Tunnelling and relaxation for a two-proton system

JOURNAL

OF MAGNETIC

RESONANCE

7,359-363

(1972)

Tunnellingand Relaxation for a Two-Proton System M. .I. R. HOCH Physics

Department,

University

of the Witwatersrand,

Johannesburg,

South

Africa

AND B. DE K. BUDKE* Physics

Department,

University

of Natal,

Pieteunaritzbuug,

South

Africa

Presented at the Fourth International Symposium on Magnetic Resonance, Israel, August, 1971 A semiclassical theory of spin-lattice relaxation for a system in which protons tunnel between two equivalent potential wells is outlined. Expressions for the laboratory (T1) and rotating frame (T,,) relaxation times are given. The theory predicts that the phase-interruption rate of the coherent tunnelling motion plays an important role in relaxation. At relatively high temperatures where the phase-interruption rate exceeds the tunnel rate the expressions tend to conventional forms. The theoretical predictions are compared with available experimental evidence for solid hydrogen sulphide. INTRODUCTION

Atomic tunnelling has been detected in a number of systems, using NMR methods, at temperatures down to those of liquid helium. Apaydin and Clough (I) have given a theoretical treatment of the line shape for systems containing tunnelling CH3 groups, References to experimental line-shape work are given by these authors. Relaxation-time measurements are generally more difficult to interpret than are line shapes and less work has been carried out in this area. Allen and Clough (2) have discussed the CH3 problem using an exchange model and Wallach and Steele (3) have ,treated an isolated tunnelling two-spin system. There has, however, been a lack of a general theory of relaxation applicable to such systems and also a lack of decisive experiments. The calculation which we outline in this paper represents an attempt at calculating the laboratory (Tr) and rotating frame (Tip) relaxation times in a general way which takes into account interactions between all spin pairs. It is applicable to systems in which each spin can occupy either of two sites with-equal probability. Some experimental evidence which supports the main conclusions is discussed but further experimental measurements are required. Basically the calculation is similar to that of Look and Lowe (4) with a modified form for the correlation function. THEORY

We assume that each H-atom tunnels between two sites through a hindering potential barrier. The tunnel frequency in a particular vibrational state is given by the wellknown tunnel splitting. The nuclear Zeeman Hamiltonian has eigenstates la>, j b), . , . * Now at Alcan Aluminium, Copyright AU rights

Pietermaritzburg,

0 1912 by Academic Press, Inc. of reproduction in any form reserved.

South Africa. 359

360

HOCH

AND

BUDKE

while the lattice states may be designated if), if’), interaction, which is treated as a small perturbation, for spin-lattice relaxation to proceed. The Hebel-Slichter relation is

. . . . The magnetic dipole-dipole provides the necessary coupling

where W,, is the transition rate between spin states la) and lb) which have energies E, and Eb, respectively. We may write WC&= c fYf) f,f’

wa, hf’,

PI

where W,,, bf is the transition rate from spin-lattice state /a,f) to state jb,J“) and P(f) = (l/Z)exp(-E,/kT). W,, may be evaluated using a perturbation-theory approach (5). The lattice states f andf’ must differ in energy by Awe, where w,, is the Larmor frequency. For a number of systems the tunnel splitting is of this order. Lifetime effects broaden the vibrational levels giving a spread in the splittings of these states. The quantum mechanical calculation for a many-spin system is difficult and instead a semiclassical calculation has been performed for a model which attempts to preserve the main features of the problem. We have adapted the general approach of Look and Lowe (4) which allows interactions between all spin pairs to be taken into account and which also excludes physically undesirable terms. Our calculation assumes that the spins in different molecules move independently, but allows for coherence of such motions. Essentially the problem reduces to calculating the correlation functions

where ij, ,ff’ means that spin i is in vibrational state f and spin j in statef’ and the FCq” (4 = 0, &l, i2) are closely related to the second-order spherical harmonics. Evaluation of the correlation function for a site model involves probabilities such as P{ (u,z:,r) which denotes the probability that spin i is in its zi site, andf’-th vibrational state at t = T given that it was in its u site andf’-th state at t = 0. In the absence of any interaction with the rest of the lattice a given atom will tunnel back and forth between its two wells and the probability of finding it in a given site will be a periodic function of time. In a real lattice this coherent tunnelling motion will be interrupted through ““collision” processes in which excitation to a different vibrational state occurs. By assuming that such collisions occur randomly and independently it is possible to show that the average probabilities are Pf(u, u, T) = P$(v, 2.4,T) = $[l - cos wf 7 eWrf ‘I;

Pal

PT(u, 2.4,T) = Pi(v, 21,~)= +[l + cos wf 7 eerf ‘I,

WI

and where .?‘, is the characteristic rate at which phase coherence is lost in statef. wf is the tunnel frequency. Using these probabilities it is possible to evaluate the correlation functions and hence obtain expressions for Tl and also TIP. The general forms of these expressions

TUNNELLING

FOR A TWO-PROTON

SYSTEM

361

are rather lengthy and are not given here. If it is assumed that only the groundvibrational st.ate is significantly populated it is possible to simplify the equations. We obtain

1

where the symbol & means that both + and - terms must be included. The Xip are combinations of position functions and r, = (1/2J’,) (fnow denotes the ground vibrational state). N is the number of spins labelled i, j, etc. Also 1 -=y T’P

1) 128N 4 h ,31V+

27,

a5 [ 1+4+; 2ids [XC?)

+ 1 + 4(w ,z Wf)2 T:

1

In Eq. [5] only the terms in w1 have been retained since the terms in w0 and 2w, give rise to a very small contribution. To simplify the discussion only Eq. [5] will be considered. When wf -+ 0 Eq. [5] reduces, with minor changes in the definitions of certain quantities, to the expression derived by Look and Lowe (4). For wf = w1 anomalously low TIP values may be expected at low temperatures (tunnelling resonance). For of > wr the theory suggests that a minimum will still occur for w 17, z 0.5. A further kink in ihe TIP vs Tcurve may be expected for wfr, N 1. For a powder sample, terms like X$’ must be replaced by their powder averages (X!?)) 1.l . APPLICATION Available evidence (6) suggests that tunnelling motions occur in phase III of solid H2S at a rate sufficiently rapid to prevent the line width achieving its rigid lattice value even at 4°K. Although only a single line-width transition is observed below the phase III-phase II transition temperature, two TIP minima have been detected in this region (7). Figure I shows the experimental TIP data (plotted points) together with theoretically fitted curves. The double minimum suggests that two types of motions are present. From available evidence it appears unlikely that two conventional (classical) types of motions set in since this would, for example, lead to two line-width transitions. The following model is chosen for the molecular motions. At temperatures below 70°K the H,S molecules undergo 180” flips about their axes of two-fold symmetry, via a tunnel mechanism, at a rate exceeding 1O5seconds-’ (see Footnote 1). At temperatures between 70°K and the transition point, 103.5”K, 90” flip motions occur, about the 1 Tunnelling calculations are described in Ref. (6) for a cosine barrier. Secondary minima reduce the area under the barrier and increase the tunnel rate.

will

362

HOCH

AND

BUDKE

two-fold axis, to secondary minima positions. The ideas of this paper suggest that 180” flip tunnel motions can contribute to relaxation at lower temperatures than the flip motions. The intermolecular dipole-dipole interaction is involved in the 380” relaxation process while the intramolecular contribution is primarily responsibie the 90” flip relaxation.

the 90” flip for

0.020-

';;

0.010

-

I

3 0.005-

'

'I, I-0,002-

0.001

-

; i I

1000/T

( OK ) (0)

0,020-

O.OlO-

0.005-

o~co2-

0.001

I I I I

0.0005

I 11.00

9.00

e 13.00

lOOO/T(“K) (b)

FIG. 1. Rotating frame spin-lattice-relaxation time (T,,) v IOOO/T(“K) for various rf field values. Plotted points are taken from the data of Look, Lowe and Northby (7). Theoretical curves are described in the text.

Look and Lowe (8) have derived the following expression for the relaxation rate in a powder sample in which 90” flip motions of spin pairs are occurring between unequal wells I

,

TUNNELLING

FOR

A TWO-PROTON

SYSTEM

363

where Y is the proton-proton separation in a molecule, and AE is the potential energy difference between the two types of wells. The temperature dependence of ~6is given by T; exp (E;IkT) T’ = 1 + exp (AE/kT) ’

Combining Eqs. [S] and [6], and taking 7, = T,,exp(E,/kT), it is possible to obtain reasonable fits to the Tip data by using available structure information and choosing values for the parameters El, E;, AE and the pre-exponential factors. The results are shown in Fig. l(a) and (b). In Fig. l(a), El = 3.6 kcal/mole, E; = 5.2 kcal/mole and AE = 0.73 kcal/mole, while in Fig. l(b) El = E; = 4.0 kcal/mole and AE = 0.70 kcal/ mole. It can be seen that the fits obtained in Fig. l(a) are somewhat better than those in l(b) in which El = E,. The reason for two different barriers is not obvious since both should represent the depth of the deeper well. There is some X-ray evidence (9) of a small change in crystal structure at about 90°K but it is unlikely that this would lead to a significant change in the barrier. It is possible that for the coherent tunnelling motion to be interrupted a molecule need only be excited to a higher vibrational state, below the top of the barrier, in which the tunnel rate is comparable with the reciprocal of the lifetime of the state, while for significant population of the secondary minima positions over the barrier motions are required. Clearly further experimental work on this and other tunnelling systems is required, (A difficulty in such experiments is the long Ti values which are usually encountered.) While the calculation given here is oversimplified it is hoped that it will help in planning further work. REFERENCES 1. F. APAYDIN AND S. CLOUGH. J. Phys. C. 1,932 (1968). 2. P. S. ALLEN AND S. CLOUGH, Phys. Rev. Lett. 22,135l (1969). 3. D. WALLAGH AND W. A. STEELE,J. Chem. Phys. 52,2534 (1970). 4. D. C. LOOK AND I. J. LOWE, J. Chem. Phys. 44,2995 (1966). 5. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Chapter 8, Clareodon Press, Oxford, 1961. 6. B. DE K. BUDKE, M. I. GORDON, AND M. J. R. HOCH, Proceedings of the XIVth Colloque Ampere, 396, North Holland, 1967. B. DE K. BUDKE, Ph.D. thesis, University of Natal, 1969 (unpublished). 7. D. 6. LOOK, I. J. LOWE, AND J. A. NORTHBY, J. Chem. Phys. 44,344l (1966). 8. D. C. LOOK AND I. J. LOWE, J. Chem. Phys. 44,3437 (1966). 9. J. HARADA AND N. KITAMURA, J. Phys. Sot. Japan 19,328 (1964).