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Hydrogen bond dynamics intetrafluoroterephthalic acid studiedby NMR and INSA.J. Horsewill a , A. Ikram a & I.B.I. Tomsah a ba Department of Physics, University of Nottingham,University Park, Nottingham, NG7 2RD, UKb Faculty of Education, Department of Physics, University ofKhartoum, PO Box 406, Khartoum, SudanPublished online: 20 Aug 2006.

To cite this article: A.J. Horsewill , A. Ikram & I.B.I. Tomsah (1995) Hydrogen bonddynamics in tetrafluoroterephthalic acid studied by NMR and INS, Molecular Physics: AnInternational Journal at the Interface Between Chemistry and Physics, 84:6, 1257-1272, DOI:10.1080/00268979500100871

To link to this article: http://dx.doi.org/10.1080/00268979500100871

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MOLECULA~ PHYSICS, 1995, VOL. 84, NO. 6, 1257-1272

Hydrogen bond dynamics in tetrafluoroterephthalic acid studied by NMR and INS

By A. J. HORSEWILL, A. IKRAM and I. B. I. T O M S A H t

Department of Physics, University of Nottingham, University Park, Nottingham NG7 2RD, UK

(Received 10 January 1995; accepted 6 February 1995)

The quantum and classical dynamics of hydrogen atoms in the hydrogen bonds of tetrafluoroterephthalic acid have been studied using nuclear magnetic resonance (NMR), and inelastic neutron scattering (INS). The temperature dependence of the correlation time for the motion has been investigated using measurements of the spin-lattice relaxation of both hydrogen and fluorine nuclei in the temperature range 20 K ~< T ~< 300 K, and quasi-elastic neutron scattering measurements above 250 K. It is shown that the spin-lattice relaxation is governed by modulations of both homonuclear (IH-~H) and heteronuclear (1H_ 19F) dipolar interactions and the magnetization recovery is multi-exponen- tial. Expressions for the elements of the spin-lattice relaxation matrix are derived and, together with a simple theory for the hydrogen bond dynamics, a good fit and satisfactory account of the experimental NMR and INS data are obtained. The low temperature dynamics are governed by incoherent tunnelling, and a comparison is made with the behaviour of the closely related molecules, benzoic acid and terephthalic acid. There is an exponential dependence of the tunnelling rate with oxygen-oxygen distance in the hydrogen bond in these three molecules, and this provides strong supporting evidence for tunnelling at low temperatures.

1. Introduction

A wide range of materials exhibit dynamic disorder of hydrogen atoms in the hydrogen bond (O---H--O) . This disorder is connected with the motion of the hydrogen atoms between two sites associated with the two oxygen atoms and, to a first approximation, they experience a double minimum potential. This system is of interest to fundamental studies of molecular dynamics because the low mass of the participating atoms means that quantum tunnelling will play an important role in the dynamics. Such quantum effects have been observed in a number of carboxylic acids at low temperatures and the relative simplicity of the potential offers the opportunity to study and model these in detail. This motion of a hydrogen atom in the hydrogen bond is an example of one of the simplest chemical reactions, namely proton transfer, and so the system is important as a model for many dynamical processes in chemical and biological sciences.

Hydrogen bond dynamics in carboxylic acid dimers have been much studied. Here (figure 1) two molecules are linked by the two hydrogen bonds to form a dimer, and motion of the two hydrogen atoms converts the dimer between two tautomeric forms labelled 1 and 2. It is evident that the reaction coordinate involves not only the hydrogen atom positions but also the coordinates of the heavier framework

~" Present address: University of Khartoum, Faculty of Education, Department of Physics, PO Box 406, Khartoum, Sudan.

0026-8976/95 $10.00 �9 1995 Taylor & Francis Ltd.

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1258 A.J. Horsewill et al.

F F (1) F F

'

~ 1 7 6

F F F F I t

F F 0 H I I 0 F F

F F (2) F F

(a)

r

I.H

. . . . . . ! . . . . I , . x

(b)

F F

O--H- H ~ O "~

\_ I'- ~ " ~ 0 ~ 1 2 "Jlz I-

(c)

Figure 1. (a) The two tautomeric forms of the TFTA molecule labelled (l) and (2), (b) a representation of the asymmetric double minimum potential experienced by the hydrogen atoms as a function of position in the hydrogen bond, and (c) a detail of the geometry of the TFTA molecule.

atoms. Techniques in nuclear magnetic resonance (NMR), inelastic neutron scattering (INS) and optical spectroscopy have been employed to study the quantum and classical aspects of the dynamics and the shape of the potential experienced by the hydrogen atoms in a range of carboxylic acids in the solid state [1-8]. It is one of the few systems where translational tunnelling of atomic particles has been observed directly [9]. Experiments show that, in general, the double minimum

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Hydrogen bond dynamics 1259

potential is asymmetric and this arises from the asymmetry of inter-dimer interactions. In the present paper we report NMR and INS measurements of hydrogen bond dynamics in tetrafluoroterephthalic acid (TFTA, C6F4-1,4-(COOH)2). This was chosen for several reasons. First, it is a closely related analogue of terephthalic acid (C6H4-1,4-(COOH)2), a material which exhibits strong quantum tunnelling effects [2, 3]. Second, we set out to investigate the effects of heteronuclear 1H-IaF dipole-dipole interactions on the spin-lattice relaxation. Finally, we sought to investigate the influence of the fluorine substitution on the potential and dynamics and to relate the tunnelling rate to the molecular structure. A NMR and diffraction study [10] has investigated the crystal structure of TFTA and dynamical disorder of the hydrogen atoms was observed.

Hydrogen bond dynamics may be studied using established experimental techniques such as INS and NMR. In INS the width of the quasi-elastic line provides a direct measure of the rate for hydrogen transfer; however, even with the use of backscattering INS spectrometers the available resolution often limits the application of this technique. In NMR the motion of the hydrogen atoms modulates both inter- and intramolecular dipole-dipole interactions in the crystal and provides a mechanism for spin-lattice relaxation of the magnetic nuclei. Measurements of the proton spin-lattice relaxation time T~ enable the dynamics to be determined over a wide range of temperature. In the simplest case the proton T~ is determined by homonuclear dipolar interactions and so, according to well established theory, the motional spectrum of the hydrogen is sampled at the proton Larmor frequency co H and at 2con. When a second magnetic nucleus is present then extra terms relating to the heteronuclear dipolar interactions are introduced into the spectral density function which determines T r Consequently, the motional spectrum of the hydrogen nuclei is sampled additionally at frequencies which include the sum and difference of the Larmor frequencies of the proton and the second nucleus. In this study we have chosen a system which has fluorine nuclei incorporated in the molecule in addition to the protons in the hydrogen bond. The fluorine nucleus has spin 1/2 and its magnetic moment differs by only 6~ from that of the proton. This means that the difference in Larmor frequency COd = (COIl -- COF) between the two nuclei is a small quantity, and T 1 samples the motional spectrum of the hydrogen nuclei additionally at the low frequency co d . In this respect the experiment has similarities with measurements of spin-lattice relaxation in the rotating frame.

2. Theory

2.1. The correlation time characterizing hydrogen bond dynamics

We shall assume that the motion of the hydrogen atoms in the hydrogen bond takes place under the influence of an asymmetric double minimum potential (see figure 1). The asymmetry is characterized by the energy A and in common with previous investigations we shall assume that there is just one coordinate describing the molecular rearrangement between the two tautomeric forms labelled 1 and 2. In this approximation the motion of the two hydrogen atoms in the dimer is coordinated on all timescales. Following Skinner and Trommsdorf [11] we shall assume that the dynamics at high temperature are Arrhenius and that at low temperature the motion is dominated by phonon assisted tunnelling. A more detailed account of this model has been presented elsewhere [1 l, 2], and here we present only the main results.

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1260 A.J. Horsewill et al.

If V is an activation energy for thermally activated hopping from the lowest well then we can write the rate constants for left-to-fight and for fight-to-left motion in the high temperature regime as follows,

( k~2 = zf f ' exp - ; k~, = Zo ' exp k~-7; / '

where 3o is the correlation time at infinite temperature. Hence the overall rate in the high temperature region may be written as

/m = k Uz + k2",. (2)

At low temperatures the dynamics are governed by incoherent tunnelling and the rate is given by

A /~ = k o coth 2ka-- ~ , (3)

where ,3 2 .4 (4)

ko = cp~ (hwo)2 h '

co D is the Debye cut-off frequency, ,] is an effective tunnelling matrix element, and Cp is a parameter determined by the coupling between the hydrogen bond system and the phonon bath. The dynamics in the intermediate temperature range are obtained by simple interpolation, and we can write the correlation time for the motion as,

1 (5) ~c =_rn + rL"

2.2. Heteronuclear spin-lattice relaxation

2.2.1. Evolution of the nuclear magnetization Initially we shall use the approach of Abragam [12, 13] to derive the equations

describing the evolution of the magnetizations. The spectral density functions will then be calculated for the specific case of the hydrogen bond in these systems. Consider a system consisting of two sets of identifiable nuclear spins labelled I and S both of spin 1/2. Motion of spins I gives rise to a modulation of the homonuclear ( I - I ) and heteronuclear ( I -S) dipolar interactions. We may write the dipolar Hamiltonian as

2

~ = ~. A("). F ("), (6) h i = - - 2

where A (") and F ~') are second rank tensors representing the spin and spatial variables, respectively. The time rate of change of the longitudinal magnetizations <I~> and <Sz> are given by a pair of coupled differential equations and these can be written in matrix form as

__

L ~ < s ' > J - ~ s -Os k(<s~> So)] (7)

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Hydrogen bond dynamics 1261

Here, I o and S o are the longitudinal magnetizations at thermal equilibrium with the lattice. The first matrix on the right hand side is the relaxation matrix. The relaxation processes are driven by modulations of the dipolar interactions and the elements Ps, Pt, tXs and 61 are inverse relaxation times given by the following expressions:

2 - ,

Pt - [J~~ -- cot) + 18J")(oi) + 9j(2)(Ws + cot )] (/~o/4n) 2 + pn, 16

)~2. 2 J,_2 l YSrl

61 -- - - [-- J(~ -- col) + 9J(2)(cos + coi)](#o/4~) 2, 16

(8)

6S -- [-- J(~ - cos) + 9J(2)(co! + cos)](//o/47t) 2, 16

Ps - - - [J(~ -- COs) + 18J(X)(COs) + 9J(2)(COl + C0s)](/~o/4~) 2. 16

cot and cos are the Larmor frequencies of spin systems I and S respectively and yt and "~s are the magnetogyric ratios. The term pn relates to homonudear dipolar interactions between spins I, and the rest describe the heteronuclear interactions.

The terms j(m)(CO) are the spectral density functions and are Fourier transforms of the correlation functions of the spatial parts, F ~"), of the dipolar Hamiltonian,

J(")(CO) = (F(m~*(t + z)F(m)(t))exp (ior) dz. (9) --~t3

The solutions of coupled equations (7) describe the time evolution of the mag- netizations (I~) and (S~). These are hi-exponential and can be expressed in the form of magnetization recovery ((M~) - Mo)/Mo (M = I, S) as follows:

( M z ) - Mo = c~ ) exp ( - R l t ) + c~ ) exp ( - g 2 t ) . (10) M0

Rx and R E a r e decay constants which can be measured experimentally. These have the following functional form obtained by diagonalization of the relaxation matrix:

R1.2 -- l [ (p l -F Ps) + {(Pl q- Ps) 2 - 4(plPs - tTias)} 1]2] (11)

Consequently in a spin-lattice relaxation experiment we expect that the time evolution of both (I~) and (S~) will be characterized by the weighted sum of two exponential functions. The weighting coefficients cOQ (M = I, S) will be determined by the initial state of the system at time zero, namely the initial spin temperatures of the two Zeeman reservoirs. The decay constants R 1 and Rz will be determined by the elements of the relaxation matrix and hence by the spectral density functions. We shall now derive the relaxation matrix in terms of the motional characteristics of the system.

2.2.2. The spectral density functions Andrew and Latanowicz [14] have derived expressions for spin-lattice relaxation

in a hydrogen bond system where the nuclei experience homonudear dipolar interactions only. We shall use a similar approach here but extend their treatment to include heteronuclear spin interactions and the molecular arrangement of carboxylic acid dimers.

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1262 A.J. Horsewill et al.

Figure 1 sketches the configuration of the key magnetic nuclei in the two tautomeric forms. Configuration 1 is of lower energy than 2, and the difference in energy between them is the asymmetry energy A of the double minimum potential. We shall associate the nuclei of the hydrogen atoms (1H) with spins I and the fluorine nuclei (19F) with spins S. We shall assume that the interactions can be described in a pairwise fashion so that the dipolar relaxation is the sum of a single heteronuclear contribution (t9F-1H/,) and a homonuclear contribution ( IHI-IH2) . Further hetero- nuclear contributions, for example (19F-1H2) and those between the hydrogen nuclei and other 19F nuclei, will be accommodated at a later stage. This approach neglects the fact that the motion of the two hydrogen atoms is correlated.

The probability of configuration 1 at time t is given by

a P(l , t) = - - , (12)

l + a

and the probability of configuration 2 is

1 P(2, t) = - - , (13)

l + a

where a = exp (A/ka T). In order to evaluate the correlation functions we shall need to know the probabilities of occupation of the two sites at subsequent times. There are four cases to consider: if the system is in configuration 1 at time t, it may be in 1 or 2 at time t + z, or if the system is in configuration 2 at time t, it may be in 2 or 1 at time t + z. Thus we may write the correlation function (F~'*(t + r)F~'J(t)) for the heteronuclear interaction between the fluorine and the hydrogen nuclei as a sum of four terms including the conditional probabilities P(s, t + fir, t):

(F~m~*(t + z)F~m~(t)) = ~ (F~m~*F~m~)P(s, t + z[r,t)P(r, t), (14) r , s = l , 2

where s stands for state 1 or 2 at time t + z, and r stands for state 1 or 2 at time t. The conditional probabilities have been given by Look and Lowe [15]:

1 ( a ( _ r ) ) , P(1, t + zll, t) = ~ + exp

, (2 , t + z , 2 , t ) = l + a l ( l + a e x p ( - - ~ r '

P(1, t+z [2 , t ) = a (1 e x p ( ~ ) ) l + a

(15)

z~ is the correlation time for the interchange between the two configurations. We now substitute equations (12), (13) and (15) into (14) and in turn into (9) to obtain the spectral density function

a [ (F~m'*F~m') -- (F~m'*F~m')-] 2zc (16) Jtm)(~176 (1 + a) 2 L-(Ft2m'*F~ 'm) + (Ft2'~'*F~m)).J 1 + to2r 2"

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Hydrogen bond dynamics 1263

As it stands this quantity is dependent on the orientation of the system in the magnetic field. We have studied powder samples, however, so we shall assume that the spin-lattice relaxation may be evaluated from the isotropic averages of the spatial part of the dipolar Hamiltonian (reference [14] gives explicit expressions for these quantities). Applying these to equation (16) and introducing heteronuclear dipolar interactions between i fluorine nuclei and j hydrogen nuclei we obtain the elements of the relaxation matrix as follows:

40 P l = c H - F - -

(1 + a) 2

4a GI = c H - - F - -

(1 + a) 2

4a a S = N C H - F _ _

(1 + a) 2

4a P S = N c H - F - -

(1 + a) 2

[L(ogv - OgH) + 3L(OgH) + 6L(OgF + (-OH)] + PHH,

[-L(o9v - (on) + 6L(OgF + O98)],

[ -L(Og. - ~F) + 6L(O9n + OF)],

[L(o9n - toE) + 3L(O9F) + 6L(o9n + COF)],

(17)

where N is a numerical factor of order 0-5 determined by the relative number of hydrogen nuclei to fluorine nuclei in the unit cell, and

and

L(o9) - 3o 1 + o92z~'

cH--F -- ~ 2 ~ h 2 E ['r/J 6 "~ %6 + r , S 3 r J ( l _ 3 cos 20i j l2) ] 40 i. j

(18)

Here the labels i and j refer to the particular 1Hg and 19Fi nuclei, Oij~2 is the angle subtended by the two 19Fi-XHj internuclear vectors in the two configurations 1 and 2 and ro~, ro2 are the corresponding internuclear distances (see figure l(c)).

In equation (17) the term Pna accommodates the relaxation terms arising from the homonuclear dipolar interactions between the two hydrogen nuclei in the dimer. This has been calculated by Andrew and Latanowicz [14] and others [3, 16] and has the following functional form:

where

4%[ 1 PHH = cH-U (1 1 + O9H2 rr + I -[----240)8 zr '

Cn_H 97~hZ sinZ ~t (#o~ 2 4o r6_. LUg/

(19)

In the above, rH_ H is the internuclear distance between the protons in the dimer and ct is the angle between the internuclear vectors in the two configurations (see figure l(c)). In equations (17) and (19) the factor 4a/(1 + a) z accommodates the de- population of the energetically less favourable well with decreasing temperature and the multiplier 4 has been incorporated to ensure that this factor is unity when the asymmetry A is zero.

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1264 A.J. Horsewill et al.

2.3. Scattering law for neutron scattering

The scattering law for neutron scattering from a system which exhibits hopping between two energetically non-equivalent sites may be written as [5]:

(%In) ~ (20) SI.,(Q, o~) = exp [ - 2 W ( Q ) ] (1 - S,)6(eo) + S, (1 + (o~z,)2)/

Sin~(Q, co) consists of an elastic line with intensity (1 - $1) and zero width, and a quasi-elastic line with intensity S 1 and Lorentzian lineshape with a half-width of z c 1. Both of these components are multiplied by the Debye-Waller factor exp [ - 2 W(Q)]. The intensity S 1 is Q dependent but the fuctional form is not given here as we shall not be concerned with this property in this paper.

3. Experimental details

3.1. Neutron scattering

Neutron scattering measurements were conducted at the Institut Laue-Langevin, Grenoble, using the backscattering spectrometer IN13. The resolution function of this instrument has a F W H M of approximately 10 I~eV. The powdered sample was mounted in a cylindrical aluminium cell and occupied a 1.5 mm annulus. The cell was mounted in a variable temperature helium cryostat and the temperature stability was better than 0"5 K. Spectra were recorded at the following temperatures 297"7 K, 275"9 K, 256-4 K, 110 K and 5 K. At each temperature, spectra were recorded at 27 scattering angles simultaneously, but these were later grouped together to provide better statistics. The spectra were normalized using a vanadium standard.

3.2. N M R spin-lattice relaxation times

The measurements of nuclear magnetization were made using a home-built pulsed N MR spectrometer. The magnetic field was provided by an Oxford Instruments superconducting magnet operating in persistent mode and this possessed a con- tinuous flow helium cryostat to control the sample temperature. The latter was stable to within _+0"3 K. The measurements of 1H and 19F magnetizations were both made at 26 MHz, the magnetic field being set to establish resonance for the particular nucleus of interest.

The spin-lattice relaxation of hydrogen and fluorine nuclei were measured using a modification of the saturation-recovery pulse sequence. Magnetization recovery curves of both 1H and 19 F nuclear spin systems were recorded as a function of sample temperature. Inspection of equation (10) shows that the recovery curve is dependent on the values of the weighting coefficients c~ and these in turn are determined by the initial spin temperatures of both nuclear species. In order that the initial state of the system as a whole was always prepared in a consistent way the following procedure was adopted: (i) the sample was allowed sufficient time to relax so that both (I~) and (S~) were at their thermal equilibrium values I o and So, (ii) the magnetization of one of the nuclei was reduced to zero by applying a comb of 90 ~ pulses on resonance, and (iii) the subsequent recovery of this nucleus was then followed by sampling the magnetization at time intervals r which were linear in the logarithm of time. A separate measurement was made of the equilibrium

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Hydrogen bond dynamics 1265

magnetization at each temperature. Under the initial conditions established by steps (i) and (ii) the values of the weighting coefficients may be written as (M = I, S),

C ~ ) _ R 2 - - PM and c~ ~ _ R1 -- PM (21) R E -- R 1 RE -- R 1

4 . R e s u l t s

4.1. Quasi-elastic neutron scattering spectra

The spectra obtained at the three highest temperatures showed quasi-elastic components. The quasi-elastic neutron scattering (QNS) component was narrower than the resolution function at the two lower temperatures; in fact these spectra were identical with the instrument resolution function. The 110 K data were used to fix the baselines when analysing the spectra, which displayed quasi-elastic broadening. In accordance with the scattering law (equation (20)) the spectra were fitted to a delta function (elastic line) and a Lorentzian (quasi-elastic component) both convoluted with the instrument resolution function. A representative spectrum, along with the fit, is shown in figure 2. The values of the Lorentzian half-width, averaged over all recorded Q values, were 21.0 + 0.8, 16.3_ 0"9 and 13"9 __+ 0.8 ~teV at the temperatures 297.7, 275"9 and 256.4 K respectively.

200

150 r-- -3

F, 100

v

3 0 m 50

I i I i I i I i

t 111 II \~

�9 ~I

1 L

0 I t I i I I I i I

-100 -50 0 50 100 neutron energy transfer (peV)

Figure 2. Eneqgy spect rum of T F T A at T = 256-4 K measured on INI3 . M o m e n t u m transfer Q = 4'27 A - 1. The con t inuous curve is the result of the fitting procedure explained in section 4.1, and the quasi-elastic componen t is indicated by the dashed curve.

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1266 A.J . Horsewill et al.

4.2. Magnetization recovery and spin-lattice relaxation of 1H and 19F

The magnet izat ion recovery curves of both hydrogen and fluorine nuclei were observed to display multi-exponential behaviour, particularly at low temperatures. Representative data are shown in figure 3, where the magnet izat ion recovery of both nuclei is plotted as a function of recovery time following saturation. These data were recorded at a temperature of 40 K. Attempts to fit the IH data to the bi-exponential form of equat ion (10) were unsatisfactory, in fact a satisfactory fit could be obtained only when the magnet izat ion recovery was modelled as a tri-exponential function (with M = I ) :

(Mz) -- M~ - ~M"ll~exp(--glt) + c ~ e x p ( - - R 2 t) + c ~ e x p ( - - g 3 t ) , (22) Mo

where a third decay constant has been introduced. The solid line in figure 3(a) is the fit of this function to the data and the best fit parameters are reported in table 1, where the relaxation times R7 x are quoted. Tri-exponential recovery of the ~H magnet izat ion was observed at all temperatures below 60 K.

-p, A

=>

8

E

0.1

0.01

g E

u .

i i ~ i i i i i ,m i |

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

0.1 1 10 100

0.1

0.01

0.01 0.01

i | l l = l i i =Hm I m i | l nn I = i 1~== , ,

0.1 1 10 100

recover/time, = (seconds) recovery time, T (seconds)

(a) (b)

Figure 3. Magnetization recovery of the (a) ~H and (b) 19F nuclei recorded at 40 K in TFTA. The curves are best fits to the tri-exponential decay function (equation (22)) with the parameters reported in table 1.

Table 1. The best fit parameters describing the tri-exponential recovery (figure 3) of 1H and 19F magnetizations in TFTA recorded at T = 40 K. See text for details.

M c~ ~ c~ ~ c~ ~ R ~- 1/s R 2 l/s R~- l/s

I -= IH 0-25 +_ 0.02 0"33 + 0-02 0-42 + 0-02 16'8 + 1 1-9 + 0-3 0"8 + 0"1 S -- 19F 0'61 + 0'04 0"16 ___ 0"05 0-22 + 0"05 16-6 + 0"6 3"6 + 0"8 1-0 + 0'2

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Hydrogen bond dynamics 1267

The recovery of the 19F magnetization was observed to be either bi-exponential or tri-exponential in the temperature region below 60 K. An example of the recovery of fluorine magnetization recorded at a temperature of 40 K is presented in figure 3(b). In this particular case a satisfactory fit could be obtained only with the tri-exponential function (equation 22 with M = S) and the best fit parameters are presented in table 1. Analysis of table 1 shows that there is satisfactory agreement between the ~H and 19F relaxation times at 40 K.

For the X9F nuclear species it was rarely possible to obtain fits to the tri- exponential function and usually the data fitted a bi-exponential recovery law within experimental error. Analysis of the 40 K data in table 1 suggests a reason for this. It would appear that the coefficient r 2) has such a small value that the presence of this relaxation component has low statistical significance when experimental errors are taken into account. In order to be systematic in the analysis of the temperature dependence, therefore, we fitted all of the 19F data below 60 K to the bi-exponential function.

In figures 4(a and b) we present the temperature dependence of the fitted relaxation times, 7~ ~ = Ri- ~ for 1H (i = I, 2, 3) and 19F (i = l, 2), respectively, plotted as log~0 ln~ versus 1/T. At temperatures above approximately 60 K the recovered magnetization of both nuclear species follows a mono-exponential recovery law. At temperatures below this value the recovered magnetization is multi-exponential.

The tri-exponential behaviour of some of the magnetization data was unexpected according to the theory presented in section 2.2. We should look at the assumptions made there to identify the underlying reason for this discrepancy. First, one of the assumptions we made was to use the isotropic averages of the spatial part of the Hamiltonian to represent the spin-lattice relaxation of a powdered sample. We can assume that spin diffusion within an individual crystallite comprising the powder is relatively fast (timescale of T2); however, spin diffusion between crystallites is expected to be much slower. This being the case the spin-lattice relaxation time of an individual crystallite is dependent on its orientation in the magnetic field. Consequently, the spin-lattice relaxation of the powder sample as a whole will be determined by the powder sum of the magnetization of each crystallite and will in general have multi-exponential components resulting from the orientation dependence. The use of the isotropic average may therefore be an oversimplification. A second assumption in the model was that the motion of the two hydrogen atoms is uncorrelated, if correlation were admitted then this would introduce cross-correlation effects which may introduce additional multi-exponential components.

4.3. Data analysis

It is notable from figures 4(a and b) that the values of the longest relaxation times R[ 1 are equal for the ~H and 19F sets of data within experimental error. This suggested a way forward. The short and medium relaxation times R31 and R21 characterizing the recovery of the IH magnetization were averaged at each temperature, and the corresponding values of the weighting coefficients were added together. This analysis led to the data in figure 4(c). Now a much closer association is observed between the relaxation times and the weighting coefficients determined for the two nuclear species. The curves presenting the temperature dependence of the weighting coefficients c~ are shown in figures 5(a and b).

The dynamics of the hydrogen atoms (z~ -~ versus 1/T) has been modelled

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1268 A.J . Horsewill et al.

tOO

8 ID

r 10

T

0.1 0

, l , l l

[ ] [ ]

D o []0 [ ]

[ ] Q

v v v V

0

~176176 ~ % []

, i I

[ ] [ ]

[ ] [ ]

9

v [ ] v [ ]

0 [ ]

! i

-1 [] R 1

-1 R 2

-1 o I:13

i I i i , I , , ,

0.02 0.04 0.06

Inverse temperature (~1) ( a )

R -I

1 0 0 ~ []

= 10 o "

~ [] .

0 0.02 0.04 0.06

inverse temperature (1("1) (b)

, ~ -

v 0 0 -

=~ lo

0 0 -1 -~ - t "

0 0.02 0.04 0.06

Inverse temperature (K "1)

(c)

Figure 4. The temperature dependence of the spin-lattice relaxation times R, 1 recorded for (a) 1H (tri-exponential) and (b) 19F (bi-exponential) nuclei in TFTA. In part (c) the fast and medium relaxation times recorded on the ~H nucleus have been averaged. The continuous lines in (b) and (c) are the predictions of the theory based on a best fit to the experimental data using the parameters reported in table 2 (see text for details).

according to section 2.1, and best fit parameters have been determined by fitting to the experimental data. Two of these parameters, namely t o 1 and k o, are rates which determine the dynamics in the high temperature (Arrhenius) and low temperature (tunnelling) regions, respectively. A further two, namely V and A, are defined by the potential experienced by the hydrogen atoms. In addition the dipolar constants C n-F and C "-H determine the spin-latt ice relaxation. Initial values of the parameters could

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Hydrogen bond dynamics 1269

i

"0

1 . 0 ~ - ~ = ~ = Cl - " I i I

0

0.5

0

. , I . , i I . . .

0.02 0.04 0.06

~ o I , . �9 I , , , l ,

0 0.02 0.04

Inverse temperature (K "I) Inverse temperature (I( "I )

(a) (b)

1.06

Figure 5. The temperature dependence of the weighting coefficients c~ for (a) :H and (b) 19F. The coefficients corresponding to the fast and medium components in the hydrogen data have been averaged. The continuous lines are the predictions of the theory using the parameters reported in table 2 and equation (21).

be obtained from the experimental data in certain limits. For example, an initial value of A is provided by the gradient of log T 1 versus 1/T in the low temperature limit. The gradient of the same function in the limit of high temperature provides a value for I/". Additionally, the depth of the T1 minimum gives initial values for C "-F and C "-n. The QNS data were useful also because they provided a value for Zo 1. The six parameters were refined by fitting to the temperature dependence of the NMR and QNS data. The best fits are represented by the continuous lines in figures 4(b,c), 5(a,b) and 6. These were calculated from the model (correlation time T c from equations (1)-(5), relaxation times R71 from equations (10), (11), (17)-(19) and the weighting coefficients c~ from equation (21)). The best fit parameters are given in table 2. The modelled dynamics are presented as the solid line in the plot of log r~- x versus 1/T in figure 6, where the experimental values of z~-~ determined directly by QNS are also drawn.

5. Discussion

A striking feature of the nuclear spin-lattice relaxation of many carboxylic acid dimers is the asymmetry of the log T: versus 1/T plot. The gradient on the low temperature side of the T 1 minimum is often observed to be much shallower than that on the high temperature side. This has been explained by the incorporation of tunnelling into the dynamics in the low temperature regime, and this gives a very satisfactory explanation for the T1 data. In figure 6 the onset of tunnelling is characterized when the rate r~-1 levels out at low temperatures so that the motion proceeds at a much faster rate than would be expected for a system exhibiting classical Arrhenius hopping between two sites. A comparison of the dynamical parameters with those of the closely related materials benzoic acid (BA; C6HsCOOH ) and terephthalic acid (TA) is instructive (see table 2). The value of ko (which governs

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1270 A. J. Horsewill et al.

1011 ' ' ' I ' ' ' I ' ~ '

101~

"7. ol

": ~ 109

1 0 e

1 0 ~ I I I I I I | I a ~ a

0 0 . 0 2 0 . 0 4 0 . 0 6

Inverse temperature (K "1)

Figure 6. The curve is the temperature dependence of the inverse correlation time z~-1 calculated from the best fit to the experimental data using the parameters reported in table 2 and equation (5). The three data points represent the measurements of ~- 1 made using QNS spectroscopy on INI3.

the rate of tunnelling) in T F T A is an order of magni tude smaller than that in BA which, in turn, has a ko value which is ten times smaller than that in TA. The functional form of ko is given in equat ion (4). Using a similar approach to the pressure s tudy of dynamics in BA [ 17-1 we remove the dependence on asymmet ry A by defining the dimensionless quant i ty k a = koh/A. Now, given the reasonable assumpt ion that ogi~ and %, relating to the phonon properties, are approximate ly the same in these materials with very similar molecular structure, we may reduce equat ion (4) to the form

k A - k~ - Kz~ 2, (23) A

where the p h o n o n terms have been absorbed into the constant K. To a first approximat ion, therefore, the value of k A therefore is determined wholly by the tunnelling matrix element,

,d = [- ~b*~b, dx. (24) J

Table 2. TFTA: the parameters are derived from the fitting procedure to the NMR (figures 4 and 5) and QNS data. BA and TA: the data are from references [2, 3, 11, 16].

cH-H/S -2 cH-F/S -2 ko/s-I Zo/S (I//kB)/K (A/kB)/K N

TFTA 2'5 _ 0"5 1"8 + 0"6 1-6 ___ 0-8 8"5 +__ 1"0 820 + 40 82 -t- 5 0"5 +__ 0-1 x 10 a x 107 x 107 x 10 -t2

BA 2"84 x l0 s - - 2"8 x 10 a 6"0 x 10-12 500 80 - - TA 3"5 x 10 s - - 2"1 X 10 9 1"0 X 10 T M 600 130 - -

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Hydrogen bond dynamics 1271

10 -s

104

10 -s

2.60 i i i I i i i I I I I I I I I

2.62 2.64 2.66 2,68

~ 1 ~ 1 ~ 1 ~

ro o (A)

Figure 7. A log-linear plot of the tunnelling rate parameter k^ = koh/A as a function of oxygen-oxygen distance, ro-.o, in the hydrogen bond: ~7, terephthalic acid; E3, benzoic acid; and �9 TFTA. The line is the function k A = const x exp ( - bro...o) with b = 75'8 A- 1.

The oxygen-oxygen distance ro... o parameterises the distance between the wells of the double minimum potential, and in figure 7 a log-linear plot is presented of the experimental values of k A in TFTA, BA [2, 11, 16] and TA [2, 3] as a function of ro... o. The latter are known from the crystal structures. Remarkably, there is a linear dependence so there is an exponential rise in tunnelling matrix element with decreasing ro... o and hence with decreasing well separation. In general the tails of the eigenfunctions in the right and left wells q~r and ~bz have an exponential dependence on distance in the region under the central barrier region. Consequently we expect their overlap ,~, and hence kA, to increase exponentially as a function of decreasing distance between the potential wells. This is the behaviour observed in figure 7 and while significant approximations are implied, the data provide strong supporting evidence for tunnelling at low temperature. Behaviour of a very similar nature has been observed recently in the pressure dependence study of hydrogen bond dynamics in BA [17]. Further investigations on derivatives of benzoic acid are planned to augment the limited data in figure 7.

The values of the dipolar constants C H-H and C H-F are determined by the molecular geometry in the two configurations (equations (18) and (19)). The crystal structure of TFTA has been determined by X-ray and neutron diffraction [10], and we have used the relevant internuclear distances to estimate values for C H-H and C H-F. The value of C H-n is dominated by a single pairwise intramolecular dipolar coupling between the two hydrogen nuclei in the dimer. The relevant geometrical terms are rH_ H = 2"37 A and ~ = 39 ~ Substituting these into equation (19) gives

H--H Ctheory = 2"86 x 108 s -2, and this compares very well with the experimental value from table 2 of n-n C~xpt = 2"5 + 0.5 x 108 s -2. The value of C n-F is determined by intra- and intermolecular XH-19F dipolar interactions and no single nuclear pair dominates the lattice sum. Substituting the internuclear distances and angles into

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1272 A.J. Horsewill et al.

H--F equation (18) and summing the leading eight terms we obtain Cthcory = 1.28 x l07 s -2. H--F Again there is satisfactory agreement with the measured value of Ccxpt =

i.8 + 0"6 x 10 v S - 2

6. Conclusion

Despite the unexpected complexity encountered with the tri-exponential nature of the spin-lattice relaxation at certain temperatures, a very satisfactory account of the hydrogen bond dynamics has emerged from this study. Furthermore, the magnetic resonance parameters are consistent with the known molecular structure. One important new feature revealed by this study is the correlation between the tunnelling rate parameter k A and the oxygen-oxygen distance in the hydrogen bond (figure 7). This is under further investigation.

I.B.I.T. held a postgraduate research scholarship sponsored by the Sudanese government for the duration of this work and A.I. holds a postgraduate research scholarship sponsored by the Indonesian government. We are grateful for the able assistance of Dr J. Williams and Dr A. Heidemann of the ILL, Grenoble, in connection with the neutron scattering experiments.

References

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ANDERSON, |., 1986, Phvsica B, 136, 161. [-6] HORSEW1LL, A. J., HEIDEMANN, A., and HAYASHI, S., 1993, Z. Phys. B, 90, 319. [-7] TAKEDA, A., TSUZUMITANI, A., and CHATZ1DIMITRIOU-DREISMANN, C. A., 1992, Chem.

Phys. Lett., 198, 316. [-8] STOCKLI, A., MEIER, B. H., KREIS, R. MEYER, R., and ERNST, R. R., 1990, J. chem. Phys.,

93, 1502. [-9] OPPENLANDER, A., RAMBAUD, C., TROMMSDORE, H. P., and VIAL, J.-C., 1989, Phys. Rev.

Lett., 63, 1432. [-10] SCHAJOR, W., POST, H., GROSESCU, R., HAEBERLEN, U., and BLOCKUS, G., 1983, J. Magn.

Reson., 53, 213. [-ll] SKINNER, J. L., and TROMMSDORE, H. P., 1988, J. chem. Phys., 89, 897. [12] ABRAGAM, A., 1961, The Principles of Nuclear Magnetism (Oxford: Clarendon Press),

p. 294. [13] HORSEWILL, A. J., and TOMSAH, I. I . I., 1993, Solid State nucl. magn. Reson., 2, 61. [14] ANDREW, E. R., and LATANOWICZ, L., 1986, J. magn. Reson., 68, 232. [15] LOOK, D. C., and LOWE, L. J., 1966, J. chem. Phys., 44, 3437. [-16] NAGAOKA, S., TERAO, T., IMASHIRO, F., SAIKA, A., HIROTA, N., and HAYASHI, S., 1983,

J. chem. Phys., 79, 4694. [17] HORSEWILL, A. J., McDONALD, P. J., and VIJAYARAGHAVAN, D., 1994, J. chem. Phys.,

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