Free precession decay of nuclear pairs in solids

JOURNAL

OF MAGNETIC

RESONANCE

6,466-474

(1972)

Free PrecessionDecay of Nuclear Pairs in Solids B. A. VAN BAREN, S. EMID, CHR. STEENBERGEN, AND R. A. WIND Department

of Applied Lorentzweg

Physics, Delft University 1, Delft, the Netherland.

of Technology,

Presented at the Fourth International Symposium on Magnetic Resonance, Israel, August, 1971 The nodes in the free precession decay of proton pairs in a rigid lattice, with dipolar interaction between the protons of a pair, are determined by both the distance between the protons and the possible motion of the pairs. Interaction between the pairs does not alter the place of these nodes. When using the Fourier transform of a cw lock-in experiment, the amplitude of the modulation field also appears to have no influence on the nodes, whereas they are only slightly altered because of saturation of the spin system. From the nodes, proton-proton distances of gypsum, cyanamide and methylene chloride have been determined. INTRODUCTION From the line shape or the free precession decay (f.p.d.) of proton pairs in a rigid lattice the distance between the protons in the pair can be obtained (I, 2). The interaction between the pairs has been taken into account simply by introducing a broadening function. We shall do this in a physically more adequate way. The result is that the spin-spin distance within a pair still can be determined from the nodes of the f.p.d. We studied the f.p.d. from the Fourier transform of a continuous wave (cw) lock-in experiment. Alternatively, one can use spin-echo equipment and observe the f.p.d. more directly after a 90” pulse, but then problems arise such as defining the end of the pulse and the correction for the bandwidth of the receiving coil (3). Both effects influence the shape of the decay and especially the position of the nodes from which the protonproton distance is obtained. Furthermore, the full f.p.d. cannot be measured because of the dead time of the receiver after the 90” pulse. The influence of the modulation amplitude and saturation on the f.p.d. is investigated. We use the theory for the determination of the proton-proton distances of gypsum, cyanamide and methylene chloride. FREE PRECESSION

DECAY

OF NUCLEAR

PAIRS

The f.p.d. of randomly oriented spin-$ pairs in a rigid lattice has been derived by Look et al. (2) using the Fourier transform of the line shape given by Pake (I). We shall ,derive the f.p.d. in a somewhat different way because this will deepen the insight into the way in which the interaction between the pairs can be taken into account. We first consider a fully isolated spin-+ pair in a static field B0 = w,,/y along the z direction. If only dipolar spin-spin interaction is involved, the spectrum of such a pair Q 1972 by Academic

Press, Inc.

466

FREE

consists of the well-known w. - tc (I), where

PRECBSION

DECAY

OF NUCLEAR

467

P.4IRS

doublet, with resonance lines at frequencies w. -C a and

a=(3y2ki,Ir3)(1

-3cos*i9)=p(l

[II

-3cosz@,

using r and 0 as polar coordina:es of the dipolar axis. If thi: pair is not fully isolared, the Hamiltonian H,, of the pair (spins i,j) becomes HP = fiwo(Zzi + Zzj)

t

Hz +

Ci,{Zzi Zz, - $(l+i I-, + Z-i Z+k)}

1 k( # id)

+

1

k(fiJ)

Cjk(,‘zj

Izk

-

$cz+j

lk

f

I-j

PI

I+k)},

where Chk = +y* %(I - 3 cos* dhk)r ;;‘,

11= Lj.

[31

The first two parts on the right hand side of Eq. [2] are, resJectively, the Zeeman and the secular dipolar Hamiltonia I, responsible for the doublet. The last two parts represent the dipolar interaction of rpin i andj with the other spins in the sample. Equation [2] can be rewritten as Hp zr hWo(Zzi+ Z:j) + Hi $

1

S(Cik $ Cjk)(Zzi i Zzj) Zzk

k(i’i,j) -

,,z

j) ii(Cik

+

cjk){(zA

i +

ILj)

I-k

+

(l-i

+

I-j)

I+k)

c41 Consider the right hand side of Eq. [4]. The third part is rc,sponsible for a local field dw, along the z axis at the site of the pair and it commutes Rith the first two parts. The fourth part is responsible for a local field d mfr in the x-y plane at the site of the pair. It does not commute with the first two parts and in order to preserve the doublet structure it must be small compared to H,!j, so that it can be car sidered as a perturbation, as will be assumed hereafter. This implies that r&, rjk s r3, which means that the last two parts of Eq. [4] can be neglected because Cik % Cjk. Retu -ning to the original doublet we look at the influence of Aw, and Aw,,. The local field Aw, causes a shift of the lines, so that they appear at frequencies w. + CY+ Aw, and w. - cc+ Au,. The local field dwf,. causes flip-flop transitions that limit the lifetime of a spin in a given state (4). This effect broadens the doublet. The Fourier transformation of these lines gives the f.p.d. G,(r) for this pair: G,(t)=2coscctcosAw,tA(Aw,,,t)

ISI

The function A(Ac+,, t) is the Fourier transform of one of the broadened resonance lines with the center taken as zero frequency. As the distri3ution functionsf’(cc) and ,f(Aw,, Aw,,) are independent (this follows from the above assumption), integration over all M, Aw, and Aq, yields G(r) = F(t) H(r).

161

468

VAN

BAREN

ET AL.

Note that F(t) is the Fourier transform off(a), but that H(t) is not generally the Fourier transform of,f(dw,,dW,,). (This would only be the case if ilw,, was not taken into account.)

If the dipolar axes of the spin pairs are randomly ,f’(cr)doc=f(@dO = +sin0&, F(t) = F(pt) = d~r/6pt{cosptC~(3pt)

oriented.

one finds. using

+ sinprS,(3pr)}.

[71

where C2(3pt) and S,(3pt) are the Fresnel integrals (5). The first two nodes of F(pt) are found forpt = 2.17 and 5.63. It follows from Eq. [6] that G(t) is a product function of the decays due to the interaction within the pairs and between the pairs, respectively, so that p, and hence r, still can be obtained from the measured nodes. This result was also obtained by Look et al. (2), but this was implicitly assumed by writing the line-shape function as a convolution. It is not possible to determinep from the second moment of G(t) because MzG = 4p2/5 + MzH, MZH being unknown. A criterion for the validity of Eq. [6] can be obtained by comparing the second moments of P(pt) and H(t). Because the interaction within the pair must be large compared to the interaction causing dw,,, this criterion becomes

PI

$p2 s $MzH

in which only the parts of MzH corresponding to the fourth and the sixth parts on the right hand side of Eq. [4] have been taken into account. Now if the distances within a compound are roughly known, the values in Eq. [8] can be estimated and after the measurement of G(t) they can be compared more precisely. INFLUENCE

OF

THE

MODULATION

FIELD

We now consider the influence of the modulation field, used in the case of cw lock-in detection. We take an (arbitrary) line shape g(w,), with w, = w - wO, in the region of nonsaturation. Using a sinusoidal modulation, the signal S,(wJ after selective lock-in detection is given by Andrew (6) :

m (yb,)24’’ S.&f(%) = 2 224q!(q + I)!g(2*+‘Y%)> q=o where b, is the amplitude of the modulation field. After integration and Fourier transformation of Eq. [9] the measured f.p.d. G,(t) is found to be

G,(t) = yb, G(t) 2 q+z;yi (+<)” = yb, G(t) M(t).

= ybmG(t)P-‘~Wmr)/rbmtl IlO1

The function G(t) is the f.p.d. corresponding to g(w,); J,(yb,t) is the Bessel function of the first order. Up to q = 2 the series expansion in Eq. [lo] has also been given by Fornes et al. (7). From this equation Andrew’s correction for the second moment, &y’bz, is easily found.

FREE PRECESSION

DECAY

OF NUCLEAR

PAIRS

469

From Eq. [lo] it is obvious that the signal-to-noise rati can be optimized for a rather large modulation field (e.g., in the case of a spin pair system, optimal signal-tonoiseratiointheneighborhoodo’thefirstnodeisobtainedfor:/b, = 1.84p/2.17 =0.85p). The nodes of G(t) are not chenged by the modulation. Hence in the case of spin pairsp still can be determined frljrn them. If b, is known, G(t11can be found of course by dividing the measured decay b) M(t). Moreover, Eq. [lo] F’rovides us a possibility of determining the amplitude of : he modulation field. If the ,implitude has been made large enough, the first node of W(t), given by yb,t = 3.83, becomes observable. From this node b, can be calculated. If this is not possible, measurements with at least two modulation amplitudes with a given ratio are necessary. This method has the advantage that no extra equipment is needed. This is useful especially at low temperatures where one has to measure within a criostat. SATURATION

OF

THE

SPIN

SYSTEM

As it is sometimes hardly possible to avoid saturation, especially at low temperatures where the spin-lattice relaxation time can be several hours or more, it is of interest to investigate its influence on the measured line shape. The low temperature region is often necessary to reach rigid lattice conditions. Furthermore, in order to obtain a high signal-to-noise ratio a rather large rf field is needed, which may cause saturation of the spin system. For the description of this effect we use the Provotoroc theory of saturation (8). where two spin-lattice relaxation times are involved, namely T,, for the Zeeman part of the Hamiltonian and T,, for the dipolar part. Using this t ieory one finds for lock-in detection and small yb, (9) a signal proportional to

The quantity D represents the local field and w, the amplitude of the rf field. If wigTID < 1 and w:gT,, < 1, the spin system is not saturated and Eq. [l l] turns into Eq. [9] for q = 0. After integration, g(w,) and its Fourier tnnsform G(t) are found. In order to get some information about the influence of saturation, we shall treat the special case that T,, s T,D. One frequently deals with this case at low temperatures. Assuming nwfg(O)T, D < 1 and defining Z = Tw:g(0)Tlz we get from Eq. [ 111

&(w.J= hbm WI&ld4l[l + -%/g(~))l. After integration and Fourier transformation

1121

the decay GS(l) becomes

For small Z, G,(t) corresponds with the f.p.d. as can be easily seen from the series expansion of the logarithm. If Z is not small, G,(t) will change. This however is not independent of the shape ofg(w,). If, for instance, g(wS) is a rectangle (symmetrical with respect to w,,, width 26) G,(t) will be proportional to sinbt/bt for all Z, but if g(w,) is a-more complicated function, G,(t) will change.

470

VAN

BAREN

ET AL.

We are interested, in particular, in the behavior of F(pt) (Eq. [7]), so we take am to be the corresponding line-shape function (I) and calculate F,(pr) for different values ofZ. The results are given in Fig. I. The first node of c&pt) appears earlier for increasing Z whereas extra nodes arise if Z is large enough. One also sees that the second moment of the curves (being the second derivative for t = 0) grows for increasing Z. The increase of p determined from the first node is for instance about 3 ‘:i; for Z muI. so that the decrease of r will be about 1 %. Hence it follows that saturation of the spin system does not affect the measured r very much. Moreover, correction is possible. 1 F,lpt)

I 05

0

FIG. I. The f.p.d. of randomly oriented isolated spin pairs for several values of the saturation factor Z.(a)Z-l;(b)Z=2;(c)Z=lO;(d)Z=1000. MEASUREMENTS

Figure 2 gives the Fourier transforms G(t) of the integrals of the lock-in detection signals in the case of nonsaturation and small b, of the protons in gypsum (CaS0,.2H,O), cyanamide (H,NCN) and methylene chloride (CHQ) measured at, respectively, 77,294 and 77 K. The first two samples were powdered at room temperature. This was not possible for methylene chloride, which is a liquid at this temperature, but it is still assumed to be polycrystalline at 77 K. Together with these curves, the function F(pt) corresponding to it (adapted to the first node) and curves H(t) obtained by dividing G(t) by F(pt) are shown. The latter functions are then the decays due to the interaction between the pairs. The function F(pt) fits the results for gypsum and methylene chloride quite well [G(t) and F(pt) have approximately the same second node]. The second node of F(pt) does not fit so well for cyanamide. Perhaps the second node is caused by H(t). Motion of the amino group is not likely; this will be discussed below. In context of this it is worth noting that the measuring error of the second node (approximately 10 %) is considerably larger than that of the first one (5 %). For a justification of the spin-pair hypothesis for our compounds we give in Table 1 the ratio of $p’ and $M,, (see Eq. [8]), as determined from the measured curves. The influence of the nitrogen atom in the amino group of cyanamide can be neglected.

FREE

PRECESSION

DECAY

OF NUCLEAR

PP.IRS

471

FIG. 2. Theoretical and experimental decays of (a) gypsum, (b) cyanamide and (c) methylene chloride. Dots: the measured decay G(t); Full curve: F(pt) adapted to the first node of G(r); Circles: W )l~%t ).

The calculated proton-proton distances from the first node together with some values obtained from literature are given in Table 2. It should be noted that the values from Myers and Gwinn (20) and Tyler et al. (II) are calculated from 7, all other values being from rT, which can account for some of the discrepancies (I.?).

472

VAN

RAREN

TABLE

ET AL.

1

JUSTIFICATION OF THE SPIN PAIR HYPOTHESIS

Compound

$P=I~MH

5.5 5 4.5

CaS04.2Hz0 CH,CI, HzNCN TABLE PROTON-PROTON

Compound

r

DISTANCES

2 FOR SEVERAL COMPOUNDS

from the first node (2 % error) 1.62 8, 1.93 8, 1.81 8,

CaSO, -2Hz0 CH2CII H,NCN

from literature _____ 1.58 8, (2%) (1) 1.80 8, (2 %) (IO) > 1.62 8, (II), (12) r

-__

The decay H(t) of gypsum is almost exactly gaussian, in contrast to the corresponding decays of the other compounds. From this decay we calculated the standard deviation /3 in the frequency domain. The value of /3 is found to be 11 kHz (2.6 gauss) in contrast to the value 6.6 kHz (1.54 gauss) given by Pake (I). This is because his value is obtained from a special orientation of the single crystal. 1 M (yb,t

t

FIG. 3. Theoretical and experimental modulation decays. Line: theoretical yb, = 13 kHz; Crosses: ybm = 20 kHz; Circles: yb, = 43 kHz.

~(&,t);

plusses:

In order to investigate the influence of the modulation amplitude b,, the decay of cyanamide at 294 K is determined for different values of b,. Analogous to the determination of H(t), these measurements were divided by G(t), the decay measured for vanishingly small b,. The resulting curves are adapted to the theoretical curve M(yb,,, t) given

FREE

PRECI ISSION

DECAY

OF NUCLEAR

473

PA IRS

by Eq. [lo], which gives us the vStlues of yb, (see Fig. 3). Especially for large values of b, the experimental and theoreti :a1 curves fit remarkably wel:. Table 3 gives the values of the modulation amplitude measured with an extra pick-up coil, calculated from the first node of N(r) (if visible), cr determined from adaptation of the theoretical and experimental M(t). TABLE MEASUREI~ENTS

yb, from the pick-up coil 1011 kHz 23%2kHz 43;4kHz

3

OF MODULATION

AMPLITUD

ybm from the first node of M(r)

23&2kHz 44+2kHz

3s

yb, from adaptation to M(r) 13f2kHz 20&2kHz 43+2kHz

The influence of saturation on the decay was measured Ibr cyanamide at 150 K. At least up to room temperature T,,/TID is large. Figure 4 gives the relative shift of the first node t, for increasing 2, both theoretical and experimenal. Within the measuring error the results fit rather well. ilt, t1

f

02

01

FIG. 4. Theoretical and experimental relative shift of the first node in cyanamide as a function of the saturation factor 2.

Because the second node of the f.p.d. of cyanamide did nc t agree very well with the spin pair theory we measured the position of the first node as a function of temperature in order to investigate whether motion of the pairs could account for this effect. We expect this node to shift if motion is present. However, at low I.emperatures T,, becomes very large and the measured shift in the first node is almost certainly caused by saturation of the system, so that motion of the pairs has not been detected.

474

VAN

RAREN

ET AL.

CONCLUSlONS

The calculation of the f.p.d. of a two spin system has been modified. This enlarges the understanding of the physical background of the line broadening due to interaction between the pairs. The proton-proton distance within a pair has been determined from the first node of the f.p.d. for gypsum, cyanamide and methylene chloride with rather good results. The nodes are not affected by modulation. Saturation of the system may shift the nodes, but this can be corrected for. Moreover, a method has been described for the determination of the modulation amplitude. Note. During the Symposium in Israel we were informed about the existence of two articles by Svanson (14, IS) in which the free precession decay of spin pairs is discussed [using the same assumptions as Look et al. (2)] and a similar expression for the modulation influence on the free precession decay has been derived. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13.

14. 15.

E. PAKE, J. Chem. Phys. 16,327 (1948). C. LOOK, I. J. LOWE, AND J. A. NORTHBY,J. Chem. Phys. 44,344l (1966). E. BARNAAL AND I. J. LOWE, Rev. Sci. Instrum. 37,428 (1966). BLOEMBERGEN, E. M. PURCELL, AND R. V. POUND, Phys. Rev. 73,679 (1948). ABRAMOWITZ AND I. A. STEGUN, “Handbook of Mathematical Functions,” New York, 1965. E. R. ANDREW, Phys. Rev. 91,425 (1953). R. E. FORNES, G. W. PARKER, AND J. D. MEMORY, Phys. Rev. B 1,422s (1970). B. N. PROVOTOROV, Sov. Phys. JETP 14,1126 (1962). M. GOLDMAN, J. Phys. 25,843 (1964). R. J. MEYERS AND W. D. GWINN, J. Chem. Phys. 20,142O (1952). J. K. TYLER, L. F. THOMAS, AND J. SHERIDAN, Proc. Chem. Sot. 155 (1959). W. G. MOULTON AND R. A. KROMHOUT, J. Chem. Phys. 25,34 (1956). W. G. BOVBE, C. W. HILBERS, AND C. MACLEAN, Mol. Phys. 17,75 (1969). S. E. SVANSON, Acta Chem. Stand. 16,2212 (1962). S. E. SVAN~~N AND B. STENL~F, Ark. Phys. 34,227 (1967). G. D. D. N. M.

p. 300,

Dover,