Self-diffusion Studies in Solids Using Nuclear Magnetic Resonance Techniques BY ROY E. GORDON AND JOHNH. STRANGE Physics Laboratory The University Canterbury Kent CT2 7NR Received 31st July 1978 N.m.r. measurements of spin-lattice relaxation time T,,and spin-spin relaxation time T, give valuable information on diffusive motion in a wide variety of solids when self-diffusion provides the dominant relaxation mechanism. These measurements however have a limited useful range. As the mean time 7 between jumps increases the effects of paramagnetic impurities can dominate TIand T2reaches a constant rigid lattice value. These limitations can be overcome by measuring the spin- lattice relaxation time in the presence of an r.f. field TIP, or the spin-latticerelaxation time in the dipolar field TID.Recent theoretical developments often allow z to be evaluated with reasonable accuracy and if the jump distance is known then the self-diffusion coefficient D,can be calculated. Paramagneticim-purity effects can also cause anomalous results in the solid in the fast diffusion range. The technique and its limitations will be illustrated by recent work on 19F-diffusion studies in fluorite lattices. Mea-surements of these relaxation times provide information at the microscopic level on the atomic motion and thus can indicate the dominant diffusion mechanism. Diffusion coefficients can be measured by macroscopic methods and in particular by using the pulsed field gradient n.m.r. spin echo technique in which pulsed magnetic field gradients are applied to the sample during a T2measurement sequence.Since it is a direct measurement of D this technique also serves to complement the information from relaxation studies and can provide a useful check on existing theoretical interpretation. Comparisons with the results of other experimental methods such as radio-tracer or ionic conduc- tivity for ionic solids are similarly informative. F-diffusion in barium fluoride and lead fluoride together with new results for strontium fluoride are discussed. N.m.r. measurements are seen to be particularly useful in the regions of fast ion diffusion above the high temperature phase transitions. The phenomenon of self-diffusion in solids has attracted considerable interest in recent years. Theoretical models for defect assisted mass transport in various classes of solid have been proposed and experiments performed in attempts to verify them.One area where this type of study has recently assumed much greater importance is the study of ionic motion in solids. Fast ion conductors in particular have attracted considerable attention primarily because of the increased technological importance of solid-state electrochemistry. Solid-state electrodes in batteries hydrogen storage and electrochromic display devices for example require solids exhibiting fast ion conduction. The diffusion mechanisms and point defect structures of such solids are not well understood. Electrical conductivity measurements often combined with radio-tracer diffusion studies have been widely used to investigate ionic transport in solids.Powerful alternative methods which are capable of providing information at a microscopic as well as macroscopic level are to be found in the techniques of nuclear magnetic resonance. These techniques have been applied to a wide variety of materials both solid and liquid and are particularly valuable when radio-tracer methods fail due to the lack of a suitable isotope. In this paper we shall describe the n.m.r. relaxation methods by which the (micro- scopic) mean residence time of atoms in their lattice position can be determined and the n.m.r. spin-echo techniques (macroscopic) by which diffusion coefficients are ob- I54 SELF-DIFFUSION IN SOLIDS USING N.M.R. tained directly. To illustrate the application of these techniques to solids we will pre- sent previously unreported measurements on fluorine ion diffusion in strontium fluoride and compare themwith recent work on other fluorites BaF and PbF2.Although they have no direct technological importance themselves these solids undergo an order- disorder phase transition before they melt and exhibit fast ion diffusion. The simple fluorite structure allows n.m.r. theory to be rigorously tested and also permits theoreti- cal model calculations of defect formation and migration parameters relevant to fast ion diffusion in disordered systems to be made. Experimental investigations of ionic motion at both microscopic and macroscopic levels are important for the under- standing of fast ion conductors.In 1839 the " conducting power " of PbF was first investigated by Michael Faraday.' Although behaving as an insulator at room tem- perature when introduced into a simple electrical circuit Faraday noted that "being heated it acquired conducting powers before it was visibly red hot in daylight ". The problem of understanding ionic conduction and self-diffusion in solids is not new. Much has been learnt and we shall show how n.m.r. has recently contributed to this interesting field of study. INDIRECT DIFFUSION MEASUR EMEN T-S PI N RELAXAT1ON Although other relaxation mechanisms may occur the modulation of nuclear magnetic dipole-dipole interactions by the relative translational motion of atoms is the relaxation mechanism most commonly encountered in diffusion studies.The general perturbation theory of this spin relaxation mechanism is well established. For inter- actions between like spins the spin-lattice relaxation time T, is given by T; = + cc[J")(coo) + J'2'(2m0)] (1) where J(q)(co) are the spectral densities of the dipolar interaction fluctuations at fre- quency o,coo is the Larmor frequency and a = y4h21(1+ 1). y is the gyromagnetic ratio of the nuclei with spin quantum number I. A similar expression applies to the spin-spin relaxation time T2when T2& T2r.,.,the rigid lattice value of T2 TT1 = &Cc[J'O'(O)+ lOJ"'(W0) + J'"'(2W,)]. (2) The spin-lattice relaxation in the presence of a resonant rotating magnetic field HI in addition to the steady field Ho has a characteristic time TlPgiven3 by Tlpl = 3 or[J'0'(2io,) + lOJ"'(co,) + J'2'(2coo)] (3) where co = yH,.Eqn (3) is valid when Ho 9 H 9 Hdand Hdis the local dipolar field of the nuclei. When H 5 Hd the perturbation approach fails and provided that the mean time between atomic jumps z is >T2r.,.,a strong collision theory4 can be used. In the limit of H = 0 the spin-lattice relaxation time (in the local di- polar field) TID,is given by where (a -p) is a calculable5 dipolar order parameter. A theoretical treatment which unites the perturbation and strong collision approaches has recently been developed by Wolf and Jung6 The main theoretical problem with the interpretation of relaxation time data is the precise calculation of J(q)(w)or (a -p) in terms of the mean residence time of an atom on a lattice site.R. E. GORDON AND J. H. STRANGE A variety of model^**^-'^ has been used for this purpose. In particular Eisenstadt and Redfield proposed an " encounter model " to take account of the special n.m.r. correlation effects arising with defect-assisted self-diffusion. This model has recently been significantly refined and extended by Wolf 5,11 and extensively tested experiment- ally for anion diffusion on a fluorite latti~e.'~,'~ Using calculations of J(q)(u)for a specific lattice and diffusion process zvalues obtained can be related to the diffusion coefficient D,using the Einstein-Smoluchowsky relation modified to include effects of correlated motion where (r2) is the mean square atomic jump distance and f is the relevant geometrical correlation fa~t0r.l~ The correlation effects mentioned above are sensitive to the mechanism of diffusion and their determination can allow identification of the dominant diffusion mechanism.The general form ofJ(4)(u) is such that it has a maximum when uz 21 1 resulting in characteristic minima for Tl and Tlpwhen measured as functions of temperature. When co,z < 1 theory predicts Tlp= T2and when cooz < 1 Tl = T1 = T2. Relaxation time measurements of 19Fin a pure single crystal of SrF are shown in fig. 1 and exhibit clearly defined minima in Tl and Tlp. On either side of the minima the relaxation times exhibit an Arrhenius behaviour. T2decreases with temperature until the temperature independent rigid lattice value is reached. Below about 750 K another relaxation mechanism influences T, believed to be due to a low level (-1 p.p.m.) of residual paramagnetic impurity.Such impurities which are difficult to 10 10' loo 10-.-E c c -2 .c 10 c -8 2! -3 10 I 1 I I I I 1 I I 1 1.0 1.4 1.8 2.2 1000 KIT FIG. l.-19F relaxation times at 10 MHz for nominally pure SrF, TI 0;T2 0;TI at HI = 30 G A. TI data for SrF + LaF3 (225 p.p.m.),*. The solid lines illustrate the 19F relaxation behaviour found l8 in nominally pure PbF2. SELF-DIFFUSION IN SOLIDS USING N.M.R. remove often limit the useful range of relaxation time measurements in ionic solids. TIPis valuable in following the diffusion process to lower diffusion rates. From eqn (3)it can be seen that the spectral density at CO~(CL)~< coo) makes TlPsensitive to much slower motion than Tl.DIRECT DIFFUSION MEASURE M E NT-S P I N ECHOES When self-diffusion is sufficiently fast a direct n.m.r. method of determining D is available. Aspin echo produced by a 90"-180" pulse sequence applied to a sample in a magnetic field gradient will be attenuated if the nuclei move in the field gradient during the experiment. Attenuation is the result of a loss in the phase coherence of the precessing nuclei. The nuclear spin state of the diffusing atom serves as a label and under favourable experimental conditions permits a convenient and direct measure- ment of D. Larger field gradients permit the measurement of slower diffusion but ulti-mately result in signals being difficult or impossible to observe.These technical problems have been elegantly overcome and greater precision obtained by applying large field gradients '' as pulses during the time when signals were not being observed. The relevant theory of this experiment is to be found in the literature.'' The tech- nique has hitherto been primarily restricted in its application to liquids. Recently an apparatus suitable for operation with solids at high temperatures has been described16 and has been used to measure F-diffusion in fluorites over a wide temperature range.17p1s The slow diffusion limit for this technique is primarily set by the T2of the spin system under study and the maximum pulsed magnetic field gradients that can be obtained. Asmotion slows T2decreases until it is no longer possible to achieve dis- cernible spin echo attenuation during the time available (T,).Modifications 19-,1 to the original pulse sequence have circumvented this limitation to some extent and dif- in suitable sytems. fusion coefficients as low as 3 x m2s-' have been mea~ured'~ COMPARISON 0F TECH N 1Q U ES-EX PER I MEN T S 0N F L UO R I T E S 17922 Recent F-diffusion studies in the fluorite crystals BaF, PbF2 and SrF illustrate well the application of n.m.r. techniques and allow comparison with transport data obtained by the completely independent techniques of ionic conductivity and radio-tracer diffusion. Due to the short half-life of the most convenient fluorine radioisotope "F tracer experiments are very difficult are limited to very fast diffusion rates and have been attempted only for BaF,.Tracer diffusion is also subject to spatial correlation effects. Electrical conductivity measures the movement of charged defects (e.g.,vacancies or interstitials) whose motion is usually uncorrelated. The movement of defects results in a (correlated) motion of ions. The measured conductivity CJ at a temperature Tcan be related to the diffusion coefficient [eqn (5)] using the Nernst-Einstein equation. For anion diffusion on a fluorite lattice pro- ceeding by a monovacancy mechanism this relationship is D =f(k/Ne2)aT (6) where N is the ion density e the effective charge of the defect and k is Boltzmann's constant. The spatial correlation factorI4 f is 0.653 in this case [eqn (5) and (6)].Conductivity measurements have been made22,23 on the same samples used in the n.m.r. investigations and can therefore be used for comparison. The range of applicability of the various techniques is illustrated in fig. 2. In principle the n.m.r. methods can be used throughout the ranges shown by full and broken lines. In the fluorites a practical lower limit is determined by paramagnetic R. E. GORDON AND J. H. STRANGE impurities and is marked by the full lines. Upper limits are determined by the chemi- cal reactivity of samples and the problems of sample containment. Relaxation time measurements of T, T, TIPand TI for 19F have been ma,de from 300 to 1200K in pure and doped single crystals of Ba ,,Fand provide some of the most comprehensive diffusion studies by relaxation.In the pure crystal intrinsic diffusion I f--tracer ----> I I 1 I 1 I 1 I 1 1 I I 1 1 Idle 1o-IL 10-l0 D t m2s' FIG.2.-Ranges of applicability of the various techniques for studying F-diffusion in fluorite crystals. The upper scale is z the mean residence time for ions on a lattice site. The lower scale is D the diffusion coefficient. The scales are related by eqn (5). due to the random motion of thermally generated point defects was followed from 475 to 1200 K. Measurements on this material also serve to illustrate the strong dependence of relaxation rates on crystal orientation.'2*24 The interpretation of relaxation time data obtained from crystals of unknown orientation or from poly- crystalline samples should be treated with caution especially for TIPwhen IO~T< 1 < cL)oz.Relaxation measurements for SrF, shown in fig. 1 exhibit the same general features that were found for BaF,. Also shown on fig. 1 are previously r2ported" relaxation time data for PbF from 300 K to its melting point 1095 K. Both materials have metal ions with weak nuclear magnetic moments but their influence on 19F relaxation appears negligible. The PbF2 data below the temperature of the TI minimum are again qualitatively similar to that for BaF and SrF,. Activation energies naturally differ and the T minimum occurs at a much lower temperature showing correspond- ingly faster F-diffusion in PbF,. The PbF2 data obtained above the TI minimum differ markedly from the expected classical behaviour found in BaF and SrF,.This anomalous behaviour when first observed25 was attributed to the presence of highly correlated modes of ionic motion associated with the phase transition at 705 K. Extensive linewidth26 and subsequent n.m.r. diffusion and relaxation measurements have demonstrated that the anomaly is probably due to paramagnetic impurities. The intrinsic point defects in the fluorites are believed to be predominantly anion- Frenkel pairs.27 The activation enthalpy for anion migration may be expressed as (&/2 + 11,~). h is the formation enthalpy of the Frenkel pair and IT, the migration enthalpy of the defect of type ,j (vacancy or interstitial) providing the dominant diffusion n~echanism. The two most probable mechanisms for the F- diffusion are the vacancy IT,^) and non-collinear interstitialcy (11,~) mechani~ms.~~ SELF-DIFFUSION IN SOLIDS USING N.M.R.The intrinsic ionic diffusion can be modified by the incorporation of aliovalent impurities into the lattice. Doping difluorides with trivalent or monovalent cations can create F-interstitials and vacancies respectively. Defects so produced can deter- mine the dominant diffusion mechanism and their concentration will determine the temperature range over which extrinsic behaviour extends. The activation energy in this region will be given by hmjand doping experiments can therefore be combined with the results for the intrinsic range to obtain values for h and hmjseparately. A typical effect of doping on TI relaxation can be seen in the results for SrF containing 225 p.p.m.LaF shown in fig. 1. The temperature dependence of T is reduced corre- sponding in this case to an activation enthalpy hmi. Similar effects are observed' in T2,TI and T, and reliable values for defect parameters can be obtained. Table 1 lists the values obtained for the fluorites studied by n.m.r. and also the values obtained from theoretical calculations.'* TABLE 1.-F-DIFFUSION PARAMETERS IN FLUORITES defect enthalpies /eV for F-diffusion fluorite melting order-disorder lattice point transition parameter n.m.r. studies* theoretical valuesz8 TmIK temperature TcjK rlA /I r /hV huii 1117 h,ur Ilmi SrF BaFZLL PbF,'" 1723 1560 1095 1450 1235 705 2.8998 3.101 2.969 2.58 kO.18 1.80&0.11 0.88 * 0.10 0.59 50.02 0.6250.05 0.29 * 0.02 0.80& 0.04 0.77*0.01 - 2.38 0.43 1.98 0.46 0.80 0.72 - * The defect enthalpies quoted for SrF should be considered as preliminary since studies using different levelsof alioval- ent dopant are still in progress.The intrinsic point defect concentration lid can also be estimated from doping studies. The extrinsic and intrinsic regions meet at the temperature where nd is equal to the (known) concentration of dissolved aliovalent dopant. By studying crystals with various concentrations of dopant the temperature dependence of nd can be obtained. Such information at temperatures Tc of the order-disorder phase transition is particularly interesting as there is currently considerable debate on the extent of anion disorder at this tran~ition.'~.~' Self-diffusion coefficients D,derived from relaxation data using Wolf's method of analysis and eqn (5) are presented in fig.3. A monovacancy mechanism was assumed for all pure samples as this is thought to be the dominant mechanism for most of the temperature range studied. Values for D obtained from the conductivity measurements using eqn (6) with J' = 0.653 appropriate to a monovacancy mechan- ism are shown together with those measured by n.m.r. directly. Agreement between the results of the different methods is very satisfactory particularly in BaF where measurements span eleven orders of magnitude and strongly suggest a diffusion mechanism controlled by point defects even above Tc. The region of the phase transition in each material deserves special attention.The n.m.r. pulsed field gradient measurements in BaF and PbFz show a marked decrease in the activation enthalpy in the fast diffusion region above Tcand in both materials the phase transition apparently occurs when D e2 x m2s-'. Further studies of SrF and CaF are in progress to establish whether this is a general feature of fluorites. For PbF in particular the agreement between conductivity and n.ni.r. diffusion measurements above Tc is such that extensive disorder and a high occupancy of anion interstitial sites as suggested previo~sly,~~ seems unlikely. An anion exchange mechanism of diffusion '* also appears to be inoperative. The picture of anion diffusion in fluorites above Tc that is currently emerging is R.E. GORDON AND J. H. STRANGE TIK 1000 714 500 417 1000K/ T FIG. 3.-Temperature dependence of F-diffusion coefficients in fluorite crystals. SrF2 0 ~ n.m.r. relaxation; 2 n.m.r. pulsed field gradient; ionic conductivity. BaF 0 n.m.r. relaxation; A n.m.r. pulsed field gradient; --ionic conductivity. PbF 0n.m.r. relaxation; A n.m.r. pulsed field gradient; -. -. -ionic conductivity. very rapid hopping on regular lattice sites with defect concentrations <10%. The anion radial distribution function g(r) has recently been obtained by molecular dynamics calculation~~~*~~ above Tc for CaF,. The spectral density functions for TI relaxation by translational diffusion in a liquid may be expressed33 in terms of g(r) using where N is the spin density and Tr is a differential operator.It will be interesting to apply this " liquid " model to the fluorites we have studied when g(r) data become available. One might expect a markedly different result to be obtained for TI than is found using the lattice diffusion model that we have found to be so satisfactory. The n.m.r. pulsed field gradient experiments measure displacements arising from many atomic jumps whereas conductivity measures any corresponding motion of charged defects. In contrast the n.m.r. relaxation methods probe the " local motion " on an atomic scale and diffusion coefficients can only be calculated by adopting a suitable model for the motion and employing a rather complex theoretical treatment. 160 SELF-DIFFUSION IN SOLIDS USING N.M.R.The agreement obtained between these three different approaches for the fluorites adds considerable confidence to the understanding of the atomic diffusion and defect structure of these solids. The authors gratefully acknowledge the financial assistance of the S.R.C ' M. Faraday E.vp-periiiietitn1Researches iti Electricity (R. and J. E. Taylor London 1839) vol. 1 para. 1340 426. 'A. Abragani The Priiiciples of Nriclenr Mngtietism (Clarendon Oxford 1962) chap. VIII. D. C. Look and I. J. Lowe J. Cheni. Phys. 1966 44 2995. C. P. Slichter and D. C. Ailion Phys. Rev. A 1964 135 1099. D. Wolf Phys. Rev. B 1974 10 2724. D. Wolf and P. Jung Phys. Rev. B 1975 12 3596. H. C. Torrey Phjls. Rev. 1953 92 962. * M.Eisenstadt and A. G. Redfield Phys. Rev. 1963 132 635. C. A. Sholl J. Phys. C 1974 7,3378. lo C. A. Sholl J. Phys. C 1975 8 1737. D. Wolf PIiys. Rev. B 1974 10 2710. l2 D. Wolf D. R. Figueroa and J. H. Strange Phys. Rev. B 1977 15 2545. l3 D. R. Figueroa J. H. Strange and D. Wolf Phys. Rev. B in press. l4 K. Compaan and Y. Haven Trans. Fnradny Soc. 1958 54 1498. l5 E. 0.Stejskal and J. E. Tanner J. Phys. Cheni. 1965 42 288. l6 R. E. Gordon J. H. Strange and J. B. W. Webber J. Phys. E 1978 in press. R. E. Gordon and J. H. Strange Proc. XIXrh Coiigress Anipere (Heidelberg 1976) p. 495. R. E. Gordon and J. H. Strange J. Pliys. C 1978 11 3213. l9 J. E. Tanner J. Cheni. Phys. 1972 56 3850. *O K. J. Packer C. Rees and D. J. Tomlinson Mol.Phj-s. 1970 18 421. 'I I. Zupanic J. Pirs M. Luzar R. Blinc and J. W. Doane Solid State Coiizti~.,1974 15 227. 22 D. R. Figueroa A. V. Chadwick and J. H. Strange J. Phys. C 1978 11 55. 23 V. M. Carr. A. V. Chadwick and R. Saghafian J. Phys. C 1978 11 L637. 24 D. R. Figueroa and J. H. Strange J. Phys. C 1976 9 L203. 25 J. B. Boyce J. C. Mikkelsen and M. O'Keeffe Solid State Coimz. 1977 21 955. 26 R. D. Hogg S. P. Vernon and V. Jaccarino Phys. Rev. Letters 1977 39 481. ''A. B. Lidiard Crystnls M.ith the Fliiorite Strrtctiire ed. W. Hayes (Clarendon Oxford 1974). 28 C. R. A. Catlow M. J. Norgett and T. A. Ross J. Phys. C 1977 10 1627. 29 C. R. A. Catlow R. T. Harley and W. Hayes J. Phys. C 1977 10 L559. 30 A. S. Dworkin and M. A. Bredig J. Phys. Chetii. 1968 72 1277. 'I A. Rahman J. Clieni. Phys. 1976 65 4845. 32 M. Dixon and M. J. Gillan J. Phys. C. 1978 11 L165. 33 L.-P. Hwang and J. H. Freed J. Chetii. Phys. 1975 63 4017.