NMR spectra of reorienting nuclear pairs in solids. II. Unequal residence times, and conformational order-disorder

JOURNAL

OF MAGNETIC

RESONANCE

9, 108-113 (1973)

NMR Spectra of Reorienting Nuclear Pairs in Solids. II. Unequal ResidenceTimes, and Conformational Order-Disorder E. R. ANDREW Department

ofPhysics,

University

of Nottingham,

Nottingham,

England

Received May 30,1972 The nuclear magnetic resonance spectrum is calculated for a solid containing relatively isolated pairs of nuclei where there is rapid reorientation of the pairs between two fixed directions. The calculations have been extended to include the situation in which the proportions of time spent in the two orientations are unequal. Expressions are given for the profile of the spectrum for polycrystalline material as a function of both the angle between the two fixed directions and the proportion of time spent in each orientation. The second moment of the spectrum is derived and also the line width between singularities for each case. Reorientation between two directions with unequal residence times is encountered in conformational orderdisorder phenomena in solids, and the theory is considered in connection with solid cyclobutane. INTRODUCTION

In a previous paper (I), to be referred to as I, an investigation was made of the changes in the NMR spectrum of a solid containing relatively isolated nuclear pairs when rapid reorientation of the pairs takes place between two fixed directions. The spectral profile, the second moment, and the line width between singularities were all evaluated for polycrystalline specimens as a function of the angle 2y between the two fixed directions. Equal residence times were assumed, each pair vector spending equal times along its two equally probable directions. Although the argument was quite general, it was prompted by consideration of conformational motion (2) in puckered ring molecules in solids. If such molecules contain methylene groups, interconversion between conformers of the kind known to take place in the liquid state would cause the proton pair vectors to reorient between two fixed directions in the crystal, and it proved possible to account for the motional narrowing of the NMR spectra of the low temperature phase of solid cyclobutane (3) in terms of such conformational motion of the molecules (I). It has been suggested (4) that on heating from a low temperature there is a progressive decrease of conformational preference leading to a cooperative conformational order-disorder process, mirrored in the specific heat of the solid. As the critical temperature of the orderdisorder process is reached, the preference for the two conformations becomes equal (4> and the theory based on equal residence times (1) is applicable. At lower temperatures one molecular conformation has greater preference than the other on a given crystal site, leading to complete order at the lowest temperatures. Each nuclear pair spends a fraction p of its time along one orientation and a fraction (1 - p) along the other. As the disordering process progresses, p changes from 1 at the lowest temperatures to + at the critical temperature. Copyright 0 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.

108

NUCLEAR

PAIRS

WITH

UNEQUAL

RESIDENCE

TIMES

109

The purpose of the present paper is to extend the earlier calculations of spectral profile, second moment, and line width, from the special case of p = 1 -p = 3 to the general values ofp encountered in the course of the disordering process. The argument will however not be confined to this particular application, and we shall deal with the general case of a solid containing reorienting nuclear pairs, each spending a fraction p of its time along one direction and a fraction (1 -p) along another. CALCULATION

OF SPECTRA

Consider a solid containing relatively isolated identical nuclear pairs of spin 3. The directions in the system are indicated stereographically in Fig. 1. Each nuclear pair has two orientations RI and R2, the angle between them subtending 2~. The two initial sites of the nuclei in each pair and their final sites are not necessarily coplanar.

FIG. 1. Stereogram showing the relevant directions. RI and Rz are the two directions taken by each nuclear pair vector with probabilityp and (1 - p), respectively. H is the direction of the applied magnetic field.

The pole C of the stereogram will be selected for convenience presently, but lies on R, R,. The angles between the direction of the applied magnetic field H and RI, R,, and C are 8,, 02, and CI,respectively; 4 is the azimuth angle of H. The angles between RI or R2 and Care y1 and y2, respectively, with y1 + yz = 2~. The NMR spectrum is determined (5) by treating the nuclear dipolar interactions as a perturbation on the Zeeman interaction between the nuclei and the applied magnetic field. For one nuclear pair in the system the truncated dipolar Hamiltonian is Zd = +yl yz h2 r -“(I1 I, - 31,, I& (3 co.9 B - l),

Ill

where yl, y2 are the magnetogyric ratios of the nuclei in the pair, r is the internuclear pair vector, and 8 is the angle between r and H. If r moves rapidly between the two directions R, and R,, spending a fraction p of its time along RI and a fraction (1 - p) along R2, and none along any other direction, we are required (6, 7) to replace L& by its average G??‘,,over the two orientations before calculating the energy eigenvalues of the system and evaluating the spectrum in the vicinity of the unperturbed resonance

110

ANDREW

frequency. Since 19is the only time-dependent variable in Eq. [l] we find the spectrum for an array of similarly oriented pairs consists of two lines of equal intensity (5, 6) at field strengths h = j&(3 cosz e - l), PI where h=H-H*, [31 and H* is the unperturbed resonant magnetic field strength for the frequency of the applied electromagnetic radiation, taken to be fixed. In Eq. [2] ho = $pre3,

[41

where p is the magnetic moment of both nuclei in each pair if they are identical. If the two nuclei in each pair are not identical, then ho = pre3,

[51

where p is now the magnetic moment of the nonresonant nuclei. From Eq. [2] we find that the two spectral lines occur at h = i-h&(3

co2 O1- 1) + (1 - p) (3 cosz e2 - 1))

Fl

= &ho(3 cos2 oz{pcos2 y1 -t (1 - p) cos2 y2}

+ 6 sin a cos cccos 4{p sin y1 cos y1 - (1 - p) sin y2 cos y2} + 3 sin2 a cos2 4{p sin2 y1 + (1 -p) sin2 y2) - 1).

171

We now choose the pole C of the stereogram such that p sin y1 cos y1 = (1 - p) sin y2 cos y2, or tan2yl = (I -p)sin4y/[p

+ (1 -p)cos4y],

PI

since y1+y2=2y.

[93

With this choice, [7] reduces to

where

h = +(h, cos2 CL+ h2 sin2 TVcos2+ - h,,),

WI

h, = 3h& cos2 y1 + (1 - p) cos2 y2),

[111

h, = 3ho(p sin2 y1 + (1 - p) sin2 y2).

WI

The physical significance of this choice of pole C is as follows. The dipolar interaction is represented by a second-rank axially symmetric tensor. In forming the average dipolar interaction we are adding one tensor of weightp with its axis along R, to another similar tensor of weight (1 -p) with its axis along R2. The resultant tensor is in general not axially symmetric. As chosen, C is one of the principal axes of this resultant tensor, D is another, and the third is normal to the plane of RI R2 and C. In the special case p = 1 -p = 5, Eq. [9] shows that y1 = y2 = y, and C is the bisector of the two orientations R, and R2. At the other extreme, whenp = 1, y1 = 0 and yz = 2y, and RI becomes the pole of the stereogram as one expects.

NUCLEAR PAIRS WITH UNEQUAL RESIDENCE TIMES

111

First we calculate the second moment M, of the spectrum for polycrystalline material. From Eq. [lo] we have (h, cos2 a + h2 sin2 cccos2 4 - hJ2 dq5 = +h;{ 1 - (h, h,/3h;)}.

P31 In absence of reorientational motion the second moment of the isotropically distributed identical pairs in the polycrystalline material is readily found to be

The reorientational motion therefore reduces the contribution second moment of the spectrum by a factor M2l442.0

I

= 1 - (h, h,/3h:).

of the pairs to the lJ51

a

-2

J

*6 P

P

FIG. 2. (a) Variation with residence parameter p of the second moment M2 of the NMR spectrum of polycrystalline material containing nuclear pairs reorienting between two directions subtending an angle 2~. The variation is shown for values of y from 0” to 45”; the behavior for values of y from 45” to 90” is identical with that for 90’ - y. (b) Variation with residence parameterp of the line width dH between singularities of the NMR spectrum for selected values of y.

For given values of y and p this factor is readily evaluated using Eqs. [8], [9], [l l] and [12], and is plotted in Fig. 2(a). For a given value of y the reduction is greatest when p = 3. The reduction factor for y and for (7r/2) - y is the same, and it is therefore only necessary to show curves for y up to rr/4. We now derive the spectral profile for a polycrystalline assembly of reorienting pairs. From each crystallite we obtain a pair of lines given by Eq. [IO] with Eqs. [I l] and [12]. We consider first the positive sign of Eq. [lo]; the negative sign generates an exactly similar spectrum mirrored in h = 0, obtained by replacing ho by -ho throughout, Then following an argument exactly similar to that in I we find that the normalized spectral shape is given by for-h,
[I61

112

ANDREW

g(h) = (274-l (h + h&l’2 (h, - hJ-1’2K(k-l),

I171

where

WI

kZ = (h + h,) (h, - h,)/h,(h, - ho - h),

and K(k) is the complete normal elliptical integral of the first kind of argument k. In addition, there are two similar equations to Eqs. [16] and [17] in which ho is replaced by -ho. For convenience it has been assumed that h1 > h,; however, if h, < h, we merely interchange hl and h,. The choice of C as reference direction in Fig. 1 has enabled the spectral profile to be cast in the same form as in I, but with the more general definitions for hl and h2 given in Eqs. [l l] and [ 121. p *5 *6 .7 -8 .9

I x=30°

FIG. 3. Calculated NMR spectra for polycrystalline material containing nuclear pairs reorienting between two directions subtending an angle 2y of W, for values of residence parameter p from 3 to 1; the spectra for values of p from 0 to 4 are identical with those for (1 - p).

Examples of spectral profiles computed from Eqs. [16] and [17] together with Eqs. [8], [9], [ll], [12] are shown in Fig. 3. Singularities occur at &t(h, - h21. If the further broadening of these spectra by interactions between neighboring nuclear pairs is not too great, the spectra will exhibit maxima close to these singularities. Defining a spectral linewidth AH as the interval between the two singularities we thus find AH/AH,

= 1 -p = 11- 3{psin2y1 + (1 -P)sin2y2}l, v91 0i I where AHo is the line width in the absence of reorientational motion, equal to 2ho. The singularities coincide, whenever p sin2 y1 + (1 -p) sin2 yz = 3.

WI.

Forp = 3, Eq. [20] with Eqs. [8] and [9] shows this occurs for y = sinW1(1/1/3) = 35”16’. The variation of dH/dH, with p for selected values of y is shown in Fig. 2(b). It will be noticed that for small values of y the variation of second moment [Fig. 2(a)] and line width [Fig. 2(b)] with the parameter p is very small in the vicinity of the minimum. Thus for a moIecule such as cyclobutane with y = 17” (I) the earlier calculations assuming equal residence times (p = 3) will be almost exactly correct just

NUCLEAR

PAIRS

WITH

UNEQUAL

RESIDENCE

TIMES

113

below the phase transition temperature, 146 K, even if the disordering process is not quite complete. Indeed for values ofp between 0.4 and 0.6 the variation in line width is only 1%. REFERENCES

1. E. R. ANDREW AND J. R. BROOKEMAN,J. Magn. Resonance 2,259 (1970). 2. E. R. ANDREW, Magnetic Resonance and Related Phenomena, Proc. XVI Ampere Congress, Bucharest, 11 (1971). 3. M. J. R. HOCH AND F. A. RUSHWORTH,Proc. Phys. Sot. 83,949 (1964). 4. E. R. ANDREW, Phys. Lett. A 34,30 (1971). 5. G. E. PAKE, J. Chem. Phys. 16,327 (1948). 6. H. S. GUTOWSKY AND G. E. PAKE, J. Chem. Phys. 18,162 (1950). 7. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Oxford University Press, London, 1961.