摘要:

DISCUSSIONS OF THE FARADAY SOCIETY No. 23, 1957 MOLECULAR MECHANISM OF RATE PROCESSES IN SOLIDS THE FARADAY SOCIETY A g e n t s for the Society’s Publications : The Aberdeen University Press Ltd. 6 Upper Kirkgate AberdeenThe Faraciay Society I-esei-ves the copyright OJ all Coinmunications published iri the L' Discrtssions '' PUBLISHED . . . 1957 PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS A B E K D E E NA GENERAL DISCUSSION ON MOLECULAR MECHANISM QF RATE PROCESSES IN SOLIDS A GENERAL DISCUSSION on the Molecular Mechanism of Rate Processes in Solids was held at the invitation of, and in conjunction with the Koninklijke Nederlandse Chemische Vereniging at the Koninklijke Institut voor de Tropen, Amsterdam, during April 15th-l8th, 1957. Members and visitors, who included scientists from America, Australia, Belgium, France, Germany, Italy, Netherlands, Sweden, Switzerland and the United Kingdom were welcomed at an official re- ception given by the Burgomaster of Amsterdam at the Rijksmuseum and at Koninklijke Institut by the President of the Faraday Society, Mr. R.P. Bell, F.R.S., and the President of the Koninklijke Nederlandse Chemische Vereniging, Mr. Gerding, who shared the duties of Chairman during the Discussion. 3A GENERAL DISCUSSION ON MOLECULAR MECHANISM QF RATE PROCESSES IN SOLIDS A GENERAL DISCUSSION on the Molecular Mechanism of Rate Processes in Solids was held at the invitation of, and in conjunction with the Koninklijke Nederlandse Chemische Vereniging at the Koninklijke Institut voor de Tropen, Amsterdam, during April 15th-l8th, 1957.Members and visitors, who included scientists from America, Australia, Belgium, France, Germany, Italy, Netherlands, Sweden, Switzerland and the United Kingdom were welcomed at an official re- ception given by the Burgomaster of Amsterdam at the Rijksmuseum and at Koninklijke Institut by the President of the Faraday Society, Mr. R. P. Bell, F.R.S., and the President of the Koninklijke Nederlandse Chemische Vereniging, Mr. Gerding, who shared the duties of Chairman during the Discussion. 3CONTENTS PAGE INTRODUCTION. By C. J. F. Bottcher . . 7 A. RELAXATION PROCESSES- A Mechanical and Thermodynamical Theory of Non-Linear Re- By H. C. Brinkman and F. Schwarzl laxation Behaviour of Solids. 1 1 The Mechanism of Low Temperature Mechanical Relaxation in Deformed Crystals.By A. Seeger, H. Donth and F. Pfaff . . 19 Dielectric Relaxation in Solid Hydrogen Halides. By R. H. Cole and S. Havriliak, Jr. . . 31 Dielectric Relaxation Processes in Lithium, Sodium and Potassium Halides. By J. S. Dryden and R. J. Meakins . . 39 Dielectric Relaxation and the Electrical Conductivity of Ice Crystals. By H. Granicher, C. Jaccard, P. Scherrer and A. Steinemann . 50 Dielectric Loss in Insulating Solids caused by Impurities and Colour Centres. By J. Volger . . 63 GENERAL DrscussIoN.--Dr. Y. Haven, Dr. H. C. Brinkman and MI-. F. Schwarzl, Dr. J. Wood and Dr. A. Suddaby, Prof. A. R. Ubbelohcie, Prof. F. C. Frank, Prof. J. S. Koehler, Dr. A. Seegzr, Dr. J. Volger, Dr. J. S. Dryden, Prof. J. H. de Boer, Dr.H. Granicher, Prof. F. G. Fumi, Dr. A. B. Lidiard, Dr. R. A. Sack, Dr. A. Steinemann, Prof. R. M. Barrer, Dr. M. Eigen, Dr. J. L. Meijering, Dr. J. Meinnel . . 72 B. STEADY-STATE PROCESSES NOT INVOLVrNG LATTICE --ARRANGEMENT- Introduction. By J. S. Koehler and F. Seitz . . 85 Lattice Calculations on Point Imperfections in the Alkali Halides. By F. G. Funii and M. P. Tosi . . 92 Diffusion Coefficients in Solids, Their Measurement and Significance. By J. Crank . . 99 Some Fundamental Aspects of the Mechanism of Diffusion in Crystals. By I<. Conipaan and Y . Haven . . 105 Thermoelectric Power of Ionic Crystals. By R. E. Howard and A. B. Lidiard . . 113 Dislocations and Point Defects. By F. C . Frank . . 122 56 CONTENTS Abnormal Values of A and E in Migration Processes.By A. R. Ubbelohde . . 12s Diffusion of Nickel-63 in Nickel Oxide (NiO). By R. Lindiier and A. akerstrom . . 133 Diffusion in Tonic Crystals and the Process of Sintering. By W. Jost and H. J. Oel . . 137 Point Defects Produced by Irradiation and their Annealing in Tonic and Covalent Crystals. By F. P. Clarke . . 141 GENERAL DIscuSSIoN.--Prof. F. G. Fumi, Prof. R. M. Barrer, Prof. R. A. W. Haul, Dr. J. Crank, Dr. L. E. J. Roberts, Dr. Y. Haven, Dr. A. Seeger, Prof. J. S. Koehler, Dr. P. Ruetschi, Prof, W. Jost, Dr. A. B. Lidiard, Prof. A. R. Ubbelohde, Prof. J. H. de Boer, Prof. F. C. Frank, Dr. G. Saloman, Mr. J. F. Laurent, Dr. D. A. Young, Dr. A. G. Maddock, Mr. F. P. Clarke, Mr. G. N. Walton 155 C. STEADY-STATE PROCESSES INVOLVING LATTICE RE-ARRANGEMENT- Tntroduction. By J. H. de Boer . . 171 Mechanism and Kinetics of the Allotropic Transformation of Tin. By W. G. Burgers and L. J. Groen . , 183 Orientational Effects in the Polymorphic Transformations of Sulphur. By C. Briske and N. K. Hartshorne . . 196 Azide Decompositions. By F. C. Tompkins and D. A. Young . . 202 The Chemical Effects of Radiative Thermal Neutron Capture. Part .?.-The Kinetics of a Radical Recombination Process of Solids. By M. M. de Maine, A. G. Maddock and K. Taiigbd . . 211 GENERAL DIscussIoN.-~r. P. W. M. Jacobs, Dr. K. H. Lieser, Dr. J. Meinnel, Dr. R. P. Rastogi, Dr. L. J. Groen, Dr. W. J. Dunning, Dr. N. E-I. Hartshorne, Dr. G. Salomon, Prof. J. H. de Boer, Dr. F. C. Tompkins and Dr. D. A. Young, Dr. J. Y . Macdonald, Dr. T. C . Waddington and Dr. Peter Gray, Dr. A. B. Lidiard, Dr. F. S . Stone 220

ISSN:0366-9033    

DOI:10.1039/DF9572300001

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

MOLECULAR MECHANISM QF RATE PROCESSES IN SOLIDS A. RELAXATION PROCESSES INTRODUCTORY PAPER BY C. J. F. BOTTCHER This introduction may be started with a few remarks concerning the title and the three sub-titles of the present discussions. In defining a rate process we usually limit the term to a system proceeding from one equilibrium state to another, under the influence of a change of the external parameters, in such a way that energy dissipation, and hence entropy production, takes place. If one thinks of the problem in terms of potential barriers, then it is first neces- sary that certain particles receive sufficient energy from their surroundings to overcome a barrier and then-on reaching a new condition-they must loose kinetic energy, this loss being experimentally represented by the appearance of energy dissipation as heat.Whether such a picture of potential barriers is appro- priate for the particular problem or not, there must still be, in any rate process, the element of an experimentally observable time, in which energy received from the external disturbance is dissipated as heat. Concerning the mechanism of such a dissipation process a few remarks shouid be made. The molecular models used for its description have always the dis- advantage that macroscopic concepts are used on a microscopic scale. A striking example is Debye’s model of a molecular dipole, rotating in a viscous continuum, to which Stokes’s law is applied. In essence this model is merely a very small edition of the well-known mechanical Voigt or Kelvin model, which is, in this case, so to speak, electrically actuated.Unfortunately such a use of macroscopic concepts in molecular models seems to be almost inevitable at the present state of knowledge of the statistical inter- pretation of energy dissipation and entropy production. Even in the derivation in Brinkman and Schwarzl’s paper 1-although in many respects more realistic from the molecular point of view-concepts like friction and diffusion had to be introduced. Another example is that sometimes heat conduction is used as the underlying mechanism of energy dissipation. In that case the notion of two intermixed sub-systems of different temperatures is introduced, these being, for instance, in the acoustic relaxation of gases the translational and the internal degrees of freedom.Similarly Casimir and du Pre, in their theory of paramagnetic relaxation,2 intr0duce.d the idea of different spin and lattice temperatures. As a last example of such a use of macroscopic concepts I might mention the model of the scattering of an “ ordered ” elastic wave at dislocations as described in the paper of Seeger and collaborators.3 In this model the energy of the ordered wave ultimately disappears into the continuum of the “ disordered ” elastic waves, determining the thermodynamic state of the crystal. Such an order-disorder interpretation may well be closer to a real molecular theory, but quantitative treatments in specific cases will be necessary before one can judge how fruitful such an approach will be. In the programme, section A has been described as “ relaxation processes ” and the sections B and C as “ steady-state processes ” without and with lattice rearrangement, respectively.I think this classification does not represent the 78 INTRODUCTION actual differences between the papers of group A and those of B and C. Before discussing this I should first remark that there seems to be no single definition for a steady-state process. A chemical engineer would say it is a process in which after a transient period all external conditions of the essential part of the system are maintained constant. A mathematical physicist, however, would refcr to a periodic form of solution of his equations in which transient terms have dis- appeared. In Ubbelohde’s paper 4 the term “ steady-state ” refers to the case where after the build-up of an activated state of the solid a steady migration is established. By either of these three definitions, and perhaps by any other definition of steady-state processes, only part of the experiments of the sections B and C are of a steady-state type.Moreover there is not a border line between relaxation and steady-state processes ; as in most relaxation phenomena it is possible to work under steady-state conditions, provided the developed heat is taken away. There is, however, another way to distinguish between section A and the other sections : in the sections B and C a net translational migration is essential, whereas in A this is only the case in the conductivity measurements of Granicher and col1aborators.s After these general remarks I might return to my task to introduce section A on relaxation processes.In order to demonstrate some characteristic features of relaxation phenomena I choose the example of a polar dielectric between the plates of an ideal plane condenser to which a uniform static field is applied. We imagine two fundamental experiments, represented in fig. 1 and fig. 2. E D P t -- t FIG. 1 .-Dielectric relaxation under prescribed E. D E FIG. 2.-Dielectric relaxation under prescribed D. In experiment I we keep the condenser plates at a constant potential difference during a certain period. Before and after that period we take care that the poten- tial is zero. Thus the electrical field E is prescribed as shown in fig. 1. The re- sulting curve for the dielectric displacement D is shown at the right-hand side: the electronic and atomic polarization are established immediately ; the dipoles, however, need some time before their new orientational equilibrium distribution is reached.When the potential drops to zero it takes some time before a random orientation of the dipoles is re-established. In experiment I1 we do not prescribe the potential, but the true charge of the plates and hence D. As the contribution of the dipole orientation to the field E has the opposite sign to the contribution of the true charges, the field E gradually decreases during the first period until the new equilibrium distribution of the dipoles is reached. When the true charges are suddenly taken away, for instance by removing the condenser plates, a field of the opposite direction remains, due toc.J . F . B ~ T T C H E R 9 the dipole polarization. It gradually disappears while the random orientation of the dipoles is re-established. At room temperature such time-lags are usually of an order of magnitude of 10-10 sec, but in very viscous materials, and in particular at low temperatures they can be much larger, even seconds or minutes. It should be mentioned that if the D-curve of fig. 1 would be forced, the E-curve of fig. 1 would result, according to the superposition principle. We can compare the first parts of these two experiments with two ways of extending a fibre : the first part of experiment I corresponds to an elongation of the fibre under constant stress : the first part of experiment I1 shows the stress as a function of time for a prescribed constant elongation.In such mechanical examples the two cases are always distinguished by applying the term retardation to the first part of experiment I and the term relaxation to the first part of experiment 11. In electrical problems, however, only the term relaxation is used although most experiments correspond with what in mechanical terms is called retardation. As it would be very difficult to change this generalized use of the term relaxation, I would suggest it is accepted as the general name for all after-effects and to dis- tinguish, where necessary, between (i) relaxation under prescribed stress, or retardation ; (ii) relaxation under prescribed strain, or relaxation in the strict sense of the In many experimental studies of relaxation phenomena an alternating external disturbance is applied.In that case usually a distinction is made between resonance and relaxation, for instance in the paper of Seeger and collaborators. Strictly speaking such a sharp division is not necessary because the cases where the term relaxation is used could also be described as resonance with over-critical damping. The other extreme would be the ideal case of resonance without energy dissipation. In announcing this conference the stress had been laid on the molecular mechan- ism of the rate processes. Consequently, in this introduction to section A, par- ticular attention must be paid to this question: to what extent can results of relaxation experiments lead to conclusions concerning the underlying molecular mechanism? In this respect it must be emphasized that from relaxation measure- ments on one single system hardly any conclusion can be drawn about the mole- cular mechanism.A wide variety of mechanisms can account for the same macroscopic behaviour. A striking example is that of dielectric relaxation where it is even impossible to distinguish between the frequency-dependent part of ionic conductivity losses and dipolar losses. Another example of the same limitation is the fact that from relaxation measurements on a single system only, one cannot de- cide whether the relaxation is caused by a chemical reaction or by a physical process. Some indication, however, can be gained from the magnitude of the relaxa- tion time or times, calculated from the slope of the curves in non-periodic cases and from the frequency dependence of the phenomenon for a periodical disturbances and also from the activation free energy, determined from the ex- ponential temperature dependence of the relaxation time.The experimental values of these two quantities, relaxation time and activation energy, can be used to test a hypothetical molecular mechanism in the few cases where it is possible to predict their approximate values from the model. If no large deviations are found the model niight be accepted as one of the possible explanations. The most satisfactory way, however, to arrive at more or less definite conclusions concerning the molecular mechanism is to examine experimentally how the phenomenon depends on the physical and chemical structure; in other words, physical and chemical modulation of the solid must be an essential element of the experiments.word.10 INTRODUCTION This final remark is not restricted to section A but applies to all the rate pro- cesses. In solids it means that we must examine how the rate processes depend on crystal structure and lattice imperfections, on the chemical composition of the solid-not overlooking the impurities-and, in some, on the chemical structure of the particles. Even then a profound knowledge of other physical properties of the solids is required. A classical example is the well-known work of Snoek 7 on the mechanical relaxation of cc-iron crystals, with carbon as an impurity. Snoek and his collaborators showed that in order to arrive at a definite con- clusion about the molecular mechanism it was necessary to vary both the main component and the impurity by replacing, for instance, carbon by nitrogen and part of the iron by manganese. Needless to say an accurate knowledge of the crystal structure was indispensable in this case too. The papers of our conference present many similar examples and it might even be concluded that the striking common element of most of these papers is that they show that variation of the physical and chemical structure of the solid is the essential experimental condition for obtaining conclusive information concerning the molecular mechanism of rate processes. 1 Brinkman and Schwarzl, this Discussion. 2 Casimir and du Prk, Physica, 1938,5, 507. 3 Seeger, Donth and Pfaff, this Discussion. 4 Ubbelohde, this Discussion. 5 Granicher, Jaccard and Steinemann, this Discussion. 6 see, e.g., Bottcher, Theory of electric polarisation (Amsterdam, Elsevier Publ. Co., 7 Snoek, Physica, 1939, 6, 591 ; 1941, 8,711,734. 1952), chap. 10.

ISSN:0366-9033    

DOI:10.1039/DF9572300007

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

A MECHANICAL AND THERMODYNAMICAL TMEORY OF NON-LINEAR RELAXATION BEHAVIOUR OF SOLIDS BY H. C. BRINKMAN AND F. SCHWARZL Centraal Laboratorium, T.N.O., Delft, Holland Received 15th January, 1957 A theory is worked out describing relaxation as a diffusion process of molecular groups over potential barriers. Mechanical and thermodynamical properties resulting from this model are calculated. On the basis of this theory deviations from linearity (i.e. proportionality between stress and strain) may be related to properties of the potential holes and barriers in which the diffusion process takes place. The reaction of a viscoelastic system on external forces shows retardation. It has often been suggested that such a behaviour may be interpreted on a molec- ular scale by the migration of inolecular groups from one equilibrium position to another.1 Such a picture is essentially a one-particle model analogous to the models introduced in the theory of liquids.The movement of certain kinds of molec- ular groups is considered. Their migration is assumed to be responsible for the mechanism of retarded deformation. The molecules surrounding these groups are not included explicitly in the calculation. Their influence is described by a potential field V(q) in which the movement of the groups takes place. The rate of migration of the molecular groups between two potential holes, separated by a barrier, is calculated. Up to now this has always been done by the theory of reaction rates under the very questionable assumption of thermal equilibrium between the particle density in the holes and on the barrier.In our theory the rate of migration is calculated on the basis of a diffusion theory. The consequences of this picture are discussed in detail. Even while starting from a very general model quite definite conclusions are reached due to the re- strictions imposed by thermodynamics. The model leads to a single retardation time for the resulting creep deformation; at larger stresses it shows non-linear viscoelastic behaviour, viz., a stress dependence of the retardation time and of the ultimate compliance. On the other hand, one must expect that our one particle model will describe the most simple retardation phenomena only. For the description of more com- plicated processes a distribution of retardation times is necessary. For linear behaviour, this is obtained by means of a distribution function of the parameters of the model.However, the combination of different retardation processes presents a difficult problem outside the region of linear behaviour which is as yet unsolved. On the other hand even our simple molecular picture shows traits also occurring in experiment, e.g. a decrease of the retardation time with increasing external force. We are confident therefore that the mechanical and thermodynamical properties resulting from our one particle model give some indication of the behaviour, even of complicated systems. THERMODYNAMICAL DISCUSSION OF THE RELAXATION PROCESS A viscoelastic bar under the action of a constant longitudinal force shows retarded deformation to a new equilibrium position.This creep may be inter- preted on a molecular scale as the migration of certain molecular groups to new 1112 NON-LINEAR RELAXATION BEHAVIOUR equilibrium positions. Before an external force is applied, the molecular groups are distributed according to a Boltzmann distribution in the potential field Y originating from the surrounding molecules. V(q) is a function of the position vector q. This potential field may show holes, separated by barriers. It is the migration of the molecular groups over these barriers which causes the phenomenon of retarded deformation. The external force tends to establish a new equilibrium distribution of the molecular groups, which are responsible for the creep of the material. These new equilibrium positions may be described as a new Boltzmann distribution of the molecular groups in a potential field V(q) - v(o, q).For a one-dimensional model v would be equal to uq ; in our case v is an as yet unknown function of the external force o and the position vector q. Some information on v may be ob- tained on the basis of statistical thermodynamics. Let the spatial distribution of the molecular groups considered be given by the density p(q, u, T, t ) . For the new equilibrium distribution p is given by where C is determined from the normalization condition b' pdq = 1 . The length of the bar I is related in some way to the distribution function p. It may be defined quite generally where g is an unknown function of CT and q. This means that all temperature influences apart from those acting via the Boltzmann factor in (1) have been neglected.For an experiment under a constant external force u the obvious thermodynamic functions to be defined are the entropy S and the enthalpy H : It should be noted that V, v and g are assumed to be independent of T. +a s =- FJ p logpdg, H = J (V-vlpdq. -a +a -cQ (4) When comparing two states of equilibrium for different o (and T ) the basic thermo- dynamic relation TdS = dH 4- Zdo (5) should be obeyed. By substitution between v and g is of (1) in the definitions (3), (4) and (5) the following relation obtained g = avpo. (6) The internal energy U is +a U = H + oZ= [ -03 ( Y - v + o2)pdq. a. (7) We are interested in materials for which the instantaneous deformation due to the process considered is small as compared with the retarded deformation, i.e.a change of Z is mainly due to a change of p.H . C. BRINKMAN A N D F . SCHWARZL 13 If we restrict ourselves to this case g in (3) is a function of q only and the follow- ing expressions are obtained for g, v and U +a g = f ( q ) ; v = of(q) ;,U = 1 Vpdq. (8) -a THE DIFFUSION THEORY The migration of molecular groups in the field V - v is described as a diffusion process, governed by the Smoluchowski diffusion equation.:! In order to simplify the resulting formulae the calculations will be restricted to a one-dimensional model (co-ordinate 4). No essential features are lost by this restriction. The Smoluchowski equation reads In our special case the force Fis defined by whilefis a friction factor, defined as the ratio between Fand the systematic velocity of the molecular groups.This friction factor ensures the thermal coupling be- tween the molecular groups considered and the rest of the system which does not appear explicitly in the calculations. Iff is large enough, temperature equilibrium (i.e. a Boltzmann distribution of the momenta of the molecular groups) remains established in a first approximation, even while the diffusion process is set in motion.2 The diffusion process for our special model was treated by one of us 3 with the following results. Two states, 1 and 2, are defined, corresponding to two potential holes 1 and 2. The numbers of particles in these states are +a n1 = J@ pdq and 112 = / pdq, -a 4t where qr indicates the value of q at the top of the potential barrier, separating the two holes.This definition of two separate states has a physical meaning only, if the height of the barrier is large compared with kT. The following equations for n1 and 112 may be derived from (9) 3 (12) 0 . - ni = n2 = K1nl - K2n2, where the reaction rates Ki (i = 1, 2) are found as 3 By means of (12) and (13) the change of 121 and 122 may be calculated if V and v are known. The derivation of (13) is based on the assumption that the height H of the barrier is so large as compared with kT that the particles in each hole are approximately distributed according to Boltzniann’s formula, while the density p on the barrier is so low that it may be neglected. These assumptions amount to v - v exp (- I-) 11 in hole 1 , =p 0 on the barrier, (14) exp (- --) v - v in hole 2 , 1214 NON-LINEAR RELAXATION BEHAVIOUR where nl and 122 are solutions of (12) normalized by the relation y11 -f n2 = 1, while 11 and 12 are defined as 4 =I exp (- V - V -+ (i = 1,2).hole i Substitution of (14) in (3)-(5) yields the length and the thermodynamic pro- perties of the bar as functions of time. EXPLICIT CALCULATION OF THE DEFORMATION AND THE THERMODYNAMIC FUNCTIONS FOR A SPECIAL MODEL The general formulae of the previous sections are now applied to a simple, rectangular model consisting of two holes and a barrier (cf. fig. 1). For the deformation v of this field the following assumption is made C ~ O in hole 1, v = 2 Cp L on the barrier, C20 in hole 2, where C1, Cz and Ct may be functions of q, but not of D (cf.(8)). FIG. 1.-Potential energies of holes and barrier. The introduction of functions of q would be physically justified only for a model in which the detailed structure of the holes as a function of q is considered. Therefore the C are treated as constants in our rectangular model. The mechan- ical and thermodynamical functions may now be expressed in the constants of the model (cf. fig. 1) and n1 and 122. In the results the following abbreviations, related to the molecular entropies and free energies in the states 1 , 2 and t, are used : S1 = k log d l , 5’2 = k log d2, S* = k log b, F1 = - TS1, F2 = - TS2 + h, Ft = - TSt + H, + = (F2 - F1)/2, $ = Ft - (F1 + F2)/2. (17) For the sake of clearness the free enthalpy, Fi - Ciu, is given in fig.2 as a function of the co-ordinate q in the deformed state. For the undeformed state an analogous figure is obtained by putting (T = 0. It should be observed that the molecular free energies and enthalpies, introduced above, do not include the terms related to the particle density 121 or 112.H . C . BRINKMAN AND F . SCHWARZL 15 The following expressions are obtained for the total internal energy and the total entropy U = nzh, S = - k(n1 log nl + n2 log n2) + nlS1 + n2S2. 121 = nlo + (ny - n10) (1 - exp (- t/r}, (1 8) (1 9) where nlo, n y are the values of nl in the undeformed, deformed state of equilibrium. The differential eqn. (12) yields an expression for nl, t F-CC F,-C,a F,-c*c s-C,a 4 ? a d, q-- b FIG. 2.-Free enthalpies of holes and barrier in the deformed state (undeformed state for CJ = 0).The retardation time 7(T-l = K1 + K2) and the final elongation A1 of the fb2 exp ($ - ua/kT) bar are found as (20) 7 -7 2 kT cash ($ - wa/kT) ’ where cc = c, - (Cl + C2)/2, w = (C2 - c,>/2. (2 2) Using (22), the following expressions for n10 and n? are found : (23) exp ( d / W . $2 = exp [($ - wa)/kT] 2 cosh ($/kT) ’ 2 cash [(# - i t ~ ) / k T ] ’ n10 = The retardation time r contains the friction factor f, originating from the dif- fusion theory. The expression # - ua in the exponent gives the difference between the free enthalpy of the transition state t and the mean free enthalpy of states 1 and 2. The expression C$ - wa in the denominator gives the difference between the free enthalpies of states 1 and 2. It is interesting to compare (20) with the expression for the reaction rate result- ing from the so-called transition state method 4 where K = 1[r = (kT[h) exp (- AF/kT), AF -- Ft -- F1.(24)16 NON-LINEAR RELAXATION BEHAVIOUR Eqn. (24) is derived by assuming thermodynamic equilibrium between the top of the barrier and a single hole. On the contrary, our equation is derived from a discussion of a diffusion process, while the questionable assumption of equilibrium between hole and barrier is avoided. Moreover, our treatment includes the forward as well as the backward diffusion process. DISCUSSION OF THE RETARDATION TIME AND OF THE FINAL ELONGATION A viscoelastic material shows linear behaviour if the retardation time r is independent of cr, and the elongation A1 is proportional to cr.A development of expressions (20) and (21) for r and gives r =TO 1 + k-T tanh - - - [ ""( kT w where + (gr{-& --; + (tanh&)l- U -tanh ['-" + ($)2 tanh kT + + W W (cosh +/kT)2 kT A1 = f@ exp ($lkT) T O = 2kT cosh (+/kT) A1 with respect to (J kT -A} + . . .]. (25) is the retardation time for a = 0." Now a material for which the retardation time has a term linear in cr (cf. (25)) does not even show linear behaviour for small values of a. Moreover, it behaves differently for extension and for compression. A linear region may be defined if the factors uw/kT in (25) and (crw/kT)2 in (26) vanish, i.e., if 4 = O and u = O . (28) As may be seen from fig. 2, the condition = 0 means that the free enthalpies of the two holes in the undeformed state (a = 0) are equal. This may be realized by various combinations of length and breadth of the potential holes of fig.1. The introduction of the free enthalpies in fig. 2 greatly simplifies the discussion as it combines depth and breadth of the holes in a single variable. The condition u = 0 means that the free enthalpy change caused by the deformation has the same absolute value but the opposite sign for holes 1 and 2. A model obeying conditions (28) will be called a symmetrical model. The development of r and A1 is much simplified for such a model A / = w(aw/kT+ . . . ), (30) is the retardation time under zero stress for a symmetrical model." The value of u for which crw/kT = 1 (32) may now be called the limit of linearity. a the material shows non-linear behaviour.Indeed for this order of magnitude of * This retardation time may be determined by means of recovery measurements after arbitrary stress histories.H. C. BRINKMAN AND F . SCHWARZL 17 In fig. 3 and 4 the retardation time 7 and the final deformation Al are given as functions of (T for zi = 0 (antisymmetrical free enthalpy change, cf. fig. 2) and for various values of q4 (different values of initial enthalpy difference of the holes, -4 -3 -2 -1 0 2 3 4 5 R’ COMPRESSION - -TENSION FIG. 3.-Retardation time as a function of external force for various free energy differences of the holes. 1 A 1 W - 0 -1 -2 ........ .... 2 1 ... ._.. 2 ._.-.-. -.- ,C.- .: 0 -1 - 2 / . / ./. /’ ./. - wu k T 7/. - . -.- _ _ __--/ I I I I I 1 I I - 4 - 3 - 2 - 1 0 1 2 3 4 5 =9 kT COMPRESSION - - TENSION FIG.4.-Final elongation as a function of external force for the cases of fig. 3. cf. fig. 2). From fig. 3 it may be observed that for positive $ the value of T in- creases with increasing u. This behaviour may be understood by observing that for positive 41 the retardation is mainly determined by the larger migration rate K2. This rate decreases with increasing CT and the retardation time increases. Fur negative $ the argument applies to KJ and the retardation time decreases.18 NON-LINEAR RELAXATION BEHAVIOUR The final deformation (fig. 4) shows a curious dissymmetry. The reason is that for positive q5 the particles are mainly in hole 1, even for the undeformed state. Compression does not change much in this situation. For negative 4 the particles are mainly in hole 2 and they remain so at extension. Obviously phenomena related to the asymmetry of the holes are observable for anisotropic materials (e.g. fibres) only. The authors are indebted to their colleagues Ch. A. Kruissink, A. J. Staverman and especially D. Polder for many discussions. 1 cf. Burte and Halsey, Textile Res. J., 1947, 17, 465. This paper contains a model similar to ours. However, the diffusion theory is not used for the calculation of the migration rates, while no thermodynamical discussion of the model is given. 2 Kramers, Physica, 1940, 7, 284. 3 Brinkman, Physica, 1956, 22, 29, 149. 4 Pelzer and Wigner, 2. physik. Chem., By 1932, 15, 445. Eyring, J. Chem. Physics, 1935, 3, 107. Evans and Polanyi, Trans. Faraday Soc., 1935, 31, 875.

ISSN:0366-9033    

DOI:10.1039/DF9572300011

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

THE MECHANISM OF LOW TEMPERATURE MECHANICAL RELAXATION IN DEFORMED CRYSTALS BY A. SEEGER, H. DONTH AND F. PFAFP lnstitut fur theoretische und angewaiidte Physik der Technischen Hochschule, S t ut tgart Max-Planck-Institut fur Metallforschung, Stuttgart, Germany Received 30th January, 1957 A brief discussion of the three types of mechanisms by which dislocations cause energy losses in crystals (hysteresis, resonance, relaxation) is given. The experimental evidence, in particular for the relaxation mechanism, is surveyed. A qualitative description is presented of the mechanism for the so-called Bordoni relaxation peak, which is thought to be due to dislocations overcoming the Peierls stress by thermally activated kink formation. The results of detailed calculations on the rate and the activation energy of such a process are also reported.The treatment does not employ the conventional Arrhenius equation, but derives the relaxation time from the theory of stochastic processes. It takes explicit account of the thermal stresses and the radiation losses governing the dislocation movement. The theory is compared with experimental results on metals and with dielectric measurements on quartz. 1. INTERNAL FRICTION MECHANISMS INVOLVING DISLOCATIONS In 1940, Read 1 in a series of internal friction experiments on slightly deformed single crystals of copper and zinc was able to demonstrate convincingly that dis- locations give rise to mechanical energy losses in crystals. Fig. 1 shows the ampli- tude dependence of the internal friction of copper single crystals at room tem- perature as a function of prestress.Since the number and the arrangement of the dislocations vary with prestress, these results can be interpreted in terms of energy losses caused by dislocations. Although this work and similar investigations on the dependence on crystal orientation and on the recovery of stress-induced internal friction of crystals had very early established the importance of dislocations (see Seitz’s book,2 1943), it was not until recently that a detailed picture of at least some aspects of the dis- sipating mechanism due to dislocations began to emerge. In Zener’s classical book (1948) on anelasticity of metals 3 dislocations are barely mentioned as con- tributing to energy dissipation, and even in 1952 Wert4 stated that the internal friction measurements on plastically deformed crystals have failed to tell us how the dislocations involved were arranged and anchored, and with what types of dislocations we are dealing.Two main reasons are held responsible for the slowness of progress in the field. First, the internal friction of single crystals is sensitive to static stresses even below the critical shear stress 70, as shown by fig. 1. (For copper crystals of the purity employed by Read : 1 TO rn 100 g/mm2). As reported by most experi- mental investigators internal friction measurements on metal single crystals are therefore extremely sensitive to handling damage. Secondly, the internal friction depends on the amplitude 2 of the alternating strain employed in the measurement (fig.1) well below the critical shear strain EO (in metal crystals, EO = 10-5-10-4). Since the theory of dislocations was developed mainly to account for the plastic properties on a stress level T 2 70, the internal friction experiments involved dis- location properties on which no detailed theories were available. On the other 1920 LOW TEMPERATURE MECHANICAL RELAXATION hand the damping experiments provide an excellent method of studying dis- locations on a stress level to which there is no other easy access. According to Kempe and Kroner5 the mechanical energy losses caused by the movement of dislocations which have been discussed in the literature, can be grouped as follows : (i) hysteresis losses (Nowick,6$7* 8 Weertman 9), (ii) resonance losses (Koehler lo), (iii) relaxation losses.I 2 5 Strain amplitude F FIG. 1.-Internal friction decrement A of copper single crystals (99.998 %) at room temperature after compressional prestressing (resolved shear stresses T). The loads had been applied for 1 min.1 The great majority of the energy losses by dislocation processes, including those shown in fig. 1, is of the hysteresis type. Typical hysteresis processes are, e.g., those in which dislocations are torn away irreversibly from pinning points by the applied stress. Such pinning points may be provided, e.g., by foreign atoms or other point defects,lo-l4 * and by other dislocation lines intersecting the area swept out by the moving dislocations.16-18 Tt can undergo resonance if the frequency f of an applied alternating shear stress coin- cides with the characteristic frequencyfo of the dislocation string.Plotted as a function of frequency the internal friction should go through a maximum at .fo (or near to the maximum of a distribution of resonance frequencies). The dis- * The mechanism by which impurity atoms pin dislocation lines is still under discus- sion, since-as pointed out by Weertman and Salkovitz 11-the interaction energy between individual substitutional foreign atoms and dislocation Iines is in general too small to pin the dislocation lines effectively at room temperature, at which most experiments have been made. Furthermore Baker 15 has recently shown by internal friction experiments employ- ing a small superimposed biasing static stress that some of the " break-away ¶ ¶ effects in the amplitude dependence of the decrement cannot be due to impurity pinning.A dislocation line fixed at two points behaves somewhat like a string.A. SEEGER, H. DONTH AND F. PFAFF 21 location loops which are thought to be present in crystals are so short, however, that fo is in general larger than the frequencies accessible with present megacycle techniques.19 There is, however, experimental evidence 20 that in attenuation experiments in germanium up to f ;= 300 Mc/sec one is approaching the resonance frequency fo. A typical relaxation process in crystalline solids is the Snoek process of carbon atoms in a-iron (see, e.g., ref. (4), (8)). Its characteristic features are as follows : in thermal equilibrium the physical system under consideration passes with a tem- perature-dependent mean frequency v from one configuration of minimum energy through a saddle point configuration to one of several other minimum configara- tions.An applied stress favours in general some of these minimum positions with respect to others. If we apply an alternating stress of frequency f, well- known arguments show that the energy dissipation is a maximum if the two frequencies f and v are equal. We know of only one process in crystals which involves dislocations only and which shows all characteristic features of a relaxation mechanism, namely that discovered in various f.c.c. metals by Bordoni.21. 22 We shall discuss the experi- mental evidence for this dislocation relaxation mechanism in 5 2. In passing, we mention that there are also relaxation mechanisms involving both dislocations and impurity atoms, for instance the one found by the Ke-method at about 220" C in cold-worked iron containing carbon or nitrogen.23--27 2.EXPERIMENTAL EVIDENCE FOR THE DISLOCATION RELAXATION MECHANISM The Bordoni peak, i.e. a maximum in the internal friction measured at a fixed frequency f as a function of temperature T, has been observed in various mono- crystalline and polycrystalline f.c.c. metals, namely, copper,22.28929~ 17 alu- minium,30.22 silver,22 and lead,319 329 33.349 22 For the frequency range normally employed (f- 3 x 104 c/s) the temperature of the peak ranges from about 35" K (lead) to about 100' K (aluminium). The shift of the peak with frequency cor- responds to activation energies H ranging from about 1000 cal/mole (lead) to about 4000 cal/mole (aluminium).This is of an order of magnitude which is rather unusual in solid-state physics. The frequency factor f0 in the relation f =f" exp (- H/kT) between the frequency f employed in the experiment and the absolute temperature T of the peak is also of an unusual magnitude, namely, about 109 sec-1. The height of the peak may be undetectably small in well-annealed, undeformed single crystals 153 17 but increases rapidly with small prestrains (see fig. 2, 3, 4). At larger strains the height of the peak appears to decrease somewhat with further straining.29 It does not anneal out at temperatures where point defects are known to anneal out, but only at temperatures at which the specimen presumably re- crystallizes (fig.4). As Niblett and Wilks 29 point out this shows that the peak is caused by dislocations. The experimental fact that the position of the peak is independent of prestrain (see fig. 2, 3, 4) and independent of the impurity con- tent (fig. 5) shows that we are dealing with an '' intrinsic " dislocation effect 35 which does not depend on a particular length of dislocation lines or on their interaction with impurity atoms. The measurements of Niblett and Wilks revealed that in both polycrystals 28.29 and single crystals 36 of copper the Bordoni peak is accompanied by a second, smaller peak occurring at lower temperatures (see fig. 2, 3,4, 5). The most natural explanation for this is that the two peaks are caused by two types of dislocations.* The relaxation character of the Bordoni process is borne out by the way in * Presumably both peaks are due to dislocations running along the (1 10)-direction, one of them belonging to screw dislocations and the other one to dislocations with an angle of 60" between the Burgers vector and the direction of the dislocation line.3522 LOW TEMPERATURE MECHANICAL RELAXATION which the internal friction depends on temperature for a fixed frequency (see fig.2, 3, 4), particularly in the form of eqn. (2.1), which is found to hold approxim- ately if the measurements are made in a sufficiently narrow temperature range. Further, it is in agreement with the relaxation character that both the height and the position of the peak depend only slightly, if at all, on the strain amplitude.36~ 17 0 50 100 I so 2 0 0 2 5 0 Temperature [OK] FIG. 2.4nternal friction of polycrystalline copper (99.999 %) prestrained 0.1 %, as a function of temperature, showing two Bordoni peaks (f= 1100 clsec).The resonance coefficient Q is related to the logarithmic decrement A, the energy loss per half cycle AE, and the total vibrational energy E according to l/Q = A/n = AE/2rE. Fig. 2 to 5 are due to Niblett and Wilks. 0 5 0 100 i 50 200 2SO Temperature la#] FIG. 3.-Internal friction of pure polycrystalline copper (99-999 %) after prestraining and annealing (f= 1100 c/sec). It may be seen from fig. 2, 3,4 and 5 that the relaxation peaks are superimposed on a background which also depends on the prestrain, but in a different way. The annealing behaviour and the response to neutron-irradiation 36 differ also for the peaks and the background.The background is presumably due to a disloca- tion-hysteresis mechanism the details of which, however, are not yet known. In the analysis of Nowick's 7 data on this background in copper single crystals, an activation energy of about 1500 cal/mole is found. In view of the implicationsA. SEEGER, H. DONTH AND F. PFAFF 23 for the theoretical interpretation it would be very important to check experiment- ally whether there is a close relation of this value to the activation energy of the Bordoni process as is suggested by the numerical values. $A;:/z/ 'after annealing lh at 350°C 1 0 S O 100 150 2 0 0 2 50 Temperature PK] FIG. 4.--Internal friction of pure polycrystalline copper (99.999 %) after prestraining and annealing (f = 1100 cisec).I 0 5 0 I00 I50 200 2SO Temperature ["K] FIG. 5.--Internal friction of impure copper as a function of temperature. 5.5 % ; impurity content 0.0026 % Bi, 0.032 % P; f = 1100 c/sec. Strained We close with the remark that in crystalline quartz a similar internal friction peak has been observed at about 20" K 37 and has been attributed to the same cause as in metals.34.37 Since no deformation experiments have been made on quartz this correlation is still open to experimental verification. We shall return to a short discussion on quartz in 4 5. 3. QUALITATIVE THEORY OF THE DISLOCATION RELAXATION MECHANISM A dislocation relaxation mechanism which is in accord with the experimental facts summarized in 5 2 has been described elsewhere.38~ 39.35 We may therefore confine ourselves to a brief summary. Consider a dislocation line which runs24 LOW TEMPER AT U R E ME C IH A N I C A L RELAX AT I 0 N parallel to one of the close-packed directions of its glide plane.In order to move such a dislocation line by one interatomic distance without the aid of thermal fluctuations a resolved shear stress T must act on the dislocation line which is equal to the Peierls stress 7; at 0" K. (For discussions of the Peierls stress or, as it is often called, the Peierls-Nabarro force, see ref. (40), (41), (42), (38).) At finite temperatures the dislocation line will no longer lie in only one valley of its potential energy surface but will contain kinks,43,44 i.e. it will change occasionally from one valley to a neighbouring one (fig.6). Jf a stress T < T; is applied the X b +.- -. - 'Y - - . _ I t - I - - i - - - + - + - - + + - - - , - - - + - * -3a -2a -a , a 2a 30 * FIG. 6.-The potential energy surface of a dislocation line due to the Peierls stress T;. The figure is not drawn to scale, since in reality Eo $ (Tiab/n) and w & a. dislocation line can move forward by two processes : sideways movement of the kinks (which requires very small stresses only), and formation of new pairs of kinks. The formation of a pair of kinks of opposite sign requires thermal energy and occurs therefore with a temperature dependent frequency. If the frequency f of the applied stress is large compared with the frequency v with which a pair of kinks in a dislocation line is formed, kink formation contributes nothing to the strain and to the energy dissipation. I f f is very small compared with the frequency of thermal formation of kinks the kinks are always in thermal equilib- rium and no energy losses occurs either.Only if the two frequencies are approxim- ately equal do we get a contribution to the internal friction of the crystal. This process therefore fulfils all the requirements of a relaxation process, and we believe that it is responsible for the Bordoni peaks. The quantitative treatment of this dislocation mechanism, however, is much more difficult than that of the Snoek mechanism mentioned in $2. The difficulty is due to the fact that the system passes from the initial state (dislocation line lying in a single valley) to the final state (dislocation line containing two kinks of opposite sign which are sufficiently separated in order to be stable) not throughA.SEEGER, H . DONTH AND F. P F A F F 25 a well-defined saddle point configuration. Since the dislocation line is flexible there is a variety of ways by which the final state can be reached. The standard derivation (see, e.g. ref. (45)) of the Arrhenius eqn. (21) from the Eyring-Zener theory of reaction rates is therefore not applicable. In not too complicated situations there are two ways of treating rate processes.* The first one employs Boltzmann distributions of free energies. The action of thermal stresses and the damping due to the radiation losses 46 is then summarily included.The latter statement is illustrated by the standard treatment of the Snoek process by the rate theory.45 There is, for instance, no need to introduce explicitly the radiation of waves from the carbon atom oscillating in its potential well since these waves are part of the thermal equilibrium and are therefore already allowed for in the Boltzmann expression. The second way introduces explicitly the forces exerted by the thermal stresses on the dislocation line (in the problem under discussion) and must then also include the damping mechanism, in our particular case the radiation of elastic waves from accelerated dislocation lines. Donth 47 has treated the present problem of thermally activated kink formation in this way, which in complicated cases is the only reliable one.For the details of Donth’s method which is believed to be useful also in other problems, reference must be made to Donth’s paper.47 We shall give a brief outline in 5 4. 4. SUMMARY OF THE QUANTITATIVE THEORY The shape of the dislocation line y(x, t ) (see fig. 6, t = time), lying in the xy-plane (nearly parallel to the x-direction) is to a good approximation determined by the differential equation (for more details see ref. (35)) s y 32y 2VY 3x2 3t2 a Eo- - m- = bri sin- - br. (4.11 (Eo and in = energy and mass per unit length of the dislocation, b = dislocation strength, r = resolved shear stress, n = period of the dislocation energy in the crystal, ri = “ ideal ” Peierls stress in the absence of thermal or quantum- mechanical activation). Starting from eqn.(4.1) it can be shown 35 that the energy of a kink is that the kink width w (see fig. 6) is given by and that the separation d,, of (unstable) equilibrium of the kinks under the stress ~ ( r < ri) is A study of the solution of eqn. (4.1) for7 = 0 shows 47949*50 that there are “ normal modes ” of the dislocation movement which can be characterized by their energy W (per wavelength). They are ordinary harmonic waves if W e 2W,, but are related to pairs of kinks of opposite sign if W w 2Wk. The exchange of energy between these modes can be represented by a model in which particles (labelled by the co-ordinate W ) diffuse under the influence of thermal shear stresses and * One of the authors (A. S.) gratefully acknowledges discussions on this topic at and after the Lake Placid (1956) Conference on Dislocations and Mechanical Properties of Crystals, in which particularly G.Leibfried and K. Lucke participated. Further results, particularly on higher terms in eqn. (4.4) and allowing for a second sin term in eqn. (4.1), are given elsewhere.4826 LOW TEMPERATURE MECHANICAL RELAXATION radiation daniping.47 The generalized diffusion equation representing this process is solved, subject to the boundary condition that kinks of opposite sign separated by a distance larger than d,, are independent of each other, since they are torn apart by the applied stress T. The final result is an expression (which cannot be written in closed form) for the mean frequency v with which a dislocation leaves the potential wells of the E(y)-surface shown in fig.6 . FIG. 7.-The function Fl(r ; a) of eqn. (4.5). It is related to the function n(r) employed by Donth 47 by the equation Fl(r ; E) = - In 2vn(r). The frequency v and activation energy H can be found as a function of tem- perature from the equations : In (v/B) = Fl(r; a), (4.5) and (4.5a) (4.6) Here v is the appropriate sound velocity (for screw dislocations in an isotropic medium, the velocity vt of transversal waves) ; fi(r ; a) are functions shown in fig. 7 and 8, which depend on the parameters r = 2Wk/kT, (4.7) and (Tn most cases a w 1.) (4.8)A . SEEGER, H . DONTH AND F . PFAFF 27 Using rate theory in a similar way as Mason 33 it can be shown that the decre- ment Amax at the temperature of the relaxation peak is given by 48 where (4.9a) Here NO denotes the number per unit volume of dislocation loops (of average length L) contributing to the relaxation process; A is the area swept out by one dislocation during the process, and G the shear modulus of the crystal.Putting / / employed A = La gives us a lower limit forp and (Q-l)max (provided L/a is sufficiently large), since some of the dislocations may move by more than one interatomic distance. An upper limit for (Q-l)max can be obtained from the assumption that each dislocation loop sweeps out the largest possible area which is compatible with the applied stress, the line energy Eo, and the loop length L. The upper limit is (4.10) 5. COMPARISON wm EXPERIMENTAL DATA The only unknown parameter in the theory of § 4 is the " ideal " Peierls stress T;. In principle, the precise measurement of the temperature of the Bordoni maximum at a given frequency suffices to determine 7;.There are, however, practical difficulties, since the frequency depends exponentially on temperature and since the temperature of the peak is often not well defined. It is therefore sometimes preferable to use eqn. (4.6) and to determine the activation energy H28 LOW TEMPERATURE MECHANICAL RELAXATION (which itself is weakly temperature dependent) from the slope of the usual lnf against l/kT plot, if a sufficient number of experimental points is available. in order to test the theory as critically as is possible at present we have cai- culated, in table 1, T; separately from each measurement on metals which is known TABLE 1 metal and shear modulus G (d y neslcmz) lead 0.8 x 1011 aluminium 2.7 X 1011 silver 2-9 x 1011 copper 4-5 x 1011 -EVALUATION OF EXPERIMENTAL DATA ON THE BORDONI PEAK IN METALS author Bordoni 22 Welber 32 Bommel (after Mason 33) Bordoni 22 Hutchison- Filmer 30 Bordoni 22 Niblett- Wilks 28 Bordoni 22 f = v (sec-1) 10,334 10,163 10,348 20,500 29,500 50,000 10.1 x 106 26.5 x 106 38,665 39,600 5 x 106 31,332 400 400 1,100 1,100 30,300 30,450 28,260 28,600 37,992 T (“W 36 36 36 43 47.5 50 120 140 112 97 155 62.5 61 65 68 74 82 80 91 87.5 97 r 7.3 7.3 7.3‘ 7-05 6-9 6.7 4.5 4.1 7.6 7-5 5.4 7.1 9-25 9.3 8.85 8.85 7.4 7.35 7.5 7.45 7.35 H mole (e!) 1.2 1.2 1.2 1.4 1.5 1.5 2.4 2.6 3.9 3-3 3.7 2.0 2.6 2.8 2.8 3.0 2-75 2-7 3-1 3-0 3.0 % (23 0.28 0.28 0.28 0.37 0.44 0.46 1.2 1.3 2.8 2.1 2-75 0.7 1.5 1.7 1.7 2.0 1.7 1.6 2.2 2.0 2.1 T ; ~ G x 104 3.5 3.5 3.5 4.6 5.4 5.6 14.6 16.5 10.3 7.5 10.0 2-3 3.3 3.7 3.7 4.4 3.8 3.5 4.8 4.4 4-6 to us.* It will be seen that the various measurements give about the same ri- values for both copper and aluminium.Best values for copper appear to be ri/G = 4.0 x 10-4 and w/a = 48. There appears to be a systematic deviation in the sense that the measurements at higher frequencies give larger Peierls stresses. At the moment we do not know whether this discrepancy is partly due to experimental inaccuracies. There are, however, several possibilities which might give rise to deviations from the theory as pre- sented here; 35 one of them is that higher harmonics may be required for a satis- factory representation of the potential energy surface E(y) (cp.§ 4). From the decrement at the peak information on the number of dislocation per unit volume NO and on the loop length L can be derived, in particular as a function of prestrain. If we confine ourselves to very small deformations (see fig. 2), we find that the observations can be accounted for by values of the order of No = 10+12 cm-3 and L = 2 x 10-5 cm. These are reasonable magnitudes. The magnitude of NO shows that only about 1/100 of the total number of loops in the crystal contributes to the peak. This is in agreement with our model, which assumes that only dislocations lying nearly parallel to a close-packed direction take part in the relaxation process. * We used rn = Eo/vt2 and EO = Gb2/2. The latter relation was found to hold ap- proximately for the present situation after a lengthy calculation for screw dislocations in copper carried out by H.Donth along the lines indicated by Schoeck and Seeger.51 In f.c.c. metals : (2/2/3)a = b = distance between nearest neighbours. For lead, the agreement is less good.A. SEEGER, H . DONTH AND F . PFAFF 29 We can summarize the discussion on metals by saying that the theory accounts rather well for the majority of observations. It allows one to deduce numerical values of ri from internal friction experiments. The magnitude of 7-i is found to be in rather good agreement with theoretical predictions.52~ 38 We conclude with a few remarks on quartz, which exhibits also a low-temper- ature mechanical relaxation, as already mentioned in § 2.Its interpretation as a Bordoni process 37 along the lines which have been shown to be rather successful for metals, presents some difficulties, however. Experimentally 37 eqn. (2.1) seems to be much better satisfied than our theory can account for. Since it seems unlikely that prestraining experiments will decide this problem in near future it is tempting to resort to measurements of dielectric loss. Dielectric measurements on clear, unirradiated quartz crystals do not reveal a relaxation peak corresponding to the mechanical peak.53 Natural crystals of smoky quartz and both natural and synthetic crystals irradiated with X-rays or electrons, however, do show a dielectric relaxation peak (in the kilocycle range) in the temperature interval at or below 20" K (so-called p-peak).s49 57 Both the position and the relaxation strength of the peak vary from crystal to crystal.Volger and Stevels56 suggest that the peak might be due to electrons or holes which are trapped in colour centres and which undergo electrically active transi- tions from one location in the centre to another one. Another possibility is that there exists a connection with the mechanical relaxation process mentioned above, for which eqn. (2.1) is reported 37 to hold with H = 155 cal/mole andf" = 2 x 10s sec-1. The corresponding values for the dielectric measurements are somewhat uncertain but seem to be of the same order of magnitude.55 This suggests that both processes are due to the same mechanism. Further experiments are required in order to confirm or refute this view.They might also help to clear up the open question whether the mechanical losses in quartz are of the Bordoni type and whether the Peierls stress in quartz is really so surprisingly small as would be implied by such an interpretation. The authors would like to thank Prof. U. Dehlinger for his encouragement and for his interest in their work, and Dr. P. Haasen and Dr. J. Diehl for valuable comments on the manuscript. 1 Read, Trails. Amer. Inst. Min. Met. Eng., 1941, 143, 30. 2 Seitz, The Physics of Metals (McGraw-Hill, New York, 1943). 3 Zener, Elasticity and Anelasticity of Metals (Univ. of Chicago Press, Chicago, 1948). 4 Wert, Modern Research Techniques in Physical Metallurgy (American Society for 5 Kempe and Kroner, Z. Metallknnde, 1956, 47, 302.6 Nowick, Physic. Rev., 1950, 80, 249. 7 Nowick, A symposium on the plastic deformatioii of crystalline solids (Pittsburgh, 8 Nowick, Progress in Metal Physics, vol. 4 (London, 1953), p. 1. 9 Weertman, Physic. Rev., 1956, 101, 1429. 10 Koehler, Imperfections in nearly perfect crystals (Wiley & Sons, New York. 1952), 11 Weertman and Salkovitz, Acta Metal., 1955, 3, 1. 12 Granato and Lucke, J. Appl. Physics, 1956, 27, 583. 1 3 Thompson and Holmes, J. Appl. Physics, 1956, 27, 713. 14 Weinig and Machlin, J . Appl. Physics, 1956, 27, 734. 15 Baker, Ph.D. thesis (University of Illinois, 1956). 16 Birnbaum and Levy, Acta Metal., 1956, 4, 84. 17 Sack, Conference on Ultrasonic Energy Losses in Crystalline Materials (Providence. 18 Kamentzky, Thesis (Cornell University, 1956).19 Granato and Lucke, J. Appl. Physics, 1956, 27, 789. Metals, Cleveland, 1953), p. 225. 1950), p. 155. p. 197. 1956).30 LOW TEMPERATURE MECHANICAL RELAXATION 20 Granato and Truell, J. Appl. Physics, 1956, 27, 1219. 21 Bordoni, Ricera Sci., 1949, 19, 851. 22 Bordoni, J. Acoust. Soc. Amer., 1954, 26,495. 23 Snoek, Physica, 1941, 8, 711. 24 West, Trans. Amer. Inst. Min. Met. Eng., 1946, 167, 192. 25 K6, Trans. Amer. Inst. Min. Met. Eng., 1948, 176,448. 26 Ktjster, Bangert and Hahn, Archiv Eisenhuttenw., 1954, 25, 569. 27 Kiister and Bangert, Acta Metal., 1955, 3, 274. 28 Niblett and Wilks, Confe'rence de physique des basses tempe'ratures (Paris, 1955), 29 Niblett and Wilks, Phil. Mag., 1956, 1, 415. 30 Hutchison and Filmer, Can. J. Physics, 1956, 34, 159. 31 Bommel, Physic. Rev., 1954, 96, 220. 32 Welber, J. Acoust. Soc. Amer., 1955, 27, 1010. 33 Mason, J. Acoust. SOC. Amer., 1955, 27, 643. 34 Mason, Deformation and FIow of Solids (Colloquium, Madrid, Sept., 1955 ; Springer, Berlin-Gottingen-Heidelberg, 1956, p. 3 14. 35 Seeger, Phil. Mag., 1956, 1, 651. 36 J. Wilks, private communication. 37 Bommel, Mason and Warner, Physic. Rev., 1955, 99, 1894. 38 Seeger, Theory of Lattice Defects, Encyclopedia of Physics, vol. 7, 0 72 (Springer, 39 Seeger, Deformation and Flow of Solids (Colloquium, Madrid, Sept., 1955, Springer, 40 Cottrell, Dislocations and Plastic FZow in Crystals (Oxford University Press, Oxford, 41 Read, Dislocations in Crystals (McGraw Hill, New York, 1953), chap. 2, 7. 42 Nabarro, A&. in Physics, 1952, 1, 269. 43 Shockley, Trans. Amer. Inst. Min. Met. Eng., 1952, 194, 829. 44 Mott and Nabarro, Report of a Conference on the Strength of Solih (Physical 45 Zener, Imperfections in Nearly Perfect Crystals (Wiley and Sons, New York, 1952), 46 Eshelby, Proc. Roy. Soc. A, 1949, 197, 396. 47 Donth, Thesis (Stuttgart, 1957), to be published. 48 Pfaff, Diplomarbeit (Stuttgart, 1957). 49 Seeger, Donth and Kochendorfer, 2. Physik, 1953, 134, 173. 50 Seeger, 2. Naturforschung, 1953, 8a, 246. 51 Schoeck and Seeger, Report Conference on Defects in Solids (Physical Society, 1955), 52 Dietze, 2. Physik, 1952, 131, 156. 53 Stevels and Volger, Chem. Weekblad, 1956, 52, 469. 54 Stevels, van Amerongen and Volger, 2. physik. Chem., 1955,3, 382. 55 Volger, Stevels and van Amerongen, Philips Res. Report, 1955, 10, 260. 56 Volger and Stevels, Philips Res. Reports, 1956, 11, 79. p. 484. Berlin-Gottingen-Heidelberg, 1955). Berlin-Gottingen-Heidelberg, 1956), p. 322. 1953), chap. 3. Society, London, 1948), p. 1. p. 289. p. 340.

ISSN:0366-9033    

DOI:10.1039/DF9572300019

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

DIELECTRIC RELAXATION IN SOLID HYDROGEN HALIDES BY ROBERT H. COLE AND STEPHEN HAVRJLIAK, JR. Metcalf Chemical Laboratories, Brown University, Providence, R.I., U.S.A. Received 3 1st January, 1957 Dielectric constant and loss measurements of solid hydrogen chloride, and hydrogen and deuterium bromide and iodide, are summarized and discussed in relation to order- disorder phase transitions. Electrostatic calculations indicate the importance of dipole interactions for the structures and rates of molecular orientation in the low temperature phases, but specific effects show the significance of other factors as well. In the less ordered higher temperature phases, the dispersion behaviour is characteristically different, but more complete dielectric measurements and structural evidence will be needed for a proper understanding.Of the rather limited number of solids which exhibit appreciable dipole orientation in an electric field, the hydrogen halides are of considerable interest because the molecules and the solid structures, to the extent they are known, are comparatively simple. The molecular " shapes " are not far from spherical, and X-ray evidence 1 indicates that the various solid phases have halogens ar- ranged in face-centred structures reminiscent of cubic close packing of spheres. One can hope that dielectric relaxation behaviour of these systems will be more easily analysed in terms of molecular orientation and interaction than more com- plicated molecules and structures. The discussion of the experimental evidence will show that the behaviour is by no means simple; at the same time, there are features often encountered in other cases which have so far resisted molecular explan- ation. The purpose of this report is to summarize the available evidence, indicate respects in which it is incomplete, and discuss the relation to structure and theory.The different solid phases of hydrogen and deuterium halides and the degrees of dipole orientation possible are conveniently shown by plots of static dielectric constant €0 against temperature as in fig. 1. Hydrogen fluoride is omitted, because the small dielectric constant of the solid indicates essentially " frozen " orientations at all temperatures, and no other evidence of phase transitions is known to us (the relevance of X-ray structure evidence to properties of the other halides is considered later).Hydrogen chloride has a first-order solid phase transition at 99" K ; above this, the dielectric constant is large and has the inverse temperature dependence of freely orienting molecules. The low-temperature phase exhibits a much smaller but measurable dipole polarization at temperatures near the transition. In the bromides and iodides, there are two (and for hydrogen bromide three) A-type specific heat transitions, with a pronounced dielectric constant change at the lowest temperature transition in the form of a rapid rise to a large peak value (over 200 for the bromides). The low-temperature transitions clearly involve considerable changes in molec- ular order with greatly increased freedom for dipole orientation, and much of the accompanying dispersion processes has been accessible to audio and radio frequency measurements.LOW TEMPERATURE DISPERSIONS Representative results for the low temperature phases a few degrees below the transitions are shown as complex plane diagrams in fig. 2. The relation of 3132 RELAXATION I N HYDROGEN HALIDES E' to e r r is represented over considerable frequency ranges by the empirical formula 4 (1) where €0 and €1 are the static and limiting high frequency dielectric constants of the dispersion, LL) is radian frequency, TO the relaxation time, and cc an empirical parameter proportional to the depression angle of the circular arc. = (€0 - d / [ 1 + (iw70)1-~1, E* = E r - iE" ZOO I00 5 0 2 0 10 5 I I I I 8 0 1 2 0 T"K I 6 0 2( FIG.1 .-Static dielectric constants (logarithmic scale) of hydrogen and deuterium halides plotted against temperature. Data of Swensonz for HCI, of Brown3 for HBr and of S. Havriliak for the other substances. The arc function has been found to represent dielectric relaxation of a variety of otherwise quite unrelated substances; of these, the present examples seem the simplest with respect to molecular structure and arrangement. It is also of interest that the arc function is an accurate fit within the rather small limits of experi- mental error (this question has been considered in detail elsewhere for HBr). The simple arc function does not represent the entire dispersion behaviour, except possibly for the iodides. This is apparent from the fact that the high frequency limit €1 is significantly larger than the induced polarization value n2 from refractive index estimates, or any somewhat larger figure making reasonable allowance for atomic polarization.There must therefore be a further dispersion process or processes somewhere between radio and infi-a-red frequencies. Such a dispersion was observed in both HBr and DBr at temperatures sufficiently low for the frequencies to lie in the experimental range (up to 500 kclsec). These high frequency portions are illustrated in fig. 3, from which it is clear that the form is approximately a circular arc, the amplitude very much less than for the " primary ", lower frequency process, and the second high frequency limit stillR . H. COLE AND S. HAVRILIAK, J R .33 perhaps a little larger than reasonable estimates of induced polarization values. Present measurements do not extend to sufficiently high frequencies to determine whether a similar situation exists in the iodides. O L 0 :'r 0 0 P I I I 2 0 3 0 € 1 4 0 700c DBR. 78' K 25' 1.0 HI, 60.4'K E'li j - - E ; 0 1 7 O f' 3 . 1 4 - 2 12' FIG. 2.-CompIex plane diagrams for low-temperature phases. 2 E " I 0 3 4 E' FIG. 3.-High-frequency dispersion of HBr, DBr in their low-temperature phases. B34 RELAXATION I N HYDROGEN HALIDES DISPERSION IN THE HIGHER TEMPERATURE PHASES A rather limited amount of information about the dispersion processes above the low-temperature transition is available, because measurements have not yet FIG. 4.-Complex plane dia- grams for DBr, HI, DI in their intermediate solid phases.The solid curves represent the range of measurements ; the dashed curves are possible continua- tions to higher frequencies dis- cussed in the text. been extended above 500 kc/sec. The results show a distinct change in character from the low temperature behaviour, as shown in representative complex plane diagrams (fig. 4) for HI and DI. The results are quite incomplete, especially at 6 higher temperatures, but do show defi- nitely that the low-frequency part of the ; relaxation is consistent with a Debye g semi-circular locus rather than the de- $ pressed cireular arcs found below the < transition (fig. 2). 0 Another way of confirming the be- haviour when the frequency range is too low to show much dispersion is by plotting absorption conductance (pro- portional to WE") against frequency, as for Debye behaviour and WTO << 1 one has lo-' lo-' WE" (€0 - E1)T0W2.(2) The plots in fig. 5 of loglo WE" against loglof, where f = ~ / 2 7 ~ , are fitted by lines of slope 2, and therefore show that the limiting low frequency behaviour conforms to a Debye-type equation (i.e. eqn. (1) with cc = 0). I 0- ' The highest temperature phases of the F rcqoc'ncy - K.C bromides -and iodides and the high- FIG. 5.-Loga&hmic plots of absorption temperature phase of hydrogen chloride conductance (- WC") against frequency. exhibit no measurable relaxation effects The straight lines are drawn with slope two.R . H . COLE A N D S . HAVRILIAK, J R . 35 below 1 Mc/sec, and one can only conclude that the characteristic frequencies must be well above 100 Mc/sec.TEMPERATURE DEPENDENCE OF RELAXATION TIMES Characteristic relaxation times for the measured dispersions are calculated from the frequency of maximum absorption when the data are sufficiently complete. If only conductance dispersion was measurable, values of TO (sec) were estimated from eqn. (2). These require an assumption about €1 which is unknown (as indeed DBR --- /‘ / I I I 10 I2 I4 ~ o o / T”K temperature (“K). FIG. &-Plots of loglo 70 (sec) for the several dispersions against reciprocal is the validity of the Debye equation for higher frequencies); for simplicity we have set €1 = 0. This is certainly not correct as €1 must exceed n2 and may well be ~ 0 / 3 or more, but the error should not significantly affect comparison of rate plots of loglo TO against reciprocal temperature.The rate plots are shown in fig. 6, and values of activation energy E, (kcal/mole) and frequency factor A (sec-1) for the equation (3) are listed in table 1. The activation energies all fall in the range 1.5 to 3.6 kcal/mole with no obvious relation to the halogen substitution, the times are longer for the deuterium isotope in the ranges studied by factors from 1.3 to 4, and the rates decrease on cooling through the low-temperature transition by factors 1.6 to 5. loglo TO =- log10 A + EJ2.3 RT TABLE 1 .-CONSTANTS OF RATE LAW DESCRIBING THE OBSERVED DISPERSIONS range of T log10 A Ea HC1 100-63 - 13.00 2-6 HBr (1st) 89-63 - 11.2 2.7 (2nd) 89-63 - 12.1 1.6 DBr (1st) 95-63 - 16.2 1.5 (2nd) 95-63 - 13.5 3.6 111-100 - 15.4 1.5 HI 68-62 - 13.6 2.2 80-70 - 17.2 3.6 DI 75-62 - 13.3 2.2 100-78 - 15.6 3.1 (“K) (sec-1) (kcaI/mole)36 RELAXATION IN HYDROGEN HALIDES STRUCTURE EVIDENCE AND THEORY Interpretation in molecular terms of the various relaxation processes will ultimately require a better knowledge of structures and changes in structure of the solid phases than is now available.The picture of the various transitions as co-operative changes to states of increasing molecular disorder in the higher temperature phases is generally accepted, but there are no certain conclusions about the ordered structures, and the intermediate states of order for dipole orientation are quite unknown. For the highest temperature phases stable just below the melting point, there is little doubt that the structures are face-centred cubic and hence isotropic.The static dielectric constant values confirm the X-ray evidence indirectly, as the ob- served values 2, 3 are quite close to those calculated from gas dipole moments by Onsager’s formula,6 which cannot be expected to hold if there is net short-range correlation of dipole orientations, and, in fact, has no relation to the behaviour in the more ordered low-temperature phases. The high-temperature transitions of the bromides and iodides show slight changes in dielectric constant of the magnitude expected from density differences, but the large effects at the low-temperature transitions and at the first-order transition in hydrogen chloride (fig. 1) indicate very large changes in dipole order.Tisza 7 has pointed out that the shift of the lower transition to lower temperatures on going from HBr to HI and of the upper ones to higher temperatures are related qualitatively to the smaller dipole moment and larger polarizability of HI, and so suggested that dipole interaction forces dominate the lower transition and van der Waals’ polarizability forces the upper. Powles 8 has pointed out that estimates of dipole-dipole energies predict much larger changes in the transition temperatures than are observed, and came to the conclusion that anisotropy of repulsive forces, corresponding to deviations of the molecules from spherical shape, were dominant in the ordered structures. What is now known of these structures indicates, however, that neither point dipole interactions nor repulsive forces are adequate for an explanation, but that the former probably have an important role.The X-ray evidence for the low-temperature phases is not conclusive but in- dicates face-centred orthorhombic or tetragonal arrangement of the halogens. Infra-red studies of the vibrational frequencies for solid films of hydrogen chloride and bromide by Hornig and associates 9s 10 limit the possibilities for orientations of the hydrogens and molecular dipoles in the unit cell, and the simplest structures found to be compatible with the observed splittings are the two shown in fig. 7. Both have zigzag chains of dipoles in layer planes normal to the C-axis of the four molecule unit cell, the difference lying in relative placement of alternative layers of oppositely directed chains.The structure A is very similar to one of two possible structures found by X-ray determinations 11 for crystalline hydrogen fluoride,* which gives some basis for the further consideration. The structures in fig. 7 are substantially different from ones found stable for dipole-dipole forces or suggested on the basis of repulsion and molecular shape, and seem more indicative of short-range incipient hydrogen linkages. However, it should be recognized that arrays considered for calculations of point dipole interaction energies 129 13 have involved head-to-tail strings (+-+), with adjacent strings parallel or antiparallel and at various angles with cubic crystalline axes. Corresponding calculations for the structures A and B of fig.7 have been made by summing the dipole interaction energy where p is the dipole moment, the angle between dipoles i andj, and Or and Oj are the angles each makes with the connecting distance vector I-,. * The structure preferred by these workers placed the hydrogen atoms randomly in the chains (F-F-F angle l2Oo), that is, as FH-F or as F-HF, but they stated that the alternative similar to A was almost as acceptable. vi, = p2 (cos 0, - 3 cos 0, cos e j y r , 3 , (3)R . H . COLE AND S . HAVRILIAK, JR. 37 The calculations (made assuming cubic symmetry and 90" dipole angles for simplicity) show that structure B is very unfavourable for this potential, but that A has quite low energy. For interactions summed over the first eight shells of neighbours (140 molecules) the result is zqj =- 1.17 N2p2, where N is the dipole concentration.This figure is not as good as Luttinger and Tisza's most favourable value - 1.808 N2p2, but it does show that dipole interactions may play a significant role. FIG. 7.-Possible structures of ordered phases of hydrogen chloride and bromide. Open and shaded circles represent halogens in adjacent layers. Kirkwood 14 and Kruger and James 15 have considered nearest-neighbour (4) and shown that a second-order transition results from considering only the first term, which is part of the dipole potential (3), while fist- and second-order transi- tions are possible for different relative values of the coefficients A and B. None of the assumptions about the potentials accounts for the most probable structure A or the actual sequences of phase transitions, and the potential (4) has the diffi- culties of representing only part of the angular dependence of dipole and quadru- pole interactions as well as being confined to first neighbours.The consequences of the second term in cos2 8, are interesting in suggesting the importance of quadrupole interactions (if expansion in higher order poles is a workable approach), and also because such terms can give rise to a secondary minimum in the potential for rotation of a single molecule through 180" from the stable position. This possibility could lead to a potential barrier problem with depths and populations of the minimum changing with temperature as more dipoles can occupy " wrong positions ".Unfortunately, our knowledge of quadru- pole moments is too meagre to permit any estimate of the reality of the possible consequences. The available evidence on structure and states of order is evidently somewhat frustrating as a starting point for correlation with the dielectric evidence. Struc- tures similar to that of fig. 7a may well represent the ordered phases of hydrogen chloride and bromide, with the situation for hydrogen iodide in doubt, but the nature of the disordering processes can only be guessed. interaction potentials of the form Y;:~ = A cos e, + B cos2 eii, RELATION OF STRUCTURE AND RELAXATION PROCESSES One is tempted to identify the activation energies E, of the dielectric relaxation processes with heights of intermolecular potential barriers to reorientation.The magnitude of barrier which dipole forces offer to rotation of one molecule with others fixed as in fig. 7a is 1.15 kcal/mole, on using pHC1 = 1.1 D, which is38 RELAXATION I N HYDROGEN HALIDES somewhere near thz magnitude of E, observed. On the other hand, there is no indication. in the observed values of a dependence on p2, which should be pronounced if dipole forces are a dominant factor ( p ~ a = 1-12, ,UHB~ = 0.85, The occurrence of two relaxation regions in the low-temperature phases may well reflect the anisotropy of the structures. That they are anisotropic is well established, and any arrangement like fig. 7a would be expected to give large differ- ences of dielectric constants measured parallel and normal to the dipole layers.The measurements are of mosaic crystals with presumably random orientations, but different rates of relaxation of the principal dielectric constants would be resolved in time by measurements over a range of frequencies. There is suggestive evidence for this in the fact that the amplitudes of the two relaxation processes of the low-temperature phases of deuterium bromide showed hysteresis effects depending on previous thermal history, which could well result from changes in crystallite orientations from stress relaxation effects. The necessity for more than a single simple dispersion in the intermediate phases of hydrogen and deuterium iodide has already been mentioned. The continuation of the observations could lead to finding a second dispersion of the form indicated by the dashed curve (a) of fig.4, and anisotropy could be advanced 1s an explanation. On the other hand, there is precedent for a continuation of the form indicated as curve (6) or in still other ways. The need is obviously for measurements in a considerable range of frequency above 1 Mc/sec, and work is in progress toward this end. Definitive measurements at the much higher fre- quencies necessary for the disordered highest temperature solid phases are a formid- able problem, which is unfortunate because knowledge of the nature of the re- laxation processes for the simple structures would be valuable in itself and for interpreting the results at lower temperatures. The change in character of the so far accessible regions of dispersion near the low-temperature transitions are quite remarkable because of the change from circular arc loci to a dispersion which, at least in its beginnings, has the simple Debye behaviour of relaxation theories incorporating a single relaxation time. The difference is brought about by definite changes in arrangement and order of neighbouring molecules, and more knowledge of the structures ought to bring the lines of evidence to such a state that each can contribute to it better under- standing of the molecular re-orientations.PHI = 0.40 D). This work has been supported by the Office of Ordnance Research, U.S. Army, and by the Office of Scientific Research, Air Research and Development Command. One of us (R. H. C.) is indebted to the John Simon Guggenheim Memorial Founda- tion for a Fellowship in 1955-56 held at the University of Leiden, and to Prof. C. J. F. Bottcher for hospitality during his stay there. 1 Natta, Mem. Accad. Italia Chem., 1931, 2, no. 3, 5 ; Gazz. chim. ital., 1933. 2 Swenson and Cole, J. Chem. Physics, 1954, 22, 284. 3 Brown and Cole, J. Chem. Physics, 1953, 21, 1920. 4 Cole and Cole, J. Chem. Physics, 1941, 9, 341. 5 Cole, J. Chem. Physics, 1955, 23, 493. 6 Onsager, J. Amer. Chem. Soc., 1936, 58, 1486. 7 Tisza, Phase Transformations in Solids (WiIey, New York, 1951), p. 28. 8 Powles, Trans. Faraday SOC., 1952,48,430. 9 Hornig and Osberg, J. Chem. Physics, 1955, 23, 662. 10 Hiebert and Hornig, J. Chem. Physics, 1952, 20, 918. 11 Atoji and Lipscomb, Acta Cryst., 1954, 7 , 173. 12 Luttinger and Tisza, Physic. Rev., 1946, 70, 954. 13 Sauer, Physic. Rm., 1940, 57, 142. 14 Kirkwood, J. Chsm. Physics, 1940, 8, 205. 15 Kruger and James, J. Chern. Physics, 1954,22, 796.

ISSN:0366-9033    

DOI:10.1039/DF9572300031

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

DIELECTRIC RELAXATION PROCESSES IN LITHIUM, SODIUM AND POTASSIUM HALIDES BY J. S. DRYDEN AND R. J. MEAKINS Division of Electrotechnology, C.S.I.R.O., Sydney, Australia Received 14th January, 1957 The dielectric properties of the lithium, sodium and potassium halides containing divalent cation impurity have been studied at frequencies and temperatures where the relaxation absorption curves are clearly separate from the background loss due to d.c. conductivity. In several cases two distinct absorptions were observed, the higher fre- quency one being present only at higher concentrations of impurity. Activation energies have been obtained for the lower frequency absorption which is due to orientation of the dipole formed by association of a divalent cation and metal ion vacancy. These energies agree with those obtained from d.c.conductivity measurements for the movement of cations provided these are correctly interpreted, but are lower than the values generally accepted. A method is suggested for calculating these energy barriers using elastic constants which gives excellent agreement with experiment. A mechanism is suggested for the second absorption. It has been demonstrated that there is dielectric absorption present in alkali halides containing divalent cation impurities 1 9 2 and that this absorption can be obtained quite distinct from the loss due to d.c. conductivity.3 Dielectric ab- sorption in alkali halides is important in presenting a simple case of a dielectric relaxation process and the one in which it appeared most likely that some theor- etical calculations could be made of the activation energy. I t was decided, there- fore, to extend the investigation to a number of alkali halides.The activation energies for the movement of cations in alkali halide crystals have been measured previously from d.c. conductivity and diffusion experiments. However, the measurement of a frequency of maximum dielectric absorption appeared to have advantages over these other methods, particularly after pre- liminary experiments demonstrated that it was not necessary to have large single crystals from which to obtain samples. EXPERIMENTAL MATERIALS.-The criterion of purity required of the alkali halides used in this work was that they should show no dielectric absorption in the region of the absorption peak which occurred after the addition of divalent impurity.With NaC1, NaBr, KCl, KBr and KI the commercial A.R. or reagent-grade materials were found to be suitable, provided that the crystals for measurement were chosen from the clear region of the mass solidified from the melt. LiCl was prepared from A.R. Li2SO4. H2O and A.R. BaC12.2H20. For the bromide and iodide, the sulphate was first reacted with A.R. K2CO3 to give Li2CO3 and the latter purified by conversion to the chloride and reprecipitation. LiBr was then prepared by heating the Li2CO3 with A.R. NHdBr, and LiI was obtained by bubbling HI into a suspension of Li2CO3 in water. NaI was prepared similarly from A.R. Na2C03. For NaF, the A.R. Na2C03 was heated with a slight excess of A.R. NH4F. The divalent impurities were the best commercial grades available and, where con- venient, were added as the halides,P_and in other cases as the carbonates in the reaction mixture. PREPARATION OF SPECIMENS FOR MEASUREMENT.-with LicI, NaCI, NaBr, KCl and KBr the crystals were prepared simply by melting the material (20-40 g, according to density) in a platinum crucible in a furnace and allowing it to cool slowly. For NaF 3940 RELAXATION IN HALIDES and KF, the air in the furnace was replaced by H2, and for LiBr and KT, atmospheres of H2 and HBr and H2 and HI, respectively, were used.NaI and LiI were melted in silica ware. The powdered material was introduced into the bottom of a silica tube 2.5 cm int. diam. sealed at the bottom and having a con- striction about 7-8cm from the bottom.It was then heated under vacuum for about 4 h to remove moisture and the portion of the tube below the constriction sealed off under vacuum and detached. The material was then melted and solidified in the furnace and the silica vessel transferred to a dry box to be broken. In each case on solidification, the mass separated into a clear, glassy region at the outer edge and a granular, poorly-crystalline region at the centre. The former gave small cubic crystals when broken up with a probe and these were split with a sharp blade to give flakes about 1 mm in thickness and a few mm in length and breadth. The number of splits required for a sample varied from about 10 to 35 according to size and, as the thickness had to be determined at this stage by estimate, there were appreciable variations throughout the sample.To reduce the amount of water absorbed by the freshly split surfaces, the breaking-up of the mass was begun when its temperature had decreased to about 130" C, and this operation and the splitting were performed on a hot plate. As an additional precaution with LiBr, NaI and LiI the hot plate was housed in a dry box. DIELECTRIC MEASUREMENTS.-A three-electrode assembly was used, consisting of two brass electrodes 2-5cm in diam., enclosed in a brass cell. Silica gel was placed in the bottom of the cell and the whole assembly heated for an hour or more at 130" C. The crystal splits were then taken from the hot plate and placed between the electrodes while they were still hot, the dry box being again used for the very deliquescent materials. After cooling a little, the joints in the assembly were completely sealed with Apiezon sealing compound.The temperature of the sample during measurement was controlled by immersing the electrode cell in a liquid bath contained in a dewar. The bridge used for the measurements has been described recently by Thompson.4 The capacitance between the electrodes was usually about 10pF of which about two-thirds was contributed by the alkali halide dielectric. ANALYSIs.-The concentrations of calcium impurity in KCl and in NaCl were estimated spectrophotometrically using a method described by Watson and Scott.5 RESULTS AND DISCUSSION No dielectric absorption other than that arising from the presence of d.c. conductivity was detected in any of these compounds in the absence of divalent impurity.This confirms the conclusions of Haven 1 and of Jacobs 2 for NaCl and of Burstein et al.6 for KCl. In the presence of a divalent impurity with an ionic radius approximately the same as that of the cation of the alkali halide, two absorptions may be detected. That occurring at the higher frequency is present only when the impurity con- centration is above a certain level. It was considered desirable, in most cases, to eliminate the higher absorption or reduce it to a low level. If the first sample prepared was found to contain an appreciable amount of the upper absorption, a portion of it was diluted with more alkali halide and recrystallized. This process was repeated until only the main absorption remained.MAIN ABSORPTION.-The main absorption will be discussed first. In all cases the dielectric loss factor E" could be described, as a function of frequency, by a Debye curve where T is the relaxation time and f the frequency. This signifies that the ab- sorption is characterized by a single relaxation time. As examples, the variation of dielectric loss factor with frequency is given in fig. 1 and 2 for NaBr and for LiCl. The latter is chosen as an example of compounds with a low activation energy. In this case, comparatively low temperatures had to be used before the absorption peak was clear of the background.J . S . DRYDEN AND R . J . MEAKINS 41 Tn all cases the frequency fmaX at which maximum absorption occurs obeys the following expression as a function of temperature (2) The values of A and of AE obtained for the different compounds are listed in table 1 (columns 2 and 4).fmaX = A exp (- AE/kT). 3.6- g 3 . 4 - 53 2 - 0 Y 3.0- lo9 frequency (c/s) FIG. 1 .-Dielectric absorption at various temperatures for sodium bromide with calcium impurity. t lo9 frequency ( c / s ) FIG. 2.-DieIectric:absorption at variousztemperatures focdithium chloride with magnesium impurity. In most of the alkali halides only one divalent impurity was added but in KC1 three impurities, Sr, Ca and Ba, were used in the measurements (see fig. 3). Some experiments were made with NaCl containing Cd and Mn. No absorption was observed with Cd and with Mn a complex absorption curve was obtained. It is possible, however, that both these impurities could have been added under different conditions to give similar dielectric results to those obtained for other divalent impurities.42 RELAXATION 1N HALIDES Activation energies from other dielectric absorption measurements are listed in column 5.Those of Breckenridge7 are not included because his AE values were determined from measurements at one temperature only and in many cases his absorption curves are not well defined. '015- c .- c . - c 3 ZI L 2 -!? ,010 v 0 I J 0 L - I 1 I I I I f 0 I 2 3 4 5 b loq frequency ( c / s ) FIG. 3.-Change in frequency of maximum absorption with radius of divalent cation; KCl containing Ba2+, Sr2+ and Ca2+ ; temperature 41" C. A sec-1 compound and impurity LiF (Mg) 7 x 1012 ref. (3) LiCl (Mg) 4 x 1012 LiBr (Mg) LiI (Mg) NaF (Ca) NaCl (Ca) 1.6 x 1013 NaBr (Ca) 8 x 1012 NaI (Ca) 6 x 1012 KF (Sr) 8 x 1012 KCl (Ca) 8 x 1012 KCl (Sr) 8 x 1012 KCI (Ba) 8 x 1012 KBr (Sr) 7 x 1012 KBr @a) 7 x 1012 KI (Sr) 9 x 1011 TABLE 1 AE from dielectric absorption eV this liter- paper ature 1.2 x 1013 0.653 0.42 0.40 0.35 8 x 1012 0-87 6 x 1012 0.68 0.7 1 4.5 X 1012 0.62 3 x 1012 0.56 068 4.5 x 1012 0.64 4.5 X 1012 0.67 0.7 6 4.5 x 1012 0.70 3 x 1012 065 3 x 1012 0.68 3 X 1012 0-58 AE from d.c.conductivity eV AE from AE from nuclear diffusion magnetic recalc. eV resonance liter- from eV ature resuits in literature 0658 - 0.7 9 0.5910 0-418 0.5610 0-398 0-388 0.45 - 0.8 9 0.7915 0.8615 0.7 15 0.7816 0.9415 0.7 1543 J . S. DRYDEN A N D R . J . M E A K I N S MECHANISM OF ABSORPTION For every divalent cation present in the crystal there will be an alkali metal ion absent.Some of these divalent cations, which have a net positive charge and metal ion vacancies, which have an effective negative charge, will be associ- ated. The absorption in these compounds arises from the orientation of the dipoles so formed. The vacancy has twelve possible equivalent positions around the impurity ion (fig. 4) and the orientation of the dipole can change either by @ Metal ion @ H a l i d e ion 0 Divalent impurity 0 Metal ion vacancy FIG. 4.-A sodium chloride lattice containing a divalent cation impurity and a metal ion vacancy on adjacent sites. (i) an alkali metal ion in one of the positions marked 1, 2, 3, 4, moving to the vacant site, or (ii) by an exchange of places between the vacancy and the divalent ion.The relaxation time, determined from these experiments, of dipole orienta- tion is equal to 1/2(wl + waj, where w1 and w2 are the probabilities per second of transitions (i) and (iij occurring. For these transitions the ions must pass over energy barriers El and E2 respectively, and w1 and ~ ' 2 will be proportional to exp (- AEl/kT) and to exp (- k * / k T ) . If the two energy barriers differ by more than a few per cent, one of the transition probabilities will be so much lower than the other that the activation energy determined experimentally will be the smaller of the two. It is not possible to say which of these energy barriers will be the smaller. For the lithium and sodium salts and of Ba in potassium salts, where the radii of the impurity ion and of the alkali metal ion are close to the same size the energy barriers will differ only in so far as the extra charge on the impurity affects it.For Sr and Ca in KC1 the divalent cations are 15 % and 25 % smaller respectively in radii than potassium and it might be thought that the barrier to movement of the impurity in these cases will be considerably lower than that of the potassium.44 RELAXATION I N HALIDES However, this may not be so since the net positive charge on the impurity ion will result in the nearest negative ion neighbours being attracted closer to the impurity and there will be a contraction in this part of the crystal. This is supported by the experimental results which show only a slight decrease in energy barrier with ionic radius (see table 1).COMPARISON OF AE WITH THAT ESTIMATED BY OTHER METHODS There are two established methods for studying the movement of vacancies in ionic compounds, viz., d.c. conductivity and diffusion. More recently, nuclear magnetic resonance studies have been used.17 The activation energies, where they have been determined by these methods, are included in table 1. The energy barriers for movement of a cation vacancy are equal to (in KF) or smaller '4 '11 FIG. 5.-D.c. conductivity in KCl (Sr) and KC1 @a) as a function of 117'. The full line is taken from curves given by Kelting and Witt.15 than those for an anion vacancy in all of these compounds, and hence the activa- tion energies obtained from conductivity measurements should be those of the cation.Since the fundamental process in both the conductivity and the dielectric mechanisms is the jumping of a cation from one position to an adjacent vacant site the activation energies should be approximately the same. The only differ- ence should arise from the presence of a divalent ion on a next nearest lattice site in the dielectric case. However, it can be seen that with the exception of those of Havens on LiF, LiCl and LiBr the activation energies from d.c. conductivity are higher than those measured from dielectric absorption to the extent of about 0.2 eV. An important difference between Haven and other workers is that Haven calculated his activation energies at higher temperatures. When some of the other workers' curves for the variation of conductivity with 1/T are re-examined, activation energies close to those from dielectric absorption are obtained.An example of this is shown in fig. 5 in which the conductivity curves 15 for KC1 containing Sr and Ba are redrawn. These curves consist of three parts: (a) a linear portion from which the activation energy is usually calculated, (c) the region in which the curve is much steeper and the slope of which is taken to be the sum of the energy to create a defect and the activation energy for the movement of a defect and (6) a region between these which may be present and which is linear over a small temperature range only. The energies listed in column 7 of table 1 have been obtained from region (b) and it can be seen that there is good agreement with the energy barriers from dielectric relaxation (columns 4 and 5).It can be concluded therefore that the energy barrier to movement of a cation is not in- fluenced to any significant extent by the presence of a divalent impurity at a next nearest lattice site. The difference between the energies calculated from the slope of part (a) and those determined from dielectric measurements is presumably the energy of association of the divalent cation-cation vacancy complex and the temperature at which the curve changes shape ((a) to (b)) that at which dissociation is complete.J . S. DRYDEN AND R . J . MEAKINS 45 When the divalent impurity and the metal ion are nearly equal in radii the associ- ation energy is therefore 0.15-0.20 eV but can be lower when the divalent impurity is smaller ; for example, in KCl containing Ca it is approximately 0.09 eV only.There have been several attempts to calculate the association energies in alkali halides 18 but the calculations yield values of 0.3-0.4 eV, i.e. higher than experi- mental by a factor of two. ESTIMATION OF THE ENERGY BARRIERS FROM ELASTIC PROPERTIES * Examination of the lattice structure suggests that an alkali ion moving from a position A to a vacant site A1 faces the smallest energy barrier if it moves via the interstitial position Z (fig. 6(a)). In passing along such a route AA1 it passes P P FIG. 6.--Illustrating path of a cation in moving from one lattice site to an adjacent vacant site. (b) and (c) show the distortion necessary to allow an ion to be in positions X and Z respectively.twice through a position in which it is coplanar with three halogen ions. Such a position is shown as X at the centre of an equilateral triangle LMN (fig. 6(b)), using NaCl as the example. The distortion necessary for a cation to be in the interstitial position Z is illustrated for NaCl in fig. 6(c). The energy is greater when the cation is at X than at Z . Therefore a calculation of the energy difference between positions X and A will give a value of the energy barrier which a cation has to overcome in moving from A to A1. A rough method can be devised to calculate this energy difference. If a volume vo of a substance whose compressibility is /3 is subjected to a uni- form volumetric compressive (or dilational) strain A then the stored energy is given by E = (1 /2/3)A2~0.(3) * This section is a result of collaboration between the authors and Dr. P. G. Harper and Dr. J. J. O'Dwyer of this laboratory.46 RELAXATION I N HALIDES Referring to fig. 6(b) we can write a linear strain associated with the position of the alkali ion at X as and calculate the corresponding volumetric strain as ( 5 ) Using vo = Z03, the values of /3 selected by Hojendahl19 and ionic radii of _ - vo 10 Zachariasen 20 the calculated activation energies are given in table 2. TABLE 2 pound LiF c1 Br I NaF c1 Br I KF c1 Br I KC1 Br dijt. A 2.01 2-57 2.75 3.02 2.3 1 2.8 1 2.98 3.23 2.67 3.14 3.29 3.53 energy difference 2 A V / V ~ ref. (19) calc. from A cmzdyne-1 eqn. (3) B nearest corn- neighbour S:Fzf eV 2.01 1.64 0.37 0.83 1 .5 0 ~ 10-12 0.61 2.49 2.10 0.39 0.66 3-34 0.38 2.64 2.25 0.39 0.64 4.23 0.3 5 2.87 2.46 0.41 0.59 5-89 0.28 2.31 1.89 0.42 0.81 2.07 0-66 2.79 2.30 0.49 0-77 4-18 0.54 2.94 2-44 0.50 0.75 4-98 0.51 3.17 2-64 0-53 0.72 6.94 0-43 2-66 2.18 0.48 0.81 3.25 0.65 3.14 2-56 0.58 0-83 5.53 0.64 3.29 2.68 0.61 0.83 6.56 0.63 3-52 2.88 0.64 0.82 8.37 0.53 0.64 0.63 e;;t. 0.65 1.05 0.42 1.10 0.40 1-15 0.35 1.25 0-87 1.33 0.68 1.25 0.62 1-2 0.56 1.3 0.65 1.05 Sr 0.68 1.05 0.67 1.05 0.58 1.1 calc. I Two main objections can be raised to this calculation. First, the formula holds only within the elastic limit while the volumetric strains in this model are of order 0.5. However, the strain is spread over a much larger volume than that used in the model and throughout most of this volume the elastic limit may not be exceeded.If then A+) dr equals A%, for some suitable volume integration the formula would be correct. Secondly, the strain is more two-dimensional than three-dimensional at its source but again this should be rapidly transmitted into a volume effect. For these reasons the excellent absolute agreement between the calculated and experimental values must be regarded to some extent as fortuitous. Of greater significance is the uniformity in the ratio between these energy barriers. The ratios for the potassium halides while they agree to within 5 % amongst them- selves are lower than the ratios for the lithiurn and sodium halides. This is partly because the divalent impurity, strontium, is smaller than potassium as has been discussed earlier.J FREQUENCY FACTORS So far, the discussion has been concerned with energy barriers. The other quantity which is obtained from these experiments is the frequency factor A . These frequency factors are listed in table 1 , column 2. The frequency at whichJ . S . DRYDEN A N D R . J . MEAKINS 47 maximum absorption occurs, is equal to w~/T, where w1 is the probability of a cation jumping into a vacant site in unit time. On the simple two position model, w1 =(w0/2n)exp (- AE/kT), provided there is no entropy of activation.21 However, in the case being con- sidered here, each cation vacancy has four positions into which it can move so that an extra factor of 4 must be included in eqn. (6). Therefore, provided that no entropy differences have to be taken into consideration, and that the lower energy barrier is for univalent cation movement rather than divalent cation movement, the frequency factors in the alkali halides should be equal to 4fol.r.This quantity is listed in column 3 of table 1 for those compounds in which fo is known? It can be seen that the agreement between 4fo/n and A is good to within a factor of three or better. This is almost within experimental error. MAGNITUDE OF THE ABSORPTION The disadvantage of using a large number of small crystals in these measure- ments instead of one larger section of a single crystal is that the magnitude of the absorption cannot be obtained accurately. However, it is possible to estimate the magnitudes from the area and thicknesses of the different slabs.For a sample of NaCl containing 0.075 mole % Ca2+ the maximum loss factor has been estimated to be 0.013. On the assumption that all the divalent impurities and cation vacancies are associated the value calculated for this concentration by means of the following equation 23 is 0.025 : where n = the number of divalent cations per cm3, a = half the lattice constant and e is the electronic charge. In view of the approximate nature of the cal- culation of ezrn from the experimental result no significance can be attached to these two values except that they are of the same order. SECOND ABSORPTION As mentioned earlier, a second absorption is detected in many of the alkali halides when the concentration of divalent cation is greater than a certain limit. This absorption was studied more thoroughly in NaCl than in the other com- pounds.In fig. 7 the dielectric loss factors are shown in three samples of NaCl containing different amounts of Ca2f, the scales have been adjusted so that the main absorption is the same magnitude in each. It can be seen that for a con- centration of 0.075 mole % Ca the loss factor curve is close to a Debye curve and there is oiily a trace of the second absorption present. As the concentration increases beyond this value the magnitude of the second peak increases until at 0.95 mole % it is comparable with the main absorption. The upper absorption region, unlike the main absorption, is always wider than a Debye curve. In NaCl the activation energy for the two regions is the same to within 4 %. The difference in the frequencies of maximum absorption arises from the difference in frequency factors, the value being 1 x 1015 sec-1 for the second absorption compared with 1.3 x 1013 sec-1 for the main absorption.The activation energy was not measured in any other compound but from the fact that the two absorptions moved with temperature at a similar rate it was con- cluded that the activation energies were nearly the same. An exception to this occurred in NaF containing Ca in which an absorption with a lower activation energy was sometimes detected. Since the activation energies are the same or nearly the same it is concluded that the second absorption is also associated with the movement of a cation vacancy. In considering possible mechanisms of this absorption the X-ray studies of Miyake and Suzuki24 on NaCl + CaC12 solid solutions are relevant.48 RELAXATION I N HALIDES These authors report that when crystah were grown from melts containing more than 0.5 mole % of Ca2+, extra diffuse reflections were obtained on X-ray photo- graphs. These authors did not analyse their crystals but by analogy with Kelting and Witt's experiments on the solubility of divalent impurities in KCl 15 it can be assumed that the concentration of Ca in the solid is about one-tenth that in the melt.This means that the extra reflections appear when the concentration of Ca in the crystals is about 0-05 mole %, which is in reasonable agreement with the concentration at which the second dielectric absorption appears, viz. 0.075 mole %. loq frequency ( c / s ) FIG.7.-Dielectric absorption in sodium chloride containing various proportions of calcium impurity. In a later paper, Suzuki25 proposes that the extra X-ray reflections are due to the existence of platelets having a crystal structure resembling that of CaC12 but retaining coherency with the structure of the NaCl crystal. At the surfaces of the platelets, where there is a change from one structure to the other, divalent cations and cation vacancies are on adjacent lattice sites, thus forming dipoles, and it is possible that the dielectric absorption under discussion is associated with the movement of these dipoles. The environment in which these dipoles move will be different from those in the bulk of the crystal, thus accounting for differences in the entropies of activation for the two cases.VARIATIONS IN RELATIVE MAGNITUDE OF THE TWO ABSORPTIONS WITH HEAT TREATMENT It was found that when samples were heated for some hours at 130" C or higher the magnitude of the second absorption in NaCl containing calcium was decreased and that of the main absorption increased. No experiments were conducted to determine if this process was reversed after storing at room temperature. Miyake and Suzuki24 report changes in their X-ray patterns after heat treatment but their experiments were carried out at rather higher temperatures (400" C). The authors wish to acknowledge the assistance of Mr. J. S . Cook in the preparation of some of the compounds and in the analysis of the NaCl and KC1 samples ; also Mr. P. Buss and Mr.Cook for the construction of a furnace suitable for the crystallization of the various compounds in different atmospheres. APPENDIX In view of the success in calculating energy barriers for the movement of cations from eqn. (3)-(5) it seemed desirable to extend the calculations to the anions. Unlike the cations the path which presents the lowest energy barrier to the transition of an anionJ. S. DRYDEN AND R . J . MEAKINS 49 from an occupied to a vacant site is not the same throughout this group of compounds. In KF where the two ions are of the same size the energy barrier for the transition of an F- ion is the same as that already worked out for Kf but in all the other compounds the negative ion is the larger. The strain at position X (fig. 6) is the same whether the larger or the smaller ion is moving, but it is no longer true that the interstitial position Z is lower in energy than X.In addition to the path shown in fig. 6 the ion can go via the shortest path AAI passing midway between ions M and N. Calculations show that for NaF and for KC1 the path via the interstitial position presents the lower energy barrier whilst in the other cases the lower barrier is via AAI. The values are set out in table 3 to- gether with the experimental results which are available. compound Li F c1 Br I NaF c1 Br I KF c1 Br I TABLE 3 E in eV Calc. expt. 1-84 1-28 1-26 1.10 1.55 1 *80 1 -67 26 1-70 1.18 13 1 a 5 0 0.65 1-54 1-97 1.91 1 Haven, J. Chem. Physics, 1953, 21, 171. 2 Jacobs, Naturwiss., 1955, 21, 575. 3 Dryden and Rao, J. Chem. Physics, 1956,25, 222. 4 Thompson, P.I.E.E., 1956 (in press). 5 Watson and Scott, J. Chem. Physics, 1956, 24, 619. 6 Burstein, Davisson and Sclar, Physic. Rev., 1954, 96, 819 (abstract only). 7 Breckenridge, J. Chem. Physics, 1948, 16, 959 ; 1950, 18, 913. 8 Haven, Rec. trav. chim., 1950, 69, 1471. 9 Lehfeldt, Z. Physik, 1933, 85, 717. 10 Ginnings and Phipps, J. Amer. Chem. SOC., 1930, 52, 1340. 11 Etzel and Maurer, J. Chem. Physics, 1950, 18, 1003. 12 Mapother, Crooks and Maurer, J. Chem. Physics, 1950, 18, 1231. 13 Schamp and Katz, Physic. Rev., 1954, 94, 828. 14 Phipps, Lansing and Cooke, J. Amer. Chem. SOC., 1926,48, 112. 15 Kelting and Witt, Z. Physik, 1949, 126, 697. 16 Wagner and Hantelmann, J. Chem. Physics, 1950, 18, 72. 17 Reif, Physic. Rev., 1955, 100, 1597. 18 Bassani and Fumi, Nuovo Cimento, 1954, 11, 274. 19 Hojendahl, K. danske vidensk Selksab, 1938, 16, no. 2. 20 Kittel, Introduction to Solid State Physics (Wiley, New York, 1953), 1st ed., p. 40. 21 Frohlich, Theory of Dielectrics (O.U.P., London, 1949), 1st ed., p. 68. 22 Barnes, 2. Physik, 1932, 75, 723. 23 Lidiard, Report on Coi:ference of Defects in Crystalline Solids (Bristol, 1954), p. 283. 24 Miyake and Suzuki, J. Physic. SOC., Japan, 1954, 9, 702. 25 Suzuki, J. Physic. SOC., Japan, 1955, 9, 794. 26 Patterson, Rose and Morrison, Phil. Mag., 1956, 1 (8), 393.

ISSN:0366-9033    

DOI:10.1039/DF9572300039

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

DLELECTRIC RELAXATION AND THE ELECTRICAL CONDUCTIVITY OF ICE CRYSTALS B Y H. GR;I;NICHER c. JACCARD P. SCHERRER AND A. STEINEMANN Physikalisches Institut der Eidg. Techn. Hochschule Zurich Schweiz. Received 5th February 1957 The results of previous and recent extended dielectric investigations on pure hexagonal ice crystals and on mixed-crystals containing hydrofluoric acid are summarized. Measure-ments of the d.c. conductivity were performed using electrodes of different nature. Quantitative electrolysis experiments demonstrate the ionic character of the electrical conductivity. The theory of the dynamic behaviour of dielectrics is applied to ice assuming that polarization changes only occur at lattice imperfections of two types orientational defects (i.e. vacant and doubly occupied hydrogen bonds) and ionized states (i.e.H3O+ and OH-). For both cases the possible proton jumps are computed statistically. The theory gives the correct dispersion behaviour and shows that the static dielectric constant of pure ice (- 100) is due to proton jumps at molecules adjacent to doubly occupied bonds. The theory is extended to mixedcrystals with HF. It can be concluded that at medium high HF concentrations the predominant dispersion mechanism takes place by proton transfer at H3O+ ions whereas at the highest concentrations the vacant bond mechanism prevails. Finally it is shown that the present theory of lattice imperfections in ice is consistent with all known structural and physical facts inchding the zero-point energy. 1. RESULTS OF DIELECTRIC INVESTIGATIONS The dielectric properties of pure ice crystals are of great interest and have been studied repeatedly since 1924.1 The most reliable results were obtained in 1952 on polycrystalline ice2 and with single crystals an anisotropy of the dielectric constant of about 15 % was determined.3 Two features of the dielectric pro-perties are astonishing : (i) Ice shows an ordinary Debye dispersion with an extremely low dispersion frequency (cod = 2 x 104 c/sec at - 10” C) compared with that of the liquid (od - 1011 c/sec) and of other polar substances.The temperature dependence of the relaxation time is given by The values 2.3 B = 5.3 x 10-16 sec and E = 13-2 kcal/mole which corresponds to 0.575 eV have been verified in the present investigation. (ii) The static permittivity of ice is about 100 and is thus even higher than that of liquid water.The measurements of the real and imaginary part of the dielectric constant E’ and E” fit the semicircles of the Cole plot E”(E’) well except for very low frequencies where deviations occur even under the conditions of highest purity. For low temperature measurements the data give the appearance of fitting the Cole plot better because the measurements have not been extended to sufficiently low frequencies. In this case the static permittivity obtained by extrapolation of the Cole plot is too large due to the d.c. conductivity q. (2) The imaginary part E” of the Debye dispersion should vanish at low frequencies since it is proportional to the frequency. Actually the deviations are such that E” diverges as 1/0.This behaviour might be interpreted as due to a frequency-r d = B exp (EJkT). (1) €;(go) = E i true + 2CJ‘O7/EO. 5 1-1. GRANICHER c . JACCAKD P . SCHERRER AND A . STEINEMANN 51 independent d.c. conductivity 00 but this explanation does not account for the observed increase in E' (fig. 1). Tt has been known for a long time that the deviations from the ideal Debye dispersion are favoured by impurities.4.5 A systematic investigation of these effects as a function of frequency temperature impurity content crystal thickness and ageing time has now been completed. The choice of the added impurities was based on the following two considerations.6 (i) Reproducible results are only to be expected if mixed crystals with ice are formed and if the impurity anion is able to replace an oxygen atom in the lattice.This requires that the size of the I o4 I 0' E'(V)-€b, to' 10 I I I I I I 10' I 10 I o2 10' s e ~ ' 10' I o5 frequency V FIG. 1.-Frequency dependence of E' for 5 nearly pure ice crystals at - 10" C. atoms be the same. (ii) As shown below the dielectric properties and the elec-trical conductivity are connected with the occurrence of lattice defects in ice. Therefore the addition of impurities should give rise to lattice defects (i.e. vacant proton sites) in a predictable manner. Hydrofluoric acid proved to be ideally suited for this purpose 6 and was used for most of the experiments in the concentration range from 10-7 to 10-1 M HF. The water used had a conductivity of - 5 x 10-7 L?-1 cm-1.At the lower limit of concentrations the behaviour became random due to the remaining impurities of the water and to natural lattice defects. No clear homogeneous mixed crystals were obtained for concentrations above 10-1. Fig. 1 shows that the permittivity at very low frequencies increases with the impurity content. This increase turns out to be due to the occurrence of a second dispersion regiun.7 This is evident from the behaviour of the loss factor tan 8 curves of fig. 2 where two maxima can be distinguished. The maximum at high frequencies is due to the Debye dispersion and is unaffected by not too high impurity concentrations. The additional maximum at low frequencies is significant for a new dispersion which is shifted to higher frequencies with increasing im-purity content.With this dispersion is associated an a.c. conductivity UF which is proportional to the square root of the HF concentration. From the behaviour of tan 8 the static limit of E' can be calculated and is found to be proportional to U F ~ though it is not directly measurable. For these nearly pure crystals in the frequency range of about 1 c/sec the increase of E' is steeper (cc OF) and leads to values of the order of 104 (fig. 3). However a critical concentration o 52 RELAXATION OF ICE CRYSTALS 3 x 10-4 M HF (corresponding to OF = 5 x 10-7 Q-1 cm-1) exists above which the permittivity E’ drops abruptly to values of about 25. This is evidence that the polarization mechanism above and below the critical concentration is completely different, 10 -tan b w l -I Olr / / I I //”V I I I 1 lo-’ 10 10’ 10’ 10’ rec-‘ 10’ ’ lo6 frequency V FIG.2.-Frequency dependence of tan 6 for the same crystals as in fig. 1 at - 10” C . I I 1 I ‘ 2 ‘ 3 ’ 4 ‘ 5 ‘ 6 I ! FIG. 3.-DieIectric constant E’ as a function of the a.c. conductivity OF (= const. x N F ~ ) at - 10” C for different frequencies as a parameter. In the low concentration range the Debye dispersion-especially its relaxation time-is not changed by the addition of HF. The Debye dispersion as observed in pure ice and the low frequency dispersion are therefore of different origin. The low frequency dispersion is fairly well described by the theory of Macdonald,s which is an extension to a.c. fields of the theory of Jaffe.9 The main assumptions of this theory are as follows a cd [To face page 5 H .GRANICHER c. JACCARD P . SCHERRER AND A . STEINEMANN 53 The crystal of a high frequency permittivity EZ contains NI neutral centres which are partly ionized by thermal activation. The charge carriers thus formed are free to move in the crystal but are not discharged (blocking electrodes). Con-sequently under the influence of an electric field the charge carriers build up a space-charge which is denser near the electrodes and which gives rise to an addi-tional polarization. Its frequency dependence closely resembles a Debye dis-persion. Among the relations which can be compared with the experiment the proportionality between the static value of the impurity contribution AEF and the thickness of the crystal capacitor is well fulfilled.The a.c. conductivity, however does not depend on the thickness. This gives direct evidence that the low frequency dispersion is caused by space-charge effects. The dependence of c;'.(w) on the HF concentration for nearly pure crystals-as described above-is in agreement with the theory for very low frequencies and for the static limit. Since not all assumptions made by the theory 8 are appropriate for ice other properties investigated are only in qualitative agreement with the theory. For HF concentrations above the critical value only the dispersion region of the Debye type is observable. The relaxation time is much shorter in this region (e.g. o d = 5 x l o 5 c/sec at - 10" C for a crystal with OF = 5 x 10-6 Q-1 cm-1) and its activation energy has dropped from the value of 0.575 eV for pure ice to ETL = 0.23 eV.The static limit of the Debye contribution is now only about 25 After a second minimum it increases to values of about 70 toward the highest accessible HF concentrations. These results strongly suggest that the mechanism leading to the Debye dispersion in pure and nearly pure ice breaks down at the critical concentration. The properties of ice with high M F content are to be explained by a different polarization mechanism. 2. ELECTRICAL CONDUCTMTY MEASUREMENTS The occurrence of the space-charge polarization focuses interest on the electrical conductivity and the nature of the charge carriers in ice. Conductivity measurements are rare and have so far been made only with a.c.fields.lo.11 As shown before the frequency behaviour is very complex. It is therefore difficult to compensate capacitive effects in a.c. conductivity measurements. The determination of the d.c. conductivity is a tricky problem too. Space-charge effects in d.c. fields have been observed.12 They lead to considerable changes in the potential distribution and to a time-dependence of the current in accordance with theoretical work? The potential irregularities were measured with clear ice rods in static fields.13 The space-charge effects were marked unless the water had been outgassed carefully before freezing. In our experiments water of conductivity of 10-6 to 10-7 Q-1 cm-1 was used and further purified after solidification by a zone-refining process.The electrodes were thin metal foils of Au Pt or Ag frozen on the crystal surface. Even under the conditions of highest purity and with the greatest care the current always decreased with time. The decay was proportional to 1 - (t/to)* at the beginning and became exponential later. Completely reproducible results could not be obtained. However if Au electrodes were applied by evaporation in a high vacuum while the crystals were kept at liquid air temperatures time-independent currents were observed in static fields. The conductivity was 10-8 to 10-9 Q-1 cm-1 at - 10" C with an activation energy of about 0-6 eV. Further experiments in particular to lower temperatures are in progress. By analogy with a useful method in semiconductor technique so-called " sand-wich '' electrodes were tried.The surface of a pure ice crystal was covered by a thin layer of ice containing HF in high concentration. Pt or Au electrodes were then frozen on the contaminated layers. Such " sandwich " crystals always showed ohmic contact and a time-independent conductivity at all temperatures. The potential distribution was measured using 2 potential probes and found t 54 RELAXATION OF ICE CRYSTALS be linear. Thus one is sure to measure the true conductivity. With an applied tension of about 100 V the conductivity between 250" and 125" K is represented by (11, with E = (0.325 f 0405) eV. C varies for different crystals from 3.6 to 5.0 Q-1 cm-1. The corresponding values of o(T) are 1-0 to 1-4 x 10-6 Q-1 cm-1 at 250" K. The diffusion of fluorine atoms into the bulk of the crystal was found to be negligible.The nature of the charge carriers might be electronic or ionic. Though the intrinsic absorption lies at 1670 A which corresponds to a forbidden energy gap of 7-42 eV extrinsic conduction seems possible. The results of thermolumin-escence 14.15 and paramagnetic resonance experiments 16 on irradiated ice specimens show the occurrence of electronic processes. The non-linear behaviour of the conductivity of halide-contaminated ice was interpreted as electronic but later reported to have been ionic in all cases.17 Using pure ice crystals with " sandwich " electrodes quantitative electrolysis experiments at - 10" C were performed. Measuring the volume of hydrogen formed at the cathode of the crystal one finds within experimental errors (1-2 %) that the conduction is entirely ionic by a proton transfer mechanism.Further evidence for ionic conduction is obtained from the current/voltage characteristic. With tensions below about 1 V which is the dissociation potential of water. practically no current is observed. o(T) = Cexp (- E&T) 3. CRYSTAL STRUCTURE OF ICE Under ordinary conditions water crystallizes in a hexagonal structure of space-group D& - P63/mm~. Each oxygen atom is surrounded by four oxygen atoms at a distance of 2-76 A. The tetrahedra thus formed are practically ideal. Bernal and Fowler 18 suggested the following three rules for the hydrogen positions : (i) the H atoms lie on the lines connecting neighbouring 0 atoms; (ii) there is only one H atom on each such linkage; (iii) each 0 atom has two H atoms at a short distance (0.99& and hence water molecules are preserved.The special type of bond -OH . . . 0- is called the " hydrogen bond ". Pauling 19 calculated the number of possible configurations which are com-patible with these three rules and found (3/2)" Assuming that these configur-ations have practically the same energy and thus equal probability (Pauling hypothesis) he successfully explained the observed zero-point entropy of ice SO = 0.82 cal/mole deg. = R In (3/2) as a result of the disorder in the H arrange-ment. The time and space average of a crystal with such a disorder is described by half-hydrogens lying on the two possible sites of each bond. Very accurate neutron diffraction studies with D20 single crystals 20 proved that this " half-hydrogen " model is essentially the correct long-range description of the ice structure.Completely polar structures (i.e. of the Rundle type 21 and the inter-pretation of the microstructure as antiparallel twins of a polar structure 22) cannot be maintained. If the three rules must be strictly fulfilled configurational changes by a trans-lational motion of protons along the hydrogen bonds or by rotations of molecules would require a simultaneous process on a chain through the crystal or on a closed ring of 6 or more molecules. Such processes are very unlikely in view of the high activation necessary. In addition movements on closed rings give no change in polarization and therefore cannot contribute to the permittivity.An ice crystal obeying the Bernal-Fowler rules and having no imperfections such as those described below and no dislocations stacking faults etc. is henceforth called ideal H . G R ~ N I C H E R c . JACCARD P . SCHERRER AND A . STEINEMANN 55 because it would show no electrical conductivity and a static dielectric constant of only 3.2. 4. CHARACTERISTIC LATTICE IMPERFECTIONS IN ICE Bjerrum 23 pointed out that the electric properties of ice cannot be explained unless two types of lattice imperfections are assumed : (i) Orientational defects are generated by the rotation of a H20 molecule around one of its four bonds preferentially around one of the two bonds with a close-lying proton. The two normal bonds (B) -OH . . . 0- thereby give one doubly occupied bond (0) -OH.. HO- and one vacant bond (L) -0. . . . 0-. The reaction equation for this process can be written 2B+ D + L. (1) The mass action law then gives for equal numbers of D- and L-defects * JvbM=lN~2 = (ND/NB>~ = a2 exp (- Eo/kT) = ‘W02. (2) The number NB of normal bonds is twice the number N of 0 atoms per unit volume. The constant a as yet undetermined takes into account the lattice disturbance in the surroundings of a lattice defect. W D = WL is the probability of finding a doubly or a vacant bond on any linkage. By rotation of a molecule adjacent to a D- or an L-defect the defect moves to a neighbouring linkage and is thus able to diffuse in the crystal. It is important to note that with orientational defects the molecules only change the direction of their dipole moments.(ii) Ionized states. If a proton moves along a hydrogen bond to the neigh-bouring molecule an H3O+ and an OH- ion are formed. By subsequent trans-lational motions of protons both sorts of ion states can move from one molecule to another This dissociation is formulated as a reversible reaction as it is usual for liquid water 2H20 + H3O+ + OH-. (3) In this case the mass action law gives for thermal equilibrium where EI is the activation energy for pair formation and b is a constant similar to a. As the lattice imperfections diffuse in the crystal their concentration will be homogeneous in the bulk of the crystal and imperfections of the same type re-combine according to the reaction equations. But it must be emphasized that no recombination is possible between orientational defects and ionized states.However one feature is common to both types when a lattice defect has moved past a certain molecule-either due to ordinary diffusion or due to a field-induced motion-the molecule is left behind in an orientation depending on the path and the nature of the defect. This means that a change in polarization has occurred. Since in all processes only one proton per molecule actually is shifted it is much more convenient to consider the polarization change brought about by the in-dividual shifts rather than to compute the dipole moments of all the molecules. 5. DYNAMIC THEORY OF DIELECTRJCS The general theory 24 of the dynamic behaviour of a dielectric containing par-ticles of charge e with two equilibrium positions separated by a distance 6 and a * Upright and cursive capitals are used for temperature-independent and temperature-dependent quantities respectively 56 RELAXATION OF ICE CRYSTALS potential barrier of height E between them leads to the differential equation d dt - (Jv; - 4) =- (W12 + W21) (Jy; - Jl) + (W12 - W21)(M2 + m.(1) ./v; and M2 are the number of particles in position 1 and 2 respectively. Wg are the probabilities for a transition from position i to j . In the absence of an ex-ternal field F the probabilities are all the same namely, (2) but become different under the action of the field. Since e6F< kT one obtains Wo = V r exp (- E/kT), W12 = Wo[l f (e6F/ZkT)] 21 and W12 f W21 = 2W0, and Thus eqn.(1) can be written W12 - W21 = Wo eGF/kT. and can be solved for a periodic field F = FoIexp iwt by (3) (4) ( 5 ) Comparing this equation with the formula for a Debye dispersion and considering c* = E' - i~" one obtains and Td = 1/2w0, (.A5 + XI) being the total number of particles. 6. APPLICATION TO HEXAGONAL ICE The foregoing theory holds only if interactions among the charged particles are negligible. This assumption is fulfilled for ice. One easily calculates 25 that only every 107th molecule has to reorient under the action of a field of 1 V/cm in order to produce the polarization corresponding to a static permittivity of 100. In the following only the case of the polarization mechanism by motion of doubly occupied bonds is outlined; the calculations for vacant bonds are completely analogous.If a doubly occupied bond is present and with sufficient thermal activation one of its two protons is able to jump to an adjacent normal bond. By this process the D-defect has moved to a previously normal bond and a polarization change arises. Since every molecule has four outgoing bonds pointing in different direc-tions the probability of finding a D-defect on a given bond e.g. parallel to the z-direction is W~14. With the value of WD from eqn. (4.2) the probability which must be introduced in eqn. (5.10) is in this case (1) and thus the relaxation time is (2) W6 = (1/4)W~ vr exp (- E:,D'/kT) = (1/4) av exp -(+Eo + E$:?)/kT, TD = 1/2W6 = (2/vp) exp (+Eo + c t i ) / k T H. GRANICHER c. JACCARD P. SCHERRER AND A .STEINEMANN 57 Assuming e.g. that the electric field is in the z-direction the following possibilities for proton jumps (by rotation) exist if the doubly occupied bond is on one of the bonds parallel to z the defect is transferred to one of the two neighbouring bonds which had no close-lying proton before. Similarly a D-defect on one of the linkages oblique to the z direction is able to move over to a bond parallel to z. The shift component in the z direction is for all jumps z = (413)ro~ = (4/3) 0.99 A. These favourable jumps are only a fraction of the total of 24 possible jumps which can take place at the 4 different molecules of the elementary cell. Therefore the quan-tity 82 in eqn. (5.9) has to be replaced by a statistical average 62. This average is defined by the fraction of favourable jumps times the sum of the squares of the shift components in field direction of the individual jumps.The fraction of favourable jumps in the case under consideration being 1/4 one obtains The calculation of €I - EL with eqn. (5.9) is more complicated. The same result is found for fields in the x- or y-directions thus no anisotropy appears. However for fields perpendicular to z it seems plausible to exclude jumps of protons from the z direction which would be at a high angle to the field direction if another proton is in position to jump in the xy-plane. By this restric-tion 7/48 rn 1/6 of the jumps are rejected. Defining the anisotropy as a frequency and temperature independent quotient A one finds 1 / ''IlC - E.T ' c = 14.6 %.A = I Esllc - ECO TABLE 1 A O/O It c I C theor. 106.4 91-3 14.6 theor. 124.0 106.0 14.6 expt. e i at 0" C expt. 105 91 13 + 4 E at - 40" C 129 104 19 It 4 dc6 theor. 0*44/deg. 0.38/deg. -dT expt. 0.6 5 0.1 0.3 5 0.1 - _ In table 1 the theoretical results for + E& = E its temperature coefficient de;/dT and the anisotropy A are compared with the experimental values.3 The quantities introduced in eqn. (5.9) are + ( N 2 +Nl) =N=3-08 X 1022 molecules per cm3 computed from the X-ray density E&= 3.2 and the 0-H distance t - 0 ~ = 0-99 A. The theoretical results are estimated to be accurate to f 1/2 % the uncertainty arising mainly from the 'OH value which is accurately known for D20 20 but not equally well for H2O. The ab-solute accuracy of the experimental values is t 4 % but the relative accuracy for crystals of the same orientation is 5 1 to 2 %.With these values the limits of error given in table 1 for dc/dT and A are obtained. The theoretical values for the polarization mechanism by doubly occupied bonds are in good agreement with the experiment.3 A systematic deviation only occurs for eiC at low temperatures where the omission of a conductivity correc-tion (1.2) leads to too high ci values. The computation for a vacant bond mechan-ism gives exactly the same results. A difference between the two mechanisms arises only in the relaxation time since it is to be expected that the activation energies Erot are different in general. Both mechanisms are present and hence the measured relaxation time depends on the sum of the dispersion frequencies.As is shown in 3 7 there is evidence that W D > OL. The observed dielectric properties of pure ice crystals must be ascribed to the doubly-occupied bond mechanism 58 RELAXATION OF ICE CRYSTALS Since the lattice dimensions and the bond energies are only slightly affected by the replacement of H by D the dielectric properties of D202 are very nearly the same as for H20 except for a change of d2 in the pre-exponential factor B (eqn. (1.1)). This is to be expected by this theory because a 42-change occurs in the oscillation frequency v,. Similar statistical calculations were made for the translational mechanism of H3O+ and OH- ionized states. One finds and thus the contribution of the translational mechanism Ae;rms to the static permittivity is 23 only.In principle since the orientational and the translational mechanisms are independent the true static permittivity should be 125. The fact that down to frequencies of 1 cJsec only the rotational contribution is observed shows that the dispersion frequency of the translational mechanism is very low. 7. EXTENSION OF THE THEORY TO CRYSTALS CONTAINING HF As mentioned in $ 1 HF forms mixed crystals with ice and one has good reasons to assume that each HF molecule replaces one H20 molecule in the lattice. As for each such replacement one H atom is missing a number of vacant bonds equal to the number NF of fluorine atoms is present. In eqn. (4.2) Jvz must now be replaced by the total number N L tot = c/y;. therm + NF-This leads to The thermal defect concentrations are thereby very much reduced and for high HF content the concentration MDtot = c/Ybthem of doubly-occupied bonds tends toward zero whereas the number of vacant bonds becomes equal to NF.A further consequence of the presence of HF is an increase of the H3O+ con-centration according to the dissociation equation (3) The mass action law gives (4) and hence the number of H3O+ ions is proportional to N$. (5) Only the case of low dissociation is considered so that NF can be regarded as con-stant (NF > N:). The H3O+ ions formed by dissociation of HF influence the dissociation equilibrium of the water molecule (eqn. (4.3)). One has to combine eqn. (4.4) with (7.4) and to introduce the total number of H3O+ states The result-similar to- the case of orientational defects treated above-is that the OH- concentration diminishes with increasing NF and Ntot starting from the value JV+ rises with NI; for low dissociation up to the limiting value given by eqn.(5). The expression (5.8) for the dielectric constant has now to be written in the following more generalized form : HF -1- H20 +- H3O+ + F-. NJ&/(NNF) = c2 exp (- EF/kT), M$ = c(NNF)~ exp (- EF/%T). in both equations .N;’,t = .N+ + A’$ (6) where the dispersion frequencies ar I f . GRXNICHER C . JACCARD P . SCHERRER AND A . S T E l N E M h N N 59 wrot = COD -+- COL 1= v,.("/CrD ,,,/ZNB) exp (- E$s'/kT) td2NB) exp (- E%i/kT) (8) and Wtrans = V t ( 4 W N ) exp (- E & d W - (9) * Since permittivity measurements are limited for experimental reasons to fre-quencies above 0.1 c/sec mechanisms of polarization with a dispersion frequency o d below 1 c/sec due to a small number of the respective defects (Jvb NL .N+ or N-) cannot be observed.In all measurements of ice with or without added im-purities the space-charge contribution is always present. Therefore only the I I I I I I I I I I - - - + + + \+ + ' \+ ++ x x' E,(pure) - 0 5 7 5 eV + \ c 0 -I - 3 O C v v I I I I I I I I I I 16' 2 9-' cm-' Id' (NF =number o f f ~ u O f l n $ a r O n l s ) ' 2 - . ~ F (-IOoc) FIG. 4. Debye dispersion with the highest can be observed. Its relaxation time Td is plotted as a function of the a.c. conductivity OF in fig. 4. Mechanisms with values of wd 5 Wspaw&arge are always masked by the space-charge polarization.In very high concentration crystals the conductivity leads to tan 8 ;t 103 and prevents even the measurements of the space-charge polarization at lower frequencies. One is now in position to discuss the expressions (7)-(9) as a function of in-creasing HF content. The observed static permittivity E w 100 corresponds to the value A& as computed in 5 6. Hence one concludes that wtrans << 1 c/sec. In nearly pure crystals Jvb decreases whereas Jyt tot increases with rising HF content. The fact that at the critical concentration the static limit of the Debye contribution drops from - 100 to - 25 shows (i) that in pure ice wg > WL or the predominant,polarization mechanism is (ii) that in concentrations above the critical value the translational mechanism In this concentration range-between the two minima of E; (fig.3)-A'& is prac-tically equal to J$ according to eqn. (5). The proportionality between wtraw the one involving doubly-occupied bonds and E!,q' < E$&) ; by H3Of states becomes effective. * A second term with can be omitted since it is never important 60 RELAXATION OF ICE CRYSTALS and NF* is actually fulfilled (fig. 4). NL tot however increases as NF and one has to expect that at still higher concentrations the rotational mechanism by vacant bonds will overtake the translational mechanism. This explains the new increase of ei at the highest HF concentrations above the second minimum and the change of the slope of the T~(Q) curve. Theoretically one can predict that E would rise to - 125 if crystals of sufficiently high HF content could be produced.8. INTERPRETATION OF THE CONDUCTIVITY AND DISCUSSION OF THE ACTIVATION ENERGIES It is quite clear that the d.c. conductivity in pure ice crystals requires proton transfers along the hydrogen bonds since the ionic character of the conduction has been ascertained. The conductivity will therefore depend on the activation energy Ez for ion pair formation and on EZans and EGans for translational motion. If a crystal had in the mean no polarization before a d.c. field is applied, a H3O+ state can move by proton transfer on a chain of molecules through the crystal. The molecules on the chain are left in an orientation which corresponds to a polarization opposite to the field.The existence of a time-independent current requires that after a certain time the same chain is again able to conduct another ion-state. This is possible only if the molecules have changed their polarization from opposite the field to a polarization parallel to the field. These molecular turns which are necessary to reactivate the chains for conduction are possible only with the aid of orientational defects-doubly occupied or vacant bonds. Hence the concentration of these defects and the activation energies for rotation Ei$) and E:;? have to be considered too. Because of the many processes involved it is not yet possible to decide which processes are rate-determining. But it is hoped that the conductivity experiments with H20 and D20 crystals which are in progress will give a clearer insight, The a.c.conductivity of HF-contaminated crystals is much easier to interpret because with high HF concentrations only H3O+ states and vacant bonds occur. As shown in 6 7 the number NL, is then practically equal to NF whereas the H3O+ concentration is given by eqn. (7.5). Hence a~ = const. N F ~ exp - ( ~ E F + Eiki + E&,,)/kT. The activation energy of UF determined by dielectric measurements is 0-32 eV. For the d.c. conductivity with " sandwich " electrodes the same mechanism prevails. In the electrode layer the H3Of concentration is much higher than in the bulk of the crystal and the diffusion of the ions would soon be counteracted by a diffusion potential since a negative space-charge of the immobile F- ions remains in the surface layer.However the surface also contains a high number of L-defects which have a tendency to diffuse into the bulk. The molecules which permitted this diffusion by molecular rotations are left behind in a state of polar-ization pointing toward the surface. Thus the combined diffusion of H3Of ions and vacant bonds does not give rise to a resultant diffusion potential. One there-fore has to assume that the bulk of" sandwich " crystals is homogeneously crowded by defects which were generated by the HF content of the electrode layers. This explains the much higher conductivity of " sandwich " crystals compared with crystals with evaporated electrodes and the occurrence of the same activation energy (0.325 eV) as in the a.c. conductivity of HF contaminated crystals.The activation energies are now summarized and compared with the experi-mental values. For the dielectric relaxation time Td in pure ice we found (eqn. (6.2)) ETPuTe = *Eo + = 0.575 eV. (2) In the fluorine contaminated crystals where the translational mechanism prevails, combining eqn. (7.9) with (7.5) gives GF = +EF + E&ans = 0.23 eV (3 H . G R ~ N I C I - I E R c . JACCARD P . SCHERRER AND A . STEINEMANN 61 For the a.c. conductivity of these crystals and the d.c. conductivity of " sandwich " crystals we found (eqn. (1)) (4) The difference between eqn. (3) and (4) leads to I?:; = 0.095 eV. It was shown in Q 7 that > E!,q>. Therefore the energy for formation of orientational defects must be within the limits 0.96 < EO < 1 .1 5 eV. The theoretical values 23 are 0.9 < EO < 1 . 0 eV. The interpretation that the L-mechanism is predominant only for HF concentrations above the second minimum of E permits the estimate < 0.02 eV. A further separation of the activation energies is not yet possibIe. With the limits for E D and the empirical value of the pre-exponential factor B (eqn. (1.1)) the number of orientational defects in pure ice turns out to lie between 2 x 1015 and 4 x 1013 per cm3 at - 10" C. The ratio between normal and defect bonds is 107 to 109. Thus the vital assumption of the dynamic theory of dielectrics that the interaction among the jumping particles can be neglected is well fulfilled. It may be worthwhile to note that the degrees of dissociation of H20 and HF according to the reactions (4.3) and (7.3) are considerably smaller in ice than in liquid water.The conclusion drawn in $ 7 that Wtrans < 1 means that the ion concentration in pure ice is smaller than the values given above for orientational defects by at least a factor 103. This is to be expected since the permittivity of the bulk effective for dissociation is 3.2 in ice but is 81 in water. It must be emphasized that it is immaterial for the present theory whether the molecular rotations or the translational motions of protons along the hydrogen bonds are classical activation processes or whether they are of quantum-mechanical nature (tunnelling). The distinction will be made by comparison of the behaviour of H20- and D2O-ice. EUF = +EF + ELans + EiL; = 0.325 eV.9. NEW CONCEPT OF THE LATTICE DISORDER IN ICE In order to explain the observed zero-point entropy of ice Pauling 19 postulated that the (3/2)N possible arrangements for the H atoms in the hexagonal ice lattice are of practically equal energy and equal probability. In the past few years this assumption was criticized,23 * because electrostatic energy calculations showed that different configurations may have energy differences comparable with kT at the melting point. Recent computations 26 taking into account higher order interactions than considered previously 23 led to considerably smaller energy differences. However from the present point of view these configurational energies are not essential for the explanation of the zero-point entropy. The experimental facts-in particular the zero-point entropy the neutron diffraction data,20 the results of nuclear magnetic resonance 27-28 and the electric properties as interpreted by the present theory of lattice imperfections in ice-are consistent with the concept summarized in the following statements.(i) The Pauling (half-hydrogen) model is an essentially correct description of the average (long-range) structure of ice. Local ordering of protons along a certain axis leading to a spontaneous polarization may be possible to an amount not exceeding 20 x.20 One easily verifies that ordered (polarized) regions of opposite orientation or ordered with disordered regions can be joined with each other without violating the Bernal-Fowler rules for the hydrogen positions. It is not possible to speak of twin boundaries between such regions in the usual sense.(ii) An ice crystal obeying strictly the Bernal-Fowler rules is called ideal, though it has a statistically disordered hydrogen arrangement. In such a crystal changes among the (3/2)N possible configurations do not occur. Such recon-structive transitions would require to break a considerable number of bonds and * for further references see ref. (22) 62 RELAXATION OF 1CE CRYSTALS hence need a high activation energy. They are therefore very unhkely even near the melting point. Hence the ideal crystal remains in the same incidental con-figuration established during freezing and therefore possesses zero-point entropy. (iii) The following types of lattice imperfections account for the electric pro-perties of ice : orientational defects i.e.doubly occupied bonds and vacant bonds : ionized states i.e. H3O+ and OH- ions. The diffusion of these lattice imperfections causes configurational changes among the ( 3 / 2 ) N possibilities. These changes are however only of local (short-range) character the number of lattice imperfections being so small. In addition, molecules on the diffusion path of an imperfection are left behind in a certain orientation (5 4) which is not necessarily the one of least configurational energy. Since the formation and diffusion of these lattice defects require thermal activa-tion their concentrations decrease exponentially with lowering temperature or if the crystal is quenched the imperfections freeze in and are no longer able to diffuse.Therefore at very low temperatures no configurational changes occur at all. (iv) The final conclusion is that the temperature where thermal equilibrium among the ( 3 / 2 ) N hydrogen arrangements is established lies well above the melting point,22 if the values for the configurational energy of 23 and others are assumed to be correct. Recent calculations26 lead to smaller energy differences which are compatible with freezing-in in the range of liquid-air temperatures. The observation of a zero-point entropy of the value So = R In ( 3 / 2 ) shows that the ideal as well as the real crystal is disordered with respect to the hydrogen arrange-ment and that the configurations are frozen-in at temperatures where the spread in configurational energy is still smaller than kT.Detailed papers on the present investigation will appear in Helu. phys. Acta. 1 Errera J. Physique Rad. 1924 5 304. 2 Auty and Cole J. Chem. Physics 1952 20 1309. 3 Humbel Jona and Scherrer Helw. phys. Acta 1953 26 17. 4 Granier Compt. rend. 1924 179 1314. 5 Smyth and Hitchcock J. Amer. Chem. SOC. 1932,54,4631. 6 Granicher Scherrer and Steinemann Helv. phys. Acta 1954 27,217. 7 Griinicher Jaccard Scherrer and Steinemann Helv. phys. Acta 1955 28 300. 8 Macdonald Physic. Rev. 1953 92,4. 9 Jaffe Ann. Physik 1933 16 217 ; J. Chem. Physics 1952 20 1071. 10 Johnstone Proc. Trans. Nova Scotian Inst. 1912 13 126. 11 Murphy Physic. Rev. 1950 79 396. 12 Joffe The physics of crystals (McGraw Hill,.New York 1928) chap. 7. 13 Oplatka Helv. phys. Acta 1933 6 198. 14 Grossweiner and Matheson J. Chem. Physics 1954 22 1514. 15 Ghormley J. Chem. Physics 1956 24 1111. 16 Matheson and Smaller J. Chem. Physics 1955,23 521. 17 Workman Truby and Drost-Hansen Physic. Rev. 1954,94 1073. 18 Bernal and Fowler J. Chem. Physics 1933 1 515. 19 Pauling Nature of the Chemical Bond (Cornell University Press Ithaca N.Y., 20 Peterson and Levy Acta Cryst. 1957 10 70. 21 Rundle J. Physic. Chem. 1955 59 680. 22 Granicher Helv. phys. Acta 1956,29 213. 23 B j e m Dan. Mat. Fys. Medd. 1951 27 nr. 1. 24 Frohlich Theory of Dielectrics (Clarendon Press Oxford 1949) chap. 3. 25 Debye Polar Molecules ('The ChemicaI Catalog Co. New York 1929) p. 102. 26 Piker and Polissar J. Physic. Chem. 1956 60 1140. 27 Bloembergen Purcell and Pound Physic. Rev. 1948,73 679. 28 Pake and Gutowsky Physic. Rev. 1948,74,919. 1948) p. 301

ISSN:0366-9033    

DOI:10.1039/DF9572300050

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

DIELECTRIC LOSS IN INSULATING SOLIDS CAUSED BY IMPURITIES AND COLOUR CENTRES BY J. VOLGER Philips Research Laboratories, N.V. Philips' Gloeilampenfabrieken, Eindhoven, Netherlands Received 28th January, 1957 Experiments at low temperatures on solids with various lattice defects have revealed the existence of dielectric relaxation phenomena due to these defects. The relaxation times are governed by activation energies far smaller than those normally found with diffusion or migration of ions. Typical measurements are given and discussed qualita- tively in relation to models of some lattice imperfections including colour centres. The study of dielectric properties of solids may, in some cases, give information concerning the imperfections of the lattice. The purpose of this paper is to report on a few investigations of dipolar relaxation due to imperfections of various types.It seems logical to distinguish between effects due to irregularities in the atomic arrangements only and those connected with " extra " electrons (or holes) in the solid. Though one cannot be very rigorous in this distinction, it has been used as a frame for this paper. One type of losses, however, will not be discussed, viz., the migration losses, i.e. both conduction and dipolar relaxation losses due to movements of ions over interatomic distances. These phenomena are related to diffusion. Migration losses are found, e.g., in alkali halides and glass, if the temperature is sufficiently high. The activation energy Q needed for the ionic jumps is of the order of magnitude of 0.5-1 eV.At low temperatures the ionic mobility is suppressed and migration losses can be excluded. Other dielectric loss mechanisms, however, may still exist. This is indicated by the experiments which will be discussed in the following. DEFORMATION LOSSES IN CRYSTALS Low-temperature losses have been studied in single crystals of clear quartz, mainly at frequencies of the order of 104 c/s. At room temperatune quartz exhibits a small loss factor, about 10-4, probably due to some mobile impurities such as Na+ ions. At lower temperatures the latter are frozen in and tan 6 decreases. Below 100" K, however, the tan 6 against T curves show sharp peaks depending upon the concentration and the nature of the impurities in the samples. Fig. 1 gives a few results obtained with crystals of different origins (cf.also ref. (1)). These low temperature losses can be characterized by specific relaxation times or, if we rely on the well-known formula (1) by specific sets of values TO and Q. These can be derived from an analysis of a number of tan 6 against T curves, each taken at constant frequency. Though this method is inferior to an analysis of tan 6 against o curves taken at constant temperatures, it quickly gives a valuable survey of the phenomena. The classical 7 = 70 ~ X P (QJkT), relaxation formulae in the approximation for small A€ € =€a + ~ 1 + w W Ae,k,, are applicable : (2) A€ or Em 1 +ow' tan 6 = - - (3)64 DIELECTRIC LOSS is the dielectric constant of the matrix in which dipoles with concentration N and moment p are embedded, giving rise to an increase A6 of the dielectric constant at zero frequency. The orientational polarizability depends on the internal field, affecting the factor f in but if N is small fequals 1 and AE < em.Then (em + 2)24nNp2 27kT ' E~ - EW = AE = FIG. l.-Sumey of tan 6 against T curves for a few monocrystals of clear quartz; measuring frequency 32 kc/s. ' A, specimen natural quartz from Brazil ; B, specimen synthetic quartz from Bell Telephone Lab. ; C, specimen synthetic quartz from General Electric Co., England. By comparison of results obtained with various crystals and using spectro- chemical analysis one can try to identify the different peaks. In fig. 1 has been indicated which impurity is thought to be responsible for the effects considered.To the temperature scale along the abscissa in fig. 1 an energy scale has been added, indicating the activation energies determining the relaxation times. This is permissible because the temperature T,,, where a maximum of the tan8 against Tcurve is found, is proportional to Q according to TO is found to be 10-13 sec for all processes found in clear quartz. p is of the order of 1 D. A further discussion of these relaxation processes, governed by such small values of Q (of the order of 0.1 eV or smaller) requires some model of the atomic situation at the lattice defect or built-in impurity. As has been remarked already,J . VOLGER 65 the ionic mobility in the usual meaning of the word is suppressed at the low temperatures applied. Our hypothesis is, however, that certain ions in the dis- turbed lattice maintain (e.g.by variation of valency angles) a possibility for dis- placements, though far more restricted and not exceeding 1 A, say, within the shell of their surroundings. Closely situated positions of relatively lowest energy might still occur. If the poten- tial barrier between these is not too high, dielectric loss may be expected at low temperature, provided the tran- sitions are electrically active. A proper name for such losses is perhaps defornzation losses, as they are as- sociated with slight local deformation rather than rearrangement of the lattice. It seems to us that in the experiments discussed we are dealing with such deformation losses, the more so as the dipole moments are small.Zt is useful to compare the di- electric investigations with recent measurements of the internal mecha- nical damping in a synthetic quartz crystal by Bommel (e.g. ref. (2)). Fig. 2 reproduces the published curve. The peak at about 55" K has been analysed as due to a relaxation pro- cess which is characterized by TO = 10-13 sec and Q = 0-056 eV and thus may be regarded as originating from the same lattice impurity which caused the dielectric effect characterized by these quantities, i.e. the peak at 38" K in fig. 1. lo+ 3 FIG. 2.-Internal damping in synthetic quartz crystal, measured by Bommel, e.g. re€. (2); measuring frequency 5 Mc/s. The effect at about 55" K is related to the peak at about 38" K in fig. 1. The effect at about 20" K is not considered in our paper.Theoretical work on the structural features of lattice defects (cf. also ref. (3)) and on the relaxation mechanisms, both dielectric and mechanical, in the disturbed lattice would be of a great value. DEFORMATION LOSSES IN GLASSES An experimental investigation of the low-temperature dielectric losses of various glasses, especially silica glasses, has recently been carried out by Stevels and the author.3 In these experiments ionic migration could again be excluded. At temperatures below 100" K the glasses showed losses of the deformation type, characterized by small activation energies, spread over a broad distribution function, in contrast to what had been found in quartz crystals. A typical tan 6 against T curve is shown in fig. 3. It has been proved that, generally, the deformation losses in glasses are related to the alterations of the SiO2 network, e.g.by the incorporation of extra oxygen or the substitution of silicon ions. Even fused silica shows deformation losses, probably caused by the presence of aluminium, which is an almost inevitable impurity. Boro-silicate glasses and glasses containing no silicon at all, such as phosphate glasses, also exhibit the low temperature loss under consideration. The same holds for silicates with structures like organic polymers, such as dimethylsiloxane (silicone) in the low-temperature glassy condition.4 Here the C66 DIELECTRIC LOSS deformation losses might be due to orientational relaxation of the side-chains of the molecules, which process may be governed by a small activation energy, as is so often used in restricted molecular rotation models.It should be noted that internal mechanical damping, analogous to dielectrical deformation losses, has been found in glasses, e.g. by Fine and co-workers 5 in fused silica and by Marx and Sivertsen 6 in commercial silicate glass. It is useful to try an analysis of the tan 6 against T curves in terms of distribution functions of activation energies. Let us return to fig. 3. According to a theory 6 0 0 5 01 015 0 2 0 2 5 0 3 035 0 4 045 0 5 0 5 5 Q (ev) FIG. 3.-Tan 6 against T of a glass specimen containing 70 mole % Si02 and 30 moIe % BaO ; measuring frequency 32 kc/s. The curve may also be considered as the distribution function f (Q) for the activation energies Q present. given by Gevers and du Pr&7 such a curve may be regarded as a representation of the distribution function f ( Q ) , as f ( Q ) is proportional to tan 6 with the T-axis interpreted as a Q-axis.The proof is based on the assumptions that f( Q) is a broad flat function and that p and TO do not depend on Q, such as to enable the integration to be carried out simply as follows : = nkTAf(Q)/2 with Q = - kT In 070. (7) In theory, A is inversely proportional to T. In order to adjust the Q-scale along the T-scale the value of WTO, i.e. of TO should be known. It appears that in glasses TO is again of the order of the inverse lattice frequency, perhaps somewhat greater than 10-13 sec which, however, is the value used, as it is hard to determine it exactly. Three remarks can be made : (i) As Imf(Q)dQ = N, the value Np2 can be derived from .oJ . VOLGER 67 For this glass eW = 6.1 and one finds from the curve after correction for the migra- tion losses at the high temperature side along the dashed extrapolated line, p2N = 8 x 1019, where p is expressed in D and N in cm-3. So if N = 1022 cm-3, which seems a reasonable figure for this glass, p would be of the order of 0.1 D only. (ii) The mean Q-value is 0.155 eV with a standard deviation of 0.090 eV, but the mode of the distribution function is 0.105 eV. (iii) Hence the distribution function is far from Gaussian. It is not logarithmic- normal either, but the 2/Q values are distributed almost normally. It is not yet clear whether much information can be derived from this feature. If, however, Q might be considered as being proportional to the square of a configura- tional co-ordinate x, characterizing the positions of the neighbouring atoms, it would mean that x is distributed normally around a mean value XO.DIELECTRIC LOSS IN INHOMOGENEOUS CONDUCTORS Solids may be rather good insulators for d.c. but may still contain insular regions of good conductivity. It has long been known (Maxwell, Wagner) that inhomogeneous materials show dielectric relaxation. This symposium on the molecular mechanism of rate process in solids is perhaps not the proper place for a discussion of these somewhat trivial effects. It might be useful, however, to give some consideration to this point, as it might elucidate certain conclusions of investigations with polycrystalline or ceramic substances.The disordered zones between adjacent crystallites are often barriers of rela- tively high resistivity. A simple linear barrier layer model with one thickness parameter proves to be satisfactory as a basis for a phenomenological theory.s9 9 Its results are that both overall conductivity and apparent dielectric constant vary with frequency according to simple Debye-like dispersion formulae, and that where G is the electrical conductivity, and where the subscripts s and denote values for d.c. and a.c. of infinite frequency respectively. €0 = 8.86 x 10-12F/m if Giorgi units are used, and If the intrinsic dielectric constant is the same for the barrier layer (1) and the bulk material (2), if the relative thickness dl of the layer is very small compared with unity, and if strong dispersion is found for (T, (9) and (10) can be written as with The ratio q/q is small and can be neglected, so (11) is almost identical with the usual relaxation loss formulae.An analysis of the dielectric loss angle in these inhomogeneous conductors gives in fact information on the conduction inside the well-conducting regions (grains). The curve for tan 6 is flattened and its maximum value is reduced if a distribution of T-values is present. In fig. 4 recent measurements of the loss factor of ceramic Fez03 are given. The sample had kindly been prepared by Mr. van Oosterhout of this laboratory. It contained 10-2 % Ti and had been fired in such a way that barrier layer resis- tance occurs. From the shift of peak temperature with measuring frequency it is derived that 0 2 1 ( T Z ~ exp (- Q/kT) with Q -= 0.20 eV.68 DIELECTRIC LOSS COLOUR CENTRE DIPOLE LOSSES Many a solid becomes coloured after irradiation by X-rays or fast electrons.The colour centres formed are spots of irregular charge distribution in the lattice where an electron or an electron hole is trapped. What dielectric properties do such centres exhibit? On the basis of a hydrogen-like electron orbit model the polarizability of the centre can be calculated, as has been done, e.g. for donor centres in germanium,lo but this is essentially " atomic " polarizability and orientational dipolar relaxation is not involved. However, irradiated quartz crystals (smoky quartz) investigated at low tem- peratures show marked dielectric losses of the relaxation type, which are neither manifest before irradiation nor after thermal bleaching of the specimens and which are most probably due to colour centres formed from impurities already present in the crystal before irradiation.1 Fig. 5 gives an example of the tan 6 against T curves found.A magnetic field of 3000 oersteds did not affect these losses. 4 0 0 D O E 200 100 0 0 FIG. 4.-Tan 6 against T and E against T of ceramic Fez03 containing some excess Fe; measuring frequencies 1 kc/s and 32 kc/s. We do not yet know with satisfactory precision how the static polarizability of the centres, nor how the relaxation time T varies with temperature. If we try to characterize r by a Boltzmann exponential the activation energy is of the order of 0-01 eV and the pre-exponential 70 is many orders of magnitude greater than 10-13 sec.From estimations of the concentration of centres it would follow that the dipole moment is far greater than in the case of the deformation losses dis- cussed in the preceding sections ; it might be about 10 D. An interesting feature is the partial disappearance of the loss peaks found in the clear crystals, after irradiation, as is clear from fig. 6 where the 38" K peak in the curve for the ir- radiated sample seems to be considerably reduced. Natural smoky quartz exhibits essentially the same effects. From the properties mentioned it is obvious that the nature of the colour centre dipole losses is quite different from that of the deformation losses. Our hypothesis is that we are dealing here with effects mainly due to the trapped electrons or holes themselves and that the dipolar relaxation indicates deviations from the simple hydrogen atom model for these.11 That such deviations occur is also known from paramagnetic resonance experiments.The trapped electron may in some cases, e.g. in F-centres in KCl, be considered as attached in turn to one of the cations which in a perfectly symmetrical way surround the anion vacancy.12 This leads to a model in which the electron or hole, at least at low temperature, is supposed to find energetically equivalent positions around the attracting centre proper, which positions could be stabilized by slight deformationsJ. VOLGER 69 of the lattice as a result of Coulomb forces, especially so when polar binding of the lattice occurs.The electrically active transitions between these equivalent positions can then be considered as a rate process (with small activation energy, apparently), though a tunnel effect may still play some role. FIG. 5.-Effect of irradiation upon low-temperature dielectric losses of monocrystal of natural quartz from Brazil. The peaks at 50°K and 60°K are probably due to the presence of alkali ions (lithium). t I I 1 I80 FIG. 6.-Effect of irradiation upon low temperature dielectric losses of monocrystal of synthetic quartz. The peak at 38" K is due to the presence of aluminium.70 DIELECTRIC LOSS In glasses, though colour centres of the same kind and in appreciable con- centration can be formed by irradiation, no trace of colour centre dipole losses has been found.However, the essential feature of the model above, i.e. the perfect symmetry, is absent in the glass. DONOR CENTRE DIPOLE LOSSES IN SEMI-CONDUCTORS In semi-conductors the free electrons or holes are at low temperatures trapped at the donor or acceptor centres and here again we may look for deviations from the spherical charge distribution. Recent dielectric investigations at low temperature FIG. 7.-Tan 6 against T curves and E against T curves at two measuring frequencies for Fe203 with some excess Fe (left-hand curves for 32 kc/s ; right-hand curves for 100 kc/s). The variations at about 30" K correspond to the dispersion phenomenon shown in fig. 8, and are due to electrons bound at the donor centres. FIG. 8.-Tan 6 and E at 20" K as functions of the frequency, for Fez03 with some excess Fe.The dashed curve is a single Debye curve with the same maximum value as the experimental one. on various semi-conductors indeed seem to indicate such deviations. As an ex- ample, results of measurements on Fez03 with excess Fe, will be discussed. A ceramic sample has been prepared by Mr. van Oosterhout and was, apart from its non-stoichiometric composition, very pure. The effects found in it were not shown by a sample of stoichiometric composition fired from the same raw material.J . VOLGER 71 Fig. 7 gives tan 6 against T curves and E against T curves for two different frequencies. It is seen that E varies in the dispersion regions in accordance with the dispersion formulae (2), though somewhat flattened.AE decreases with increasing temper- ature. (At about 100” K a new phenomenon sets in, but this is an inhomogeneity effect due to the increased conductivity, such as has been dicussed earlier.) From the curves it would appear that AE does not vary exactly proportionally to T-1, but rather like T-4. A provisional analysis of the r-values on the basis of (I) gives a Q-value of about 0.005 eV and TO rn 2 x 10-7 sec. This result may throw some light on the conduction mechanism in oxide semi-conductors like Fe2O3. It has often been suggested that a conduction electron in Fe2O3, once loosened from the donor centre “jumps ” from the potential well at one iron ion to that of the next ion, every time changing an Fe3f into a Fe2+ and vice versa. This, however, as is indicated by our dielectric measurements, seems also to be the case for the trapped electron in the vicinity of the centre, with this difference that the trapped electron is not free to move away into the lattice.Its transitions may be regarded as embryonic conduction. The mobility of the loosened conduction electron can be estimated on the basis of this picture. If the relaxation time for the transitions of the trapped electron may be used and if numerical factors of the order of magnitude 1 are disregarded, the mobility p is Fig. 8 gives tan 6 and E as functions of the frequency at 20” K. p =$/ekTr, (1 3) which is a strongly temperature-dependent quantity. p would be very small (about 10-5 cm2/V sec at room temperature) but this result is not unreasonable in comparison with the conclusions of electrical investigations by Morin.13 The extrapolation of our T-values to higher temperatures, however, is subject to uncertainty . Thanks are due to Mr. D. Hofman for his most valuable help withthe measurements. 1 Volger, Stevels and van Amerongen, Philips Res. Reports, 1955, 10, 260. 2 Bommel, Mason and Warner, Jr., Physic. Rev., 1955, 99, 1894 ; 1956, 102, 64. 3 Volger and Stevels, Pliilips Res. Reports, 1956, 11, 470. 4 Volger, Supplhtent au Bulletin de l’lnstitut Infernational du Froid, aniiexe, 1955, 2, 89. 5 Fine, van Duyne and Kenney, J. Appl. Physics, 1954, 25,402. 6 Marx and Sivertsen, J. Appl. Plzysics, 1953, 24, 81. 7 Gevers, Philips Res. Reports, 1946, 1, 197, 297, 361, 447. 8 KOOPS, Physic. Rev., 1951, 83, 121. 9 Volger, Physica, 1954, 20, 49. 10 d‘Altroy and Fan, Physic. Rev., 1956, 103, 1671. 11 Volger and Stevels, Philips Res. Reports, 1956, 11, 79. 12 Kahn, and Kittel, Physic. Rev., 1953, 89, 315. 1; Morin, Physic. Rev., 1954, 93, 1195. Hutchinson and NobIe, Physic. Rev., 1952, 87, 1125.

ISSN:0366-9033    

DOI:10.1039/DF9572300063

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. Y. Haven (Philips’ Res. Lab., Eindhoven) said: In order to compare the diffusion theory of Brinkman and Schwarzl with transition-state theory it seems appropriate to compare their eqn. (13) with the corresponding equation in the transition-state theory, which reads as follows, if we confine ourselves to classical statistical mechanics and apply it to the special case under discussion : - 1 v - v Ki = R$ I vX I - a 1 exp (- F ) d q / exp (- q ) d q . (13A) top of the barrier hole i Here IzI = an average velocity, A = integration length at the top of the barrier and R = coefficient, accounting for the deviation from “ equilibrium ” (a kind of reflection coefficient). The factor R is < 1, but according to transi- tion-state theory, by hypothesis, R = 1, in zeroth approximation.Apart from a slight difference introduced by the integration over the barrier, the two eqn. (13) and (13A) are completely equivalent, with the following cor- respondence, where A,ff is an effective width of the barrier. Therefore, with respect to applica- tion, the diffusion theory of Brinkman and Schwarzl and the transition-state theory will yield the same results. The transition-state theory, however, has the advantage over the diffusion theory in as far as in the approximation R = 1, the quantities on the right of eqn. (13A) can be determined from theory, so that this equation is liable to experimental verification. Dr. H. C. Brinkman and Mr. F. Schwarzl (Delft) (cornmunicated): In reply to Dr. Haven we wish to point out that the difference between our eqn.(13) and Haven’s eqn. (1 3A) may be considerable, as our factor, , and Haven’s factor, 1 exp (- Y - v /kT) dq, have a different origin and may give quite different results. Furthermore in our diffusion theory the forward as well as the backward diffusion is taken into account. This is not possible in a logically consistent way in the transition state method. Dr. J. Wood and Dr. A. Suddaby (Sir John Cass College) (cornmunicated) : We should like to make two comments which are relevant to Dr. Brinkman’s paper. First, the Smoluchowski equation follows from the Planck-Fokker equation when terms of the order (l/y)2 and higher are neglected. When the restoration of thermodynamic equilibrium arises by means of intermolecular forces which may be represented by a friction constant, Kirkwood 1 has shown that the Planck- Fokker equation follows as a consequence of the molecular random motion in a liquid.In doing this he has related the friction factor 7 to the intermolecular forces. Secondly, when in these circumstances the Planck-Fokker equation is applied to a molecular rate process we have shown2 that the rate constant is given by exp ( V - v /kT) dq {J ( ) I-’ 0 w kT Q complex y h Q initial k j = - - exp (- AE/kT), when terms of 0 (l/y)2 are neglected, and AE > kT. The symbols apart from 1 Kirkwood, J. Chem. Physics, 1946, 14, 180. 2 Wood and Suddaby, to be published. 72GENERAL DISCUSSION 73 o and y are identical with those employed in the transition state method. y, the friction constant (units sec-1) depends on the intermolecular forces, w (also sec-1) is the curvature at the maximum of the plot of energy against reaction path.The more exact expressions when AE is comparable with kT, and when terms 0 (l/y)2 are included have also been obtained by us. Prof. A. R. Ubbelohde (Imperial College) said: A problem of fundamental importance with respect to dislocations is to determine whether any entropy is to be attributed to them, or whether they introduce purely energy terms in the overall energy of a condensed state. If each of the various kinds of dislocations introduces only energy terms, the ideal crystal when it has fully attained equilibrium will squeeze them out at all temperatures. Kinetically this may require a con- siderable lapse of time, especially at low temperatures. If, on the other hand, an entropy as well as an energy can be attributed to any kind of dislocation, then the familiar considerations about the increase of point defects with rise in temperature in thermal equilibrium might apparently apply also to dislocations which are co- operative defects.Theoretically it is difficult to calculate entropy contributions from dislocations. It may be suggested, however, that one way of proceeding is to calculate the dis- turbances in the vibrational spectrum of the crystal, arising from the presence of dislocations. For the various modes, wherever the vibrational spectrum traverses frequencies of vibration at which the specimen exhibits marked relaxation peaks, strong absorption is found. This contribution aims to draw attention to the con- sequence that the vibrational spectrum of the condensed phase will be traversed by absorption bands at the relaxation frequencies.In a way analogous to the constitution of Bloch-Brillouin zones in metals, these absorption bands of forbidden frequencies distort the vibrational spectrum of the ideal crystal. The consequent modification of the vibrational entropy, c,ibd In T corresponds with an entropy contribution from each dislocation. When this contribution is positive. as may happen for certain types of dislocations, the crystal containing the appropriate proportion of these dislocations will be thermodynamically more stable than the ideal crystal free from dislocations, at temperatures above absolute zero. Studies of the interaction between mechanical relaxation frequencies and the presence of dislocations may help to throw light on the possible thermodynamic stabiliza- tion of dislocations, which is otherwise difficult to establish experimentally.Prof. F. C. Frank (Bristol) (communicated): The basic answer to Prof. Ubbelohde’s enquiry about entropy contributions from dislocations is that there can be no coupling between dislocation strains and vibrations where Hooke’s law is obeyed. Hence, at low temperatures, this coupling which can increase the entropy exists only within a small radius from the dislocation line, and can have only a finite effect, whereas the strain energy density has a long-range dis- tribution giving a logarithmically divergent integral. Hence, at low temperatures, the crystal with a low concentration is certainly unstable with respect to the crystal having none.Prof. Ubbelohde’s statement of the situation is of course slightly erroneous. In order that dislocations shall exist in the equilibrium state, it is not sufficient that their entropy contribution SD be positive, but that TSo shall be greater than their enthalpy contribution HD. “ An appropriate proportion of dislocations ” in equilibrium is not to be expected : rather, we should expect none, or very many. Above some critical temperature the latter condition could be the stable one. This formally justifies an identification of the liquid with the very highly dislocated crystal : but this description is not a very useful one, because dislocation defini- tions become non-unique at excessively high dislocation densities.Prof. F. C. Frank (Bristol) (communicated): I think there is a basic miscon- ception in Dr. Dryden’s comment. The low concentrations of lattice defects con- templated should be without any eEect on the steady-state polarization ultimately J:74 GENERAL DISCUSSION established, which is an equilibrium quantity: they provide only the mechan- ism by which it is established and their concentration determines the rate of establishment. This simple statement assumes that one of the two quantities I have called a~ and ag greatly predominates over the other. There are two conceivable polarization states according to which predominates. The situation will be more complex if they are nearly balanced, but in no case should we expect polarization proportional to exp (- EDlkT).Prof. J. S. Koehler (1ZIimi.s) said: With regard to the paper by Seeger, Douth and Pfaff : (i) Is 7; related to the yield stress in pure metals? (ii) In what respects does this theory differ from that of Mason? (iii) Will the presence of partial dislocations influence matters ? Dr. A. Seeger (Stuttgart) (communicated) : The answers to Prof. Koehler's questions are as follows: 7; is certainly nut related to the yield stress of pure metals at sufficiently high temperatures, since thermal energy helps the dislocations to overcome the Peierls stress. At high enough temperatures it will bring down the " effective Peierls stress " to virtually zero. Provided we are justified in neglect- ing quantum effects 7; will be equal or closely related to the critical shear stress at 0" K if in the undeformed crystals all dislocation lines lie along close-packed direction or if the dislocation arrangement is such that during the operation of the Frank-Reed sources the dislocation rings can be " trapped " along the close- packed directions.If, however, the critical shear stress (at all temperatures) of pure metal crystals is determined nut by the length of the Frank-Reed sources but by the elastic stress fields of the dislocation and by the so-called dislocation point, 7; should not be related to the critical shear stress. The reason for this is that under these conditions the majority of the dislocation lines will run along other directions than the close-packed ones, and will therefore not " feel " the Peierls stress.There is considerable experimental evidence that the critical shear stress of pure metal crystal is due to the stress contributions mentioned above ; 1 therefore there should be nu connection between 7; and the critical shear stress of pure metal crystals. This conclusion is in agreement with the fact that the numerical values of 7; deduced in our paper are by an order of magnitude larger than the critical shear stress of copper 2 and aluminium 3 9 4 single crystals extrapolated to 0" K. Mason's theory and the theory presented in our paper have this in common that both of them connect the Bordoni relaxation peak with the Peierls stress. The details, however, are quite different as may be seen from the fact that the numerical values of 7; deduced in our paper are two orders of magnitude larger than those derived by Mason.The main difference between the two theories is that accord- ing to Mason's mechanism the activation energy of the relaxation process should depend linearly on the length I between pinning points due to impurities or between dislocation nodes. (This is at variance with a large number of observations ac- cording to which the temperature of the Bordoni peaks is remarkably insensitive to impurity content, on pre-strain, and neutron bombardment.) In the present theory however the partition of the relaxation peak is to a very good approxima- tion independent of the state of perfection of the crystals. As was also pointed out in a discussion between Weertman5 and Mason the mechanism of thermally activated kink formation, discussed in our paper, is thought to make the process considered by Mason impossible.1 Seeger, 2. Naturforschg., 1954, 9a, 758, 870. 2 Blewitt, Coltman and Redman, Rep. Con$ Defects in Solids (London, 1955), p . 369. 3 Sash and Koehler, Physic. Rev., 1956, 101, 972. 4 Noggle and Koehler, J. Appl. Physics, 1957, 28, 53. 5 Weertman, Physic. Rev.,GENERAL DISCUSSION 75 To a first approximation the fact that the dislocations in metals like copper and silver comist of a pair of partial dislocations should not influence the dis- cussion presented in our paper. If eqn. (4.1) holds for each of the partial dis- locations it will also hold for the extended dislocation formed by a pair of partials.The reason for this is that the sine terms on the right-hand side of eqn. (4.1) due to the partial dislocations, add to another sine term for the extended dislocation. This holds even if the individual sine terms are “ out of phase ”, i.e. if the separa- tion between the partial dislocations is not a multiple of a. If the separation between the partial dislocations depends on temperature, the parameter ri (in eqn. (4.1) for the complete dislocation) will be temperature-dependent (even at temperatures low enough to render Dietze’s mechanism 1 for the temperature dependence of 7; ineffective). Some caution is therefore necessary if the theory is to be applied to data which have been obtained over a wide range of temperatures. The preceding discussion shows that it is presumably justified to apply the fornzalfrarnework of the theory to the existing data on f.c.c.metals. With regard to the rmgrzitzide of T; there might be an influence of the extension into partial dislocations. Other things being equal, 7; is predicted to be smaller for partial dislocations than for complete unextended dislocations.29 3 Although it is true that aluminium should not in all respects be compared to the noble metals, it is tempting to associate the empirical result of T ~ / G being larger for aluminium than for copper or silver with the fact that in aluminium the dislocations are practically unextended whereas in the noble metals they are well separated into partial dis- locations. Dr. J. Volger (Eindhoven) said: As one of the authors of the paper on the dielectric measurements on smoky quartz referred to at the end of the paper by Seeger et ai., I would make the following comment.The relaxation time of the dielectric loss mechanism in smoky quartz is characterized by a very small activation energy and a somewhat unusual pre-exponential factor, of the order of 10-8sec. The same orders of magnitude are found with the low-temperature losses in Fez03 with electrons trapped at donor centres. A connection with dis- location damping may suggest itself. However, unless the colour centres in quartz (which are without doubt chemical impurities) and the donor centres in the Fe203 investigated (excess Fe) are lined up along dislocations, it is most likely that the dielectric relaxation peaks found are due to point defects.This point of view is also taken in my paper and seems to be supported by recent considerations of Frohlich.4 Dr. J. S. Dryden (Sydney) (communicated) : Seeger, in this discussion, suggests that the mechanisms of the dielectric absorptions discussed by Volger may be similar to the mechanical relaxation in deformed crystals as discussed by Seeger, Donth and Pfaff. Without wishing to comment as to whether this is correct or not I want to point out that one of Seeger’s arguments is not valid. This argu- ment is that a frequency factor of 109 sec-1 does not occur in other examples of dielectric absorption. However, to take one example, in a barium magnesium titanate which we have investigated the frequency factor associated with a di- electric relaxation involving the movement of ions is 4 x 109 sec-1. Dr.A. Seeger (Stuttgart) (communicated) : In reply to Dr. Volger and Dr. Dryden I should like to thank these authors for their comments on the low-temperature dielectric relaxation in quartz. I agree with their remarks that frequency factors of the order 109 sec--1 have been found in dielectric relaxation phenomena not connected with dislocations. As indicated in our paper the mechanical losses occurring in the same temperature region are unlikely to be of the Bordoni type. 1 Dietze, 2. Physik, 1952, 132, 107. 2 Dietze, 2. Physik, 1952, 131, 156. 3 Seeger, Handbuch der Physik, VII/l, Ziff. 72. 4 Frohlich, Discussions, Colloqiie A.M.P.E.R.E., St. Malo (France), 1957.76 GENERAL DISCUSSION Since so far no alternative explanation for the mechanical losses has been advanced, and since frequency factors of the order lO9sec-1 are very rare in mechanical relaxation phenomena in crystals an experimental investigation into a possible interrelation between the mechanical and the dielectric losses is still thought to be interesting, In order to indicate just one possibility we mention that the following inter- pretation seems to be compatible with the existing data.Both the mechanical and the dielectric losses are due to the same process, or the same group of processes. The mechanical losses show up in both unirradiated and irradiated crystals. The dielectric losses, however, occur only in irradiated crystals, due to the introduction of an electric dipole moment upon irradiation.Prof. J. H. de Boer (Geleen) said: Dr. Cole, in the presentation of his con- tribution, referred to solid solutions of rare gases and hydrogen halides. Could he, possibly, give some more information? It does not look possible that a non- interrupted row of mixed crystals of rare gases and the low-temperature forms (ordered forms) of the hydrogen halides could exist. Dr. H. Granicher (Zurich) said: The dielectric constants of solid hydrogen halides show in the low temperature phase a dispersion at low frequencies similar to the behaviour of ice crystals. Further common features are the rather large dipole moments of the molecules and structural evidence that the crystals are built up by a relatively rigid system of hydrogen bonds. It is therefore suggested that the dielectric properties of the hydrogen halides might be explained by a lattice imperfection mechanism similar to that of ice described in our paper.Dr. J. S. Dryden (Sydney) said: By the same method as used in our paper we have calculated the energy barrier for the movement of a positive ion vacancy in AgCl to be 0.31 eV. This agrees to within 30 % with the experimental value of 0.36 eV obtained from d.c. conductivity.1 I have learnt recently from Dr. Morrison (National Research Council, Ottawa) that the energy barrier for the movement of C1 ions in NaCl is nearer 1.3 eV than 1.7 eV, the value quoted in table 3 of our paper. This makes the agreement with the calculated value poorer for NaCl but improves the consistency of the ratio expt./calc. between NaCl and NaBr.Dr. Y. Haven (Philips’ Res. Lab., Eindhoven) said: Energies of activation are often obtained by drawing two intersecting straight lines through the points of measurements in a logarithmic plot of ionic conductivity against 1/T. For the ionic conductivity of alkali halides containing divalent ions, however, it is possible to account for the interaction between impurity defects and thermally produced defects (as described in the paper of Compaan and Haven in this Discussion). When doing so, one obtains lower energies of activation for jumping and higher energies of activation for producing defects, than is ordinarily found in literature. This may account for the difference between the several energies of activation cited in table 1 of the paper of Dryden and Meakins.This same correction makes the high-temperature energy of activation for diffusion of Na in pure NaCl larger, and the energy of activation for diffusion of C1 in NaCl smaller. Therefore, the energy of activation for the movement of C1- and Br- ions in NaCl and NaBr, respectively, will undoubtedly be much smaller than that cited in table 3 of Dryden and Meakins’ paper. For C1- in NaCl, this value should be smaller than 1 eV. Prof. F. G. Fumi (Universita di Palerrnu) said : Dryden and Meakins schematize the logarithmic plot of the ionic conductivity of a doped alkali halide crystal against 1/T in the impurity-sensitive range with two straight lines, of which the one referring to the lower temperature region has a steeper slope. They assume that the difference in slope of these straight lines is related directly to the binding energy of the complex between the divalent impurity cation present in the crystal and the positive-ion vacancy.In effect one should expect several distinct regions 1 Compton and Maurer, J. Physic. Chem. Solids, 1956, 1, 191.GENERAL DISCUSSION 77 in the impurity-sensitive range, and the straight line of the authors for the lower temperature region (1.4 < lO3/T < 1.6) corresponds most likely to the impurity precipitation region of the Gottingen school.1 Thus the difference between the values of AE given in columns 4 and 6 of table 1 of their paper does not give the binding energy of the impurity complex. The elastic method used by Dryden and Meakins to estimate the energy barrier for the movement of ions in an alkali halide crystal has naturally an heuristic character as the elastic method used by Buffington and Cohen2 for metals; for one thing, the choice of the strained volume TJO is rather arbitrary.I think one should be careful in assuming that an estimate of this sort really allows one to say which is the path of movement of ions in an ionic solid : in particular, electrostatic terms such as the coulombic interaction between the moving ion and the two neighbouring vacancies, which are neglected in an elastic calculation, will be important in determining the actual path of movement, and they favour the “straight” path AA’ (fig. 6a of Dryden and Meakins’ paper) against the “ interstitial ” path AXZX’A’ by something like a couple of electron volts.Dr. Y. Haven (Philips’ Rex. Lab., Eindhoven) said: When energies of activation from conductivity data in the form K = KO exp (- U/kT) are compared with energies of activation from dielectric loss data, it is convenient to compare U + kT++ E (dielectric loss) because of the relation K m nyezdzjkT, where y is the jump frequency. When doing so one obtains LiF LiCl LiBr LiI E (diel. loss) (Dryden, Meakins) 0.65 0.42 0.40 0.35 U + kT (Haven) 0.73 0-47 0.45 0.43 Thus the energy of activation for jumping is smaller if a vacancy is associated with a divalent ion, than when the vacancy is free. Previously 3 this was seen as an indication that for instance in NaCl, a Naf does not jump straight through between two C1 ions from one equilibrium site to another, but makes a detour passing more near the tetrahedral hole.In that case the repulsion of the Naf ion by the divalent ion will be smaller at the barrier than at the neighbouring lattice site, thus lowering the energy barrier. Dr. A. B. Lidiard (Harwell) said: In connection with the paper of Dryden and Meakins I would like to make two minor criticisms and one general comment. First, eqn. (6) is correct as it stands and no factor 4 should be inserted : the angular frequency at which maximum absorption occurs is 2(wl + w2), where w2 is defined as the frequency of impurity-vacancy exchange and w1 is the jump frequency for an associated vacancy going to a particular but arbitrary one of the four neighbour- ing positions common to both the vacancy and the impurity (fig.4 of their paper). This was shown explicitly a year or two ag0.4 When w2 < w1 we then obtain A = f 0 i . r and not 4fo/n. The comparison of A withf& in table 1 thus indicates the occurrence of entropies of activation for the w1 jumps of about 216. My second criticism is that eqn. (7) needs a factor 4rr on the right-hand side to be ~ o r r e c t . ~ At 300” K, I then obtain 0.055 for the maximum loss in NaCl containing 0-075 mole % CaC12, which is four times larger than the estimated experimental value of 0.013. Although, as the authors point out, this estimate may not be accurate owing to the use of many small crystals, the result is consistent with the 1 see, e.g., Zuckler, TIzesis (Gottingen University, 1949). 2Buffington and Cohen, Acta Met., 1954, 2, 660; see also Brooks, Zmpurities and 3 Haven, Defects in Crystalline Solids, Rristol, 1955, p 261.4 Lidiard, Conference on Defects in Crystalline Solids (Bristol, 1954)) Physic. SOC., Imperfections (Amer. SOC. Met., 1955). 1955, p. 283.78 GENERAL DISCUSSION previous results of Haven,l also on NaCl + CaC12, and of Teltow and Wilke 2 on AgBr + CdBr2. In both these cases the magnitude of the loss coming from the impurity-vacancy complexes is only about one-third of that predicted by the theory given in 0 2 of ref. (1). This may well indicate that the electric dipole moment of the complex is considerably smaller than the d%a assumed in that treatment; a strong attraction of the electrons on the anions neighbouring the complex to the divalent cation may well reduce the dipole moment below the value 6 e a . Some confirmation of this view is obtained from the results of Busse and Teltow 3 on dielectric loss in AgBr + AgzSe : here the complexes are Se2- ions and interstitial Ag+ ions.The maximum loss in this case was found to be only one-twentyfifth of the value expected from the simple ionic model : transfer of electrons from the Se2- ion to the interstitial Ag+ will very probably occur and will reduce the dipole moment accordingly. Dr. R. A. Sack (Cambridge) said: Dr. Dryden and Dr. Meakins are led to compare the frequency factor A and the expression 4foi.r by using eqn. (6) for the transition probability w1. Yet the Debye relation T = 1/2w for jumps between stable dipole orientations applies only if the dipole moment is completely reversed ; in the present case, if the field is along one of the crystal axes (see fig.4), two of the four possible jumps do not alter the relevant dipole component at all; the other two take it only half-way to a reversal. The product of the number of possible transitions and their mean efficiency is thus 4 x + = 1. Hence the factor A should be compared with f&; in table 1 this would improve the agree- ment for LiF and KI, but make it worse in the other cases. Prof. F. C. Frank (Buisiol) said: It happens that last weck P was at the Nottingham conference of the Physical Society, at which Schneider presented a study by paramagnetic resonance of sodium chloride containing 10-5 or 10-4 parts of manganous chloride. This gave a very clear indication of the various situations of manganous ions in the crystal, which may help in the interpretation of Dryden's observations.The spectra observed were, in order of increasing temperature, from room temperature to about 500" C , I think : (A) a broad un- resolved hump giving place to (B) and (C), a complex orientation dependent spectrum of many lines, out of which emerged (D), an orientation-independent spectrum of 6 lines. (B) and (C) can be resolved into two parts, (B) having symmetry about (110) axes, and (C) having symmetry about (100) axes about 1/10 as strong as (B). These are interpreted as : (D), manganous ions in cubic environment, normal substitutional cations ; (B) manganous ions with cation vacancies in nearest neighbour positions ; (C), manganous ions with cation vacancies in second nearest neighbour positions ; and (A), manganous ions close to other manganous ions.As no precipitate is visible in the ultramicroscope, 1 would surmise that the latter are very likely coherently segregated in monatomic layers, like Preston-Guinier zones. Without referring to these, since vacancies are trapped both in nearest and in second nearest neighbour positions to a divalent cation, there should be two different relaxation components, for jumps between two nearest neighbour positions, and for jumps between nearest and second nearest neighbour positions. Dr. J. S . Dryden (Sydney) said: The model used by Granicher, Jaccard, Scherrer and Steinemann and by Bjerrum to explain the dielectric properties of ice appears to me to contain a feature which is inconsistent with experiment.It is claimed that lattice defects must be present in the crystal before polarization, other than atomic and electronic, can occur. The number of these lattice im- perfections is a function of temperature and consequently the magnitude of the 1 Haven, Conference on Defects in Crystalline Solids (Bristol, 1954), Physic. Soc., 2 Teltow and Wilke, Naturwiss, 1954, 41, 423. 3 Busse and Teltow, Noturwiss., 1957, 44, 11 1. 1955, p. 261.GENERAL DISCUSSION 79 polarization should vary in proportion to exp (- E&T) whereas in fact (€0 - E ~ ) varies with 1/T. The same authors make use of a quantity which they call the " ax. conduc- tivity ". The measurements of tan 8 at any frequency is the same as measuring an a.c. conductance at that frequency ; for example, each point in fig.2 of the paper of Granicher et al. can be related to an a.c. conductance but beyond this the term has no meaning. However, Granicher et al. use the term having some other property in mind and I would be interested to know what this is. The dielectric absorptions discussed by Volger are very small, parts in 105 in some instances, and I would like to know what errors are associated with the curves in his figures. Dr. A. Steinemann (Zurich) said: In reply to J. S. Dryden, the conductivity correction (eqn. (1.2)) can be derived from the Debye equations by adding a d.c. term a&~o to E". It can be shown that, for the Cole plot, semicircles still exist, but have an increased radius. In the alkali halides polarization processes only occur at lattice defects and hence (6; - EL) is proportional to the defect con- centration.In ice crystals, however, in principle every water molecule is able to re-orient and to contribute to the polarization. The fraction of polarized molecules in static fields depends on the ratio eSF/kT of the electric to the thermal energy. The number of defects only affects the rate of reorientation processes, but not the equilibrium polarization. The d.c. conductivity was measured by the time-independent current in a static applied field. The a.c. conductivity results from the bridge measurements, if the complex permittivity of the crystal is interpreted as a parallel circuit of a real permittivity and a loss resistance. In $ 8 of our paper and elsewhere d.c.and a.c. conductivities are compared. With the term a.c. conductivity we then mean the low-frequency limit of the a.c. conductivity designed by OF. Prof. R. M. Barrer (Imperial College) said: R. S. Bradley has recently measured the conductivity of ice from 0 to - 25" C, and obtains CT = 23.4 exp (- 12,30O/RT) ohm cm-1. As far as the energy barrier is concerned, this agrees only moderately with Granicher's value 0.6 eV = 13.8 kcal. What accuracy can be achieved with such measurements? Is there likely to be anything like grain boundary con- ductivity in polycrystalline ice? I would also like to draw attention to work done at Basle by Kuhn and Thurkauf on self-diffusion in ice, on which a brief report appears in the 10th Solvay Corzgr. Clzern.1 They found that DzO and H2018 diffused at the same rate in ice, with D - 10-10 cm2 sec-1 at - 1 O C. This means a molecule diffusion mechanism, except in the unlikely event that a coupled diffusion of OH- and H 3 0 f occurred.Prof. Kuhn is, I think rightly, convinced that a place interchange between neighbour- ing pairs of water molecules is the controlling mechanism, and not a diffusion of holes or defects. The time needed for adjacent molecules to interchange was - 5 x 10-6 sec at - 1" C, i.e. such changes occurred with a given water mole- cule 2 x 105 times per sec. Since such events occur so rapidly I would expect them to have an influence on the dielectric relaxation properties of ice, and would like to hear the views of Granicher et al. about this. Dr.H. Griinicher (Zurich) said: In reply to Prof. Cole, we are quite aware of the problem connected with the electric field F which appears in our eqn. (5.3) to (5.6). The term e6Frepresents the change of the electrostatic energy of a proton displacement. Since actually work is delivered for such a movement, the field in question must be the externally applied field. Prof. A. R. Ubbelohde (Imperial College) said: With reference to the paper by Granicher, Jaccard and Steinemann, we have recently studied proton conductance in solid organic acids and other crystals containing hydrogen bonds.2 So far 1 1956, pp. 77 et seq. 2 6 . Pollock and Ubbelohde, Trans. Furuduy SOC., 1956, 52, 1112.80 GENERAL DISCUSSION as the preliminary findings extend, it seems likely that systems of co-operative hydrogen bonds in crystals arranged as networks, sheets or spirals, permit proton migration mechanisms that require considerably lower activation energies than when hydrogen bonds are present as isolated pairs in the crystal.This lends considerable support to a co-operative Grotthus type of conductance, in which the proton co-operative systems polarize in the external field and relax from these specific positions by one or more micro-mechanisms. The net consequence of these two operations is to lead to proton migration. Crystals such as organic acids offer certain advantages over ice in the study of co-operative hydrogen bonds because the limits of the co-operative systems are more clearly defined. The existence of polymorphs of practically the same free energy as ordinary ice makes it likely that boundaries between one crystallite and the next involve small regions where the ideal arrangement is modified ; this may confuse estimates of residual entropy based on the assumption of an ideal lattice of infinite extent. Concerning the paper by de Boer and the remarks by Dr.Mannil, we are en- gaged in various studies of crystal transformations in which a single crystal is subjected to temperature cycles, traversing the transformation temperature T,, and in which the persistence of crystal axes is followed by X-ray methods. These bring out the great importance of traces of impurities, such as water, for deter- mining the extent to which the single crystal traverses the same path in detail. It may be added that one of the best ways of removing these traces of water is to subject the crystals to very high vacua and take them through a cycle of temperatures including T,.Co-operative defects that arise in the neighbourhood of T‘, because of the coexistence of domains of closely similar structure, greatly facilitate the removal of impurities that migrate rapidly around the transformation temperature. Recent work has shown that this ‘‘ thermal shock ” cyclic treatment is effective, for example, for nitrates and for sulphocyanides. Dr. Y. Haven (Philips’ Res. Lab., Eindhoven) said: In order to discuss the di- electric properties of ice, it is appropriate to note that one can discern “ individual ” and “ collective ” properties. The relaxation of a vacancy-divalent ion-pair in alkali halides is an individual property and independent of the concentration of lattice defects.The relaxation in ice, however, is a collective property as all molecules contribute to the equilibrium. If the average jump frequency of the molecules is raised because the concentration of lattice defects is raised, the re- laxation frequency will be raised, as has been observed for the larger concentra- tions of fluoride ions. This type of relaxation is closely connected with the electrical conductivity of ice. There might be found, however, another relaxation process when fluoride ions are present. If the corresponding hydrogen vacancies associate to the fluoride ions, one obtains a dipole of the type as in alkali halides containing divalent ions with “ individual ” properties.A dielectric relaxation corresponding with the orientation of these dipoles may be expected at much higher frequencies than used by Granicher et al. (say 108 sec-1) or at much lower temperatures. Dr. M. Eigen (Giittingen) said: In general, it is not possible to determine the ionic concentration and mobilities separately by conductivity measurements, as the specific conductivity contains these quantities in form of the product c(u+ + u-). Earlier considerations of some authors resulted in the conclusion that these three quantities may be nearly the same as in water, because the conductivity of ice shows the same values as in water. Bjerrum was the first to point out that these conclusions would lead to difficulties, as the electrostatic interaction of the fast moving ions in ice cannot be described by the static dielectric constant as for water.Using a dynamical value he calculated a heat of neutralization which is appreciably higher than that of water. The results of Dr. Granicher and co-workers support these assumptions by reporting a value of ionic con- centration between 3 x 10-9 and 6 x 10-11 mole/l. (The respective value ofGENERAL DISCUSSION 81 water would be 2 x 10-8 molell.) Assuming equal conductivities of both media, the lower concentration has to correspond to higher mobility. We came to very similar conclusions by a rather different approach. We studied fast protolytic reactions in solution using relaxation methods (e.g. sound absorption, dispersion of the dissociation field-effect). In these investigations we found for the rate constant of the neutralization reaction Hf + OH- -+ H20 the very high value of 1.3 x 1011 I./mole sec which suggests that protons and hydroxyl ions recombine spontaneously if they approach to a distance of 2 to 3 H bonds.1 This is only possible if the mobility of these ions in H-bonds is higher than normal in water.In order to determine the mobility of protons in H-bonds (e.g. ice crystals) directly, we developed a special technique, which has recently been described 2 (the stationary field-method). Preliminary results show that in ice the mobilities of the proton (and also of the hydroxyl ion, or defect proton) are higher by 1 to 2 orders of magnitude, and the ionic concentration is lower by 2 to 3 orders of magnitude than in water (in agreement with Dr.Granicher’s results). The proton mobility in H-bonds differs from electron mobility in metals only by 1 to 2 orders of magnitude, approximately corresponding to the square root of their mass ratio. This result strongly suggests a quantum-mechanical mechanism (tunnel effect) for the proton transfer within the H-bond. The high value of the proton mobility mentioned should be determinable by Hall-effect measurements in very high electric and magnetic fields. Such measure- ments are in progress. On the other hand, the above-mentioned stationary field method is generally applicable to semiconductors with very low concentrations of charge carriers, and permits one to determine the mobilities in cases where Hall-effect measurements become difficult.As the properties of protons and defect-protons correspond to those of electrons and holes, interesting analogies between H-bond systems and electronic semi- conductors arise. So it was possible to test ap,n rectifier consisting of two ice crystals, one contaminated with HF (proton donor) the other one contaminated with LiOH (proton acceptor). Instead of normal metal electrodes, one has to use a special electrode system containing acid and base solutions or exchangers and unpolarizable electrodes. As H-bond systems are present in biological membranes, the above-mentioned effects may be important for biological processes. Prof. F. C. Frank (Bristol) said : I have a bad conscience about the Bjerrum theory of dielectric polarization in ice, which Granicher and his colleagues appear to have satisfactorily established.When I thought of this theory just about ten years ago, I was quite excited with it, and went and told Onsager about it, who happened to be in England at the time. He told me he had discussed the same theory in lectures two or three years before. This depressed me somewhat. Neither of us ever published it, so it is Bjerrum’s theory. One of my reasons for not publishing was that I did not know how to make it unique or exact. Let me use the letters P and N to denote hydroxonium ions and hydroxyl ions, and call these the intrinsic defects of class (A), the ionic defects, and then call D and L the orientaticnal defects with two protons or none between a pair of oxygen atoms respectively, class (B).Then if either class were absent or immobile, the crystal would behave as a polarizing dielectric. A positive current carried along any particular path by a defect of one of the classes can only be followed along the same path by a negative current carried by the same class, or a positive current carried by the other class. This is the well-known situation with regard to the Grotthus conduction mechanism-that ion-transfer and reorientation processes must both occur for steady-state conduction : if either occurs alone, there is only a polarization. Given non-polarizing electrodes, the lesser of the two quantities, OA = ( I I P ~ P +- IINPN) and UB (HDPD + HLPL), 1 Eigen and De Maeyer, 2. Elektrochenr., 1955, 59, 986. 2 Eigen and De Maeyer, 2. Elektrockeni., 1956, 60, 1037.82 GENERAL DISCUSSION where n and p signify concentration and mobility respectively, gives the steady state conductivity, while the difference between these two gives the polarizing current which determines the dielectric constant.It is easy to see with regard to the charges, by considering the effect of transferring two defects along the same path, that e p + eD = ep - eL = -eN - eL = -eN + eD = e , and I estimated approximately, eD = - eL -+e, e p =- eN - t e . However, just because the dielectric constant of ice is so large, it appeared to me that it could be interpreted by predominance of either a~ or OB, so far as the order of magnitude is concerned. An approximate calculation of the consequences of this mechanism is not difficult, but for an exact calculation there are some complex correlation problems to be solved.These are intimately connected with Pauling’s problem concerning the entropy of ice. Pauling’s problem is to estimate the number of configurations of ice without restriction as to the degree of polarization -the formal problem relating to the dielectric constant is to estimate the number of configurations corresponding to various degrees of polarization. I think it is still the case that neither problem has been solved exactly. Pauling’s two deriva- tions of R In (3/2) for the residual entropy of ice are certainly not rigorous. It is possibly in these correlation corrections that one should seek the explanation of the anisotropy in the dielectric constant of ice, rather than in the unsatisfactory explanation offered by Granicher (“ it seems plausible to exclude jumps .. . at a high angle to the field direction ”). The direction of the applied field cannot discontinuously affect the jumping frequencies : the field causes only a very slight bias in the frequencies of jumps which occur spontaneously all the time. Dr. A. Steinemann (Zurich) said: Dr. Bradley’s value 3 for the activation energy of the electric conduction in pure ice of 0.5 eV is considerably smaller than our present value (0.6). This might be due to impurities and the too small temperature range of measurements. Since impurities give rise to lattice imperfections the energy term corresponding to the formation of these imperfections does not appear in the total activation energy.Hence only the highest measured activation energy is significant for pure ice. The diffusion coefficient of fluorine in our ice crystals was about 5 x 10-11 cm2 sec-1 at - 10” C in agreement with Kuhn’s results for the self-diffusion of H2018. These diffusion processes are only possible by vacant molecular sites and (or) interstitial molecules. If interstitial molecules are considered as rather free to rotate, a dielectric constant higher than 100 would be expected theoretically. The ionic conductivity of ice, however, cannot be explained unless orientational defects and ionized states are present simultaneously as underlined by Prof. Frank in his discussion remark. Our theory is able to interpret the dielectric as well as the conductive properties of pure and impure crystals.This fact together with the evidence given by Eigen in his contribution makes it appear that a con- nection between the dielectric properties and the mechanism of self-diffusion is rather improbable. Grain boundary conductivity has been observed and inter- preted by a special surface diffusion mechanism (bipedal random walk) by Murphy.4 Dr. J. L. Meijering (Philips’ Res. Lab., Eindhaven) said: I agree with Prof. Frank that Pauling’s value R In (3/2) for the theoretical residual entropy of disordered ice is not quite correct. A better approximation yields a 5 % higher value : 0.85 cal/mole deg. (In the corresponding quadratic lattice the entropy can be computed rather accurately and one finds 63 % more than R In (3/2).) As the experimental values for the residual entropy appear to be about 0.80 cal/mole deg.(0.82 for H20 and 0-77 for D2O) one can now say that there are indications of partial order of the dipoles in ice. 3 Trans. Faraday Soc., in press. 4 J. Chem. Physics, 1953, 21, 1831.GENERAL DISCUSSION 83 Dr. H. Granicher (Zurich) said: In reply to Prof. Frank and Dr. Meijering, we agree that the computation of the number of possible hydrogen configurations in an ice crystal as given originally by Pauling is crude. A full proof of the value ( 3 / 2 ) N has been given by Bjerrum (our ref. (23)). This derivation is based on the three-dimensional ice structure model and is considered to be correct for an infinitely large crystal for which contributions of the surface can be neglected. Deviations from the value (3/2)N were not known to us before, but apparently do not exceed the inaccuracy of the experimental values.The experimental accuracy (& 0.15 cal/mole deg.) is not such as to justify Dr. Meijering's con- clusion that ice should be partially ordered. It is very probable that stacking faults occur in ice crystals. The sequence of the close-packed o-layers in hexagonal ice along the c-axis may be interrupted by one or more layers which have the orientation of cubic "diamond" ice. (Stacking faults of this type are j'ery common in synthetic crystals of ZnS.) In contrast to the suggestion of Prof. Ubbelohde, such stacking faults in ice do not necessitate any disturbance in the H-arrangement. Since " diamond " ice has the same number of possible H-configurations, such imperfections do not affect the interpretation of the measured zero-point entropy.Dr. J. L. Meijering (Philips Res. Lab., Eindhoven) (communicated): The cal- culations of the residual entropies of ice and of the quadratic pseudo-ice lattice will bc published in Philips' Research Reports. The reason why Pauling's result is too low may be sketched as follows. Consider a group of dipoles which has a plane of symmetry traversed by IZ bonds. In each bond the hydrogen has two possi bfc positions. If the number of configurations of each isolated half-group is P, then P2j'is that number for the whole group, where f stands for the probability that all IZ interconnecting bonds " fit ". There are 2" possible combinations of the H positions in these bonds.Let pi be the chance that a half-group of dipoles shows one of those coinbinations (Zpi = 1) ; then, because of symmetry, f = Epi2. This quadratic expression is a minimum (f = (+)") if all pi are equal, which is as- sumed in Pauling's treatment. This random distribution of the hydrogens is, however, excluded by the requirement that each oxygen must have two " near " and two " far " hydrogen neighbours. Inspection of small groups of dipoles shows indeed that the pi are unequal, which makes f > (t)". And in building an infinite lattice by repeatedly doubling in size the group of dipoles this excess of con- figurations cannot be compensated, becausef can never be smaller than (+)". Dr. H. Griinicher (Zurich) said: Prof. Munster's statement that a knowledge of the temperature at which configurational changes freeze-in is not essential for the explanation of the zero-point entropy itself, is certainly correct. But it is emphasized that the existence of configurational changes is of great importance for the interpretation of most of the physical properties of ice, in particular for the dielectric behaviour. Dr. A. B. Lidiard (Harwell) said: The use of eqn. (4) and (5) in Volger's paper implies the assumption of a Lorentz internal field. I do not wish to discuss this question in general, but I think it is worth pointing out that there are situations connected with lattice defects where the use of this field seems to be incorrect. I would refer in particular to the type of dipole studied in the alkali halides by Dryden and Meakins. Such a dipole changes its orientation by a jump of the vacancy which is associated with the impurity from one neighbouring position to another. In both positions it has the same cubic environment of field-induced dipoles (except for a small change caused by the difference in polarizability between an impurity ion and a normal ion) and thus its energy of interaction with this environment is unchanged. Hence the Lorentz internal field should not be in- cluded in this case, since no work is done by it when the dipolar orientation changes; is then given simply by 4n-Np2/3kT. If the situation described by Dr. Volger in quartz should be similar then somewhat larger and more " normal " moments would be inferred.84 GENERAL DISCUSSION Dr. J. Meinnel (Rennes) said: Many results found in Prof. Freymann's labor- atory (Rennes) are very similar to those of Dr. Volger. The study of ZnO (pure or doped) have shown no less than five different mechanisms with different activa- tion energies, giving rise to dielectric losses. In the study of Sic losses were found with very low activation energies (0.01 eV for gray Sic, 0.03 eV for green Sic) ; these low temperature losses (T below 50" K) may be interpreted as due to trapped elcc- trons. Dielectric studies of selenium have given valuable information on the homogeneity and on the impurity centres in this material.

ISSN:0366-9033    

DOI:10.1039/DF9572300072

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

€3. STEADY-STATE PROCESSES NOT INVOLVING LATTICE RE-ARRANGEMENT INTRODUCTORY PAPER BY JAMES S. KOEHLER AND FREDERICK SEITZ Dept. of Physics, University of Illinois, Urbana, Illinois Received 1 1 th December, 1956 Nature provides many ways of achieving non-equilibrium states in solids. For example, one may introduce a gradient in temperature, or in isotopic or chemical composition. Rate processes involving local atomic rearrangement may then occur in such a direction as to re-establish equilibrium. Initial non-equilibrium condition can also be produced by the application of electro-magnetic fields or external stresses. Some atomic motion may occur in the equilibrium state; for example, the motion which is associated with atomic diffusion occurs normally at high temperatures. In other cases, the rearrangement is strongly influenced by the deviations from equilibrium.For instance, a strong electric field may induce local ionic motion on a much greater scale than would occur in the absence of the field. Similarly, radiation may induce lattice imperfections and may also stimulate the migration which removes them. Processes which require thermal fluctuations have been studied most widely to date and are most thoroughly understood in a quantitative manner. As a result, they will receive most of our attention. It is not necessary that the thermal fluctuations be furnished by the ambient. Under appropriate circumstances, such as when the system is being irradiated by photons, electrons, or more massive particles, significant fluctuations, commonly termed " thermal spikes ", may be induced by the radiation.In general, it appears that thermal fluctuations may act in two broad ways to produce local atomic changes. In the first, the system starts from a nearly perfect state and, in a brief time, of the order of the atomic oscillation time,undergoes a transition through an activated state to a new, nearly perfect state. For example, a normal and a radioactive atom of the same species which lie at neighbouring normal lattice sites in the lattice may interchange places as the result of motion made possible by a thermal fluctuation. In such a case the imperfection in the lattice represented by the activated, or intermediate, complex is highly transient, enduring for a time of the order of 10-13 sec.In the second case, the thermal fluctuations operate through lattice imperfec- tions which may or may not be produced by thermal fluctuations. For example, an equivalent atomic rearrangement may be produced by the thermally activated motion of vacant lattice sites or interstitial atoms generated either by independent thermal fluctuations or by external agents such as bombarding particles. The second of the two types of mechanism occurs much more commonly than the first, at least in the solids which have received most attention to the present time. For this reason, an understanding of imperfections is an essential pre- requisite for the treatment of rate processes in solid systems. Conversely, the systematic study of rate processes has provided us with a more profound insight into the nature of some of the imperfections which occur in solids.In any event, any consideration of processes which induce either equilibrium or steady-state 8586 INTRODUCTION conditions in a solid system which is not at equilibrium automatically involves a discussion of crystal imperfections and their properties. TYPES OF IMPERFECTION The lattice defects which play an important role in determining the re-establish- ment of equilibrium may be divided into three categories, namely, point, line and planar imperfections. All three may be the effective agent under appropriate circumstances; however, the second two may be active principally as catalytic agents for production of the first type. POINT IMPERFECTIONS The principal point imperfections responsible for migration are vacant lattice sites and interstitial atoms.An atom neighbouring a vacant lattice site may move into the vacancy. It, therefore, has greater freedom to move than an ordinary lattice atom. The vacancy is conserved in this process so that it may be used by a large number of atoms in succession. Similarly, an atom in a normal site which is a neighbour of an interstitial atom of the same or similar species may move into another interstitial position and allow the initial interstitial atom to take its place. Thus an interstitial atom, like a vacancy, may impart relative ease of motion to the atoms in its neighbourhood, the interstitial pattern or interstitialcy, like the vacancy, being preserved in migration. Evidently vacancy or interstitialcy motion is favoured over other mechanisms of motion only if the activation energy is sufficiently low.It should be noted that an interstitial atom may migrate through interstitial sites without changing places with a normal atom. This mode of motion is probably preferred whenever the interstitial atom is much smaller than the atoms of the host lattice (e.g. H in Pd, or C in Fe) or is of a radically different species. In such cases only the interstitial atom is transported, and the typical atom of the lattice does not gain mobility from the presence of the interstitial. In contrast, a vacancy can move only by imparting motion to at least one atom in a normal position. For energetic reasons, a vacancy or interstitial may become bound to a par- ticular distorted site or foreign atom in the lattice to produce a complex which possesses properties which are distinguished from those of either imperfection when present at a typical position in an otherwise perfect lattice.LINE IMPE~CTIONS The most important line imperfections of interest for understanding processes which effect equilibrium or steady-state conditions are dislocations.1 The typical dislocation may be envisaged, geometrically, by imagining the specimen which contains it to be cut over a surface whose boundary is a well-defined closed curve which lies partly or entirely inside the crystal. The parts of the specimen on either side of the cut are displaced by a vector distance b relative to one another (the Burgers vector) and rejoined in the new position. Wherever b has a component normal to the surface so that the material would overlap or a gap would develop as a result of displacement, material is cut away or added so that the two sides are contiguous before making the join.If the vector b is an allowed translation of the lattice, the lattice will be coherent over the surface; however, there will be an accumulated strain which grows in magnitude as one approaches the bounding line of the surface. Thus the resulting stress and strain pattern may be regarded as associated with the bounding curve which is termed the dislocation line. It can be shown that the stress-strain pattern depends only upon this line and the vector b, if the latter is an allowed translation. On the other hand, the pattern evidently depends upon the surface on which the cut is made, as wellJ .S . KOEHLER AND F. SEITZ 87 as upon the bounding line and b, if the Burgers vector is not an allowed translation, for the surface will be one of misfit in the second case. A wide range of possible dislocations can result from the variation of the line and the vector b. Two important simple cases are those in which the line is straight and b is normal to and parallel to the line. The first case is that of a Taylor-Orowan or edge dis- location. The second is that of a Burgers or screw dislocation. The first may be regarded as the result of inserting an extra portion of a crystallographic plane, lying normal to b and of thickness I b 1, which has an edge terminating at the line. The second may be regarded as the result of converting the crystal, composed of a set of lattice planes perpendicular to b, into a helical screw having pitch b.The screw is formed of the lattice planes mentioned and winds about the line of the dislocation. The intrinsic energy of a crystal evidently is increased by the presence of dislocations. This energy may be divided into a part which is associated with the length and orientation of each of the segments of the dislocation lines and a part which results from the interaction of the segments. The energy per unit length is sufficiently large that dislocations are not generated by normal ambient thermal fluctuations in typical crystals. They are usually formed as a result of an accident of growth or by large applied stresses. Dislocations can move and be generated as the result of applied stresses sufficiently great to induce plastic flow.In fact, such flow can be described in terms of the motion of appropriate dislocations. Thus the migration and gener- ation of dislocations is one of the processes which acts to re-establish equilibrium in a specimen which has been pushed away from the equilibrium state by the application of mechanical stresses. In the phenomena of creep, thermal fluctu- ations play a role in aiding the migration and generation of dislocations when stresses are applied. An edge dislocation can act as a source or sink for vacant lattice sites or interstitial atoms. For example, the edge of the added plane can be extended by depositing there either interstitial atoms or atoms taken away from normal sites.The average energy required to generate a vacancy or interstitial atom at a long edge dislocation is the same as the average energy required to create either from the surface of the specimen. It is believed that dislocations are the sources and sinks of most of the vacancies or interstitial atoms produced by ambient thermal fluctuations in typical crystals. Many of these possibilities have been found in crystals. PLANAR IMPERFECTIONS The most common planar imperfections are the boundaries between differently oriented crystals. If the angular disorientation is small, of the order of 0.1 radian or less, the boundary can be represented by an appropriate array of dislocations. It then possesses properties which can be expressed in terms of those of dislocations.On the other hand, when the disorientation is large, of the order of a radian or so, the grain boundary cannot be described uniquely in terms of a simple array of dislocations, but has a more complex individuality. In general, it may act as a source or sink for point imperfections. The stacking fault is a planar imperfection produced commonly as a result of plastic flow and other processes. The simplest stacking fault can be regarded as the result of passing a dislocation having a Burgers vector that is not an allowed translation across a surface, so that the crystal is not in registry over the area of passage, although it is in registry elsewhere. A twinned region may be produced, under proper geometrical circumstances, by passing identical dislocations which would produce stacking faults across a sequence of neighbouring lattice planes.The boundaries between the twinned and normal area evidently are planar im- perfections, closely related to stacking faults.88 INTRODUCTION THE OCCURRENCE AND INFLUENCE OF POINT IMPERFECTIONS SALTS The clearest information concerning the nature of the point imperfections which influence mass transfer is available in the salts, particularly the alkali halides.;! It is known that vacancies in the positive and negative ion lattices are generated thermally in equal numbers at elevated temperatures in the alkali halides and promote diffusion and electrolysis. Each type of vacancy is confined to migrate through its own sublattice. Ions possessing valence higher than unity which are present in substitutional solution are accompanied by vacant lattice sites.The multivalent impurity ion and its vacancy may be dissociated at elevated temperatures and contribute to the transport within the crystals. The vacancies associated with the foreign ions may impart a very high mobility to these ions, as well as to the ions of the crystal. Since the concentration product of the thermally produced defects should be constant at any given temperature, the addition of an agent which enhances one type (e.g. positive ion vacancies) will suppress the other (e.g. negative ion vacancies). It is interesting to note that the halogen ion vacancies in the alkali halides possess a characteristic absorption band in the ultra-violet portion of the spectrum, termed the alpha band. The existing evidence suggests that the vacancy can be made to migrate as a result of successive absorption and emission acts, as if the highly transient temperature fluctuations which accompany absorption and emission induce the vacancy to jump.It has been amply demonstrated that the point defects which are generated most easily by thermal fluctuations in the silver halides are interstitial silver ions and the corresponding vacant sites. The two types are present in equal numbers in the ideally pure crystal; however, each may be enhanced by the introduction of divalent negative ions or divalent positive ions, respectively. It has also been demonstrated that the point imperfections play an important role in the photo- graphic process, both in transporting the silver and halogen and in trapping electrons and holes produced by the incident light. All details of the process are not yet understood, however.There is much interest in the thermally induced point defects in many other salts, such as oxides, sulphides, and divalent halides. Extension of the techniques employed in the univalent halides should eventually provide more detailed knowledge. METALS The nature of the point imperfections generated by thermal fluctuations in metals is still somewhat obscure, although the topic is receiving much clarification at the present time. To date, most attention has been focused upon the metals having partly filled or newly filled d shells, that is, upon metals in the central parts of the long rows of the periodic system.Relatively rudimentary calculations,3 particularly in copper, have indicated that atomic migration takes place preferen- tially by the generation and migration of vacancies in cases in which ambient thermal fluctuations alone furnish the defects. The calculations suggest that the direct interchange of neighbouring atoms, without the aid of a lattice defect, and the production of interstitial atoms requires substantially more energy than the production and migration of vacancies. It is possible, however, that the situation is quite different in the alkali metals and in other metals involving newly filled rare gas shells. Experiments have established that the diffusion interface between dissimilar metals which are soluble in one another becomes displaced, usually toward the metals possessing the lowest heat of sublimation, when diffusion is permitted to occur.This Kirkendall shift demonstrates4 that the diffusion does not occur by direct interchange of atoms in the corresponding systems, but involves theJ. S . KOEHLER AND F. SEITZ 89 migration of either vacancies or interstitial atoms which move preferentially from one side of the boundary to the other. It has not yet proved possible to decide between the two modes of imperfection diffusion on the basis of experiment alone. Recent experiments 5 on the residual resistivity produced in thin wires of gold by rapid quenching from elevated temperatures seem to show that it is possible to preserve the point defects formed thermally at elevated temperatures.The atom fraction of such defects proves to be in the neighbourhood of 10-4 at 950" C . The activation energy EF for formation is found to be 1.00 5 0.05 eV (23 f 1 kcal/niole). The corresponding defect can be annealed from the specimen near room temperature and requires an activation energy EM of about 0.7 eV. The sum of the two activation energies, namely, 1.7eV, is close to the activation energy ED for self-diffusion in gold, namely, 1.71 eV. The annealing is accompanied by a contraction of the specimen. In conformity with the theoretical calculations described above, the defects are interpreted as vacancies at the present time, although this designation is not absolutely certain. Additional valuable information concerning the properties of defects in metals can be obtained by studying 6 specimens bombarded by massive energetic particles such as neutrons, protons, deuterons, and electrons. It is possible, in this way, to produce interstitial atoms and vacancies as well as more complex imperfections.The simplest disorder is produced by electrons which have only slightly more than enough momentum to dislodge atoms permanently from their normal positions. The study of the annealing of such irradiated materials shows the presence of two prominent processes. One, which has an activation energy near 0.7 eV in copper, silver and gold, appears to be almost the same as that found in the specimens of gold quenched from elevated temperatures and is probably to be associated with the migration of vacancies. The other process is found to occur at remarkably low temperatures, near 30" K, and, in accordance with Huntington's theoretical analysis, is interpreted in terms of interstitialcy migration.The activation energy is in the vicinity of 0.1 eV. It is interesting to note that many rate processes in solids, such as the ordering of disordered alloys, can be accelerated by irradiation. Presumably the lattice defects produced by irradiation serve as catalysts for the transport of atoms required for the transformation. Similarly, it appears that the high local tem- peratures achieved during bombardment, particularly with heavy ions, can produce extensive disordering in localized regions of many alloys. Less information is available for the alkali metals. The table summarizes the present data.If it is assumed that the defect observed near the melting point is also responsible for self-diffusion, the energy of motion EM shown in the table can be calculated. A very low value is obtained. MacDonald9 has attempted to quench sodium without success. This result is in agreement with a very low value of EM. Detailed theoretical calculations have not yet been made for the TABLE 1 .-ACTIVATION ENERGIES FOR DIFFUSION, DEFECT FORMATION AND DEFECT MIGRATION IN THE ALKALI METALS Li Na K self-diffusion (QsD) 0.57 eV 7 0.453 & 0.01 eV 8 energy of defect formation (EF) 0.40 & 0.02 eV 9 0.395 5 0-004 eV 9 0.395 i 0.004 eV9 energy of defect motion (EM) 0-17 & 0.04 eV 0.06 & 0.015 eV EM = QSD - EF90 INTRODUCTION alkali metals. The low values of EM may result from the fact that the body-centred cubic alkali metals are not close-packed.Thus defect motion may require less energy than in the close-packed noble metals. SILICON AND GERMANIUM At the present time we possess only a few items of information on the nature of the point defects which promote atomic migration in valence crystals. Almost all the significant investigations have been made with germanium. It is known that the activation energies for self-diffusion 10 are much larger than in metals possessing the same melting temperatures (e.g. the activation energy for self-diffusion in Ge is about 70 kcal), presumably because the energy for defect formation is large. Mayberg 11 has found that a defect may be quenched from elevated temperatures with a formation energy of about 46 kcal/mole and has suggested that this defect is that responsible for diffusion.Frank and Turnbull 12 have analysed the experiments of Tweet and Gallagher 13 on the penetration of copper into germanium and have concluded that copper may be present both interstitially and in combination with the defect responsible for self-diffusion. The latter unit is relatively tightly associated and accounts for almost all the equilibrium solubility of copper. Since the noble metals act as electron acceptors and may trap as many as three electrons, it seems most reasonable to postulate that they are normally bound substitutionally in germanium and that the defects which promote diflusion are vacant lattice sites. Studies of the annealing of the defects produced by irradiation in germanium reveals the presence of an activation energy of about 41 kcallmole, which may be that for the migration of vacancies.MOLECULAR CRYSTALS Relatively little is known concerning the nature of the thermally induced point defects which promote migration in the rare gas solids and in organic molecular crystals. By analogy with the salts and metals one might anticipate that vacancies are principally responsible for diffusion. Kanzaki 14 has estimated that the activation energy for the formation of vacancies in solid argon is almost the same as the heat of sublimation. Comparable calculations for the formation of interstitial atoms are not available. THE INFLUENCE OF LINE IMPERFECTIONS We have already seen that dislocations play an important direct influence in the re-establishment of equilibrium in specimens subject to stress as a result of their ability to move and thereby cause plastic flow.In addition, they play an exceedingly important indirect role by acting as sources and sinks for vacancies and interstitial atoms. In so acting, the associated change in dislocation pattern, designated as climb, induces strain somewhat analogous to that produced by the action of dislocations in plastic flow. These changes are responsible for the displacement of the diffusion boundary in binary systems, that is, the Kirkendall shift. The alkali halides2 expand when irradiated with X-rays and other ionizing radiations. It seems well established that the expansion is the result of the production of point defects, the most prominent of which are lattice vacancies.These defects probably are formed at dislocations, possibly as a consequence of temperature spikes produced by conversion of electronic excitation energy into vibrational energy. Under appropriate conditions the associated climb and glide of dislocations may produce relatively dramatic changes at the surface of a specimen. The whiskers formed 15 on the surfaces of solids under circumstances in which internal dislocations are induced to climb are probably the most striking example.J . S . KOEHLER AND F. SETTZ 91 I t may also be noted here that the imperfection region at a place where a dis- location terminates at a crystal surface can act as a catalyst for a variety of rate processes, among which is that of growth 16 from solution or the vapour phase.There is evidence to show 17 that moving dislocations generate point defects, either as a result of geometrical factors or high local temperatures. Thus diffusion and electrolytic conductivity can be enhanced by plastic flow. Still further, there is evidence 18 that atoms and other point imperfections may diffuse more rapidly along dislocations than through the bulk material. Such difYusion may play an important role in materials containing appropriate arrays of dislocations, particularly at low temperatures where normal volume diffusion becomes slow. THE INFLUENCE OF PLANAR IMPERFECTIONS Planar imperfections may play a wide variety of roles in promoting the re- establishment of equilibrium. Naturally, small angle boundaries will behave like arrays of dislocations, and many of their properties can be discussed in terms of the component dislocations.On the other hand, typical large-angle boundaries, such as normal grain boundaries, possess more general characteristics. For example, they may act as sources and sinks for vacancies and dislocations, as paths for easy atomic or ionic migration, and as areas for easy plastic flow, par- ticularly at elevated temperatures. The systematic study of the properties of planar imperfections, like the corresponding study of the properties of crystal surfaces, is still in its infancy and is one of the regions now being opened for extensive exploitation. 1 General discussion of the properties of dislocations may be found in the book by Cottrell, Plastic Flow and Creep (Oxford Univ. Press, 1953), and that by Read, Dislocatioizs iii Crystals (McGraw-Hill Book Company, New York, 1953). 2 A summarizing discussion of the alkali halides may be found in the article by Seitz, Rev. Mod. Physics, 1954, 26, 7. 3 Huntington, Physic. Rev., 1953, 91, 1092. 4 see, for example, Seitz, J. Physic. SOC. Japan, 1955, 10, 679. 5 Bauerle, Klabunde and Koehler, Physic. Rev., 1956, 102, 1182. 6 see, for example, Seitz and Koehler, Solid State Physics (Academic Press, New 7 Holcomb, and Norberg, Physic. Rev., 1954, 93, 919 ; see also Slichter, Bristol Con- 8 Nachtricb, Weil, Catalan0 and Lawson, J. Chem. Physics, 1952, 20, 1189. 9 MacDonald, Bristol Conference on Defects in Solids (Physic. SOC., London, 1955). 10 Letaw, Slifkin and Portnoy, Physic. Rev., 1954, 93, 892. 11 Mayberg, Physic. Rev., 1954, 95, 38. 12 Frank and Turnbull, Physic. Rev., 1956. 13 Tweet and Gallagher, Physic. Rev., 1956. 14 Kanzaki (to be published in the Phil. Mag.). 15 Hardy, Prog. Metal PJzysics (Pergamon Press, 1956), 6, p. 45. 16 Burton, Cabrera and Frank, Faraday SOC. Discussions, 1949, 5. 17 Seitz, PJzysic. Rev., 1950, 80, 239 ; Advances in Physics, 1952, 1, 43. 18 Turnbull, Bristol Coiference on Defects iiz Solids (Physic. Soc., London, 1955), p. 203. York, 1956), vol. 2, p. 305. ference on Defects in Solids (Physic. SOC., London, 1955), p. 52. p. 383.

ISSN:0366-9033    

DOI:10.1039/DF9572300085

出版商:RSC    

年代:1957

数据来源: RSC