons has to be done

220

H.W.

Spiess,

Molecuhr

motionfrom IWR powder spectra.I

al

Fig. 1. Curves of constant frequency for (a) an octahedron and (b) a tetrebedronin stereographicprojection.The unique axes of the a tensors for the different positions in the molecule and the mrnxponding frequenciesare labekd ri and wi, respectively.The frequencieswi a WI t KAWare labelled after Y rangingfrom 0 to 1.

cule, described by 8 and cp,will often correspond to frequencies that are different from our sampling frequencies for the other 0 tensors in the molecule. One therefore has to take the closest samplingfrequency instead. As the samplingfrequencies are not distributed uniformly over the range from w,, to wlr the intensities calculated for a given samplingfrequency have to be divided by dw/d9’ at this frequency, where 9’ is the polar angle of Ho in the principal axes system of o. The whole procedure is, of course, only an approximation, but has the advantageof simplicity and can be generalizedeasily. The 9 and q intervalsare chosen so that one obtams a fair description of the line shape in the rigid case. Fig. 2 showssuch spectra for slowjump frequencies, Awr = 500. The full line is the ideal powder pattern, the dots are intensities calculated in the way described for a tetrahedron and a trigonal bipyramid. There are deviations from the ideal powder pattern, but they are relatively small. In order to further reduce the errors introduced by our approximate way of integration in the actual calculations for arbitrary jump frequencies, the scaled differences to the corresponding slow limit spectra were calculated, then subtracted from the ideal powder pattern and 50meuuw m00thhg applied. Typically the intensities were calculated for 20 frequency points distributed over the spectrum and for about 200 orientations specified by 0 and cp. After interpolation between the samplingfrequencies the spectrum was convoluted with a gaussianto take care of dipolar line broadening.

3. Results and discussion 3.1. Fast exchange limit

only once and does not have to be repeated if one wants to calculate the lineshapesfor different values of the jump frequency for the same molecule. We can, however, easily perform the integration approximately, if we do not require that the sampling points of frequency divide the interval AU in equal increments. Then we can vary 19and IJJlinearly. By varying a9linearly, a set of frequencieswithin the range of w1 and w,, is obtained for the u tensor the unique axis of which is along 9 = 0’. These frequencies are taken as the samplingpoints. If one now varies cp also linearly, the directions of ii0 relative to the mole-

Before discussingthe results for arbitrary jump frequencies,it is worthwhile to briefly look at the fast exchange limit, where the jump frequencies are fast compared with the NMRline splittings, thus 7-l S AU. Becauseof our way of samplingdescnied in the preceding section, our calculation will not be able to predict deviations from a lorentzian in the fast exchangelimit, as obtained for an octahedron by Alexander et al. [7]. At the present stage of actually achievableline narrowingin NMR [ 1,2], such effects could easily be missed in the experimental spectrum

H. W.Spiess, Molecular motion from NMR powder spectra. I

221

L

Fig. 2. Powder pattern due to axially symmetric shielding lensors in the limit of slow motion calculatedfor a teLrahedron and a trigonal bipyramid (for details see text).

anyway. Thereforewe willnot giveresults of numerical calculations here. In the fast exchange limit the powder spectrum will, in a first approximation, simply be given by the weighted averageof the shielding tensors for the individualpositions in the molecule that the spin can experience during the jump process. Since all IYtensors can be treated as symmetric second rank tensors (see above), the averageover different shieldingtensors will alwaysbe a symmetric tensor again and therefore GXI be dkgonalized. Thus, even in the most general case, one will observe in the fast exchange*tit an usualshieldingtensor with in general three different principal elements. An example where the a tensor in the fast ex-

change limit will actually have three different principal elements, although the shieldingtensor is axially symmetric for the nucleus at rest, is a nucleusjumping between two equivalent positions, both havingaxially symmetric shieldingtensors with elements u,, and al. Here the indices IIand 1 refer to the unique direction of the shieldingtensor one wou[d observe if the jump frequency were zero. In the rigid case the anisotropy of the shieldingwould thus be PO = u,, - uL.The two unique axes define a plane in which two of the principal axes of the effective shieldingtensor lie, one along the bisector, the second perpendicular to it, the third perpendicular to the plane (see fig. 3). For different anglesof a. which is half the angle between the unique axes, the effective principal o components are plotted in fig. 3 for convenience. Only for the trivial case, Q= 0” or a = 90”, and for a = 45’, the effective shieldingtensor is axially symmetric. Since uXX,uuu, and uzr define the edges of

Fig. 3. Efiective @elements for a nucleusjumping rapidly between two sites. both having axially symmetric shielding tensors with unique axes z t and z2 as a function of the angle between .z~and ~2.

the powder spectrum, it is clear that for only two sites

even in the fast exchange limit the powder pattern e% tends over at least half of the frequency range it would cover in the rigid case. If the system has higher symmetry, however, the width of the powder spectrum in the fast exchange limit decreasesconsiderably compared to the rigid case. The limit is reached for cubic moleculeswhich jump between their equivalent positions, where the spectrum consistsof a singleline, whose frequency is determined by f(u,,+2u,). Cubic symmetry is not possible for, e.g., five positions of the same nucleus. Five positions can occur, however, at the comers of a trigonal bipyramid, an example is Fe(CO)5.if we assume that the d tensors are all axially symmetric with the same principal elements, then in the fast exchange limit we will observe also an axially symmetric tensor. But, if in the rigid case the anisotropy of a is Au = CT,, - u,, the apparent anisotropy will then be reduced to Au’ = t$Au. 3.2. Numeriazl adculotion for arbitrmy iumpfiequenties

Numerical calculations for arbitrary jump frequencies were performed for an octahedron, a tetrahedron,

222

H. W. Spies Molecab mot& from NhfR powdw spectra. I

and a trigonal bipyramid. In the first two cases the nuclei are situated on a 4- or 3-fold axis and the e tensor therefore is axially symmetric. For the trigonal bipyramid it was assumed that ah u tensors are also axialfy symmetric, although for the three equatorial positions they are not required to be so by symmetry. The trigonal bipyramid was treated having in mind the % resonance in solid Fe(CO), studied experimentally [ 121, for whichthis assumptioncan be justified [lZ].The other experimental example [12] is the 3rP resonance in solid white phosphorus consisting of P, tetrahedra. Therefore most of the spectra discussed will be of tetrahedra and the values for the jump frequencies 7-l were mostly chosen to correspond to experimental spectra presented in the following paper. For all spectra shown the values for the dipolar linewidth was chosen to be 26 = &JALL Similar to the usual spin exchange in Iiquids [8,9] we expect that reliable information about both, the width of the spectrum in the rigid case and the jump frequency, can be obtained from the spectrum for intermediate jump frequencies, Aor z 1. This is borne out by the calculated spectra shown in fii. 4 for a tetrahedron which jumps between its equilibrium positions. Consider, e.g.. spectrum 4(d), for which Awr = 1.5. Although most of the intensity is already concentrated near the center, the edges of the powder spectrum in the rigid case both are developed already. Probably most unexpected is the observation that as the jump frequency is reduced one does not directly get the usual powder pattern after the central peak has disappeared. Instead we observe an extra hump near w,, for br = 10 [fig. 4(b)]. In order to understand these spectra qualitatively, in fig. 5 some spectra of individual molecules are plotted for the Samevalues of the jump frequency as in fig. 4. Two possible orientations of the tetrahedron were chosen, one where the spectrum consists of four lines distributed over the frequency range of the powder spectrum (fig. Sa), and one where a single line near wp is separated from the other three (fig. Sb). The spbctra are drawn on scale, so that the area underneath’the line is always the same. Let us fust look at the region near o,,: For slow jump frequencies we have well separated lines and the intensity in the wingsof each line is low. As the jump frequency increases, the lines broaden and thus contain more intensity in the wings.This extra intensity

Fig. 4. Lineshapesdue to axially symmetric shielding tensors for a jumping tetrahedronfor different jump frequencies. 1

b) . f

,

ih

AU:

Fig. 5. Cahlated spew for two orientations of a tetrahedron relative to HO for the samejump frequenciesas in f~. 4. The polar an&es of HOin a coordinate systemas ia f~.1withr~at~=Oandzaatrp=O~e:(a)~=70~.lp= 90”;and(b)6=10”,9=900.

H. W.Spiess.Mokdzr

morion fmm NMU powder spectra.I

in the wingspiles up at the frequencies close to w,,. A similareffect happens to frequenciesnear wI, where the intensity also increasesunder the influence of the motion. This extra intensity is taken away from the central region of the spectrum. Thus for slowjump frequencies there exists a range for r-1, where the intensity both near w,, and wI is h@zer than in the rigjd case, whereasit is lower than in the rigid case in the middle of the spectrum. This results il an extra hump which can clearly be seen in the experimental spectra 1121,although the loss in intensity in the central region is only about 7% of the total intensity of the powder spectrum. For higherjump frequencies(AWT= 1.5) we see that the spectra of the jumping moleculesare very similardespite of the different spectra for the molecules at rest. The highest peak is found already at the center of the powder spectrum, but appreciableintensity is still spread over the whole frequency range of the powder spectrum. ‘Thisexplains that even for relatively highjump frequencies the width the spectrum will have in the rigid case manifests itself in the spectrum (see also ref. [I 21). It is interesting to compare these lineshapeswith lineshapecalculationsfor viscousliquids. Sillescu 16] has compared the ESR lineshapesfor a spin S = l/2 and an axMy symmetric g tensor for slow reorientations described by a brownian rotational motion model and by a rotational jump model. In fig. 6 a direct comparisonof his results with the lineshapesof a jumping tetrahedron in the solid is given. For intermediatejump frequencies(fig. 6c, AW = I .9) the differences for the solid and both liquid models are quite strong and it should be possibleto observe them experimentally. This comparisonalso shows that some of the characteristicsof the solid state spectra are due to thz fact that the jumps occur rhroughfied angles,whereas the jump angle is random in a liquid. Therefore it should be possibleto distinguish between solid type and liquid type reorientational processesfrom the spectra. An experimental example for the rotational jump model has recently been established [I31 and the followingpaper [12] will giveexamples for reorientations in solidsand their effect on the powder spectra. In order to check whether one can easiIydistinguishbetween different motions ti the solid, ir, fig. 7 the powder spectra for a jumping octahzdror, tetrahedron, and trigonal bipyramid are

223

Fig.6. Lineshapes for solid type and liquid type reorientational models: ( -) jumpingtetrahedron (this work), (----)brownianro~tionalmotionmodol.and(....)cotational jump model,bath takenfromref. 161.

L

Fig.7. Lineshppes for ajumpingocttcdron (a). tctmhedron (3). andUigoti bipyramid (c). Out = 1.9 iu all threeGUCS. shown for the same value of the jump frequency (Awt = 1.9). In all three cases the spectrum is basically the same. At first sight, the spectrum for the trigonal bipyramid looks different from the other two spectra, but from fig. 4 it is clear that the difference from, e.g., the spectrum for a jumping tetrahedron, could be removed by usinga somewhatdifferent value for the jump frequency. The spectrum calculated for an octahedron is in good agreement with the results obtained by Alexanderet al. [7] using expansion techniques. It is, therefore, not likely that one can in generaldistingu’rshunambiguouslybetween different

H. W. Spies, Mole&v

224

motion

types of reorientation in solids, especially if only a limited region of jump frequencies is accessibleexperimentally. In caseswhere the rapid exchange limit is reached in the solid, however; one can distinguish between a rotation about a singleaxis, resulting always in an axially symmetric u tensor [5], and more general types of motion discussedin the beginningof this section.

from NMR powder sycttn. I Adtnowkdgemcnt The author would like to thank Professor K.H. Hausserfor his interest in this work. Many stimulating discussionswith Dr. U. Haeberlen concerning this problem were especially helpful and are gratefully acknowledged.

4. Summary and outlook It is clear that by studying NMR powder spectra in

In this appendix we want to describe a method to calculate the area of a general polygon encountered if

solids governed by anisotropic shielding,information about molecular motion can be obtained directly from the spectra. The frequency range of the motion, for which the lineshape is sensitive, is typically of the

one divides the surface of a sphere by curves of constant frequency (see section 2.2). Clearly the polygon can be divided by great circles so that most of its area is covered by spherical triangles.The problem then

order of l-100

reduces to calculatingthe area of a spherical lune, defined by a great circle between two points 1 and 2 (see fig. 8) and a smallcircle connecting the same points. After transformation the pole of the sphere can be chosen as the center of the small circle as shown in fig. 8. The area of the spherical lune is given

kliz, thus it is comparable with the

frequency range covered by T,,, TIPand in favourable cases T1 measurements.There are obviously advantagesin getting the information from the spectrum directly rather than from relaxation time measurements: The experiment is much simpler and less time

consuming.The results of relaxation studies are also often changed by paramagneticimpurities, whereas their effect on the spectra will be negligible. By analysis of the lineshape one can distinguishto some ex. tent between different types of restricted

by

motion.

me spectra contain, of course, also information about the motion, if they are governed by dipolar coupling. !t should be clear from fig. 4, however, that the line&ape is much more sensitive towards the motion if it is governed by an axially symmetric shielding tensor

than if it is dominated by dipolar coupling, where one gets the information mainly from the linewidth of a symmetric line [9]. A word of caution, however, seems to be appropriate here: Not all deviations from the ideal powder pattern that one observesare due to motion of the spins.Preferential orientation of the crystallites in the powder due to sample preparation [ 141 and partial saturation due to an angulardependence of the relaxation times [ 151 are probably the most tammon sources for such deviations. In favourable cases,however, it will be possible to do both, analysisof the spectra and relaxation studies, so that the results of ouz numerical . calculati& can be checked. This is done in the following paper [ 121.

The function QJ) can easily be obtained because it is the equation of the great circle through the points 1 and 2. The center of this great circle has the coordinates 9, and vp. withIp, = f(ql+&. We can always redefme rp.so that cp = 0 without loss of generality. We then obtain for I!(V): s(s) = arc cot(-tanfil, coslp) ,

(A-2)

where cot00 tans,

=--

and

cosA$/2

hp = IP~-P~I.

(A.3)

By substituting (A.2) into (A.1) we obtain &IQ F= &cosbg

-

1 -A&

cos [arc cot(c COS+O)] dplJA.4)

withc = -tan 19~.Becauseof the symmetry of the sphericallune we can change the limits of integration sothat

H.W.Spiess, Molecubr

motionfrom NMR powder

spectra I

225

the sphere by curvesof constaut frequency for axiahy symmetric shieldingtensors.

References [l] JS. Waugh. L. Huber and U. Haeberlen.Phys. Rev. Fig. 8. Spherical lune de&bed by the great circle connecting points 1 and 2 and the small circlefor 9 = 60 through 1and 2.

&I2 F= I&costY()

- 2 $

cos

[arc cot (c cosp)] dlpl.

0

(A.43)

The integral can be obtained in analytical form by a series of substitutions:

s cos [arc cot(c co@] dp (A-5) = -arc tan{c2 - (c2t 1) cos* [arccot(c co~+7)]}-“~, where cphas to be in the interval 0 f cpG II. With OUI limits of integration the expression for the area is further simplifiedbecause for cp= 0 (A.5) givesftr and arc cot (c cost) = 19, for 9 = N/2. We then get the foUotig expression for the area of the sphericallune: F=I~cos90-~++2arctan[c2-(c2+1)cos2bo]-f’2 (A.61 This enables us to calculate the area of the most general polygon that we can encounter by dividing

Letters 20 (1968) 160. [2] A. Pines, M.G. Gabby and IS. Waugh. J. Chem. Phyr 56 (1972) 1776. 131 D.C. Ailion, Advan. Magn. Res. 5 (1971) 177, see aho: 0. Lauer, D. Sfehhk and K-H. Rausser. J. Magn. Res. 6 (1972) 524. [4] A. Pines, M.C. Cibby and IS. Waugh.Chem. Phys. Letters IS (1972) 373. [S) M. Mchrhtg, R.G. Grtftim and J.S. Waugh, I. Chem. Phys. 55 (1971) 746. 161 H. SiBescu, J. Chem. Phys. 54 (1971) 2110. [7] S. Alcxandc:. A. Baram and Z. Luz. MoL Phyr 27 (1974) 441. [ 81 P.W. Anderson and P.R. Weiss, I. Phys. 8oc Japan 9 (1954) 316. [9J A. Abragam. The principles of nuclear magnetism. (Oxford University Press. London, 1961). [lo] J.A. We&T. Buch and I.E. Clapp, Advan. Magn. Res. 6 (1973) 183. [ 111 H. Hartmann. M. Fleissner and H. SiUcsnr. Theor. Chim. Acta 3 (1965) 347. [ 121 H.W. Spiess, R. Grosescu and U. Haeberlen. Chem. Pbyr 6 (1974) 226. [ 131 K. Hensen, W.O. Riede, H. Sikscu and A. v. WittgenStein, J. Cbem. Phyr, to be published. [ 141 I. Kcmpf. H.W.Spicss,U. Haebcrlcnand H. Zimmermarm, Chcm. Phys. 4 (1974) 269. [ 151 A. Pines. M.G. Gibby and JS. Waugh. I. Chem. Phys 59 (1973) 569.