Lecture Notes on Equilibrium Point Defects

LECTURE NOTES ON EQUILIBRIUM POINT DEFECTS AND THERMOPHYSICAL PROPERTIES OF METALS LECTURE NOTES ON EQUILIBRIUM PO...

Author: Yaakov Kraftmakher

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LECTURE NOTES ON

EQUILIBRIUM POINT DEFECTS AND

THERMOPHYSICAL PROPERTIES OF METALS

LECTURE NOTES ON

EQUILIBRIUM POINT DEFECTS AND

THERMOPHYSICAL PROPERTIES OF METALS

Yaakov Kraftmakher Bar-/Ian Universitx Israel

orld Scientific *New Jersey*London OHongKong

Published by World Scientific Publishing Co. Pte. Ltd.

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LECTURE NOTESON EQUILIBRIUM POINT DEFECTS AND THERMOPHYSICAL PROPERTIES OF METALS Copyright 0 2000 by World Scientific Publishing Co. Re. Ltd All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface Formation of point defects in solids has been predicted by Frenkel (1926). At high temperatures, thermal motion of atoms becomes more vigorous and some atoms acquire energies sufficient to leave their lattice sites and occupy interstitial positions. In this case, a vacancy and an interstitial atom (the socalled Frenkel pair) appear simultaneously. Wagner and Schottky (1930) have shown a way to create only vacancies: atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal (voids, grain boundaries, dislocations). Such vacancies are often called Schottky defects. This mechanism dominates in solids with close-packed lattices, where formation of vacancies requires considerably smaller energies than that of interstitials. Point defects are thermodynamically stable because they enhance the entropy of a crystal. The Gibbs free energy of the crystal reaches a minimum at a certain defect concentration. From thermodynamic considerations, point defects are present in a crystal at any temperature. The equilibrium concentration of point defects rapidly increases with increasing temperature. Point-defect formation in metals is a well-documented phenomenon. As a rule, the enthalpies of vacancy formation obtained by various experimental techniques are in reasonable agreement. In many cases, however, dramatic differences have been found in equilibrium vacancy concentrations. These are also governed by the formation entropies. This contradiction is especially strong for refractory metals.

V

vi

Preface

Despite the significant progress in the study of point defects in metals, some important problems still do not have unambiguous solutions. One of the most practically important questions relates to equilibrium defect concentrations. It is indeed surprising that this fundamental problem is still under debate. Nowadays, two opposite viewpoints exist on equilibrium point defects in metals. (1) Defect contributions to physical properties of metals at high temperatures are small and cannot be separated from the effects of anharmonicity. The only methods appropriate for studying point defects are positron-annihilation spectroscopy, which provides the enthalpies of vacancy formation, and differential dilatometry, which probes the equilibrium vacancy concentrations. Equilibrium defect concentrations at the melting points range from lo4 to lop3. That the formation enthalpies deduced from the nonlinear increase in high-temperature specific heat of metals are reasonable is just accidental, while the derived defect concentrations are improbably large. Therefore, this approach is generally erroneous. (2) In many cases, defect contributions to the specific heat of metals are much larger than the nonlinear effects of anharmonicity. Thus, their separation does not introduce crucial errors. This approach is quite appropriate for determining pointdefect parameters, especially, equilibrium defect concentrations. Equilibrium defect concentrations at melting points are of the order of in low-melting-point metals and of in highmelting-point metals. Strong nonlinear effects in the hightemperature specific heat and thermal expansivity of metals are caused by the formation of equilibrium point defects. Examination of these effects rules out anharmonicity as the possible origin of this phenomenon. It may turn out that calorimetric determinations provide the most reliable values of equilibrium vacancy concentrations in metals.

Preface

vii

This book discusses experimental results and theoretical considerations favoring each claim. At present, the majority of the scientific community holds the first viewpoint. Regrettably, the data supporting the second viewpoint were never displayed and discussed together, and the criticism of this viewpoint never included a detailed analysis. Important new arguments have appeared in the last decades. First, the relaxation phenomenon in specific heat, caused by vacancy equilibration, has been observed. Such measurements were proposed long ago and considered to be crucial for the determination of equilibrium vacancy concentrations. Second, new differential-dilatometry measurements on silver and copper revealed vacancy concentrations several times larger than values commonly accepted for three decades. High concentrations of thermally generated vacancies were observed in some alloys and intermetallics. Finally, thermodynamic relations favoring high entropies of vacancy formation in metals have been found. All of these results support the second viewpoint. At the same time, the weakness of the first viewpoint is now clearly seen. In essence, only two results support this opinion, namely: (i) differential-dilatometry data on low-meltingpoint metals, and (ii) low extra resistivities of quenched samples and small concentrations of quenched-in vacancies observed in high-melting-point metals by electron and field ion microscopy. In this book, the focus is on equilibrium point defects in metals and their relation with the thermophysical properties of metals at high temperatures. An attempt will be made to answer two important questions: (i) what are the equilibrium vacancy concentrations in metals, and (ii) what is the nature of the strong nonlinear increase in the specific heat of metals at high temperatures. The majority of the scientific community considers these two questions to be unrelated. As a rule, physicists studying point-defect formation in metals ignore calorimetric and other thermophysical data from high-temperature measurements.

viii

Preface

On the other hand, physicists studying thermophysical properties of metals do not take into account the expected point-defect contributions. This situation is caused by the opinion that equilibrium concentrations of point defects are too small to markedly affect the thermophysical properties. The author's intention is to show that this well-established opinion needs reconsideration. Though the author always believed that the questions (i) and (ii) above are closely related, the opposite viewpoint is also presented in this book. Along with a discussion of the experimental data and the theoretical estimates now available, some approaches are proposed that seem to be most suitable for settling the questions discussed above. I gratefully remember my teacher, the late Professor P.G. Strelkov (1899-1968), and my students and collaborators, A.I. Akimov, I.M. Cheremisina, S.Y. Glazkov, O.M. Kanel', T.Y. Pinegina, S.D. Krylov, E.B. Lanina, V.P. Nezhentsev, G.G. Sushakova, V.L. Tonaevskii, and the late A.A. Varchenko. I would like to thank in particular my following colleagues for useful discussions: the late Dr. A. Cezairliyan, Professor Th. Hehenkamp, Professor V.M. Koshkin, the late Professor I.M. Lifshits, Dr. K.D. Maglic, Professor A.A. Maradudin, Professor E.V. Matizen, Professor 1 . 1 . Novikov, the late Professor A.N. Orlov, Dr. F. Righini, the late Dr. G. Ruffino, Professor H.-E. Schaefer, Professor A.V. Voronel. I am greatly indebted to Professor A. Seeger for his constructive criticism.

Y. K. Ramat-Gan, February 2000.

Contents 1.Introduction 1.1. Point defects in solids. Formation parameters 1.2. Influence of point defects on physical properties 1.3. Strong nonlinear increase in specific heat and thermal expansivity of metals 1.4. Two viewpoints on equilibrium point defects in metals

2. Basic theory of point-defect formation 2.1. Thermodynamics of point-defect formation 2.2. Origin of the formation entropy 2.3. Temperature dependence of formation parameters 2.4. Results of theoretical calculations 2.5. Summary

3. Methods for studying point defects 3.1. Measurements in equilibrium

1 2 4 6

9 17 18 19 20

23 27 29 30

Advantages of equilibrium measurements according to Seeger. Criteria for choice of a suitable physical property. Equations to fit experimental data. Determination of formation enthalpies.

3.2. Quenching experiments

35

Extra electrical resistivity of quenched samples. Stored enthalpy.

3.3. Observation of vacancy equilibration

38

How to observe vacancy equilibration. Modulation calorimetry as a tool to study vacancy equilibration. Formulas for relaxation in specific heat. Prediction of the relaxation phenomenon in tungsten.

3.4. Summary

4. Modulation calorimetry and related techniques 4.1. Introduction 4.2. Basic theory of modulation calorimetry 4.3. Modulation of heating power Direct electric heating. Induction heating.

ix

44 45

46 51 59

X

Contents

Modulated-light heating. Electron bombardment. Separate heaters. Peltier heating.

4.4. Measurement of temperature oscillations Use of oscillations in the sample's resistance. Photoelectric detectors. Pyroelectric sensors. Thermocouples and resistance thermometers. Lock-in detection of periodic signals. 4.5. Modulation dilatometry Principle of modulation dilatometry. Differential method. Bulk samples. lnterferometric modulation dilatometer. Nonconducting materials. Measurement of extremely small periodic displacements. 4.6. Modulation measurements of electrical resistivity and thermopower Temperature derivative of resistance. Direct measurement of thermopower. 4.7. Summary

5. Enthalpy and specific heat of metals 5.1. Point defects and specific heat Why point defects affect high-temperature specific heat. What was said about calorimetric data, and the opposite viewpoint. 5.2. Methods o f calorimetry Adiabatic calorimetry. Drop method. Pulse and dynamic techniques. Relaxation method. Rapid-heating experiments. 5.3. Formation parameters from calorimetric data 5.4. Extra enthalpy of quenched samples 5.5. Question to be answered by rapid-heating experiments How to derive vacancy-related enthalpy and resistivity. Rapid-heating data for tungsten and molybdenum. 5.6. Specific heat of tungsten - a student experiment 5.7. Summary

6. Thermal expansion of metals 6.1. Point defects and thermal expansion 6.2. Methods o f dilatometry Optical methods. Capacitance dilatometers. Dynamic techniques. 6.3. Differential dilatometry Revision of Simmons-Balluffi data. Nowick-Feder example.

65

83

96

100 101 102

105

117 127 129

132 136 137 138 140

147

Contents 6.4. Equilibrium vacancy concentrations 6.5. High vacancy concentrations in some alloys and intermetallics 6.6. Lattice parameter and volume of quenched samples 6.7. Summary 7. Electrical resistivity of metals 7.1. Influence of point defects on electrical resistivity

xi 152 156 157 160 161 162

Deviations from Matthiessen's rule. Extra resistivity of vacancies and of vacancy clusters.

7.2. Resistivity of metals at high temperatures

164

How to derive formation parameters. Why measurements of temperature derivative of resistivity are preferable.

7.3. Quenched-in resistivity

169

Quenching in superfluid helium. Quenching with reduced cooling rate. Annealing experiments.

7.4. Comparison of data from two methods 7.5. Summary 8. Positron annihilation 8.1. Positron-annihilation techniques

173 178 179 180

Why vacancies affect positron annihilation. Lifetime spectroscopy. Mean positron lifetime. Doppler broadening. S-, W-, and D-parameters. Angular correlation of y-quanta.

8.2. Experimental data 8.3. Drawbacks of positron-annihilation techniques 8.4. High vacancy concentrations in some intermetallics 8.5. Summary 9. Other methods 9.1. Hyperfine interactions

192 194 195 196 197 198

Perturbed angular correlation of y-quanta. M6ssbauer spectroscopy. Nuclear magnetic resonance.

9.2. Other physical properties

207

Thermoelectric power. Thermal conductivity and thermal diffusivity. Mechanical properties. Spontaneous magnetization Current noise. Properties of superconductors.

9.3. Microscopic observation of quenched-in defects

212

Electron microscopy. Field ion microscopy.

9.4. Summary 10. Equilibration of point defects 10.1. Role of internal sources (sinks) for point defects

216 21 7 218

Contents

xii 10.2. Electrical resistivity 10.3. Specific heat

219 220

Enhancement of modulation frequencies. Relaxation phenomenon in tungsten and platinum.

10.4. Positron annihilation

229

Relaxation phenomenon in gold. Slow equilibration in some intermetallics.

10.5. Equilibration times from relaxation data

233

Comparison of relaxation times from various techniques. Are the relaxation times consistent with the vacancy origin of relaxation?

10.6. Summary I I . Parameters of vacancy formation 11.l. Equilibrium concentrations of point defects 11.2. Point defects in high-melting-point metals 11.3. Temperature dependence of formation parameters 11.4. Summary

12. Discussion 12.1. Comparison of experimental techniques 12.2. Critical vacancy concentrations 12.3. Thermodynamic bounds for formation entropies 12.4. Effects of anharmonicity 12.5. Constant-volume specific heat of tungsten 12.6. Thermal defects in alloys and intermetallics 12.7. Self-diffusion in metals 12.8. Point defects and melting 12.9. How to determine vacancy contributions to enthalpy - a proposal 12.10. Summary

13. Conclusions 13.1. Current knowledge of equilibrium point defects in metals 13.2. Actuality of Seeger's formulation 13.3. What could be done to reliably determine equilibrium defect concentrations Acknowledgments

References Index

236 237 238 245 248 253 255 256 257 259 261 265 272 275 278 28 1 286 287 288 298 300 302 303 323

Chapter 1

Introduction 1.I. Point defects in solids. Formation parameters 1.2. Influence of point defects on physical properties 1.3. Strong nonlinear increase in specific heat and thermal expansivity of metals 1.4. Two viewpoints on equilibrium point defects in metals

1

2 4

6 9

2

I . Introduction

1.I. Point defects in solids. Formation parameters Formation of point defects in solids has been predicted by Frenkel (1926). At high temperatures, thermal motion of atoms becomes more vigorous and some atoms acquire energies sufficient to leave their lattice sites and occupy interstitial positions. In this case, a vacancy and an interstitial atom (the socalled Frenkel pair) appear simultaneously. Later, Wagner and Schottky (1930) have shown a way to create only vacancies: atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal (voids, grain boundaries, dislocations). Such vacancies are often called Schottky defects (Fig. 1.1). This mechanism dominates in solids with close-packed lattices, where formation of vacancies requires considerably smaller energies than that of interstitials. Point defects are thermodynamically stable because they enhance the entropy of a crystal. The Gibbs energy of the crystal thus reaches a minimum at a certain defect concentration. From thermodynamic considerations, point defects are present in a crystal at any temperature. Their equilibrium concentration rapidly increases with temperature. The equilibrium vacancy concentration c, is C, =

exp(-GF/kBT)= exp(S',/k,) exp(-H,lk,T) = = A exp(-H,/k,T),

(1.1)

where G, denotes the Gibbs free energy of vacancy formation, HF is the formation enthalpy, S , is the formation entropy (not including the configurational entropy), k , is Boltzmann's constant, and T is the absolute temperature.

1. Introduction

...a. ...a. ....a

3

...a

Fig. 1.I. Point defects in crystal lattice: V - vacancy, I - interstitial atom, F P - Frenkel pair, D - divacancy.

The enthalpy of formation is HF = EF + p V F , where EF is the formation energy and VF is the defect volume. The term pV, becomes important when the pressure reaches a few kilobars, and usually the enthalpy and the energy of defect formation are not significantly different. The formation entropy, S,, results from vacancy-induced changes in lattice vibration frequencies. After creation of a vacancy, the lattice becomes softer, so that the vibration frequencies decrease. The formation entropies are therefore positive. For interstitials, the formation entropies are rather negative.

4

1. Introduction

Point-defect formation in metals is a well-documented phenomenon. As a rule, the enthalpies of vacancy formation obtained by various experimental techniques are in reasonable agreement. In many cases, however, dramatic differences were found in the equilibrium vacancy concentrations. These are also governed by the formation entropies. This contradiction is especially strong for refractory metals.

1.2. Influence of point defects on physical properties Point defects affect many physical properties of metals. Vacancies cause an increase of the volume and thermal expansivity (the coefficient of thermal expansion) of a crystal. The scattering of conduction electrons by point defects contributes to electrical resistivity. The enthalpy and specific heat of the crystal increase. Vacancies form traps for positrons, and this phenomenon is also utilized for studying vacancy formation. The point-defect mechanism dominates in diffusion phenomena. Extra concentrations of point defects in a sample arise after quenching, deformation or irradiation. At low temperatures, such non-equilibrium defects can be frozen in the lattice. Quenched samples possess an extra enthalpy, volume, and electrical resistivity; their mechanical properties, thermopower, and parameters of positron annihilation also alter. Quenched-in defects are observable by an electron or field ion microscope. Manifestations of point defects in physical properties of metals were observed in the 1930s but correct interpretation of these has appeared much later. In the early 1950s,the influence of point defects on electrical resistivity, specific heat and thermal expansion of metals has been understood. The point-defect contribution to high-temperature electrical resistivity has been

I . Introduction

5

discovered at the same time as the extra resistivity of quenched samples. The drawbacks of quenching experiments became clear in a short time. Nevertheless, many studies were performed using this technique. In the 1960s, many investigators employed differential dilatometry. It consists in simultaneously measuring the macroscopic thermal expansion and changes in the lattice parameter of the sample at high temperatures. The latter is available from X-ray or neutron data. A difference between the two quantities shows the difference between equilibrium concentrations of vacancies and interstitials. Equilibrium concentrations of interstitials are believed to be negligible, and this technique is now considered as being the most appropriate one to determine vacancy concentrations. Using this approach, equilibrium vacancy concentrations were determined in Na, Li, Bi, Cd, Pb, Zn, Mg, Al, Ag, Au, and Cu. In all the cases, the concentrations did not exceed at the melting points of the metals. In the 1970s, studies of point defects under equilibrium conditions have been recognized to be superior to any nonequilibrium experiments. This opinion has been clearly formulated by Seeger (1973a): “The principal advantage of equilibrium measurements lies in the fact that the pre-history of the samples is relatively unimportant and that a limited number of external parameters, of which by far the most important are temperature and pressure, determine the nature and the concentration of the point defects involved to an excellent approximation. This is to be contrasted with, say, quenching experiments, in which the nature and the concentration of the defects retained depends on additional parameters, such as quenching rate, dislocation density and specimen diameter, some of which are difficult to reproduce and control from experiment to experiment ... The basic theory required for the analysis of equilibrium measurements is in general more straightforward and

6

I . Introduction

much simpler than that required for handling situations far from equilibrium.” Two new experimental methods have been developed at that time, positron annihilation and perturbed angular correlation of y-quanta. Positrons can be captured by vacancies, and their lifetime therefore changes, as well as the parameters of the annihilation y-quanta. This approach seemed to be very promising. However, serious difficulties inherent to it are not overcome until today: vacancy concentrations are not available, and the technique is inapplicable to some metals. Moreover, even determinations of the formation enthalpies in some metals now seem doubtful. Perturbed angular correlation of y-quanta senses the interaction between a defect-produced electric-field gradient and the nuclear quadrupole moment of a probe atom. This technique is capable of discriminating defects of different structure and is potentially useful for determinations of equilibrium defect concentrations. However, no data on equilibrium defect concentrations in metals have been obtained by this technique.

1.3. Strong nonlinear increase in specific heat and thermal expansivity of metals The nonlinear increase in high-temperature specific heat of metals has been discovered long ago. It is especially strong in refractory metals (Fig. 1.2). However, a time elapsed before this fact has been commonly accepted. The measurements on refractory metals were performed by the pulse and modulation techniques not recognized at that time. Earlier, drop calorimetry was employed in such measurements, so that only enthalpy of the samples was measured directly. The relative contribution of point defects to the enthalpy is about one order of magnitude smaller than to the specific heat. Therefore, it is hardly to

I . Introduction

7

determine this contribution using the drop method. A common opinion has been established that high-temperature specific heat of metals depends linearly on temperature and the specific heat at the melting point does not differ strongly from that at room temperature. After the nonlinear increase became evident from direct measurements of specific heat, some authors began to consider it in approximations of high-temperature enthalpy.

.-Y-o o

Q)

n

0

1000

2000

3000

4000 K

Fig. 1.2. Nonlinear increase in specific heat of metals. Pb, A l adiabatic calorimetry (Kramer and N6lting 1972); W , Pt - modulation calorimetry (Kraftmakher and Strelkov 1962; Kraftmakher and Lanina 1965); Cr - drop method (Kirillin et al. 1967); Mo, Nb - dynamic calorimetry (Cezairliyan et al. 1970; Righini et al. 1985). The difference between low-melting-point and high-melting-point metals is clearly

seen.

1. Introduction

8

The strong nonlinear increase in thermal expansivity of metals is also evident (Fig. 1.3). However, the origin of both phenomena remains under debate. The problem consists in correctly separating defect contribution. The problem is that an unknown part of the nonlinear increase may originate from anharmonicity.

20

15

10

5

1000

1500

2000

2500

3000

3500 K

Fig. 1.3. Nonlinear increase in thermal expansivity of high-meltingpoint metals. Pt - modulation method (Kraftmakher 1967a); Ir - traditional dilatometry (Halvorson and Wimber 1972); Ta, Nb dynamic technique (Miiller and Cezairliyan 1982; Righini et al. 1986a); W: 1 - modulation method (Kraftmakher 1972), 2 - recommended values (Swenson et al. 1985).

A reliable method to separate defect contributions is well known: one should measure the specific heat of a sample under

1. Introduction

9

such rapid temperature changes that the defect concentration could not follow them. In this case, the measured specific heat almost corresponds to a hypothetical defect-free crystal. The only difficulty to overcome is a short relaxation time, and such data have been obtained only for tungsten and platinum. In both cases, the difference between the specific heats measured under slow and rapid temperature oscillations is in agreement with the nonlinear increase in the specific heat. This means that this increase is caused by point-defect formation. A simple empirical rule has been established: the enthalpies of defect formation in metals are nearly proportional to melting temperatures, as well as the enthalpies of self-diffusion and of vaporization.

1.4. Two viewpoints on equilibrium point defects in metals Despite significant progress in studies of point defects, some important questions have no unambiguous answers. One of the most practically important questions relates to equilibrium concentrations of point defects. It is indeed surprising that this fundamental problem is still under debate. For instance, vacancy concentrations in refractory metals based on the nonlinear increase in specific heat are of the order of On the other hand, low quenched-in electrical resistivity of these metals corresponds to defect concentrations two orders of magnitude smaller. Nowadays, two opposite viewpoints exist on equilibrium point defects in metals. (1) Defect contributions to physical properties of metals at high temperatures are small and cannot be separated from the effects of anharmonicity. The only methods appropriate for studying point defects are positron-annihilation spectroscopy, which provides the enthalpies of vacancy formation, and

10

1. Introduction

differential dilatometry, which probes the equilibrium vacancy concentrations. Equilibrium vacancy concentrations at melting points range from l o 4 to lop3. Thus the formation enthalpies deduced from the nonlinear increase in high-temperature specific heat of metals are reasonable is just accidental, while the derived defect concentrations are improbably large. Therefore, this approach is generally erroneous. (2) In many cases, defect contributions to the specific heat of metals are much larger than the nonlinear effects of anharmonicity. Thus, their separation does not introduce crucial errors. This approach is quite appropriate for determining pointdefect parameters, especially, equilibrium defect concentrations. Equilibrium defect concentrations at melting points are of the order of lop3 in low-melting-point metals and of lo-* in highmelting-point metals (Fig. 1.4). Strong nonlinear effects in the specific heat and thermal expansivity of metals at high temperatures are caused by the formation of equilibrium point defects. Examination of these effects rules out anharmonicity as the possible origin of this phenomenon. Important arguments supporting this viewpoint have appeared in the last decades. It may turn out that calorimetric determinations provide the most reliable values of equilibrium vacancy concentrations in metals. Our aim is to discuss experimental results and theoretical considerations favoring each claim. The majority of the scientific community holds the first viewpoint. Regrettably, the data supporting the second viewpoint were never displayed and discussed together, and the criticism of this viewpoint never included a detailed analysis. Important new arguments have appeared in the last decades. First, the relaxation phenomenon in specific heat caused by vacancy equilibration has been observed. Such measurements proposed long ago were considered to be crucial for the determination of equilibrium vacancy concentrations. Second,

I . Introduction

11

vacancy concentrations ( 1 o - ~ ) 1000

I

I

I

4

6

8

10

20

30

40

50

100

10

‘2 100

10

1

‘10

1 0 4 (K-’) ~ Fig. 1.4. Equilibrium vacancy concentrations in metals derived from nonlinear increase in specific heat. The concentrations at melting points are of the order of in low-melting-point metals and of lo-* in high-melting-point metals.

12

1. Introduction

new differential-dilatometry measurements on silver and copper have shown vacancy concentrations in these metals much larger than values commonly accepted for three decades. High concentrations of thermally generated vacancies were observed in many alloys and intermetallics. Finally, thermodynamic relations favoring high entropies of the vacancy formation in metals have been found. All of these results support the second viewpoint. At the same time, the weakness of the first viewpoint is now clearly seen. In essence, only two results support this opinion, namely: (i) differential-dilatometry data on low-melting-point metals, and (ii) low extra resistivities of quenched samples and small concentrations of quenched-in vacancies observed in highmelting-point metals by electron and field ion microscopy. The main conclusions that will be made are as follows. (1) The well-established opinion that equilibrium point-defect concentrations in metals are small needs revision. (2) It may turn out that just calorimetric determinations provide the most reliable data on equilibrium defect concentrations in metals. During the last five decades, many theoretical and experimental studies of point defects in metals have been carried out. A brief presentation of the long history of studying point defects in metals is presented here (Table 1.1). Many monographs, reviews, and conference proceedings have been published. A part of these is listed below.

13

1. Introduction

Table 1.1 Brief presentation of the long history of studying point defects in metals.

Item

Reference

Frenkel 1926 Prediction of vacancy-interstitial pair formation Mechanism of vacancy formation Wagner and Schottky 1930 Calculations of point-defect parameters Huntington 1942 Extra resistivity of quenched samples Kauffman and Koehler 1952,1955 Defect contribution to resistivity at high temperatures MacDonald 1953 Defect contribution to specific heat Carpenter 1953, Pochapskyl953 Vacancy parameters from thermal expansion Gertsriken 1954 Theory of defect concentrations Vineyard and Dienes 1954 Differential dilatometry van Duijn and van Galen 1957 Feder and Nowick 1958 Nenno and Kauffman 1959 Stored energy in quenched Au DeSorbo 1958 Hirsch et al. 1958 Electron microscopy of quenched samples Muller 1959 Observation of point defects by a field ion microscope Proposal to observe point-defect equilibration Jackson and Koehler 1960 Specific heat of Mo and Ta Rasor and McClelland1960 Differential dilatometry on Al, Ag, Au, Cu Simmons and Balluffi 1960-1963 Specific heat and vacancies in W Kraftmakher and Strelkov 1962 Quenching in superfluid helium Rinderer and Schultz 1964 Equilibration of vacancies in Au Seidman and Balluffi1965 Influence of vacancies on positron annihilation Berko and Erskine 1967 MacKenzie et al. 1967 Cezairliyan et al. 1970-1971 Specific heat of refractory metals Kramer and Ndlting 1972 Specific heat of low-melting-point metals Seeger 1973 Evidence of the priority of studies under equilibrium Relaxation in specific heat of Au Skelskey and Van den Sypel974 Maier et al. 1979 Positron-annihilationdata on refractory metals Miiller and Cezairliyan 1982-1991 Thermal expansion of refractory metals Relaxation in specific heat of W and Pt Kraftmakher 1985, 1990 Schaefer 1987 Vacancy equilibration in Au, positron annihilation Schaefer and Schmid 1989 Varotsos 1988 Theoretical bounds for formation entropies Kluin and Hehenkamp 1991 New differential-dilatometry data on Cu and Ag Mosig et al. 1992

14

I . Introduction

Monographs, reviews, and conference proceedings Mott, N.F., Guerney, R.W. Electronic Processes in lonic Crystals (University Press, Oxford, 1948). Damask, A.C., Dienes, G.J. Point Defects in Metals (Gordon and Breach, New York, 1963). Lattice Defects in Quenched Metals, eds. R.M.J.Cotteril1, M.Doyama, J.J.Jackson, M.Meshii (Academic Press, New York, 1965). Lattice Defects and Their Interactions, ed. R.Hasiguti (Gordon and Breach, New York, 1967). Vacancies and lnterstitials in Metals, eds. A.Seeger, D.Schumacher, W.Schilling, J.Diehl (North-Holland, Amsterdam, 1970). Point Defects in Solids, vols. 1-3, eds. J.H.Crawford, L.M.Slifkin (Plenum Press, New York, 1972). lnteratomic Potentials and Simulation of Lattice Defects, eds. P.C.Gehlen, J.R.Beeler, R.I.Jaffee (Plenum Press, New York, 1972). Flynn, C.P. Point Defects and Diffusion (Clarendon Press, Oxford, 1972). Seeger, A. Crystal Lattice Defects 4, 221-253 (1973). Seeger, A. J. Phys. F: Metal Phys. 3, 248-294 (1973). Review papers from International Conference on Point Defects and Their Aggregates in Metals. J. Phys. F: Metal Phys. 3, N 2 (1973). Doyama, M., Hasiguti, R.R. Crystal Lattice Defects 4, 139-163 (1973). Defect lnteractions in Solids, eds. K.I.Vasu, K.S.Raman, D.H.Sastry, Y.V.R.K.Prasad (Indian Institute of Science, Bangalore, 1974). Stark, J.P. Solid State Diffusion (Wiley, New York, 1976). KovBcs, I., El Sayed, H. J. Mater. Sci. 11, 529-559 (1976).

I . Introduction

15

Progress in the Study of Point Defects, eds. M.Doyama, S.Yoshida (University of Tokyo Press, Tokyo, 1977). Leibfried, G., Breuer, N. Point Defects in Metals 1. introduction to the Theory (Springer, Berlin, 1978). Properties of Atomic Defects in Metals. J. Nuclear Materials 69/70 (1978), eds. N.L.Peterson, R.W.Siegel. Dederichs, P.H., Zeller, R., Schroeder, K. Point Defects in Metals I/ (Springer, Berlin, 1980). Point Defects and Defect lnteractions in Metals, eds. J.Takamura, M.Doyama, M.Kiritani (North-Holland, Amsterdam, 1982). Maier, K., in Positron Solid-state Physics, eds. W.Brandt, A.Dipasquier (North-Holland, Amsterdam, 1983). Orlov, A.N., Trushin, Yu.V. Energy of Point Defects in Metals, in Russian (Energoatomizdat, Moscow, 1983). Kirsanov, V.V., Orlov, A.N. Sov. Phys. Uspekhi 142,219-264 (1984). Defects in Solids. Modern Techniques, eds. A.V.Chadwick, M.Terenzi (Plenum Press, New York, 1986). Atomic Transport and Defects in Metals by Neutron Scattering, eds. C.Janot, W.Petry, D.Richter, T.Springer (Springer, Berlin, 1986). Varotsos, P.A., Alexopoulos, K.D. Thermodynamicsof Point Defects and Their Relation with Bulk Properties (North-Holland,Amsterdam, 1986). Vacancies and lnterstitials in Metals and Alloys, eds. C.Abromeit and H.Wollenberger (Trans Tech Publications, 1987). Papers presented at the European Meeting on Positron Studies of Defects, Phys. Status Solidi A 102, 31-179, 481-588 (1987). Characterization of Defects in Materials, eds. R.W.Siegel, J.R.Weertman, R.Sinclair (Materials Research Society, Pittsburgh, 1987). Agullo-Lopez, F., Catlow, C.R.A., Townsend, P.D. Point Defects in Materials (Academic Press, New York, 1988). Electronic Structure and Lattice Defects in Alloys, eds. R.W.Siege1 and F.E.Fujita (Trans Tech Publications, 1989).

16

I. Introduction

Proceedings of the International Conference on Diffusion in Metals and Alloys. Defect Diffusion Forum 66/69 (1990), eds. F.J.Kedves,DLBeke. Diffusion in Solid Metals and Alloys, ed. H.Mehrer. Landolt-Bornstein, vol. 26, Part X (Springer, Berlin, 1990). Atomic Migration and Defects in Materials, eds. D.Gupta, H.Jain, R.W.Siegel (Trans Tech Publications, 1991). International Conference on Diffusion in Materials. Defect Diffusion Forum 1431147 (1997). Doyarna, M. Mater. Chem. Physics 50, 106-1 15 (1997). Defects and Diffusion in Metals, ed. D.J.Fisher (Trans Tech Publications, 1999).

Chapter 2

Basic theory of point-defect formation 2.1. Thermodynamics of point-defect formation 2.2. Origin of the formation entropy 2.3. Temperature dependence of formation parameters 2.4.Results of theoretical calculations 2.5. Summary

17

18 19 20 23 27

2. Basic theory

18

2.1. Thermodynamics of point-defect formation Point defects are imperfections of crystal lattice having dimensions of the order of the atomic size. Their main parameters are the enthalpy and entropy of formation that govern the temperature dependence of the equilibrium defect concentrations. The formation enthalpies for vacancies are smaller than for interstitials, so that vacancies are dominated point defects in equilibrium. Along with monovacancies, vacancy clusters may exist in the crystal lattice under equilibrium and after quenching. The formation of a vacancy can be considered as a removal of one interior atom from the crystal and replacement of the atom on the crystal surface. The Gibbs free energy of vacancy formation equals to the corresponding change in the Gibbs free energy for the whole crystal. Let us remove n atoms from the crystal containing N atom sites and place them on the surface. Each of the formed n vacancies is associated with an enthalpy of formation, HF, and a vibrational entropy, S,, resulting from disturbance of the neighbors of the vacancy. In addition, a configurational entropy, S,, appears that equals

S,

=

k,ln[(N

f

n)!lN!n!].

The Stirling’s approximation gives

S , = k,Nln[(N

+ n ) / N ] + k,nln[(N

n)/n]G z k,nln(iV/n).

f

(2.2)

The change in the Gibbs free energy of the crystal due to the vacancy formation is

2. Basic theory AG

=

nH,

-

TS,

-

nTS,.

19

(2.3)

The vacancy formation lowers the Gibbs free energy of the crystal until an equilibrium vacancy concentration is reached. This equilibrium concentration, c , = n / N , fulfills the requirement d(AG)/dn = 0. From this relation, one obtains

H, + k,T lnc,

-

TSF

=

0,

(2.4)

The quantity G, = HF - TS, denotes the Gibbs free energy of vacancy formation that governs the equilibrium vacancy concentration. Vacancies are stable at any temperature above the absolute zero, and their equilibrium concentration rapidly increases with temperature.

2.2. Origin of the formation entropy The formation entropy reflects changes in the vibration frequencies of the atoms surrounding the vacancy. These frequencies become lower than those before the vacancy was formed. Thus, the formation entropy is positive. This entropy equals

AS,

=

3nek,T l n ( o ' / o ) ,

(2.6)

where o and o' are the unperturbed and perturbed vibration frequencies, and E is a quantity proportional to the volume perturbed by the vacancy. Since the surface of the crystal and internal defects act as sources and sinks for vacancies, equilibrium concentrations of vacancies and of interstitials are independent. In metals, the

2. Basic theory

20

enthalpies of interstitial formation are markedly larger than of vacancy formation. In addition, the entropy of interstitial formation is rather negative. Vacancies are therefore dominated equilibrium point defects in metals. At premelting temperatures, divacancies may measurably contribute with an equilibrium concentration given by (2.7)

Here AS,,, H,, and z denote the association entropy and the binding enthalpy of a divacancy, and the coordination number of a lattice site, respectively.

2.3. Temperature dependence of formation parameters At a fixed temperature, the enthalpies and entropies of vacancy formation can be considered independent quantities. However, their temperature derivatives at constant pressure are interrelated through the thermodynamic relation

(dH/dT), = T(dS/dT),.

(2.8)

The interatomic distances increase with increasing temperature, while the rigidity of the lattice decreases. The relaxation of the atoms near a vacancy also increases leading to an increase in the formation entropy and enthalpy. Earlier, Mott and Guerney (1948) considered the formation enthalpy to linearly decrease with increasing temperature:

H,

=

H,, - a T .

(2.9)

2. Basic theory

21

This dependence gives a contribution to the pre-exponential factor in the expression for the equilibrium defect concentration, which equals exp(alkB). However, the corresponding contribution to the entropy should be negative according to the relation (2.8). Later, Vineyard and Dienes (1954) have shown that the entropy of vacancy formation depends only on the lattice vibration frequencies before and after the vacancy is created. No further contribution to the entropy arises even when the formation enthalpy remains temperature dependent. Maradudin (1966) developed the theory of the lattice vibrations in a disordered crystal lattice. In order to calculate the formation enthalpy, it is necessary to consider the interaction between the ions and a redistribution of the conduction electrons. One has to solve three main problems (Ho 1971): (i) to choose a proper potential for the description of the metallic bond; (ii) to find the atomic configuration after the lattice relaxation; (iii) to take into account the change of the lattice energy caused by the change in the volume. The potential weakly affects the atomic configuration but strongly influences the formation enthalpy. Theoretical considerations of the point-defect parameters include determinations of formation enthalpies and entropies, lattice relaxation, enthalpies of migration and energies of the binding of defects. As a rule, calculated formation enthalpies are in reasonable agreement with each other and with experiment. At the same time, strong contradiction is seen in formation entropies. Generally, evaluations of formation entropies are less accurate than of formation enthalpies. Foiles (1994) performed such calculations for copper in a wide temperature range using Monte Carlo simulations and some approximate techniques. The author has pointed out that the harmonic methods underestimate the temperature dependence of the Gibbs free energy of vacancy formation (Fig. 2.1).

2. Basic theory

22

Using molecular-dynamics simulations, Smargiassi and Madden (1995ab) have found significant temperature dependence of the vacancy formation entropy in sodium. The formation entropy is lkB-2kB at low temperatures and increases to about 4kB-5kB near the melting point. The Gibbs free energy of vacancy formation decreases nonlinearly from about 0.35 eV at low temperatures to about 0.17 eV close to the melting point. The equilibrium vacancy concentration in sodium at the melting point is thus predicted to be 5 ~ 1 0 An ~ ~unexpected . conclusion has been drawn from these molecular-dynamics simulations: it turned out that the role of interstitial$ at high temperatures cannot be dismissed.

Y-

>

3

cu

0

0.5 -

C Q)

tn

D

6

1

I

I

Fig. 2.1. Temperature dependence of Gibbs free energy of vacancy formation in Cu (Foiles 1994). 1 - quasiharmonic method, 2 - local harmonic approximation, 3 - Monte Carlo simulations.

2.Basic theory

23

Najafabadi and Srolovitz (1995) have calculated the vacancy-formation energy in copper by Monte Carlo simulations. The formation energy increases nonlinearly from 1.32 eV at 250 K to 1.64 eV at 1250 K. The uncertainty in these values rapidly increases with temperature, from 0.03 eV to 0.25 eV. Other techniques employed in the calculations (quasi-harmonic and free-energy-minimization methods) have shown much weaker temperature dependence, as well as the calculations by Rickman and Srolovitz for gold (1993). De Koning and Antonelli (1997) have found that the Gibbs free energy of vacancy formation in copper decreases from 1.27 eV at zero temperature to about 0.97 eV at the melting point. The increase in the crystal volume associated with the vacancy formation, i.e., the vacancy formation volume, V,, satisfies the thermodynamic relation

The vacancy formation volumes are thus obtainable from pressure dependence of equilibrium vacancy concentrations.

2.4. Results of theoretical calculations Only a small part of theoretical calculations of the parameters of point-defect formation in metals is listed below. Vaks et al. (1989) have deduced formation and migration enthalpies for vacancies in a-Fe, Cu, Ni, Zr, Ti, and Mg. Vacancy-formation enthalpies for Sc, Ti, Co, Ni, Cu, Y, Tc, Ru, Rh, Pd, and Ag have been evaluated by Drittler et al. (1991). Dederichs et al. (1991) have calculated enthalpies of vacancy formation in Cu, Ag, Ni, and Rh. Benedek et al. (1992) have found formation energies for vacancies in Li and At. Fernandez and Monti (1993) have derived vacancy-formation parameters, HF and S,, in HCP metals Mg,

24

2. Basic theory

a-Ti, and a-Zr. Polatoglou et al. (1993)have calculated vacancyformation energies in Al, Cu, Ag, and Rh. Frank et al. (1993) have determined properties of monovacancies in lithium and Breier et al. (1994)in sodium. Frank et al. (1996) have carried out first-principles calculations of vacancy concentrations and self-diffusion parameters in lithium. The temperature dependence of the formation entropy was calculated. The obtained vacancy concentration at the melting point appeared to be considerably smaller than that determined by differential dilatometry (Feder and Charbnau 1966).Meyer and Fahnle (1997)have obtained ab initio the vacancy-formation parameters in molybdenum. The formation energy is 2.9 2 0.1 eV, and the formation volume is VF = (0.64 O.l)Q,where C2 is the atomic volume. In a molecular-dynamics study of self-diffusion in sodium, Schott et al. (1997)observed a contribution of Frenkel pairs close to the melting point. Shimomura (1997) studied point defects and their clusters in FCC metals by computer simulations. Satta et al. (1 998) considered vacancy formation in tungsten. The authors have found that electronic contribution to the formation entropy, usually not being taken into account, makes a significant contribution. For tungsten, it amounts 1.74kB. Similar results are expected for chromium and molybdenum. From molecular-dynamics simulations in copper, Nordlund and Averback (1 998) have found the melting-point concentrations of divacancies and of interstitials to be surprisingly high, of the order of Asato et a\. (1998)calculated formation parameters for Al, Cu, and Ni. Pohlong and Ram (1998) evaluated vacancyformation entropies for BCC metals a-Fe, Mo, and W. The entropies are 1.56kB,2.25kB, and 3.2kB, respectively. Recently, Korzhavyi et al. (1999)have presented results of first-principles calculations of vacancy formation in transition and noble metals. Belonoshko et al. (2000) investigated the melting

25

2. Basic theory

of copper by means of a molecular dynamic method. Schott et al. (2000ab) calculated properties of vacancies in lithium, sodium, and potassium, and the influence of divacancies in lithium. Below, some results of theoretical calculations of the vacancy formation enthalpies (Table 2.1) and of the formation volumes (Table 2.2) are presented.

Table 2.1 Theoretical values of vacancy formation enthalpies HF (ev): 1 - Kornblit (1981, 1982), 2 - Kostromin et al. (1983), 3 - Harder and Bacon (1986), 4 - Ackland et al. (1987), 5 - Krause et al. (1989), 6 - Rosato et al. (1989), 7 - Ghorai (1991), 8 - Korhonen et al. (1995).

Metal

Rb K Na Li Pb

1

2

3

5

6

7

8

0.335 0.321 0.379 0.458 0.573 0.85 1.14

A1

As

0.66 1.021 0.962 1.191 1.458

AU

cu Ni Pd Pt V Cr Rh Ir Nb Mo Ta W

4

1.20 1.57

1.83 1.91

2.5 2.2

1.31 1.60 1.50 2.03

0.78 0.6 1.2 1.46 1.0 1.28

1.195 0.962 1.209 1.279 1.176

1.24 0.82 1.33 1.77 1.65 1.45 3.06 2.86

2.49 2.7 3.2 2.2 3.3 3.8

3.43

2.48 2.54 2.87 3.62

1.78 3.67 2.73 4.57

2.92 3.13 3.49 3.27

26

2. Basic theory

Table 2.2 Theoretical values of vacancy formation volume VF, in units of atomic volume R: 1 - estimates by Seeger (1973b) based on activation volumes of self-diffusion, 2 - Jacucci and Taylor (1979), 3 - Bauer et al. (1982), 4 - Harder and Bacon (1986), 5 - Ackland et al. (1987), 6 - Rosato et al. (1989), 7 - experimental data (Emrick and McArdle 1969; Grimes 1965; Emrick 1972).

Metal

K Na Li

In sn Cd Pb Zn

Al

Jw AU cu Ni

1

0.25 0.2-0.25 0.45 0.25

2

3

4

5

6

7

0.59-0. 71a 0.65-0. 72a 0.43-0.68a

0.5 0.4 0.35-0.4 0.65 0.75 0.65 0.8

Pd Pt V

cr Ir

0.62 0.78

0.69 0.63 0.79

0.73 0.77 0.88

w

0.52

0.7

0.74 0.84 0.75 0.77

Fth

Nb Mo Ta

0.76 0.72 0.8 0.8 0.77 0.76

0.96 0.73 0.83 0.68

a Two values show the formation volumes at the absolute zero and the

melting points.

2. Basic theory

27

2.5.Summary Point defects, vacancies (missing lattice atoms) and interstitials (atoms that occupy non-lattice sites), are thermodynamically stable because they lower the Gibbs free energy of the crystal. Equilibrium concentrations of point defects rapidly increase with increasing temperature. In metals, vacancies are the predominant point defects in equilibrium. Their concentrations at high temperatures are much larger than those of interstitials. Divacancies are the only defects whose equilibrium concentrations at high temperatures may become comparable with those of monovacancies. However, interstitials and divacancies are more mobile and therefore may markedly contribute to self-diffusion at high temperatures. 0 At a fixed temperature, the enthalpies and entropies of vacancy formation can be considered independent quantities. However, their temperature derivatives at constant pressure are interrelated. This leads to increase of the formation enthalpies with increasing temperature.

The vacancy formation volumes are obtainable from pressure dependence of equilibrium vacancy concentrations. 0 As a rule, the enthalpies of vacancy formation calculated by various theoretical methods are in reasonable agreement with each other. Theoretical calculations of the vacancy-formation volumes are in agreement with experimental data.

Generally, evaluations of formation entropies are less accurate than of formation enthalpies. Various theoretical calculations of the vacancy-formation entropies lead to very different values.

28

2. Basic theory

Recently, theoretical calculations of the temperature dependence of the formation parameters, H,, S, and V,, have appeared. All these parameters increase with increasing temperature. According to some calcutations, the Gibbs free energy of vacancy formation at melting points may be 1.5-2 times smaller than at low temperatures. These results support high vacancy concentrations.

Chapter 3

Methods for studying point defects 3.1. Measurements in equilibrium

30

Advantages of equilibrium measurements according to Seeger. Criteria for choice of a suitable physical property. Equations to fit experimental data. Determination of formation enthalpies.

3.2. Quenching experiments

35

Extra electrical resistivity of quenched samples. Stored enthalpy.

3.3. Observation of vacancy equilibration

38

How to observe vacancy equilibration. Modulation calorimetry as a tool to study vacancy equilibration. Formulas for relaxation in specific heat. Prediction of the relaxation phenomenon in tungsten.

3.4. Summary

44

29

30

3. Methods for studying point defects

3. I.Measurements in equilibrium Advantages of equilibrium measurements according to Seeger It is commonly agreed that equilibrium point defects are to be studied under equilibrium conditions. In principle, any physical property influenced by the defects is usable to determine their equilibrium concentrations. Important advantages of equilibrium measurements over non-equilibrium ones have been understood many years ago. Seeger (1 973ab) formulated these advantages as follows. (1) Temperature and pressure are the only parameters that govern equilibrium concentrations of point defects. The number of different kind of the defects created in equilibrium is quite limited. In most cases, quantitative interpretation is obtainable by taking into account only one additional type of defects, usually divacancies. (2) Equilibrium measurements are less sensitive to the microstructure and pre-history of the samples and the presence of impurities than non-equilibrium measurements. (3) The theory of equilibrium measurements is very simple and straightforward. The measured quantities are directly expressed in terms of the enthalpy and entropy of defect formation. Under equilibrium, point defects can be studied through various physical properties: enthalpy and specific heat, thermal expansion, electrical resistivity, thermopower, positron annihilation, perturbed angular correlation of y-quanta. Such measurements should provide most reliable data on equilibrium defects. However, properties of a hypothetical defect-free crystal are unknown and cannot be calculated precisely. It is therefore

3. Methods for studyingpoint defects

31

impossible to unambiguously separate point-defect contributions. In addition, one has to know the relation between the defect concentration and the contribution to the given physical property. Differential dilatometry is considered the only exception because it provides the necessary background data.

Criteria for choice of a suitable physical property Criteria for the choice of a physical property suitable for studying equilibrium point defects are obvious: (i) the magnitude of the defect contribution and the reliability of separating it; (ii) the accuracy of the measurements: (iii) the knowledge of parameters entering the relations between the defect contributions and concentrations of the defects. The most appropriate property seems to be specific heat, for which all the criteria listed above are well fulfilled: (1) It is very likely that specific heat of a defect-free crystal weakly depends on temperature. (2) There exists, at least in principle, a straightforward experiment to separate the defect contribution by observations of the defect equilibration. (3) In many cases, defect contributions are much larger than errors of the measurements. (4) The extra specific heat strictly relates to the defect concentration. The increase in specific heat is caused only by defects whose concentration reversibly follows the sample’s temperature. It is due not to the presence of the defects in the crystal lattice (this influence is much weaker) but to the temperature dependence of the equilibrium defect concentration. Clearly, studies of point defects should include measurements of other properties, e.g., thermal expansion. A special approach, differential dilatometry, has been developed to directly determine equilibrium defect concentrations. It consists in simultaneously measuring the bulk thermal expansion of the

32


sample and the dilatation of its unit cell by means of X-rays. The difference between the two quantities shows the difference between the equilibrium concentrations of vacancies and of interstitials in the sample. Since the equilibrium concentrations of vacancies are much larger than those of interstitials, the method provides the equilibrium vacancy concentrations. It is often referred to as an 'absolute technique'. Unfortunately, some reasons exist to consider data from differential dilatometry to be underestimated. Still more important, no data have been obtained by this technique for high-melting-point metals. Data on bulk thermal expansion are now available for many metals including refractory ones. Earlier, such data were inapplicable because of large inherent errors. Usually, the temperature dependence of thermal expansivity was believed to be linear. Owing to improvements in the traditional dilatometry and to development of modulation and dynamic techniques, the measurements have become much more accurate. Now the nonlinear increase in thermal expansivity is evident. Another property depending on concentrations of point defects is electrical resistivity. It can be measured more accurately than specific heat or thermal expansivity. However, point-defect contributions to electrical resistivity at high temperatures are relatively small.

Equations to fit experimental data Determinations of point-defect contributions to various physical properties require measurements in wide temperature ranges. Measurements at temperatures where these contributions are still negligible are necessary to accurately approximate properties of a defect-free crystal. Otherwise, the uncertainties in the derived defect parameters may become unacceptably large. Point-defect contributions to enthalpy, volume, and electrical resistivity of


33

metals at high temperatures are relatively small. They should be added to the regular temperature dependence of these properties, usually quadratic. These properties should be fitted by the equation

X = a + b T + cT2 + dexp(-H,/k,T),

(3.1)

where the last term represents the defect contribution. The situation is much more favorable when specific heat, thermal expansivity, or the temperature derivative of resistivity are measured directly. Point defects strongly affect these properties, whereas a linear extrapolation from intermediate temperatures is sufficient to separate defect contributions. Hence, these properties obey the relation

where the defect contribution is given by the last term.

Determination of formation enthalpies To deduce the formation enthalpy from the defect contribution AY, one plots In( T2AY) versus 1/T. The plot is a straight line with a slope -H,lk, (Fig. 3.1). This procedure was applied to the nonlinear increase in the specific heat of refractory metals and yielded plausible values of the formation enthalpies. No other data on point-defect formation in these metals were available at that time, and the values obtained were compared with the melting temperatures and the enthalpies of self-diffusion. More rigorously, all the coefficients of the equation (3.2) should be evaluated by the least-squares method. Trying various formation enthalpies, one plots the standard deviation versus the assumed value of HF. The minimum in the curve indicates the most

34


probable formation enthalpy, and the width of the curve shows the uncertainty in this value. Correct formation enthalpies confirm the validity of the above approximation. When close formation enthalpies are derived from various physical properties, one can conclude that the nonlinear changes in them are of a common origin. Another technique widely used in determinations of vacancyformation enthalpies is positron annihilation. Vacancies capture positrons and influence the positron-annihilation parameters. The method is very sensitive to vacancy-type defects. It was employed for determinations of vacancy-formation enthalpies in many metals. Regretfully, some assumptions are to be made to evaluate vacancy concentrations from the experimental data.

8.0

9e

cu

7.5

v

-t

7.0

6.5

6

8

10

12

14

Fig. 3.1. Determination of formation enthalpies from nonlinear increase in specific heat. Slope of the plot ln(?AC) versus 1/T equals -H,/k,.


35

3.2. Quench ing experi ment s Many studies of point defects in metals employed measurements of quenched-in electrical resistivity. This approach was very attractive due to the apparent simplicity of the measurements and evaluations. However, the drawbacks peculiar to all quenching experiments have become evident in a short time. Balluffi et al. (1970) have exhaustively reviewed the situation. Quenching experiments include studies of properties of samples with vacancies frozen in the crystal lattice. Such properties are extra enthalpy, changes in the volume and lattice parameter, electrical resistivity, thermopower, and parameters of positron annihilation and of perturbed correlation of y-quanta. One can study quenched samples by electron or field ion microscopy. The quenched samples are compared with wellannealed samples, defect concentrations in which are negligible. The main disadvantage of equilibrium measurements is thus completely avoided. Unfortunately, concentrations of quenched-in vacancies may be much smaller than the equilibrium concentrations at high temperatures. During quenching, many vacancies have time to annihilate or form clusters. The distribution of the defects in quenched samples is thus not a representative of the equilibrium distribution. The vacancy-induced changes in the physical properties of the samples may become much smaller. The mobility of the vacancies rapidly increases with increasing temperature, and this discrepancy grows when the temperature approaches the melting point. Due to interactions between the vacancies and other imperfections in the sample, the situation becomes still more complicated.


36

quenched-in resistivity ( 1 O-' Q.cm)

m

A

o

=

&

9

8

10

1 0 " ~ (K-')

900°C

700°C 0

I

I

I

I

2

4

6

a

10

reciprocal of quenching rate ( 1 0-5 s.K-')

Fig. 3.2. Results of quenching experiments with Au (after Mori et al. 1962). (a) Extra resistivity of quenched wires at various cooling rates: 1 - lo4, 2 - 2x104, 3 - 5 x 1 0 4 K.s-', 4 - extrapolation to infinite cooling rate. (b) Quenched-in resistivity versus reciprocal of quenching rate.


37

Extra electrical resistivity of quenched samples The extra electrical resistivity after quenching, Ap, is measured as a function of the quench temperature. The plot of 1nAp versus 1/T is a straight line with a slope -H,/k,. However, instead of a straight line one often obtains a curve with a slope decreasing at higher temperatures (Fig. 3.2). Correct values may be obtained by quenching from low and intermediate temperatures or by introducing corrections for vacancy losses during quenching. The quenched-in resistivity is usually measured at low temperatures, where it makes the main contribution. At liquid helium temperatures, it is easy to measure the extra resistivity of pure samples that amounts only I ppm of the resistivity at high temperatures. The measurements thus ensure high sensitivity allowing one to quench the samples from temperatures far below melting points and hence to reduce the losses of the vacancies during quenching.

Stored enthalpy The enthalpy stored by quenched-in vacancies can be released during annealing. Measurements of the stored enthalpy corresponding to various quench temperatures provide data on the formation enthalpies and vacancy concentrations. Clearly, results of such measurements are valuable only if the vacancy losses and formation of secondary defects in the sample do not significantly alter the enthalpy related to the equilibrium vacancies. Nevertheless, measurements of the extra enthalpy of quenched samples, combined with measurements of the extra electrical resistivity, are very desirable. Regrettably, such measurements are scarce. An important additional information is available from measurements of the changes in the volume and

38


in the lattice parameter of the sample after quenching and during subsequent annealing.

3.3. Observation of vacancy equilibration How to observe vacancy equilibration Observations of the vacancy equilibration after rapid changes in the sample's temperature make it possible to unambiguously reveal the vacancy contributions to a chosen physical property. This approach is capable of solving the main problem of equilibrium measurements. The only difficulty peculiar to it is caused by short relaxation times. They are generally short and rapidly decrease when the sample's temperature approaches the melting point. Observations of the vacancy equilibration can be grouped as follows. (1) The sample is rapidly heated up to a high temperature, kept at this temperature for an adjustable time interval and then quenched. The quenched-in electrical resistivity is measured as a function of this time interval. However, it is difficult to correctly evaluate the vacancy contribution at the high temperature because many vacancies have time to annihilate or form clusters during quenching or immediately after it. The quenched-in resistivity is measured at low temperatures. The vacancy equilibration is thus observable even at intermediate temperatures, where the extra resistivity is small but the relaxation time is sufficiently long. (2) The sample is rapidly heated up to a higher temperature (or cooled down to a lower temperature), and the vacancy equilibration is monitored through measurements of a proper physical property of the sample. This approach has important advantage that both initial and final states of the sample are well


39

defined. To monitor the vacancy equilibration, the changes in the chosen property should be rapidly measured. This technique was employed in determinations of the vacancy-related enthalpy. The vacancy equilibration was also studied by the positronannihilation technique. Owing to high sensitivity of the method, the measurements were carried out far below the melting point, so the equilibration times were sufficiently long. Using electrical resistivity as a probe, the measurements should be made at higher temperatures and hence deal with shorter relaxation times. Up-quenching experiments, when the sample is rapidly heated up to a premelting temperature, are very promising. They seem to be the most straightforward approach to determine equilibrium vacancy concentrations. (3) The sample is subjected to such rapid oscillations of its temperature around a mean value that the vacancy equilibration cannot follow the oscillations. Under such conditions, the vacancy contribution to a given physical property is almost completely excluded. This statement relates only to properties depending on changes in the vacancy concentrations during the measurements: specific heat, thermal expansivity, and the temperature derivative of electrical resistivity. When the vacancy concentration does not follow the temperature oscillations and retains a mean value, these properties practically correspond to a vacancy-free crystal. This method permits a reliable separation of the vacancy contributions to the physical properties. The only drawback in this approach arises from short equilibration times due to the high mobility of the vacancies at high temperatures and numerous internal sources (sinks) for them. The relaxation is therefore observable only with high frequencies of the temperature oscillations. The amplitude of the temperature oscillations is inversely proportional to their frequency, and such measurements require a sensitive technique.

40


Modulation calorimetry as a tool to study vacancy eq u iIibrati on Modulation calorimetry is the best tool to search for relaxation phenomena in specific heat. It consists in periodically modulating the power applied to the sample and registering the temperature oscillations in it around a mean temperature. This technique enables one to directly compare the specific heats measured at various frequencies of the temperature oscillations in the sample. Relaxation phenomena occur when the modulation period becomes comparable with the characteristic time of a process contributing to the specific heat. When the specific heat is measured at a very high modulation frequency, the result should correspond to a vacancy-free crystal. At intermediate frequencies, the result depends on the modulation frequency and the relaxation time.

Formulas for relaxation in specific heat Van den Sype (1970) has presented a frequency-dependent specific heat in the complex form C ( X ) = C + AC/(l + Ix).From this expression,

Ic(x>12= (C,2 tan A +

=

+ C2X2)l(1 + X 2 ) ,

x AC/(C,

+CX~).

(3.3) (3.4)

Here C and AC denote the specific heat of a vacancy-free crystal and the vacancy contribution, respectively, X = wz is the product of the angular frequency of the temperature oscillations and the relaxation time, C, = C + AC is the equilibrium specific heat measured when X 2 z2,can a vacancy with a positron move as easily as a free vacancy? (2)Does the trapped positron move with the vacancy or become detrapped? The same questions relate to divacancies. Gramsch and Lynn (1989) and Jensen and Walker (1990) considered the trapping model for thermal and nonthermal positrons in metals. Sterne and Kaiser (1991) have calculated, from first principles, positron lifetimes in metals. Franz et al. (1993) performed computer simulations of positron-lifetime spectroscopy in copper. Korhonen et al. (1996) have carried out first-principles calculations of positron-annihilation characteristics at vacancies in Al, Cu, Ag, Au, Fe, and Nb. Seeger (1998a) has

8.Positron annihilation

183

developed an analytical theory of the slowing down and thermalization of positrons in solids. Dryzek (1998) has presented the exact solution of the one-dimensional diffusion model for trapping of positrons at vacancies in metals. In contrast to Doppler broadening and angular correlation, lifetime spectroscopy supplies simultaneously the positron lifetime characterizing the type of trap and the trapping rate that is a measure of the trap concentration. The parameters of the trapping model are the positron lifetimes in free and trapped states and the trapping rate of the transition to the localized state. The trapping rate is proportional to the concentration of the defects that capture positrons. The proportionality factor, i.e., the trapping rate per a unit vacancy concentration, is called the specific trapping rate. The positron-annihilation techniques are applicable to studies of the vacancy formation in equilibrium and to measurements on quenched samples (Petersen 1983). This method was successfully employed in observations of the vacancy equilibration at high temperatures (Schaefer and Schmid 1989; Kummerle et al. 1995; Wurschum et al. 1995). Experimental techniques for positron-annihilation studies of defects in metals are described in many papers (e.g., Doyama and Hasiguti 1973; MacKenzie 1983; Smedskjaer and Fluss 1983). Positron annihilation is considered as being the most powerful technique for determining the enthalpies of vacancy formation. For instance, Schaefer (1987) stated that “positron annihilation has developed into the most valuable technique available for the investigation of thermal equilibrium vacancies in metals at high temperatures.” This conclusion is based on the assumption that the vacancy contributions can be separated without serious errors. In a review, Puska and Nieminen (1994) stated that “...the theory underlying positron annihilation has developed from

8. Positron annihilation

184

simple models describing the positron-solid interaction to 'firstprinciples' methods predicting the annihilation characteristics for different environments and conditions. This development has paralleled the development of electronic structure calculations, which in turn has leaned heavily on the progress in computational techniques. The conceptual basis of electronic structure calculations lies in density-functional theory, and this theory can be generalized to include the positron states." In a review of the positron-annihilation techniques, Eldrup (1995) has concluded that "the positron-annihilation technique has now become a well-established, non-destructive technique for studies of defects in bulk materials but at the same time a technique which is still being developed and used in new applications."

Lifetime spectroscopy. Mean positron lifetime In its initial stage, the lifetime spectroscopy was based upon several simple assumptions: (i) the lifetime of positrons captured by vacancy-type defects increases; (ii) the specific trapping rate does not depend on temperature; (iii) no detrapping of positrons occurs until annihilation; (iv) the positron lifetimes in the regular lattice and in the vacancy may be obtained by fitting data from low and high temperatures. If n f and n, denote concentrations of free and trapped positrons, the following equations are valid for their time rates of change:

dnddt

= -

n,h,

dn,ldt

= -

n v h v + pvc,nf.

-

p,c,nf,


185

Here h, and h, are the annihilation rates for a free and a trapped positron, respectively, c, is the vacancy concentration, and pLVis the specific trapping rate. When some positrons annihilate in trapping states and others while free, then any characteristic of the positron annihilation, F , will have the value of the weighed mean of this characteristic in both states, Ff and Fv :

F

=

F f P f + FvPv,

(8.3)

where P, and P, represent the probabilities of the annihilation in the free and trapped states. The fraction of positrons annihilating in the free state is given by

whereas the fraction of positrons that annihilate when trapped in a vacancy is

The mean positron lifetime obeys the relation

where T, = l / h , and z, = l/hv are the annihilation lifetimes for a free and a trapped positron, respectively. At low temperatures, when p,c, is small compared to h, and h , the mean lifetime approaches zf.At high temperatures, when pvcv is large, it approaches 5‘ , (Fig. 8.2). From the above equation, the equilibrium vacancy concentration is

186


Cotterill et al. (1972) have determined the positron lifetimes and the trapping probabilities in aluminum separately for vacancies and dislocations. The measurements were made immediately after quenching (mainly vacancies were present in the sample) and following annealing at 353 K (mainly dislocations). The authors have pointed out that such measurements are probably inferior to equilibrium ones because of the inherent complexity of the quenching process.

captured positrons

free positrons

Fig. 8.2. Lifetimes of positrons in metals: free positrons, positrons captured by vacancies, and temperature dependence of the mean lifetime.

Hall et al. (1974) have presented a detailed description of the lifetime spectroscopy and the data analysis. Small drops of 22NaCI in a neutral solution were dried onto two pieces of the sample. Then the pieces were clamped together and electron-


187

beam welded around the edges. The decay of **Na produces a positron and simultaneously a y-quantum of 1.28 MeV. This y-quantum signals the creation of the positron. The positron enters the sample, is thermalized and either drifts through the crystal lattice or becomes trapped in a vacancy. It eventually annihilates with an electron, producing two y-quanta of 51 IkeV. A multichannel analyzer measures the time delay between the 1.28 MeV and the 51 1 keV quanta. The time resolution is usually better than lo-" s. The data provide a histogram representing the number of events as a function of the time delay. In this investigation, vacancy formation was studied in aluminum, gold, and in an AlZn alloy. The fit of the data was significantly improved by assuming a temperature dependence of the specific trapping rate. Temperature-independent traps were attributed to dislocations. A method was developed of simultaneously fitting data from all temperatures and assuming several types of the traps. From the results on aluminum and gold, the authors have concluded that the specific trapping rate increases with temperature. Luhr-Tanck et al. (1985) and Hehenkamp et al. (1986) have carried out high-resolution positron-lifetime studies on silver and copper. Two lifetimes were seen in the lifetime spectra. The shorter lifetime, T,, is given by

When the vacancy concentration becomes significant, z, decreases and a second lifetime, z2 = 7, appears. The mean lifetime was taken to be T~ = z l I l + 7212, where I , and I2 are the corresponding intensities in the lifetime spectra. Two methods were employed to calculate the vacancy-formation enthalpies. The first method is based on the 12/1,ratio at various temperatures, whereas the second one employs the 12(hl - h 2 ) value. These two approaches provide markedly different

188


formation enthalpies. The authors have concluded that the most probable explanation of this result is the trapping of the positrons during thermalization. Recently, Suzuki et al. (1999) observed temperature variations of the short positron lifetime in Zn. It was shown that the vacancy-formation energy can be determined only from data from temperatures considerably below the melting point. The short lifetime component decreases with increasing temperature. Wang et al. (1984) measured positron lifetimes in Ni,AI samples, irradiated or quenched, and subsequently annealed. The formation and migration enthalpies have been determined. Puska and Manninen (1987) calculated the trapping rate of positrons into small vacancy clusters and light substitutional impurities in metals. Trumpy and Petersen (1 994) have found the trapping rate to be a linear function of T’.

Doppler broadening. S-, W-, and D-parameters Doppler broadening of the y-line was used in many investigations. A lithium-drifted or intrinsic germanium crystal detected the annihilation radiation. Since the detector is placed close to the sample, the method possesses high efficiency. A disadvantage of this technique is the low resolution. The samples under study are usually characterized by one of the so-called shape parameters. The shape parameters S and W are defined as follows:

s = C,N(E)/C,N(E),

(8.9)


189

511 keV I

I

I

Lc

0

a,

0

I

I

channel number

Fig. 8.3. Doppler broadening of the annihilation line: 1 - annealed samples, 2 - quenched samples. The W-parameter shows the difference in the line shape at high and low temperatures.

Here N ( E ) is the number of counts per channel corresponding to the energy E , z C , N ( E ) is the total number of counts within the annihilation line, and the intervals M , L and R are chosen to include the maxima in the difference curve of the spectra at low and high temperatures (Fig. 8.3). The vacancy formation leads to an increase of the S-parameter. The W-parameter is sensitive to the probability that positrons annihilate with high-energy core electrons and decreases when positrons are captured by vacancies.


190

Maier et al. (1979) employed this technique for studying vacancy formation in vanadium, niobium, molybdenum, tantalum, and tungsten (Fig. 8.4). The measurements were carried out in wide temperature ranges, from 4.2 K up to slightly below the melting points. The samples in the form of tubes with blackbody models were prepared using high-purity materials. The samples were annealed at high temperatures in a vacuum of Pa. Then 10-20 pCi of 22NaCI, vacuum-evaporated onto foils of the sample material, were placed into the tubes. The tubes were sealed by the electron-beam welding.

..... ring cathodes

/ window

Ge (Li) detector

optical pyrometer

n

U

Fig. 8.4. Measurement of Doppler broadening in refractory metals (Maier et al. 1979).


191

The samples placed in a vacuum system were heated by an electron beam. Their temperature was measured by an automatic spectral pyrometer focused on the blackbody model in the sample. The annihilation line was observed by a Ge(Li) spectrometer having a resolution 1.2 kV at l o 4 counts per second. Determinations of Doppler broadening are considerably faster than positron-lifetime or angular-correlation measurements. Nevertheless, only a few data could be obtained near the melting points because of the high vapor pressure of vanadium, molybdenum, and tungsten. The temperature dependence of the W-parameter was used to evaluate the formation enthalpy. In the case of vanadium, the parameter D = S - W was employed because it exhibited less scatter of the data than the W-parameter. The following formation enthalpies were determined: 2.1 f 0.2 eV (vanadium), 2.6 f 0.3 eV (niobium), 3.0 f 0.2 eV (molybdenum), 2.8 f 0.6 eV (tantalum), and 4.0 f 0.3 eV (tungsten).

Angular correlation of y-quanta In measurements of the angular correlation of annihilation quanta, the quanta are detected in coincidence by counters shielded from direct view of the source. Lead collimators in front of the detectors define the angular resolution being typically better than 1 mrad. Single-channel analyzers are tuned to 511 keV quanta and the device counts the coincidence pulses as a function of the angle between the counters. The correlation curve consists of two parts. An inverted parabola is due to annihilations with valence electrons, and a broader component is due to annihilations with core electrons having higher momentum. McKee et al. (1972) measured the angular correlation of annihilation quanta in indium, cadmium, lead, zinc, and

192

8.Positron annihilation

aluminum. The samples were spark cut from a 99.999% purity stocks, chemically etched, annealed for a day in vacuum or in argon at a temperature close to the melting points, and then reetched. The apparatus was set at a zero angle, and the coincidence-counting rate was measured as a function of the sample’s temperature. For each sample, the data were accumulated during two days. High-temperature saturation was clearly seen in zinc and aluminum. Nanao et al. (1973, 1977) studied equilibrium vacancies in copper and nickel. It turned out that the formation enthalpies obtained markedly depend on the assumed temperature dependence of the background. Triftshauser (1975) studied positron trapping in solid and liquid In, Pb, and Al. The formation enthalpies were found to be 0.48 eV, 0.54 eV, and 0.66 eV, respectively. At the melting point, the peak counting rate increases abruptly and then stays constant with temperature. No vacancy trapping was found in magnesium.

8.2. Experimental data When the positron-annihilation parameters are measured under equilibrium conditions, one needs to separate the vacancy contribution. The formation enthalpy is obtainable from its temperature dependence. The annihilation parameters of quenched samples and changes of these during thermal annealing are also informative. The positron-annihilation techniques seemed to be very promising, and many investigators used it (Table 8.1).

193


Table 8.1 Vacancy formation enthalpies from positron-annihilation measurements (from review by Schaefer 1987). A - angular correlation, D - Doppler broadening, L - lifetime spectroscopy, M - mean lifetime.

Metal

In

Sn Cd Pb Zn T1

Ms Al

AU

cu

H ,

(eV)

0.56 0.55 0.54 0.52 0.65 0.50 0.54 0.46 0.9 0.68 0.66 1.31 1.11 0.89 0.89 1.42 1.28

Method

Metal

L

Ni

A

co

D D L

Pd Pt

A

V Cr Nb Mo

A

M M L A

L M L M L M

Ta W

H ,

(eV)

1.78 1.34 1.85 1.35 1.32 2.07 2.0 2.65 3.6 3.0 3.0 2.9 2.8 4.6 4.1 4.0

Method

D A

D L

D D D D L M D M D L M D

194


8.3. Drawbacks of positron-annihilation techniques Serious drawbacks of the positron-annihilation techniques are also evident, namely: (i) vacancy concentrations are not available; (ii) in some cases, saturation in the annihilation parameters occurs far below the melting point; (iii) the unknown temperature dependence of the specific trapping rate may markedly affect the results; (iv) detrapping of captured positrons may lead to ambiguous results; (v) in some metals, the vacancy contribution has not been found. Seeger and Banhart (1987) considered the last issue as follows. In alkali metals, with ion cores small compared to the interatomic distances, positrons annihilate almost exclusively with conduction electrons irrespective of whether they are free or trapped in vacancies. The lifetime of positrons in metals has an upper limit of about 5 ~ 1 0 - 's. ~ In alkali metals, the positron lifetimes are longer than in other metals, and the predicted increase of these due to the trapping by vacancies amounts only 2-5%. This is the reason why it was not observed by present experimental techniques. Seeger and Banhart (1990) emphasized that "in order to extract reliable information on defects in crystals from positron annihilation measurements, detailed information on the thermalization, the diffusion, and the capture of the positrons by the defects acting as traps is required." In many cases, the formation enthalpies deduced from positron-lifetime spectroscopy are markedly higher than those from the mean lifetime and Doppler broadening. Dryzek (1995) has shown that the calculated specific trapping rate is extremely sensitive to the parameters of the potential used and may strongly depend on temperature.


195

8.4. High vacancy concentrations in some intermetallics Relatively low enthalpies of vacancy formation have been found by the positron-annihilation techniques in some intermetallics. These low enthalpies predict high vacancy concentrations. Schaefer et al. (1990) have found a very high concentration of thermal vacancies, 6 . 6 ~ 1 at 0~ the ~ melting point, in Fe7,,,A1,3,7. High concentrations of thermal vacancies in Fe,AI (Schaefer et al. 1990, 1992; Wurschum et al. 1995) and Fe3Si (Kummerle et al. 1995) are expected from the formation enthalpies. On the other hand, Brossmann et al. (1994) observed low concentrations of thermal vacancies in TiAI, 1 .5x1OP4 at the melting point. Wurschum et al. (1996) reported low concentrations of thermal vacancies in Ti,AI. Badura-Gergen and Schaefer ( 1997) have found low concentrations of thermal vacancies in Ni,AI. The authors observed predominant formation of thermal vacancies in the nickel sublattice and high concentration of antisite atoms. The vacancy concentration in the nickel sublattice has been estimated to be in the range ( 4 - 8 ) ~ 1 0 - at ~ the melting point. Clearly, any prediction of equilibrium vacancy concentrations requires knowledge of the formation entropy. It would be very useful to check the above predictions of vacancy concentrations in intermetallics by other methods of studying vacancy formation, including measurements of the specific heat.

196


8.5.Summary Vacancies in metals form traps for positrons and thus modify positron-annihilation parameters. Positron-annihilation studies of point defects include several techniques: lifetime spectroscopy, Doppler broadening, and angular correlation of annihilation y-quanta. 0 To make reliable conclusions about the formation enthalpies and equilibrium vacancy concentrations, some assumptions are to be fulfilled. Recently, evidence has been found that these assumptions are generally invalid. To extract reliable information from the measurements, detailed data are necessary on the thermalization, diffusion, and capture and detrapping of the positrons. In some cases, the formation enttialpies obtained by various positron-annihilation techniques are in disagreement. Nevertheless, positron annihilation is regarded as being the best tool for determining the enthalpies of vacancy formation in metals

Unexpectedly low formation enthalpies have been found in some intermetallics. High concentrations of thermal vacancies are therefore expected in these compounds. It would be very useful to check these predictions by other methods of studying vacancy formation, including measurements of the specific heat.

Chapter 9

Other methods 9.1. Hyperfine interactions

198

Perturbed angular correlation of y-quanta. Mossbauer spectroscopy. Nuclear magnetic resonance.

9.2. Other physical properties

207

Thermoelectric power. Thermal conductivity and thermal diffusivity. Mechanical properties. Spontaneous magnetization. Current noise. Properties of superconductors.

9.3. Microscopic observation of q uenched-in defects

212

Electron microscopy. Field ion microscopy.

9.4. Summary

216

197

198

9. Other methods

9.1. Hyperfine interactions These techniques employ hyperfine interactions between nuclear moments of probe atoms and extranuclear magnetic fields or electric-field gradients created or modified by neighboring point defects. The induced hyperfine interaction is a defect property, so that different defects can be recognized in new experiments. Usually, only defects within the next surrounding of the probe atom are visible. Methods based on hyperfine interactions include the perturbed angular correlation of y-quanta, the Mossbauer spectroscopy, and the nuclear magnetic resonance. Niesen (1 981) has described hyperfine-interaction methods for studying point defects in metals. Sielemann (1987) has reviewed advances in using the perturbed-angular-correlation technique and the Mossbauer spectroscopy.

Perturbed angular correlation of y-quanta In the 1970s, a new method for studying point defects and defect clusters appeared, the perturbed angular correlation of y-quanta. This technique permits measurements of magnetic fields or electric-field gradients in a neighborhood of radioactive tracer nuclei. The radioactive nuclei, introduced into a host material, decay to an excited state of the tracer nucleus, which decays to the ground state by emission of two successive y-quanta. This emission senses an interaction of the nucleus in its intermediate state with extranuclear fields that occurs during the time intervat between the two emissions. Due to the precession of the nuclear moment, the directional distribution of the emitted y-quanta becomes time dependent. In the time-differential perturbed angular correlation, the precessions of the nuclear spin of the

9. Other methods

199

excited nuclear state are monitored over its lifetime. The theory of the method is given in several sources (Steffen 1955; Steffen and Frauenfelder 1964; Frauenfelder and Steffen 1968). Details can be found in reviews devoted to this technique (Kaufmann 1981; Wichert and Recknagel 1981; Wichert 1983, 1987; Collins et al. 1990).

173 keV, 0.01 ns 247 keV, T,*= 85 ns

Fig. 9.1. Energy levels and parameters of decay of '"In employed in perturbed-angular-correlation measurements.

As a rule, the experiments employ the isotope "'Cd. The 247-keV state in '"Cd is fed from the ground state of "'In (Fig. 9.1). The half-life time of the 247-keV state, 85 ns, is well matched to hyperfine frequencies encountered in both magnetic and nonmagnetic solids and to the electronic timing methods now available. Lattice defects in the close neighborhood of probe

200

9. Other methods

atoms cause perturbation factors that differ distinctly from those of a defect-free lattice. In cubic nonmagnetic metals, 'llCd with a near-neighbor defect shows a discrete electric quadrupole interaction. The interaction is zero for undisturbed lattice sites, so that the defect-induced signal can be detected against a zero background. In magnetic metals, defect sites may be identified via both magnetic dipole and electric quadrupole interactions. Usually, the defect-induced signal is observable in a relatively narrow temperature range, far below the melting point. This is the main obstacle to determine defect concentrations at high temperatures. Hinman et al. (1964) were probably the first to attribute changes in the angular correlation to vacancies, which were created in gold by irradiation or cold work. Behar and Steffen (1972, 1973) reported results that are more definite. The authors studied polycrystalline silver foils. The tracer nuclei were created in the samples by the Ag(a, 2n)ln reaction. After the '"In ions come to rest in the lattice, they decay by electron capture to l1 'Cd. The time-differential perturbed-angular-correlation measurements revealed a quadrupole interaction with electricfield gradients. After annealing at 6OO0C for 12 h, the quadrupole interaction completely disappeared. It was therefore concluded that the most probable cause of the perturbation are vacancies and/or silver interstitials. Bleck et al. (1972) observed a similar phenomenon in cadmium. Hohenemser et al. (1975) observed vacancy trapping and detrapping in nickel during isochronal annealing in the range from 2OoC to 40OoC. Muller et al. (1976) implanted '"In ions into copper, aluminum, and platinum at 24 K and at room temperatures. In the case of platinum, the measurements have shown trapping of a single defect interpreted as a monovacancy. Wichert et al. (1978) studied annealing of defects in copper introduced by quenching and electron and proton irradiation. The

9. Other methods

201

stage-Ill recovery in irradiated samples has been attributed to mobile vacancy-type defects. Weidinger et at. (1979) studied irradiated molybdenum samples. Three different defect configurations were identified, namely: a monovacancy, a divacancy, and a tetrahedral configuration of four vacancies with '"In atom in its center. Putz et al. (1982) obtained similar results in irradiated tungsten. Pleiter and Hohenemser (1982) have summarized and interpreted their studies of defects in silver, aluminum, gold, copper, nickel, palladium, and platinum. The authors divided the defects introduced by irradiation, quenching, and ion implantation into four classes. For three of these classes, structural assignments have been made as follows: the nearest-neighbor monovacancies, divacancies or faulted loops in the { I 11) plane, and tetrahedral vacancy clusters seen only in nickel. However, some of the observed states remained with undetermined structure. Collins et al. (1983) studied gold samples, heavily deformed at 77 K, after annealing at temperatures up to 500 K. Allard et al. (1985) observed the vacancy migration and clustering in nickel after proton irradiation and deformation. Hoffmann et al. (1986) examined lattice defects in rhodium and iridium produced either by the implantation of "'In ions or by proton irradiation. The authors interpreted the defect in rhodium as a monovacancy. In iridium, two defects were found, a monovacancy and a relaxed divacancy. Collins et al. (1990) studied vacancies created by plastic deformation in gold, copper, nickel, and platinum. The authors have identified the structures of various multivacancy complexes by three methods: (i) quadrupole interaction parameters were compared with calculations of electric-field gradients for 20 structures containing from 1 to 4 vacancies; (ii) decoration of vacancy complexes with hydrogen atoms was determined and compared with calculations; (iii) annihilation of

202

9. Other methods

vacancies by mobile interstitials served to test the consistency of the identification. Muller and Hahn (1984) examined the formation of constitutional and thermal defects in the intermetallic Pdln. Thermal defects were created by quenching the samples from high temperatures. The authors observed an increasing concentration of randomly distributed monovacancies in the palladium sublattice with increasing the quenching temperature. They have concluded that vacancies in the indium sublattice are invisible because the electric-field gradients caused by them on a next-nearest-neighbor site of the indium probe are too small. Unusually high concentrations of quenched-in vacancies, of about 15% at the melting point, were found in intermetallics NiAI, CoAl, and TiAl (Fan and Collins 1990, 1993; Collins and Fan 1993). A very low vacancy concentration was observed in Ni,AI. Hanada (1993) examined diffusion processes in cadmium and in dilute Cdln alloys. Below 500 K, the author observed an irreversible loss of the defect-related signal in cadmium. He has explained this phenomenon by migration of the probe atoms to unknown trap sites. Wenzel et al. (1994, 1995) employed the perturbed-angularcorrelation technique in studies of equilibrium point defects in NiO and COO, at temperatures up to nearly 1500 K. An enhancement of the unperturbed fraction in the angularcorrelation spectra occurs at high temperatures. At the same time, the spectra of quenched samples showed well-pronounced perturbed fractions. The authors have explained this contradiction by a decrease in the vacancy-trapping probability at high temperatures. The vacancy trapping becomes thus not effective. This leads to rapid fluctuations of the electric-field gradients near the probe nuclei. This result is very important for further studies of defects in metals under equilibrium. Experimental setups for perturbed-angular-correlation studies were described in many papers. For example, Jaeger et

9. Other methods

203

al. (1 987) have developed a computer-controlled spectrometer (Fig. 9.2). The spectrometer records the time-dependent spectrum of events, in which the first y-quantum enters a detector and the second one enters another detector at a certain time later. Four detectors are arranged in a plane at 90' angular intervals with the sample at the center. Two adjacent detectors

1 1

detectors

coincidence

1

start

interface

U computer

Fig. 9.2. Simplified diagram of setup for perturbed-angular-correlation studies designed by Jaeger et al. (1987).

are tuned to the first y-quantum and start the time-to-amplitude converter (TAC), while the other two are tuned to the second y-quantum and stop the converter. The analog-to-digital converter (A/D) provides data for a computer. After an

204

9. Other methods

experimental run, the time spectrum of the perturbed angular correlation and the corresponding Fourier spectrum are stored for a further analysis. The main features of the perturbed-angular-correlation techniques are thus as follows. (1) The probe atoms are presented in very low concentrations and do not modify the properties of the sample. Only defects within the next surrounding of the probe atom induce perturbations sufficiently large for the measurements. To be visible, defects have to migrate and become trapped at the probe atoms. (2) The use of impurity probe atoms complicates the evaluation of defect concentrations because of the defect-probe binding. On the other hand, this binding significantly enhances the local defect concentration, which is necessary to obtain a measurable defect-induced signal. (3) From the parameters measured, one can discriminate defects of different structure. (4) Various defects tend to anneal out and trap at probe atoms at different temperatures. This reduces the number of configurations to be considered in any one measurement. A sufficient binding energy between the defect and the probe atom is required to retain the defect-induced signal at high ternperat ures. (5) Taking into account the defect-probe binding, absolute defect concentrations could be evaluated. However, the vacancytrapping probability at high temperatures may become too low, which will lead to a loss of the defect-induced signal. The last point is very important. Until today, no data were obtained on equilibrium point defects in pure metals. An unavoidable difficulty is the decrease of the vacancy-trapping probability at high temperatures.

9. Other methods

205

Mossbauer spectroscopy Mdssbauer spectroscopy is a technique useful for detecting interactions between point defects and neighboring atoms. Czjzek and Berger (1970) used this method in studies of FeAl alloys. The Mossbauer y-rays were emitted by 56Fe nuclei after thermal-neutron capture. The created 57Fe nuclei imparted recoil energies up to 549 eV and were displaced from their lattice sites. The authors have made qualitative estimates of the effect of vacancies and interstitials. Using the Mossbauer effect in Ig7Au, Mansel et al. (1970) measured changes in the Debye-Waller factor of gold in platinum after low-temperature neutron irradiation. The authors have attributed a reduction up to 10% in the Debye-Waller factor to irradiation-produced defects. Mansel et al. (1973) studied the influence of neutron irradiation on aluminum doped with 57C0. A new component that appeared in the spectrum after the irradiation was attributed to interstitials trapped in the immediate proximity of the cobalt atoms. Reintsema et al. (1979) used 133Xe atoms implanted in molybdenum, tantalum, and tungsten as a source of Mossbauer spectra. Four different xenon sites were identified in molybdenum and tungsten: substitutional xenon atoms and xenon atoms associated with one, two, or three vacancies. For tantalum, the results were less clear due to a considerable overlap of lines in the spectrum. Tanaka et al. (1986) studied the vacancy-antimony complexes in gold using '"Sb emission spectra. The authors investigated the behavior of the complexes during heat treatments. The spectra were measured immediately after quenching from 1073 K and after subsequent isochronal annealing at various temperatures. Wahl et al. (1988) studied vacancy trapping at impurities in quenched and irradiated tungsten.

206

9. Other methods

Nuclear magnetic resonance Minier et al. (1978, 1980) employed the nuclear-magneticresonance (NMR) technique to measure electric-field gradients caused by point defects in electron-irradiated aluminum and copper. The method allows a characterization of monovacancies independently of interstitials, divacancies or clusters. It was shown that monovacancies in these metals migrate during the stage-Ill annealing. Konzelmann et al. (1994) detected vacancy-type defects in quenched copper samples by the nuclear quadrupole double resonance (NQDOR). The samples were rapidly quenched from the melt. The electric-field gradients due to point defects cause a splitting of the energy levels of nuclei. The splitting is proportional to the field gradients and is different for crystallographically non-equivalent sites. Each atomic defect thus manifests a set of discrete level splittings. They were observed by NQDOR immediately after quenching and during subsequent annealing. By comparing the results obtained with those on irradiated samples, the authors have made following conclusions: (i) the irradiation-induced lines at frequencies above 100 kHz are due to self-interstitials; (ii) vacancy-type defects give rise to a broad distribution of NQDOR frequencies below 100 kHz; (iii) the enthalpies of migration of dumbbell self-interstitials and of divacancies are close to each other and cannot be distinguished by measurements of the activation energy.

9. Other methods

207

9.2. Other physical properties Thermoelectric power Point defects affect many other physical properties of metals. A change in the thermoelectric power (the Seebeck coefficient) after quenching was observed long ago (Gertsriken and Novikov 1960;Kedves and de Chatel 1964).The vacancy contribution is V per a unit vacancy concentration and of the order of depends on temperature. Huebener (1964, 1966) measured the thermopower of quenched samples of gold and platinum. Rybka and Bourassa (1973) investigated the vacancy-induced thermopower in aluminum. Bourassa et al. (1968)studied the effect of pressure on the thermopower at high temperatures. The authors interpreted the data for aluminum as an influence of three types of thermally activated defects: the monovacancy, the divacancy, and the impurity-vacancy pair. The thermopower of copper at high temperatures was measured (Kraftmakher 1971a) using the modulation technique.

Thermal conductivity and thermal diffusivity Vacancies influence thermal conductivity and thermal diffusivity of metals. Temperature gradients in a sample cause gradients in the vacancy concentrations, and the diffusion of the vacancies contributes to thermal conductivity (Zinov’ev and Masharov 1973; Strzhemechnii and Kal’noi 1983). Zinov’ev et al. (1968,1969) observed this phenomenon in platinum and palladium. Kraev and Stel’makh (1963,1964) have found nonlinear decrease in the thermal diffusivity of refractory metals (Fig. 9.3).This decrease is due to the nonlinear increase in the specific heat. Similar

9. Other methods

208

behavior was observed in lanthanum (Zinov'ev et al. 1973) and ruthenium (Savitskii et al. 1976).

0.35 0.30

.

w

1500

2000

2500

3000

3500 K

Fig. 9.3. Thermal diffusivity of refractory metals at high temperatures (Kraev and Stel'makh 1963, 1964). The nonlinear decrease is caused by increase in specific heat.

Formation enthalpies derived from measurements of various physical properties of metals do not contradict data obtained with basic techniques employed in studies of point defects (Table 9.1).

209

9. Other methods

Table 9.1 Formation enthalpies evaluated from thermopower, thermal conductivity, and thermal diffusivity of metals.

Property

Metal

HF (eV)

Reference

Pt

1.0 1.45

Gertsriken and Novikov 1960 Gertsriken and Novikov 1960

Thermal conductivity

Pt Pd

1.45 1.7

Zinov'ev et al. 1968 Zinov'ev et al. 1969

Thermal diffusivity

La Pd Ru

0.98 1.5 1.75

Zinov'ev et al. 1973 Zinov'ev et al. 1969 Savitskii et al. 1976

Thermopower

Ag

Mechanical properties Mechanical properties of metals also depend on point defects. In computer simulations, Stabell and Townsend (1974) have shown the effect of vacancies and interstitials on the bulk modulus of tungsten. Vacancies result in a decrease of the bulk modulus, while interstitials cause an opposite change. Gerlich and Fisher (1969) measured the elastic moduli of aluminum over the temperature range 300-930 K. The measurements were carried out on a single-crystal sample by a determination of the ultrasonic-wave velocities in different crystalline directions. The elastic moduli decrease nonlinearly with increasing temperature. This decrease may be caused by equilibrium vacancies.

210

9. Other methods

Spontaneous magnetization A decrease in the spontaneous magnetization of a ferromagnet was observed in equilibrium and quenching experiments (Nakamura 1961; Otake et al. 1981). Wuttig and Birnbaum (1966) studied electrical resistivity and magnetic after-effect in quenched samples of nickel. The magnetic relaxation was caused by a reorientation of a defect whose symmetry is lower than that of the FCC lattice. Two types of quenched-in defects have been found, monovacancies mobile at 400 K and divacancies, which become mobile at 320 K.

Current noise An interesting phenomenon was observed by Celasco et al. (1976) when measuring the current noise in thin aluminum films. The noise caused by fluctuations of the electrical resistance becomes visible when a DC current passes through the sample. The noise spectrum contained a component related to creation and annihilation of vacancies. This component is governed by the lifetimes of the vacancies and is therefore temperature dependent. The lifetimes at 435OC and 475OC were estimated to be 4.7 and 2.8 ms, respectively. The authors have evaluated the migration enthalpy and the vacancy contribution to the resistivity. With the equilibrium vacancy concentrations from differential dilatometry (Simmons and Balluffi 1960a), the contribution of a unit vacancy concentration to the resistivity appeared to be pv = 1 . 8 ~ 1 0 a.m. -~ Petz and Clarke (1985) studied the electrical l/f-noise in electron-irradiated copper films. The sample was maintained at 90 K and irradiated by the beam of an electron microscope. The authors have explained the observed difference in the recovery

9. Other methods

211

of the electrical noise and of the induced resistivity as follows. A subpopulation of mobile defects responsible for much of the added noise may represent only a small fraction of the defects. These mobile defects are deactivated, via recombination or clustering, at lower temperatures than the majority of the defects. Briggmann et al. (1994) investigated irradiation-induced defects in thin aluminum foils by measurements of the l/f noise and the extra resistivity. The defects were produced by 1-MeV electrons at 10 K. The induced noise and the resistivity were also measured after isochronal annealing. An important feature of such measurements is that they can distinguish a small minority of mobile defects among a majority of defects of low mobility.

Properties of superconductors In molybdenum and niobium, Mamalui and Ovcharenko (1989) observed an influence of quenched-in vacancies on the superconducting-transition temperature and the temperature dependence of the electrical resistivity at the transition.

212

9.Other methods

9.3. Microscopic observation of quenched-in defects Defects presented in quenched samples are observable by an electron or a field ion microscope. As a rule, such observations are accompanied by measurements of the quenched-in resistivity.

Electron microscopy During annealing, vacancies quenched-in by a rapid cooling of the samples form clusters observable by an electron microscope. Hirsch et al. (1958) were the first to observe such clusters in quenched aluminum. Siege1 (1966ab) investigated pure gold samples, in the form of narrow foils, quenched from 7OO0C or 900°C. He observed the vacancy precipitate structure in the samples annealed at temperatures 4OoC and 6OoC by a transmission electron microscope (TEM). It has been concluded that impurities of low concentrations act as efficient heterogeneous nuclei for the vacancy precipitation. From the number and size of the precipitates, the vacancy concentration in gold after quench from 900°C was (1.1 f 0.2) x l O p 4 . This figure corresponds to the vacancy concentration at the melting point of about 3 x 10-4, which is lower than the differential-dilatometry value. The resistivity of a unit vacancy concentration is p, = (1.8 f 0.4) x1OP6 a.m. In TEM investigations of a gold sample quenched from 1300 K, Fraikor and Hirth (1967) have estimated the vacancy concentration at this temperature. The obtained value was also lower than that from differential dilatometry. Eyre et al. (1978) reviewed electron-microscopy studies of point-defect clusters following quenching and irradiation. The authors concluded that fundamental differences

9. Other methods

213

exist between FCC and BCC metals concerning the formation of vacancy clusters. Rasch et al. (1980) have carried out an important investigation on tungsten samples. Thin wires, 40 to 60 pm in diameter, were used to measure their electrical resistivity. Foils of the same thickness served for TEM observations. The samples were prepared from single crystals of high purity. After preparation, the wire samples had a 'bamboo structure' with an average grain size of 0.1 mm. The authors employed the superfluid-helium quenching technique. To estimate quenching losses and to prove that helium did not enter the samples, some quenches were done in a vacuum chamber. Visible vacancy clusters (voids) were observed in samples quenched from the melting point, 3695 K, and subsequently annealed in the range from 800 to 1000 K. Quenching from below 3400 K produced no visible clusters. The voids were denuded from regions near grain boundaries. Blank experiments confirmed that the observed voids were not produced by impurities, as occurs in tungsten samples of commercial purity. After quenches from the melting point, the concentration of vacancies stored in the voids was (2 f 1) The authors have accepted the lower limit of this value. The determined formation parameters were as follows: H , = 3.67 f 0.2 eV, H M = 1.78 f 0.1 eV, S , = 2.3kB, = loT4, and p, = 6 . 3 ~ 1Q.m. 0 ~ ~A serious difficulty appeared cmP in matching these parameters with the self-diffusion data. The authors have supposed that divacancies or interstitials were probably created in equilibrium but could not be quenched because of their high mobility. Kojima et at. (1989) have shown that frozen-in vacancies form stacking fault tetrahedra, whereas interstitials form faulted dislocation loops.

214

9. Other methods

Field ion microscopy Field ion microscopy (FIM) involves atom by atom dissection of samples by pulse-field evaporation and recording images produced at each stage of the process on a cine film. Muller (1959) was the first to observe single vacancies frozen-in by quenching. This technique is applicable only to quenched samples of high-melting-point metals. The defect concentrations in the samples may thus be much smaller than those in equilibrium. Samples of submicron thickness are necessary for the observations. They are studied under gradually evaporation of the surface atoms in situ by increasing the electric field. The number of vacant sites and the vacancy concentrations after quenching are thus available. Seidman (1973) has reported such observations on platinum wires quenched from 170OoC. The quenched-in resistivity was measured at 4.2 K. The samples were prepared using electrical etching. Among about 9 ~ 1 0 lattice ~ sites, 233 vacancies have been found. The concentration of the frozen-in vacancies, about 3 ~ 1 at 0 the ~ ~melting point, appeared to be much smaller than that from the specific-heat data. The author has stressed that the quenched-in concentrations must be lower than the equilibrium ones due to the vacancy losses that occur in any real quench. For two samples, p, appeared to be 4 . 8 ~ 1 0 - and ~ 5 . 7 5 ~ 1 0 a.m. -~ The author considered these values to be upper limits of p,. Hence, they do not contradict the equilibrium value, 2 . 4 ~ 1 S2.m 0 ~ ~(Kraftmakher and Lanina 1965). Park et al. (1983) have performed a similar study on quenched tungsten samples. High-purity wires were quenched in ultra-high vacuum. The extra resistivity was measured on the same quenched wires. A cooling rate 3 ~ 1 K.s-’ 0 ~ was achieved near the beginning of the quench, where defect losses to preexisting sinks are of major concern. No appreciable stresses

9. Other methods

215

were applied to the samples during the quench, so minimizing the generation of dislocations during quench cycles. Before quenching, the samples were kept at high temperatures for only a short time, of about 1 s. For the two samples investigated, the quenched-in resistivity extrapolated to the melting point were very different, nearly lo-” and lo-’ O.m. In the FIM observations, the sample was dissected by pulsed-field evaporation, and the image was photographed after every pulse on a cine film. The film was then scanned to identify and count the quenched-in defects. Control observations of well-annealed samples served to determine the background. About 2 x 1O6 atomic sites were observed in the quenched samples and l o 6 in control samples. The following vacancy-formation parameters ~, have been obtained: HF = 3.6 eV, S , = 3.2kB, cmP= 3 ~ 1 0 - and p, = 7 ~ 1 O.m. 0 ~ ~ The authors have summarized some important conclusions, namely: (i) tungsten has a strong tendency to retain interstitial impurities, which interact strongly with point defects; (ii) due to the relatively low migration enthalpy, the vacancies migrate rather rapidly and tend to annihilate during quenching; (iii) the vacancies form their own sinks (voids) during quenching by heterogeneous precipitation with impurities; (iv) vacancy losses during quenching have complicated to a greater or lesser degree nearly all the investigations of quenched tungsten that have been carried out. Doyama et al. (1978) observed FIM images at elevated temperatures of the samples. The images were obtained by means of evaporated atoms of the samples, and this approach appeared not to be suitable for determining equilibrium defect concentrations.

216

9. Other methods

9.4. Summary Perturbed angular correlation of y-quanta is a tool capable of discrimination of various defects. The main problem concerning this technique is posed by the high mobility of point defects at high temperatures, which may cause a decrease of the vacancytrapping probability and loss of the defect-induced signal. 0

Mossbauer spectroscopy senses point defects in the crystal lattice. However, no quantitative data on equilibrium defect concentrations have been found by this technique. This conclusion relates also to the NMR technique. The vacancy-formation enthalpies derived in measurements of thermopower, thermal conductivity, and thermal diffusivity do not contradict data obtained by basic techniques employed in studies of point defects. 0

Electron microscopy and field ion microscopy are applicable only to quenched samples and thus involve all the drawbacks peculiar to quenching experiments. In gold, quenched-in vacancy concentrations observed by electron microscopy correspond to concentrations less than the differential-dilatometry value. In tungsten, the estimated vacancy concentration at the melting point is two orders of magnitude less than the calorimetric value. From field ion microscopy, vacancy concentrations in platinum and tungsten are much smaller than the calorimetric values.

Chapter 10

Equilibration of point defects 10.1. Role of internal sources (sinks) for point defects 10.2. Electrical resistivity 10.3. Specific heat

218 219 220

Enhancement of modulation frequencies. Relaxation phenomenon in tungsten and platinum.

10.4. Positron annihilation

229

Relaxation phenomenon in gold. Slow equilibration in some intermetallics.

10.5. Equilibration times from relaxation data

233

Comparison of relaxation times from various techniques. Are the relaxation times consistent with the vacancy origin of relaxation?

10.6. Summary

236

217

218

10. Equilibration of point defects

10.1. Role of internal sources (sinks) for point defects Observations of relaxation phenomena caused by the vacancy equilibration, i.e., the approach to equilibrium after a rapid change of temperature, were proposed and discussed long ago (Jackson and Koehler 1960; Korostoff 1962; Koehler and Lund 1965). The method was considered to be the most reliable one to separate vacancy contributions to physical properties of metals. The characteristic time of setting up equilibrium vacancy concentration depends on the density of internal sources (sinks) for the vacancies. This relaxation time z is proportional to the squared mean distance L between the sources (sinks) and inversely proportional to the coefficient of self-diffusion of the vacancies, D,: z = A L 2 / D , , where A is a numerical coefficient depending on the geometry of the sources (sinks). The relaxation times in various samples thus may be very different. Quenching experiments have shown that this difference amounts to several orders of magnitude. Moreover, the relaxation time depends on the pre-history of the sample. It is therefore useful to simultaneously observe the relaxation in various physical properties, e.g., in specific heat and in thermal expansivity or electrical resistivity. Pure and well-prepared samples having low dislocation densities and long relaxation times would enable one to observe the phenomena at higher temperatures, where the vacancy contributions become larger. The accuracy and reliability of the results thus could be improved.


219

10.2. Electrical resistivity In studies of the vacancy equilibration through electrical resistivity, the samples are rapidly heated up to a selected temperature, kept at this temperature for a certain time and then quenched. The quenched-in resistivity is determined as a function of the exposure at the high temperature. This technique is very sensitive permitting studies of the vacancy equilibration at moderate temperatures where the relaxation times are sufficiently long. After a rapid change of temperature, the vacancy concentration in the sample, c , follows the approximate relation ( c - c o ) / ( c , - co)

1

-

exp(-t/r).

(10.1)

Here co is the initial vacancy concentration, c, is the equilibrium vacancy concentration at the final temperature, and z is the relaxation time. When co is negligible, then

c

c,[l

-

exp(-t/z)].

(10 . 2 )

Seidman and Balluffi (1965) studied the vacancy equilibration in gold using two methods of measurements. First, a thin single-crystal slab was rapidly heated up by a jet of compressed hot air into 875-920°C range, held for a short period and then quenched. The quenched-in vacancies were detected by precipitating them as vacancy tetrahedra observable by a transmission electron microscope. Second, polycrystalline gold foils were electrically heated to a high temperature and then quenched. The extra resistivity was measured versus the exposure at the high temperature. The results have indicated that free dislocations were the predominant sources of thermally

220


generated vacancies. The half times of the vacancy equilibration were found to be 80 ms at 653OC and 10 ms at 878OC. Heigl and Sizmann (1972) have carried out a similar investigation on platinum. The wire sample was raised to a quench temperature by a discharge of a capacitor at the heating rate of about lo6 K.s-’, kept at this temperature for an adjustable time and then quenched. Relatively long equilibration times, of the order of s, were obtained for temperatures 800-950°C, far below the melting point. At these temperatures, a high efficiency of quenching is expected. The estimated extra resistivity at the melting point was 1 . 3 ~ 1 0 - *a.m. This figure is much larger than that from the most of quenching experiments 0 ~ ~ but is quite comparable with the equilibrium value, 2 . 4 ~ 1 c2.m (Kraftmakher and Lanina 1965). In studies on aluminum (On0 and Kin0 1978), the temperature of the sample was raised rapidly from an initial temperature, 16OoC, to a selected higher temperature, from 31OoC to 40OoC. Then the sample was quenched, and the quenched-in resistivity was measured in liquid helium. The results have shown the kinetics of the vacancy equilibration at high temperatures.

10.3.Specific heat The first attempts to observe the relaxation phenomenon in specific heat of metals were undertaken using modulation frequencies up to l o 3 Hz. No relaxation has been found in platinum (Seville 1974). Modulation measurements on a gold wire by Skelskey and Van den Sype (1974) have shown a frequency dependence of the quantity clR’, the ratio of the heat capacity of the sample to the temperature derivative of its resistance (Fig. 10.1).

10. Equilibration ofpoint defects

1

221

10 100 1000 modulation frequency (Hz)

Fig. 10.1. Frequency dependence of the ratio of specific heat to temperature derivative of resistance for Au at 1164 K (Skelskey and Van den Sype 1974).

This ratio appears whenever temperature oscillations are measured through oscillations of the sample’s resistance. The increase in this quantity at higher frequencies is explainable if the relative vacancy contribution to the temperature derivative of resistance is larger than to the specific heat. The measurements were carried out at a single temperature, 1164 K, so that the result could not be confirmed by the temperature dependence of the phenomenon.

Enhancement of modulation frequencies To search for relaxation phenomena in high-temperature specific heat of metals and alloys, a method of modulation measurements at frequencies of the order of l o 5 Hz has been developed

222


(Kraftmakher 1981). A high-frequency current slightly modulated by a low-frequency voltage heats a wire sample (Fig. 10.2).

lock-in amplifier

modulator and amplifier

-

high frequency amplifier

frequency synthesizer 4 A

frequency converter

heterodyne + +

frequency converter

Fig. 10.2. Block diagram of setup to observe relaxation in hightemperature specific heat of metals and alloys (Kraftrnakher 1981). Temperature oscillations of a low and a high frequency are created in the sample simultaneously, and the ratio of corresponding specific heats is measured directly.


223

High- and low-frequency temperature oscillations thus occur in the sample simultaneously. They are detected by a photomultiplier. The low-frequency component of the output signal proceeds to a lock-in amplifier. The high-frequency component selected by a resonant circuit is measured using frequency conversion and lock-in detection. An auxiliary frequency converter provides the necessary reference voltage for the lock-in detector. A plotter records a signal proportional to the difference between the amplitudes of the high- and low-frequency temperature oscillations. The measurements start at middle temperatures, where the nonlinear increase in the specific heat is negligible and no relaxation is expected. At these temperatures, the signal is adjusted to be zero. At any mean temperature of the sample, the difference between the specific heats corresponding to the two frequencies is measured.

Relaxation phenomenon in tungsten and platinum The measurements were carried out on commercial tungsten wires 8 pm thick and on tungsten filaments of vacuum incandescent lamps, 10 to 20 pm in diameter. The high frequency of the temperature oscillations was 3 ~ 1 0Hz ~ (Kraftmakher 1985). The character of the temperature dependence of the effect was always within the expectation (Fig. 10.3). This observation gained no recognition. For instance, Trost et al. (1986) concluded that “on the basis of the information available at present we cannot exclude with certainty that the observed effect is partly or even entirely due to the experimental procedure and hence not intrinsic.” In the case of platinum, the high frequency was 5 ~ 1 Hz 0 ~ (Kraftmakher 1990). The samples were cut from a 10-pm platinum foil. Due to the lower melting temperature, the power heating the sample and the amplitude of the power oscillations becomes much smaller. This results in decrease of the


224

temperature oscillations and, consequently, of the applicable modulation frequencies. The observed relaxation appeared to be in agreement with the nonlinear increase in the specific heat. As in the case of tungsten, the scatter of the experimental points increases at temperatures where X = m is close to unity. This is quite explainable because only in this range the relaxation strongly depends on X.

1.06

-

1.04

1.oo

0.5

0.6

0.7

0.8

0.9

1.o

Fig. 10.3. Ratio of specific heats measured at a low and a high frequency of the temperature oscillations in W and Pt (Kraftmakher ~ 5 ~ 1 0Hz, ~ 1985, 1990). The high frequency was 3 ~ 1 0and respectively.

Enthalpies of the vacancy formation in metals are nearly proportional to the melting temperatures, T,. This means that


225

the ratio of the vacancy contribution to the specific heat at a given temperature, AC, to its value at the melting point, ACM, should be a common function of the ratio t = TM/T for all metals:

where K = H,lkBT,. This relation allows one to compare the relaxation phenomena observed in both metals (Kraftmakher 1992b, 1994b). The effect in platinum, even at a lower modulation frequency, was observed much closer to the melting point than in tungsten. In platinum, the maximum of the relaxation is achieved at T; 0.85TM, while in tungsten at T E 0.7TM. This is probably due to a lower dislocation density in the platinum samples. However, no direct evidence exists that the phenomenon originates from the vacancy equilibration. A question therefore arises how to check this conjecture. A simple experimental approach has been proposed for this purpose (Kraftmakher 1996a, 1998a). The relaxation phenomenon should be observed during a period of time including a quench and subsequent anneal of the sample. The main sources and sinks for vacancies are dislocations and, probably, vacancy clusters. Their density drastically increases after quenching, and a certain time is necessary to anneal the sample at the high temperature. If, while the relaxation is observed, to quench the sample and then return it to the initial temperature, the relaxation phenomenon may disappear. It should appear again after a proper anneal of the sample and recovery of its structure. Such an experiment would clearly show the nature of the relaxation. The experimental setup now is much simpler. The sample is heated by a DC current with two AC components added, of low and high frequency (Fig. 10.4). The temperature oscillations of both frequencies thus occur in the sample simultaneously. The sine voltage of the reference frequency created by a lock-in

226


amplifier is used for the high-frequency modulation. The radiation from the sample falls onto a photodiode. The output voltage of the photodiode is fed to a pre-amplifier and then to two channels tuned to the low and the high frequency of the temperature oscillations. The lock-in amplifier measures the in-phase and quadrature components of the high-frequency temperature oscillations. The quadrature component senses changes in the phase of the temperature oscillations. The low-frequency signal proceeds to a selective amplifier and then to an AC/DC converter.

Fig. 10.4. New setup for observing relaxation in specific heat of metals and alloys. Measurements are performed during a period including quench and subsequent anneal of the sample.

At various mean temperatures of the sample, the irradiation falling onto the photodiode is adjusted to maintain a constant magnitude of the low-frequency AC voltage. The output voltages from both channels are stored by a data-acquisition system. The phase shift in the temperature oscillations depends nonmonotonically on X. The phase measurements are very


227

important because they confirm that the phenomenon is caused by the relaxation in specific heat. The measured quantities are monitored during a period including a quench and subsequent anneal of the sample. The amplitude and the phase of the high-frequency temperature oscillations should not alter when the sample is quenched from a low temperature, where AC > 1) even after the quench. No changes are expected also at high temperatures, where the frequency of the temperature oscillations becomes insufficient to observe the phenomenon (X2 > 1 before the quench and X 2

0

0.02

0.1

1

x=OT

10

Fig. 10.5. Expected relaxation in specific heat as a function of X = 0 7 , for ACIC = 0.1. 1 - change in specific heat, 2 - shift in the phase of temperature oscillations.


229

10.4. Positron annihilation Relaxation phenomenon in gold By means of the positron-annihilation technique, Schaefer and Schmid (1989) studied the vacancy equilibration in gold. The authors have explained the advantages of this approach as follows: (i) the positron annihilation is sensitive to only vacancytype defects; (ii) it is applicable at high temperatures; (iii) due to the short lifetime of the positrons, fast processes can be studied. The formation and equilibration processes were considered to occur by the vacancy generation at dislocation jogs and the diffusion-controlled filling up of the bulk material. The sample was first heated up to a selected temperature by an electric current. Then a superimposed capacitor discharge heated up the sample rapidly, during 0.5 ms, to a higher temperature. The time of exposure at this temperature was subdivided into seven intervals, and within each interval the positron lifetime and the Doppler broadening of the y-line were measured (Fig. 10.6). After cooling the sample, the cycle repeated. The data were accumulated during 1O6 cycles. The sample's temperature rose from 500 to 600 K, from 680 to 800 K, and from 790 to 900 K. No change in the annihilation parameters was seen in the first case because the vacancy concentrations at these temperatures are negligible. At 800 and 900 K, the relaxation times were found to be 10.7 and 3.6 ms, respectively. With a transmission electron microscope, the dislocation density in the sample after the measurements was estimated to be 8 ~ 1 0 cm-*. ~ The authors pointed out that optimizing the efficiency of y-detection and using a positron source of higher activity can improve the statistical precision of the measurements. The available temperature range could be thus extended to higher and lower temperatures.

230


channels 1 2 3 4 5 6 7

I

time

6 n

I

a 0

cn

4 A

W

2 n

>

7

A

A

A

7

2

9OOK

A

l--

... -

A

0"

A

0

0

0

800K

-

0

I

600 K

0

10

20

30 ms

Fig. 10.6. Observation of vacancy equilibration in Au by the positronannihilation technique (after Schaefer and Schmid 1989).

231


Slow equilibration in some intermetallics Kummerle et al. (1995) observed the vacancy equilibration in Fe,Si. Long relaxation times made it possible to monitor changes in the positron mean lifetime after rapid changes of the sample's temperature (Fig. 10.7).

n

v)

a

W

160

850K - 6 2 0 K

140

time

E

120 290K - 6 2 0 K

Fig. 10.7. Relaxation of the positron mean lifetime in FegSi sample observed after rapid changes of its temperature, schematically (Kummerle et al. 1995).

Using positron annihilation, Wurschum et al. (1995) studied vacancy formation and equilibration in iron aluminides. High vacancy concentrations were found, corresponding to formation entropies in the range 5.7kB to 6.5kB. A very slow equilibration

232


has been observed after rapid cooling of the samples from 770 K to temperatures in the range from 623 to 673 K. Schaefer et al. (1997) studied formation and migration of thermal vacancies in FeAl through the positron annihilation and internal friction of quenched samples. High vacancy concentrations at relatively low temperatures and low diffusivities of thermal vacancies have been found. Recently, Schaefer et al. (1999) studied relaxation in the length of FeAl and NiAl compounds after rapid changes in the sample's temperature (Fig. 10.8).

Fig. 10.8. Relaxation of the sample's length observed after rapid changes of the sample's temperature, schematically (Schaefer et al. 1999).


233

10.5. Equilibration times from relaxation data Comparison of relaxation times from various techniques It is commonly agreed that the quenched-in resistivity and changes in the positron-annihilation parameters are certainly caused by vacancies. At the same time, the relation between the nonlinear increase in specific heat and vacancy formation gained no recognition. It is therefore useful to compare results of all the relaxation experiments, even for various metals (Fig. 10.9). The relaxation time z depends on the density of sources (sinks) for vacancies. The difference between the relaxation times in gold obtained by measurements of the electrical resistivity (Seidman and Balluffi 1965) and by positron annihilation (Schaefer and Schmid 1989), which amounts to 50 times, is therefore quite understandable. Various densities of internal defects are probably responsible for such a difference in platinum samples. The migration enthalpies HIM were considered to be proportional to the melting temperatures ( H , = 7k,T,). The straight lines in the graph correspond to z = B e x p ( H M / k B T ) , where B is a proportionality factor different for various samples. Assuming a temperature-independent density of sources (sinks) for vacancies, the temperature dependence of the relaxation time is thus available from a single experimental value. In tungsten, the relaxation times have been deduced to be 5 ~ 1 O -s ~at 2600 K and 2 ~ 1 O -s ~at 2700 K. The short relaxation times are consistent with the well-known fact that dislocation densities in refractory metals are much higher than in metals such as gold or platinum. The comparison of the data is thus favorable for the conclusion that all the phenomena are of a common nature.

234


100 n

i!

10

E

1

v

.* c

.-0

5 X

-2

0.1

(CI

0.01

1E-3

A

I

.o

1.2

1.4

1.6

1.8

TM/T Fig. 10.9. Temperature dependence of equilibration times from various measurements: 1 - Au, resistivity (Seidman and Balluffi 1965); 2 - Al, current noise (Celasco et al. 1976); 3 - Au, positron annihilation (Schaefer and Schmid 1989); 4 - Pt, specific heat (Kraftmakher 1990); 5 - W, specific heat (Kraftmakher 1985). The straight lines correspond to HM = 7k,TM and constant densities of sources (sinks) for vacancies.


235

The directions of further investigations of the relaxation phenomenon are evident: (i) a use of pure and well-prepared samples, which would enable one to observe the phenomenon over a wider temperature range; (ii) measurements on metals in which low dislocation densities are obtainable; (iii) observations of the relaxation during a quench and consequent anneal of the samples.

Are the relaxation times consistent with the vacancy origin of relaxation? The very short relaxation times found in tungsten may pose a question whether they are consistent with the vacancy origin of the relaxation. Therefore, it is worthwhile to mention that the time necessary to create interstitials is much shorter. For example, Fujiwara et al. (1997)have shown that Frenkel pairs in KCI and RbCl appear in a stage that terminates within a few picoseconds after excitation. The relaxation times found in calorimetric measurements are indeed very short. By extrapolation to the melting point, the s in platinum relaxation times are expected to be about and lop8 s in tungsten. To explain the latter value, it is useful to recall the observation by Yoshida and Kiritani (1967)that small vacancy clusters may act as sources and sinks for monovacancies.

236


10.6. Summary Observations of the point-defect equilibration provide a unique possibility to reliably separate defect contributions to physical properties of metals. The only obstacle for such observations is the very short equilibration time at high temperatures. Owing to the high sensitivity of measurements of quenched-in electrical resistivity, observations of the vacancy equilibration through the extra resistivity are possible at relatively low temperatures, when the equilibration times are sufficiently long. This relates also to the positron-annihilation techniques. Observations of the vacancy equilibration through the relaxation in specific heat are most informative because they immediately show equilibrium vacancy concentrations. However, such measurements are to be made at sufficiently high temperatures, when the equilibration times are very short. A very slow vacancy equilibration has been observed in some intermetallics by the positron-annihilation technique and by dilatometric studies. Along with slow equilibration, the measurements have shown very high concentrations of thermal vacancies, of the order of lo-*. Relaxation times obtained in various experiments are very different. Nevertheless, they seem to be consistent with the supposition that all the phenomena result from a common origin, i.e., the vacancy equilibration.

Chapter 11

Parameters of vacancy formation 11.1. Equilibrium concentrations of point defects 11.2. Point defects in high-melting-point metals 11.3. Temperature dependence of formation parameters 11.4. Summary

237

238 245 248 253

238

11. Parameters of vacancyformation

11.1. Equilibrium concentrations of point defects Calorimetric measurements cannot reveal what a type of point defects prevails at high temperatures. The increase in thermal expansivity confirms the vacancy (or divacancy) formation. Quenching experiments, including microscopic observations, and positron-annihilation data confirm this conclusion. Equilibration times from relaxation measurements are compatible with the vacancy (or divacancy) nature of the defects. Except some cases, the formation enthalpies obtained by various techniques are in reasonable agreement with each other and with theoretical calculations. On the other hand, Gordon et al. (1996) have deduced the interstitial concentration near the triple point of krypton to be 7 ~ 1 0 -A ~ .positive evidence for interstitials was obtained also for Cu, Ag, Au, and Al. The authors have concluded that their results confirm the interstitialcy model of condensed matter proposed by Granato (1992). Koshkin (1998, 1999) has developed the concept of unstable Frenkel pairs to explain the properties of metals at high temperatures and the discrepancy between defect concentrations determined by various techniques. In a first approximation, the formation enthalpies are of about one half of the enthalpies of self-diffusion and one third of the enthalpies of vaporization. Contradictions in equilibrium vacancy concentrations from various experimental techniques are very strong amounting to one or even two orders of magnitude. The largest vacancy concentrations have been found in refractory metals from nonlinear increase in their specific heat. From the practical point of view, equilibrium concentrations of point defects at high temperatures are of utmost importance. Unfortunately, just this point is the weakest one until today.

239

11. Parameters of vacancy formation

The data obtained by various techniques (Table 11.I) clearly show that the situation is far from optimistic expectations. Such expectations arose every time when a new experimental approach appeared, e.g., differential dilatometry or the positronannihilation techniques. All quenching experiments may result in an underestimation of equilibrium defect concentrations. The data presented in the last three columns thus show the lower limits for equilibrium defect concentrations.

Table 11.1 Point-defect concentrations in metals at the melting points, cmP,from various techniques. L - linear extrapolation of thermal expansivity, D - differential dilatometry, C - specific heat, Q - volume and lattice parameter of quenched samples, E - stored enthalpy, M - microscopic observations of quenched samples.

cmp Metal

Na Zn Sn Cd Pb

I u Ag

Au cu Pt Mo W

L

20; 33 6; 14 24 9 20 17; 24 14 13 70; 80 190 230

D

6-9 3; 5 50.3 5 1.7 3-11 1.7; 5.2 7 2; 7.6

C

Q

E

M

30; 76 23 13 40

20; 23 11; 22

40 50 100 290; 430 210; 340

6 7 26

5; 20

3

3 1; 3

240

1I . Parameters of vacancy formation

Below, data are presented on equilibrium point defects in metals obtained by various techniques (specific heat, differential dilatometry, and extra electrical resistivity). Cs. Only a calorimetric value of the equilibrium defect concentration at the melting point is available, cmp= ~ x I O - ~ (Martin 1965). Rb. From the calorimetric data, cmp = 2 . 5 ~ 1 0 - (Martin ~ 1965). K. Two values of c have been found from specific-heat measurements, 4 . 8 1~O-3mrCarpenter 1953) and 1 . 4 1~OP3 (Martin 1965). Na. Two calorimetric values of cmpare 7 . 6 ~ 1 0 -(Carpenter ~ 1953) and 3 ~ 1 0(Martin ~ ~ 1967). A much smaller value, 7 . 5 lop4, ~ has been obtained by differential dilatometry (Feder and Charbnau 1966). In. Only a calorimetric value is available, cmp= 5 ~ 1 0 - ~ (Kramer and Nolting 1972). Li. The nonlinear increase in the specific heat observed by Carpenter et al. (1938) is too small to reliably evaluate the defect contribution. From differential dilatometry, the vacancy concentration at the melting point is 4 . 4 ~ 1 0 -(Feder ~ 1970). Sn. The calorimetric value is cmp= 1 . 3 ~ 1 0 -(Kramer ~ and Nolting 1972). From differential-dilatometry measurements, Balzer and Sigvaldason (1 979a) have concluded that cmps 3x10- 5 . Bi. From differential dilatometry, cmP= 6 . 2 1~OP4 (Matsuno 1977). Cd. Two values of cmp from differential dilatometry are 6 . 6 ~ 1 0(Feder ~ ~ and Nowick 1972) and 4 . 5 ~ 1 0 -(Janot ~ and George 1975). From equilibrium measurements, the extra electrical resistivity at the melting point is 5x10-’ L2.m (Hillairet et al. 1969). The point-defect formation in cadmium was reviewed by Seeger (1991).

I I . Parameters of vacancyformation

24 1

Pb. The calorimetric values of cmpare 2x10-3 (Pochapsky 1953) and 2 . 3 1 ~0-3 (Kramer and Nolting 1972). Differential dilatometry provided values 1 . 5 1~0-4 (Feder and Nowick 1958) and 1 . 7 1~0-4 (Feder and Nowick 1967). The defect-induced resistivity is 1.9~10-’ R.m from equilibrium measurements and 1 . 4 ~ 1 0 R.m - ~ ~ from a quenching experiment (Leadbetter et al. 1966). Zn. The calorimetric value of cmpis 2 . 3 ~ 1 0 -(Kramer ~ and Nolting 1972). Two results from differential dilatometry are 3 ~ 1 0 -(Current ~ and Gilder 1977) and 4 . 9 ~ 1 0(Baker ~~ and Sigvaldason 1979b). The point-defect formation in zinc was reviewed by Seeger (1991). Sb. From calorimetric measurements, cmp = I.2x I0-3 (Kramer and Nolting 1972). Mg. From differential dilatometry, cmp= 7 . 2 ~ 1 0(Janot ~ ~ et al. 1970). The extra electrical resistivity at the melting point is 1.9~10-’ R.m from equilibrium measurements (Mairy et al. 1967) and 6 ~ 1 0 -R.m l ~ from a quenching experiment (Tzanetakis et al. 1976). Al. The calorimetric values of cmp range from I . I x I O - ~ (Shukla et al. 1985) to ~ . ~ X I O(Kramer - ~ and Nolting 1972). Differential dilatometry has shown the vacancy concentration from 3 ~ 1 0 -(Feder ~ and Nowick 1958) to I . I x I O - ~ (Nenno and Kauffman 1959). The lower limit of the vacancy concentration determined from changes in the enthalpy during the vacancy equilibration was estimated to be 6 ~ 1 (Guarini 0 ~ ~ and Schiavini 1966). The equilibrium extra resistivity at the melting point is about 3xIO-’ R.m, which does not contradict the above vacancy concentrations. The highest quenched-in extra resistivity, (2-5)~10-’ R.m, has been obtained by BabiC et al. (1970). Ag. By means of differential dilatometry, Simmons and Balluffi (1960~)have found cmp = 1 . 7 ~ 1 0 - ~A. much higher , been reported by vacancy concentration, cmp = 5 . 2 ~ 1 0 - ~has Mosig et al. (1992). In equilibrium measurements, the extra

242

I I . Parameters of vacancyformation

resistivity at the melting point was found to be l . I x l O - ’ R.m (Ascoli et al. 1970) and 4xIO-’O R.m (Schule and Scholz 1982). Au. All experimental techniques now available were employed in studies of this metal. The equilibrium vacancy concentration at the melting point is 7 . 2 1~0-4 from differential dilatometry (Simmons and Balluffi 1962) and 4x10w3 from specific-heat measurements (Kraftmakher and Strelkov 1966a). From the stored enthalpy of quenched samples, two cmpvalues ~ are known, 5 ~ 1 (DeSorbo 0 ~ ~ 1958, 1960) and ~ x I O -(Pervakov and Khotkevich 1960). The extra resistivity at the melting point is nearly lo-’ R.m from quenching experiments (e.g., Mori et al. 1962; Lengeler 1976) and 5.3~10-’ R.m (Ascoli et al. 1970) and 1.4~10-’ R.m (Schule and Scholtz 1982) from measurements in equilibrium. Cu. The Simmons-Balluffi value of cmPis 1 . 9 ~ 1 0 - Trost ~. et . al. (1986) have confirmed this result (cmp= ~ . I x I O - ~ ) However, the value obtained by the Hehenkamp’s group appeared to be 7 . 6 ~ 1 0 - ~The . calorimetric value is cmp= 5x10w3 (Kraftmakher 1967~). Equilibrium measurements of the extra resistivity provided values from lo-’ (Hehenkamp and Sander 1979) to 2.3~10-’ R.m (Ascoli et al. 1970), while the quenched-in resistivity is in the range from 10-loto lo-’ R.m (Furukawa et al. 1976; Bourassa and Lengeler 1976; Berger et al. 1979). Ni. From specific-heat data, cmP= 1.9~10-*,while the extra resistivity at the melting point is 2 . 6 ~ 1 0 -R.m ~ (Glazkov 1987). The quenched-in resistivity at the melting point is in the range from lo-’’ S2.m (Wycisk and Feller-Kniepmeier 1976) to lo-’ i2.m (Mamalui et al. 1968). Fe. Seeger (1998b) has reviewed vacancy formation in a-iron. He has concluded that the formation enthalpy is in the range 1.61 eV to 1.75 eV. From self-diffusion data, the sum of the vacancy formation entropy and the vacancy migration entropy has been estimated to be 5kB, while the upper limit for the formation entropy should be 2.5kB.

11, Parameters of vacancy formation

243

Pd. Miiller and Cezairliyan (1980) observed a strong nonlinear increase in the specific heat. Ti. Only a value from specific-heat measurements is available, cmP= 1 . 7 ~ 1 0 (Shestopal -~ 1965). Th. Wallace (1960) observed a nonlinear behavior of the high-temperature specific heat. Pt. From the calorimetric measurements, cmp = (Kraftmakher and Lanina 1965). The relaxation phenomenon in the specific heat (Kraftmakher 1990) appeared to be in agreement with the nonlinear increase in the specific heat. No differential-dilatometry data were obtained for platinum, as well as for other high-melting-point metals. The extra resistivity at the melting point determined in quenches from relatively low temperatures has been found to be 1 . 5 ~ 1 0 R.m - ~ (Jackson 1965) and 1 . 3 ~ 1 0 -R.m ~ (Heigl and Sizmann 1972). Thus, they are - ~ (Kraftmakher consistent with the equilibrium value, 2 . 4 ~ 1 0 R.m and Lanina 1965). In other quenching experiments, the extra resistivity was much lower. Zr. Only a calorimetric value is available, cmp = 7 ~ 1 0 - ~ (Kanel' and Kraftmakher 1966). Cr. The calorimetric value is cmp = 6 ~ 1 (Kirillin 0 ~ ~ et al. 1967; Chekhovskoi 1979). V. Cezairliyan et al. (1974) and StanimiroviC et al. (1999) observed a strong nonlinear increase in the specific heat. Rh. The calorimetric value of c is while the extra mp-8 resistivity at the melting point is 1 . 4 ~ 1 0 R.m (Glazkov 1988). Nb. Two values of cmp from specific-heat measurements are known, 1 . 2 ~ 1 0 - ~(Kraftmakher 1963b) and 2 . 7 ~ 1 0 - ~ (Chekhovskoi and Zhukova 1966). The extra resistivity obtained ~ by a quench from 2600 K was very small, about 3 ~ 1 0 - lR.m (Schwirtlich and Schultz 1980b). Mo. Three calorimetric values of cmpare as follows: 4 . 3 ~ 1 0 - * (Kraftmakher 1964), 2 . 9 ~0-2 1 (Chekhovskoi and Petrov 1970), and 6 . 3 ~ 1 0 -(Choudhury ~ and Brooks 1984). From quenching

244


experiments, the extra electrical resistivity at the melting point ranges from 5xlO-’’ R.m (Schwirtlich and Schultz 1980) to 5xIO-’O R.m (Mamalui et al. 1976). Ta. Only a calorimetric value is available, cmp = 8x10p3 (Kraftmakher 1963a). Re. Taylor and Finch (1964) observed a strong nonlinear increase in the specific heat. W. Two values from calorimetric measurements have been obtained, cmp = 3 . 4 ~ 1 0 -(Kraftmakher ~ and Strelkov 1962) and = 2.1x10p2 (Chekhovskoi 1981). The relaxation phenomenon cmP in the specific heat appeared to be in agreement with the nonlinear increase in the specific heat (Kraftmakher 1985). From quenching experiments, the extra resistivity at the melting point ranges from 2xIO-‘O O.m (Schultz 1964) to 2xIO-’ O.m (Park et at. 1983). The vacancy concentrations from calorimetric measurements presented above have been calculated by the authors themselves. Some data have been derived from measurements of the high-temperature enthalpy. Such data have been obtained for chromium (Kirillin et al. 1967; Chekhovskoi 1979), niobium (Chekhovskoi and Zhukova 1966), molybdenum (Chekhovskoi and Petrov 1970), and tungsten (Chekhovskoi 1981). In several cases, when a nonlinear increase in specific heat was observed (Pd, Th, V, Re), the authors did not consider it to be caused by point-defect formation. Nevertheless, it was worthwhile to mention these observations.


245

11.2. Point defects in high-melting-point metals The situation in refractory BCC metals is most difficult to understand. The enthalpy of vacancy formation in tungsten (3.15 eV) and the equilibrium vacancy concentration ( 3 . 4 ~0-2 1 at the melting point) have been deduced from calorimetric data (Kraftmakher and Strelkov 1962). For tantalum, niobium, and molybdenum, the vacancy-formation parameters also have been evaluated. The strong nonlinear increase in the specific heat of refractory metals has been recognized only after the measurements by Cezairliyan and coworkers. This recognition relates only to the specific heat, while the origin of this phenomenon remains under debate. Schultz (1964) has performed the first successful quenching experiment on tungsten. The extra resistivity at the melting point was found to be 2xlO-’O S2.m. This figure became larger in further quenching experiments, and now the largest value is 2xlO-’ R.m (Park et al. 1983). Despite very different values of the extra resistivity, the derived formation enthalpies exhibit quite moderate scatter, ranging from 3.1 to 3.67 eV. Quenching experiments were successful also with molybdenum but the extra resistivity obtained is several times smaller than in tungsten. Later, the equilibrium vacancy concentration in tungsten has been evaluated from its thermal expansivity (Kraftmakher 1972). A linear extrapolation from intermediate temperatures was used to separate the vacancy contribution. With the formation volume VF = 0.5Q the vacancy concentration appeared to be about 1.5 times smaller than the calorimetric value. Maier et al. (1979) have determined the enthalpies of vacancy formation in refractory metals by positron annihilation. The formation entropies were postulated to be S , = 2kB. With this figure, the equilibrium vacancy concentrations at the melting

246

I I . Parameters of vacancy formation

points were calculated to be (vanadium, niobium), 3 . 5 ~ 1 0 - ~ (tantalum), 4x10p5 (molybdenum), and 2 . 5 ~ 1 0(tungsten). ~ ~ Later, Trost et al. (1986) have pointed out that the detrapping from monovacancies in these metals is so pronounced that the reported formation enthalpies may pertain mainly to divacancies. Further, the relaxation phenomenon in specific heat caused by the vacancy equilibration was observed in tungsten (Kraftmakher 1985). The relaxation is in good agreement with the nonlinear increase in the specific heat. The rapid-heating determinations of the enthalpy of tungsten (Hixson and Winkler 1990; Pottlacher et al. 1993) provide a confirmation of the vacancy origin of the nonlinear increase in the specific heat. Nowadays, the situation looks as follows. (1) The low extra resistivity of quenched molybdenum samples may result from the well-known drawbacks peculiar to all quenching experiments. The failure of the quenches on tantalum and niobium probably has the same reason. Positron annihilation may help to determine what a fraction of equilibrium vacancies survives in the sample after quenching. It would be very useful to understand how the vacancy-induced resistivity depends on temperature. (2) Above 2500 K, the errors in measurements of electrical resistivity are of the order of 1%. They are caused mainly by uncertainties in temperature measurements and in thermalexpansion data. In the case of tungsten and niobium, the defect contributions to the resistivity became evident after introducing corrections for thermal expansion. The situation in other refractory metals is probably similar. However, the above correction is insufficient to resolve the existing disagreement. With p, values of the order of low5 t2.m that are commonly accepted for refractory metals, the estimated defect concentrations remain much smaller than those from the nonlinear increase in specific heat. A possible reason of this difficulty may be a strong temperature dependence of pv (see


247

Schule 1998ab). More careful measurements of the resistivity are desirable, as well as new calculations and experimental determinations of the influence of point defects on the resistivity, including the temperature dependence of pv. A straightforward method exists to determine vacancy contributions to electrical resistivity. Under very rapid heating, vacancies have no time to appear, and the resistance should correspond to a vacancy-free crystal. With gradually changing the heating rate, the defect contribution and the equilibration time could be determined. The only necessary precaution is to avoid a superheat of the samples. Therefore, the electrical resistivity should be measured at a selected premelting temperature rather than at the melting point. (3) For refractory metals, the vacancy contributions to specific heat become visible at temperatures above about two thirds of the melting temperature. The vacancy-formation parameters deduced from calorimetric measurements should be compared with the self-diffusion data in this temperature range. However, the enthalpies of self-diffusion in the high-temperature region are larger than the sum of the enthalpies of vacancy formation and migration. At the same time, the vacancyformation parameters are consistent with the enthalpies of selfdiffusion at lower temperatures. To understand this result, one may suppose that vacancies dominate also at high temperatures. However, their contribution to the self-diffusion becomes smaller than that of other defects, divacancies or interstitials, having lower concentrations but higher mobilities. (4) Along with equilibrium vacancy concentrations based on calorimetric measurements, only a few data are available for high-melti ng-point metals. Quenched-i n vacancy concentrations have been determined by field ion microscopy only in platinum and tungsten. In platinum, the estimated vacancy concentration at the melting point was 3 ~ 1 0 -(Seidman ~ 1973). The extra resistivity at the melting point was estimated to be 2xIO-’ R.m, i.e., several times smaller than in other quenching experiments.


24%

In tungsten, the vacancy concentration at the melting point is from electron microscopy (Rasch et al. 1980) and 3 ~ 1 0 - ~ from field ion microscopy (Park et al. 1983). Nevertheless, the p, values obtained in the two experiments, 6 . 3 ~ 1 0 -R.m ~ and 7 ~ 1 R.m, 0 ~ are ~ in good agreement.

11.3. Temperature dependence of formation parameters The temperature derivatives of the enthalpy and entropy of vacancy formation must satisfy the thermodynamic relation

(dH,ldT),

=

T(dS,/dT),.

(11.1)

Two approaches are thus possible: (i) the formation enthalpy and entropy do not depend on temperature, or (ii) they depend on temperature in accordance with the above equation. The starting point in the second concept is that the atom binding in a crystal lattice weakens with increasing temperature, so the relaxation of the atoms around a vacancy increases. The formation entropy must therefore increase with increasing temperature, as well as the formation enthalpy. Gilder and Lazarus (1975) and de Vries (1975) used this concept to explain curvatures in the Arrhenius plots for self-diffusion. Experimental data on vacancy concentrations are not sufficiently accurate to reveal the temperature dependence of the Gibbs free energies of vacancy formation. In a first approximation, one can assume a linear temperature dependence of the formation entropies, S , = aT. This relation seems to be quite acceptable for temperatures far above the Debye temperature. Then

H,

=

H , + aT212,

(1 I.2)

I I . Parameters of vacancy formation

G,

=

H, - aT2/2,

249 (11.3)

where H, is the formation enthalpy at the absolute zero tem perat ure. Earlier, the enthalpies and entropies of vacancy formation were considered constant over the whole temperature range where the vacancy contributions to specific heat were measured, above about 0.STM. The derived parameters, H,* and S,*, should be related to temperatures of about 0 . 9 T M . A simple relation thus appears:

The enthalpies and the Gibbs free energies of vacancy formation can be written in a form more suitable for practical use:

HF

=

H, + A(T/TM)2,

( 1 1.5)

GF

=

H, - A(T/TM)2,

( 1 1.6)

where A = a T M 2 / 2 . It is more convenient to employ parameters HolkBTM and A / k B T Mavailable from calorimetric measurements (Table 1 1 . 2 ) . Considering these parameters to depend on the melting temperature, one can try to examine the origin of the difference between low-melting-point and high-melting-point metals. The first parameter is almost the same for all metals, regardless of their crystal structure and melting temperature, and equals 7 f 1 (Fig. 1 1 . 1 ) . In contrast, the AlkBT, ratio is nearly a linear function of the melting temperature: A l k B T M = 1.5 + 5 . 5 ~ 1 0 - ~ T , . Earlier, the formation enthalpies were believed to be proportional to the melting temperatures (Mukherjee 1965; Kraftmakher and Strelkov 1966b; Doyama and Koehler 1976;


250

Hayashiuchi et al. 1982; Varotsos et al. 1986). Now we see that this statement is correct only for H,,, whereas the ratio ( H , - A ) / k , T , decreases with increasing the melting tem perature.

Table 11.2 Parameters of vacancy formation calculated from specific-heat data assuming a linear temperature dependence of the formation entropies.

CS

Rb K Na In Sn Sb

zn Sb Al AU

cu Ni Ti Pt

zr

cr Rh Nb Mo Ta W

302 312 336 371 430 505 600 693 903 933 1337 1358 1728 1940 2041 2125 2180 2237 2750 2896 3290 3695

0.28 0.31 0.23 0.255 0.425 0.455 0.435 0.61 1.13 0.87 1.0 1.05 1.4 1.55 1.6 1.75 1.68 1.9 2.04 2.24 2.9 3.15

4.95 5.5 2.55 3.1 3.95 3.8 2.3 4.1 7.8 4.5 3.15 3.7 5.4 5.15 4.5 4.6 6.3 5.25 4.15 5.7 5.45 6.5

0.22 0.24 0.20 0.21 0.35 0.38 0.38 0.50 0.85 0.71 0.84 0.86 1.04 1.16 1.24 1.37 1.15 1.45 1.60 1.60 2.20 2.22

0.07 0.08 0.04 0.06 0.08 0.09 0.07 0.14 0.34 0.20 0.20 0.24 0.45 0.48 0.44 0.47 0.66 0.56 0.55 0.79 0.86 1.15

0.29 0.32 0.24 0.27 0.43 0.47 0.45 0.64 1.19 0.91 1.04 1.10 1.49 1.64 1.68 1.84 1.81 2.01 2.15 2.39 3.06 3.37


251

Equilibrium vacancy concentrations at the melting points are

The quantity ( H , - A)/kBTM decreases from 5.3 for T , = 300 K to 3.5 for TM= 3700 K. These two values correspond to the vacancy concentrations 5x 1OP3 and 3x 1OP2, respectively.

0

1000 2000 3000 4000 melting temperature (K)

Fig. 11.1. Parameters HolkBTM (m) and AIkBTM (0) based on calorimetric data and a linear temperature dependence of formation entropies.

252


Individual properties of metals may modify the above rule but equilibrium vacancy concentrations at the melting point generally increase with the melting temperature. The temperature dependence of the Gibbs free energy of vacancy formation in copper has been calculated by Foiles (1994). Very weak temperature dependence was obtained from quasiharmonic and local harmonic approximations. On the contrary, Monte Carlo simulations have shown strong temperature dependence of the formation enthalpies. Moleculardynamics simulations in sodium (Smargiassi and Madden 1995) provided a similar result. The temperature dependence of the Gibbs free energy of vacancy formation in copper evaluated from the calorimetric data qualitatively agrees with the Monte Carlo simulations. Hence, one has to present the formation enthalpies as functions of temperature or to indicate the corresponding temperatures. The same is true for the formation entropies and volumes. All these parameters depend also on pressure. The temperature dependence of the formation enthalpies should be taken into account in evaluations of vacancy concentrations from calorimetric data. With temperature-dependent formation enthalpies, the Arrhenius plots for equilibrium vacancy concentrations should no longer be straight lines. Nowadays, the accuracy of the measurements is insufficient to reveal this phenomenon. Unfortunately, equilibrium vacancy concentrations are measurable in temperature ranges much narrower than those involved in measurements of self-diffusion.


253

11.4. Summary Calorimetric measurements cannot reveal what a type of point defects prevails at high temperatures. The increase in thermal expansivity confirms the vacancy (or divacancy) formation. Quenching experiments, including microscopic observations, and positron-annihilation data lead to the same conclusion. The relaxation times obtained from observations of the point-defect equilibration are compatible with the vacancy (or divacancy) nature of the defects. Generally, defect-formation enthalpies from various techniques are in reasonable agreement with each other. The only exceptions are some data on refractory metals. From the practical point of view, equilibrium concentrations of point defects are of utmost importance. Regretfully, just this point is the weakest one until today. In refractory metals, the vacancy-formation parameters are consistent with the enthalpies of self-diffusion at lower temperatures. To understand this result, one may suppose that vacancies dominate also at high temperatures. However, their contribution to the self-diffusion becomes smaller than that of other defects, divacancies or interstitials, having lower concentrations but higher mobilities. Contradictions in equilibrium vacancy concentrations from various experimental techniques are very strong amounting to one or even two orders of magnitude. The largest vacancy concentrations have been found in refractory metals from the nonlinear increase in their specific heat.

254


Calorimetric data are the only source of equilibrium vacancy concentrations in high-melting-point metals. It is very important to apply other methods for studying point defects in these metals. However, it may turn out that just calorimetric measurements provide the most reliable data on equilibrium vacancy concentrations in metals. 0

The temperature dependence of vacancy-formation parameters must be taken into account. Some theoretical calculations have shown that the Gibbs free energy of vacancy formation decreases with increasing temperature. This result supports high concentrations of equilibrium vacancies in metals.

Chapter 12

Discussion 12.1. Comparison of experimental techniques 12.2. Critical vacancy concentrations 12.3. Thermodynamic bounds for formation entropies 12.4. Effects of anharmonicity 12.5. Constant-volume specific heat of tungsten 12.6. Thermal defects in alloys and inter metaIIics 12.7. Self-diffusion in metals 12.8. Point defects and melting 12.9. How to determine vacancy contributions to enthalpy a proposal 12.10. Summary

255

256 257 259 261 265 272 275 278

281 286

256

12. Discussion

12.1. Comparison of experimental techniques The advantages of calorimetric determinations of equilibrium vacancy concentrations are clearly seen from a comparison of various techniques now available. Vacancy formation strongly affects specific heat of metals. An equilibrium vacancy concentration of lo-* causes an increase in the specific heat by about 10%. The same concentration leads to the A l / l - A d a value of about 0.3%. Simultaneous measurements of the bulk dilatation and of the lattice parameter allow one to reduce errors caused by uncertainties in the sample's temperature. However, this approach does not guarantee the high accuracy of the quantities to be measured. Internal voids and vacancy clusters acting as sources (sinks) for vacancies may reduce the changes in the outer volume of the sample and lead to an underestimation of equilibrium vacancy concentrations. To exclude errors in the temperature measurements and to compare various data, one could employ All1 and A d a values measured independently and extrapolated to the melting point. In comparison with positron annihilation, calorimetric measurements are much simpler and more straightforward. Furthermore, positron-annihilation data do not provide equilibrium vacancy concentrations. It turned out that even determinations of the formation enthalpies sometimes cause doubts. The reason for this may be positron trapping during thermalization, detrapping of captured positrons, and the temperature dependence of the specific trapping rate. The only serious disadvantage peculiar to the calorimetric measurements is an unknown specific heat of a hypothetical defect-free crystal. A separation of vacancy contributions seems therefore to be impossible (see, e.g., Seeger and Mehrer 1970; Hoch 1970, 1986; Monti and Martin 1988). Hayes (1986) has

12. Discussion

257

concluded that ”the distinction between point-defect formation and anharmonicity is not always clear-cut and it is not always possible to establish the difference using certain types of measurement alone, for example specific heat measurements.” The opposite viewpoint was also presented (Kraftmakher and Strelkov 1966b, 1970; Kraftmakher 1966b, 1973c, 1974, 1977ab, 1992b, 1994ab, 1996b, 1998ab; Choudhury and Brooks 1984). Fortunately, the separation of vacancy contributions may be based on observations of the vacancy equilibration. Relaxation measurements seemed to be too complicated and failed during a long time. Now, after the phenomenon was observed, this approach should be considered more optimistically. It is worthwhile to recall that specific-heat data provided high concentrations of equilibrium point defects in some ionic crystals and molecular solids. Kanzaki (1951) has found a strong nonlinear increase in the specific heat of AgBr and attributed it to point-defect formation. Beaumont et al. (1961) observed high concentrations of equilibrium vacancies in argon and krypton, 1 . 3 1~O-* at the triple points.

12.2. Critical vacancy concentrations New theoretical calculations of the entropies of vacancy formation in metals have appeared in the last decades. For example, Wautelet (1985) has supposed that vacancies perturb the delocalized phonon spectrum via a relation 0’ =

o(l

-

ac,).

(12.1)

Here o’and o are the perturbed and unperturbed vibration frequencies, and a is a constant. With this assumption, a simple relation has been obtained:

12. Discussion

258

c,

=

exp[3a/(l

-

ac,) -H,lk,T].

(12.2)

1000

I 100

hot

Na

W .

7

OK

tAuocU

10

8

9

10

11

12

Fig. 12.1. Line of critical vacancy concentrations (-) according to Wautelet (1985) and vacancy concentrations at melting points from calorimetric data (0).

With this approach, a critical temperature appears, above which the crystal becomes unstable. This temperature may be related to the melting point. The critical vacancy concentration c* satisfies the requirement dT/dc, = 0 , which leads to a2c*2 -

(3a2

+ 2a)c* +

1 = 0.

(12.3)

12. Discussion

259

For a given a,definite values of c* and HFIT, exist. Hence, a and c* are available from experimental HFIT, quantities. The vacancy concentrations based on calorimetric data do not exceed the calculated critical values (Fig. 12.1). However, the critical concentrations depend also on the defect-defect, phonontemperature, and local defect-phonon couplings (Wautelet and Legrand 1986).

12.3. Thermodynamic bounds for formation entropies Varotsos (1988) has proposed another estimation of the formation entropies. This approach is based on the concept of the vacancy-formation parameters under constant volume and under constant pressure. It is easy to obtain the thermodynamic relation

Here S , = -(dGF/dT), is the vacancy-formation entropy under constant pressure, = -(dGF/dT), is the formation entropy under constant volume, V , = (dG,/dp), is the formation is the volume thermal expansivity, and B is the volume, isothermal bulk modulus. As has been shown by Varotsos (1988), it is very likely that

sF*

Usually, S , and VF are positive, so that S, I V,PB. In addition, S, > ?4VFPB. The formation entropies are thus restricted as follows:

12. Discussion

260

%V,PB < S , I V,PB.

(12.6)

This inequality allows one to check the vacancy-formation entropies from various techniques. It can be applied to the formation entropies based on the nonlinear increase in specific heat (Kraftmakher 1996b). This increase was observed above about 0 . 8 T , (TM is the melting point), so that the deduced parameters HF* and S,* relate to temperatures equal to about 0.9TM. Now one can calculate the upper limits for the formation entropies at these temperatures (Table 12.1).

Table 12.1 Upper limits for vacancy formation entropies in metals at T calculated according to Varotsos (1988).

P (0 9TM) Metal

Sn

K-l)

B(300 K) (loll N.m-2)

As cu

8.2

7.5

0.55 0.42 0.79 1.11 1.51

Ni

8.0 5.0 4.4 3.6 3.45 4.1 3.2

1.84 2.83 3.55 1.70 2.64 1.93 3.10

Pb Al

Pt

Ir Nb Mo Ta W

7.6 10.4 10.5

R (300 K) (lo-*’ m3)

2.70 3.04 1.66 1.71 1.18 1.10 1.51 1.42 1.80 1.86 1.83 1.58

=

0.9TM

S/k, (0.9TM) upper limit

4.7 5.6 5.8 6.5 5.6 6.8 9.0 9.3 4.6 7.1 6.1 6.6

12. Discussion

261

Generally, bulk moduli at high temperatures are not available, and the values of B at T = 0.9TM were taken to be 0.7B(300 K). To determine VF values, a simple approximation VF = 0.6Q was employed, where Q is the atomic volume at room temperatures. The upper limits for the vacancy-formation entropies are thus given by

Sup= 0.42xp(0.9TM)xQ(3O0 K)xB(300 K).

(12.7)

The melting points and atomic volumes at room temperatures were taken from a handbook (Lide 1994), and the bulk moduli at room temperatures from Vaidya and Kennedy (1970) and Fernandez Guillermet and Grimvall (1989). Taking into account possible errors in the quantities involved, uncertainties in the calculated limits amount to 3 0 4 0 % . This approach also leads to high entropies of vacancy formation, in agreement with specific-heat data. It should be remembered that experimentally obtained formation enthalpies and entropies are interrelated and uncertainties in the formation entropies range from 0.5kB to Ik,.

12.4. Effects of anharmonicity Anharmonicity is generally considered the probable reason for the nonlinear increase in high-temperature specific heat and thermal expansivity of metals. However, almost all theoretical calculations of anharmonicity predict these contributions to be mainly linear, so they cannot explain large nonlinear effects. For instance, Maradudin et al. (1961) calculated the anharmonic contributions to thermodynamic properties of solids. The constant-volume specific heat C, has been found to follow the equation

12. Discussion

262

C,

=

3Nk, + A T

-

BT-2.

(12.8)

The coefficient A may be positive or negative, as the authors have shown by numerical calculations for lead. Considering the lattice dynamics of an anharmonic crystal, Cowley (1963) has concluded that anharmonicity leads to a weak linear temperature dependence of specific heat and thermal expansivity. Similar conclusions have been made in many other theoretical studies (e.g., Wallace 1965; Leadbetter 1968; Vaks et al. 1980; Trivedi 1985; Maradudin and Califano 1993; Zubov et al. 1995). A linear extrapolation of the data from intermediate temperatures takes into account all linear contributions to specific heat, including the electronic term. MacDonald and MacDonald (1981 ) have calculated thermodynamic properties of FCC metals Cu, Ag, Ca, Sr, Al, Pd, and Ni. The authors considered the contributions to the free energy from harmonic and lowest-order (cubic and quartic) anharmonic terms. The vacancy contributions were taken according to differential-dilatometry data. It turned out that the constant-volume specific heat of the crystal lattice slightly decreases with increasing temperature. The constant-pressure specific heat, including the electronic term, linearly increases with temperature. The thermal expansivity weakly depends on temperature. The authors have concluded that much larger vacancy concentrations would be necessary to account for the observed nonlinear increase in the constant-pressure specific heat. MacDonald et al. (1984) performed similar calculations for BCC alkali metals Li, Na, K, Rb, and Cs. To fit experimental data, high vacancy concentrations should be accepted also in this case. These results support the determination of the defect contributions as nonlinear parts of specific heat and thermal expansivity .

12. Discussion

263

MacDonald and Shukla (1985) evaluated thermodynamic properties of the high-melting-point BCC metals vanadium, niobium, molybdenum, tantalum, and tungsten. The calculations failed to reproduce the rapid upward trend in the specific heat and thermal expansivity at high temperatures (Fig. 12.2).

I

I

I

I

40

__

20 '

I

I

I

2000

2500

3000

I

3500

4000 K

Fig. 12.2. Theoretical estimate of specific heat of W (MacDonald and Shukla 1985). 1 - lattice constant-volume specific heat, 2 - lattice plus electronic contribution, 3 - total constant-pressure specific heat, including vacancy contribution from positron-annihilation data (Maier et al. 1979), 4 - experimental data by Cezairliyan and McClure (1971). Accepting low vacancy concentrations, the theory failed to explain the experimental data.

The authors have pointed out that vacancies are most likely to be responsible for the high-temperature behavior in these metals. However, Fernandez Guillermet and Grimvall (1991) interpreted the strong nonlinear increase in the specific heat of

264

12. Discussion

molybdenum and tungsten as an effect of anharmonicity. Zoli and Bortolani (1990) computed the thermodynamic properties of copper and aluminum. In these metals, the anharmonic contributions lead to nearly linear temperature dependence of the constant-pressure specific heat and of the thermal expansivity. At the same time, predictions exist of a strong nonlinear increase in specific heat caused by anharmonicity and instability of the crystal lattice. Plakida (1969) considered the stability conditions for an anharmonic crystal lattice. In the limit of high temperatures or a high energy of zero vibrations, the lattice loses stability. Close to the critical temperature, the specific heat and thermal expansivity rapidly increase without limit. However, the predictions of this theory, as well as of the theory developed by Ida (1970), are far away from the observed temperature dependences. Many authors believe that vacancy contributions cannot be separated from anharmonic terms. For instance, Grimvall (1983) has shown that in a narrow temperature range the dependence X = A T + B T 2 , being specific to anharmonicity, is close to the relation X = C T + Dexp(-H,lk,T) for H,IkBT = 3.4. However, the vacancy contributions are usually determined at temperatures, where this ratio ranges from 10 to 15. To explain the strong nonlinear increase in specific heat of some transition metals, White (1988) has attributed this phenomenon to electronic effects. He stated that vacancies make no measurable contribution to specific heat at temperatures below 0.9TM, and that the anharmonic contribution linearly varies with temperature. The first statement is based on low vacancy concentrations found by differential dilatometry in low-meltingpoint metals. The second statement, ruling out anharmonicity as a possible origin of the strong nonlinear increase in specific heat, might become meaningful only if the first one were correct. Otherwise, it supports the defect origin of the phenomenon as

well.

12. Discussion

265

12.5. Constant-volume specific heat of tungsten Observations of temperature fluctuations under equilibrium conditions offer a unique possibility to determine the constantvolume specific heat of solids (Kraftmakher and Krylov 1980ab). The mean square of the temperature fluctuations in a sample, < A T 2 > , and their spectral density, , obey the relations

AT^>

=

k,T2/mC,,

= 4k,T2/P' (1

(12.9a)

+ x2),

(12.9b)

where k, is Boltzmann's constant, P' is the heat transfer coefficient, x = m C , o / P ' , and m and C, are the mass and the constant-volume specific heat of the sample, respectively (Milatz and Van der Velden 1943; Landau and Lifshitz 1980). It is very important that the constant-volume specific heat C,, rather than the constant-pressure specific heat C, enters the above expressions. The reason is that the temperature fluctuations and the fluctuations of the sample's volume are uncorrelated (Landau and Lifshitz 1980). In contrast to liquids, there is no possibility to directly measure the constant-volume specific heat of solids. Measurements of the temperature fluctuations provide such an opportunity. The main difficulty is the smallness of the temperature fluctuations, even in very small samples. To observe markedly different values of the two specific heats, C, and C,, the measurements are to be performed at high temperatures. Photoelectric sensors are the best tools to detect extremely small temperature oscillations and temperature fluctuations at high temperatures. A simple apparatus was designed for the measurements. The signal was expected to be weaker than the

12. Discussion

266

inherent noise of a photosensor. To suppress this noise, a correlation method was used (Fig. 12.3).

vacuum chamber

DC source noise source

I \ 1st 2nd channel channel

+

multiplier

' '

t-+

LF oscillator

-

recorder

Fig. 12.3. Setup for measuring extremely small temperature oscillations and temperature fluctuations (Kraftmakher and Krylov 1980a).

Two identical channels are assembled, each consisting of a photodiode, a pre-amplifier, and a power amplifier. An electrodynamometer serves to accurately multiply the signals fed into its movable coil and one of the fixed coils. The displacement of the movable coil is registered by means of an additional low-frequency current passing through it. A voltage induced in the second fixed coil depends on the orientation of the movable coil. This voltage is measured by a lock-in amplifier with a long

12. Discussion

267

time constant and recorded. An averaging of the signal over several hours thus is possible. When detecting small temperature oscillations in wire samples, a low-frequency oscillator provided the modulation of the heating power. Only one amplification channel was employed, while the second signal fed to the multiplier was the output voltage of the oscillator. In this case, the electrodynamometer operated as a lock-in detector. Using the setup, temperature oscillations in the range 10-6-10-5 K were measurable (Fig. 12.4). Probably, there is no chance to accurately measure all the quantities entering the above formulas and to deduce the constant-volume specific heat C., Modulation calorimetry helps to solve the problem. When a modulated heating power is applied to the same sample, the temperature oscillations in it are governed by the constant-pressure specific heat C,. This allows one to exclude all the unknown quantities and to determine the specific heat ratio C,/C,. To deal with spectral densities in both cases, a noise generator provides the modulation, and the temperature oscillations in the sample are of the same character as the temperature fluctuations. The only difference is that they are governed by the constant-pressure specific heat. The mean temperature of the sample is the same and the same measuring system is employed. To determine the specific heat ratio, one compares the power spectrum of the temperature fluctuations and that of the temperature oscillations caused by the noise modulation. When the modulation is performed by means of a noise generator, the power spectrum of the temperature oscillations in the sample is given by (12.10)

12. Discussion

268

where = k , T 2 / M C , where M C is the total heat capacity of the system and the reservoir. The experiment was aimed at a choice between the two theories of temperature fluctuations. Chui et al. (1992) measured the temperature-dependent magnetization of a paramagnetic salt, copper ammonium bromide, in a fixed magnetic field. A SQUID coupled to a superconducting coil wound around the salt pill detected the changes in the magnetization. The temperature fluctuations were measured at temperatures near 2 K, above the Curie point of the salt (1.8 K). To exclude another origin of the phenomenon, the fluctuations in two samples were measured simultaneously, and no correlation between them was found. Owing to the high sensitivity of the SQUID, measurements of the spectral densities of the order of lo-*’ K2.Hz-’ were possible. The power spectrum of the temperature fluctuations appeared to be in good agreement with expectations based on the known heat capacity and the determined relaxation time of the samples. Below 0.1 Hz, the spectral density was of about 10- 19 K2 .Hz-’. Estimations were made of the fluctuations in the magnetization of the samples. This effect causes an effective noise of 2 ~ l O K2.Hz-’ - ~ ~ and is much smaller than the fluctuations observed. The authors have concluded that the relation AU = CAT is applicable to the fluctuations in U and T to an accuracy of 20%. Since the heat capacity of the reservoir was

272

12. Discussion

1700 times larger than that of the sample, the results were considered a strong confirmation of the fluctuation theory leading to the formulas (12.9). Later, Day et al. (1997) discussed the fluctuation-imposed limit for temperature measurements.

12.6. Thermal defects in alloys and inter metaIIics Equilibrium vacancy concentrations in alloys found by differential dilatometry are much larger than in pure metals. For example, the melting-point vacancy concentration in Ag-8.6at.YoSn alloy appeared to be five times larger than in pure silver (Mosig et al. 1992). It would be very useful to study vacancy formation in alloys with other techniques now available, including measurements of the specific heat. Such measurements on low-melting-point alloys are possible by adiabatic calorimetry or temperature-modulated differential scanning calorimetry. High concentrations of thermal vacancies were observed in many intermetallics. Wasilewski et al. (1968) studied structural and thermal defects in NiGa by comparing the X-ray and bulk densities of quenched and slowly cooled samples. The difference in the densities was interpreted to be an indication of quenchedin thermal vacancies The fraction of thermal vacancies, after quenching from 850°C, was of the order of 1%. Berner et al. (1975) have carried out similar observations on CoGa along with measurements of the magnetic susceptibility both in equilibrium, up to 1400 K, and after quenching. Van Ommen and de Miranda (1981) studied the vacancy formation in CoGa by a dilatometric technique. The relaxation in the length of the sample was observed after changing its temperature. Very long relaxation times, in the range 102-104 s, were determined at temperatures 870-1050 K. Van Ommen (1982) observed vacancy formation in NiGa by differential

12. Discussion

273

dilatometry. A very low formation enthalpy, 0.2 eV, has been found. By differential scanning calorimetry, high concentrations of quenched-in vacancies have been observed in intermetallics NiAl (Shimotomai et al. 1985), NiSb (Jennane et al. 1992), and CoGa (Sassi et al. 1993).

quenching temperature

("C)

Fig. 12.7. Difference between bulk ( 0 ) and X-ray ( 0 ) densities shows high concentration of thermal vacancies in V2Ga5 (Waegernaekers et al. 1988).

Waegemaekers et al. (1988) have found high vacancy content in quenched samples of V2Ga,. The bulk and X-ray densities were measured after quenching the samples from temperatures in the range from 8OO0C to 900°C. The bulk density

274

12. Discussion

decreases significantly with increase of the quench temperature. The X-ray density is independent of the quench temperature, though the theory of differential dilatometry predicts such dependence. The difference between the two densities is about 2% at 8OO0C and 5% at 900°C (Fig. 12.7). The authors have concluded that thermal vacancies in this compound are created in both sublattices. Measurements of the specific heat of this compound are very desirable, as well as of other materials with high concentrations of equilibrium vacancies. Kummerle et al. (1995) have obtained a low formation enthalpy and hence a high equilibrium vacancy concentration in Fe3Si by the positron-annihilation technique. Schaefer et al. (1990, 1992) determined the formation enthalpy in Fe3AI to be 1.18 eV and the vacancy concentration at the melting point to be 6 . 6 1~O-’. In contrast, the estimated vacancy concentration in Ni,AI was 6 ~ 1 0 - ~ It .should be remembered, however, that vacancy concentrations from the positron-annihilation techniques are based on certain assumptions. Brossmann et al. (1994) observed very low vacancy concentrations in TiAI, of the order of at the melting point. At the same time, high vacancy concentrations have been found in TiAl by the perturbed-angular-correlation technique, as well as in NiAl and CoAl (Collins et al. 1983; Allard et al. 1985; Hoffmann et al. 1986). Kim (1984, 1988) has proposed a theory describing the vacancy properties in ordered stoichiometric alloys. He stated that “while the vacancy concentration in metals at the melting points is of order 0.05%, in alloys it can range up to about lo%.”

12. Discussion

275

12.7.Self-diffusion in metals Generally, different mechanisms of self-diffusion in metals are possible. However, the most probable mechanisms involve point defects (e.g., Flynn 1972; Stark 1976). With the vacancy mechanism, the coefficient of self-diffusion is given by

Here u is the lattice parameter, f is the correlation factor of monovacancies, v is the attempt frequency for jumps of atoms into adjacent vacancies, g is a geometrical factor, HF and HMare the enthalpies of vacancy formation and migration, S , and S , are the corresponding entropies, Q is the enthalpy of self-diffusion, and Do is the corresponding pre-exponential factor. Experimental data on self-diffusion permit an additional check of the vacancy-formation parameters obtained by various techniques. During a long time, the vacancies were believed to be responsible for self-diffusion in all metals, including refractory metals. The experimental data well corresponded to the Arrhenius plots with a constant slope. However, when the measurements were made over a wide temperature range, the Arrhenius plots showed significant curvature. For instance, Mundy et al. (1978) measured the self-diffusion in tungsten over the temperature range 1700-3400 K, encompassing a range of nine orders of magnitude in the coefficient of diffusion. Neumann and Tdlle (1986, 1990) have given an analysis of the existing data on self-diffusion. There exist two explanations of the curvatures observed in the Arrhenius plots (Fig. 12.8). The two-defect model supposes two competitive mechanisms of self-diffusion. With this approach, the coefficient of selfdiffusion obeys the equation

12. Discussion

276

where the activation enthalpies for the two mechanisms, Q , and Q2, are independent of temperature, as well as the corresponding pre-exponential factors, Do, and Do2.

Na

Pt Ni

Nb

w 1.o

1.5

2.0

TMfl

Fig. 12.8. A curvature in Arrhenius plots of self-diffusion in metals according to fits by Neumann and T6lle (1986, 1990). TM is the melting tem perature.

12. Discussion

277

Considering self-diffusion in FCC metals, Mundy (1987) has pointed out that the contribution of divacancies is very probable since they are more mobile than monovacancies. The one-defect model assumes a strong temperature dependence of the activation enthalpies and entropies. At present, both models describe experimental data with nearly the same accuracy, so that one cannot favor one of the mechanisms over the other. However, in both cases high activation entropies appear at high temperatures. In the two-defect model, the pre-exponential factor in the term that dominates at high temperatures is much larger than that at low temperatures (Table 12.3). The ratio of these factors corresponds to a difference in the activation entropies from 3kB to 9kB. The largest difference is seen in refractory metals. Mundy et al. (1987) measured the migration enthalpy in tungsten at high temperatures. Its temperature dependence was accepted to be linear. The migration enthalpy increases from 1.68 eV at 1550 K to 2.02 eV at 2600 K. The corresponding change in the migration entropy equals 2k,. The authors have concluded that the curvature in the Arrhenius plot for the selfdiffusion in tungsten could be explained if the formation enthalpy had similar temperature dependence. Sabochik (1989) calculated the free enthalpy of vacancy migration in tungsten using atomistic simulations. The migration enthalpy is constant below 1000 K but in the range 1500-2500 K it increases by 0.7 eV. An important question arises in treatment of self-diffusion in refractory metals. For tungsten, the activation enthalpy at high temperatures differs from the sum of the enthalpies of vacancy formation and migration now accepted. It was therefore supposed that divacancies or interstitials are responsible for the selfdiffusion in tungsten at high temperatures. However, this conjecture also causes doubts, and further investigations are necessary to elucidate the situation.

12. Discussion

278

Table 12.3 Parameters of two-component fits of self-diffusion calculated by Neumann and Tolle (1986, 1990).

K Na Li Ag AU

cu Ni Pt V

Nb

Mo Ta W

0.05 0.006 0.038 0.055 0.025 0.13 0.85 0.034 0.31 0.115 0.13 0.002 0.13

0.386 0.372 0.52 1.77 1.70 2.05 2.87 2.64 3.21 3.88 4.54 3.84 5.62

1 0.81 9.5 15.1 0.83 4.5 1350 88.6 2420 65 140 1.16 200

0.487 0.503 0.694 2.35 2.20 2.46 4.15 4.05 4.70 5.21 5.70 4.78 7.33

12.8. Point defects and melting Frenkel considered melting to be a result of softening of the crystal lattice by point defects. Many authors studied the vacancy mechanism of melting. Aksenov (1972) considered stability of an anharmonic crystal with vacancies. The instability is governed by the balance of the energy of thermal vibrations of atoms and their binding energy in a self-consistent phonon field, and by the effect of vacancies. Vacancies lead to a significant decrease of the instability point compared with an ideal lattice. The effect of vacancies reduces to a renormalization of the force constant

12. Discussion

279

governing the interaction between atoms. The instability occurs when the vacancy concentration reaches 5-8%. Following this work, Moleko and Glyde (1984) considered the stability of rare-gas solids. High vacancy concentrations have been predicted at the instability points. Chudinov and Protasov (1984) have carried out molecular-dynamics simulations of melting in copper. The temperature of the crystal and the number of various point defects were calculated as a function of the total energy per atom (Fig. 12.9). For the present discussion, the main point of these calculations is the conclusion that high defect concentrations should appear in a solid before melting. Zhukov (1985) has shown that an interaction between equilibrium vacancies can result in thermodynamic instability of the crystal. A critical temperature appears, above which the Gibbs free energy of the crystal has no minimum and decreases at all vacancy concentrations. This means that above this temperature vacancies are generated spontaneously. The critical temperature may be identified with the melting point. From the equation derived, the vacancy concentration at the critical point is e = 2.7 times larger than the value calculated without considering the vacancy interaction. The entropy of vacancy formation may increase along with the vacancy concentration (Wautelet 1985). In this case, a rapid increase of the vacancy concentration leads to melting. Fecht and Johnson (1988) have calculated the stability limit of a crystal. Their approach is based on the equality of the enthalpies and entropies of the solid and liquid phases. The authors have concluded that vacancy concentrations at the stability-limit point should be of the order of lo-’. In some rapid-heating experiments, a significant superheat above equilibrium melting points was observed. This may be due to the lack of vacancies in the crystal lattice. To answer the question, it would be probably enough to compare results of two rapid-heating experiments: (i) starting at a temperature where the

12. Discussion vacancy concentrations are still negligible, and (ii) starting at a temperature where they are sufficiently large.

1600

1200

800

400

0

1000

3000

2000

4000

60 v)

4-

0

a,

rc

a,

-0

40

'c

0

rn

A A

&

P 20

A

E

,,

rn

3

c

A A

-

A

-

I'

Fig. 12.9. Results of computer simulations by Chudinov and Protasov for Cu (1984). Caloric equation of state and number of defects: A - vacancies at the surface, 0 - vacancies in the bulk, W - unstable Frenkel pairs. The total number of atoms in the calculations was 1554.

12. Discussion

281

12.9. How to determine vacancy contributions to enthalpy - a proposal A straightforward approach has been proposed (Kraftmakher 1996a) to reveal vacancy contributions to high-temperature enthalpy of metals. After heating the sample to a premelting temperature, the initial part of the cooling curve should depend on whether the vacancies had time to arise. If they had not, they will appear immediately after the heating. At premelting temperatures, the equilibrium vacancy concentrations are set up in 10-4-10-2 s in low-melting-point metals and in 10-8-10-6 s in refractory metals. Under normal conditions, the temperature of the sample after the heating, in the time interval of interest, remains nearly constant. When the vacancies appear after the heating, the initial cooling curve should depend on the vacancy contribution to the enthalpy and the relaxation time. Both quantities strongly depend on temperature. The enthalpy necessary to create the vacancies should be measurable from the temperature drop in the sample immediately after the heating. If the heating is not sufficiently fast, then the phenomenon could be studied under gradually increasing the upper temperature of the sample. The temperature drop has first to increase with the upper temperature, reach a maximum, and then fall because of a decrease in the relaxation time (Fig. 12.10). To evaluate the expected temperature drop AT, one has to take into account the equilibrium vacancy concentration at the final temperature of the sample after the equilibration. The vacancy-related enthalpy equals A H = H , e x p ( - G , / k , T ) . For tungsten, the calorimetric data predict the vacancy-related enthalpy of about 10 kJ.mol-’ at the melting point. The heatbalance requirement is C A T = A H , where C is the specific heat not including the vacancy contribution (C z 40 J.mol-’.K-’). The

12. Discussion

202

temperature drop due to the vacancy formation can be evaluated for any temperature achieved after a rapid heating (Fig. 12.11). After heating up to the melting point, the maximum temperature drop amounts to about 160 K. This figure reduces to 105 K after heating up to 3500 K and to 50 K after heating up to 3200 K.

t

temperature

Fig. 12.10. Temperature traces expected after rapid heating of a sample to a premelting temperature (Kraftmakher 1996a). At the highest temperature, vacancies have arisen during heating.

For molybdenum, the situation is very similar. The phenomenon should be clearly seen if the equilibrium vacancy concentrations were of the order of lo-* but become unobservable if they were less than From the measurements, the temperature dependence of the vacancy-related enthalpy could be determined.

12. Discussion

283

10

8

6 4 2

3000

3200

3400

3600

3800 K

3000

3200

3400

3600

3800K

Fig. 12.11. Calculation of the temperature drop AT caused by vacancy formation after a rapid heating of a W sample to various premelting temperatures (Kraftmakher 1998b).

284

12. Discussion

The rapid-heating experiment to be made is similar to those reported earlier (Hixson and Winkler 1990, 1992; Pottlacher et at. 1991, 1993). Owing to the proposed approach, a setup for the measurements may be even simpler. Now there is no need to measure the heating current and the voltage drop across the sample. All one needs is to rapidly heat up the sample, within or s, to a premelting temperature and to observe the initial part of the cooling curve. A very important point is to completely cancel the heating at the selected premelting temperature. Any uncontrollable heating will cause ambiguities in the vacancyrelated enthalpy. The expected phenomenon should be clearly seen and amenable to a quantitative treatment. Still more important, the nature of the phenomenon would be evident. This technique offers a reliable determination of equilibrium vacancy concentrations in metals. It seems to be the simplest experiment that could be undertaken for this purpose. The vacancy formation might be also seen from the dilatation of the sample immediately after a rapid heating. It is easy to show that the temperature drop due to the vacancy formation in the sample should be accompanied by an increase in its volume. Such unusual behavior would be the best confirmation of the vacancy origin of the phenomenon. The same approach is probably applicable to determinations of vacancy contributions to electrical resistivity. To be more informative, measurements of the extra resistivity should be made along with determinations of the vacancy-related enthalpy. However, the relative vacancy contribution to the electrical resistivity of refractory metals is much smaller than to the enthalpy. An additional approach exists to check the origin of the nonlinear increase in high-temperature specific heat of metals. Under high pressures, equilibrium vacancy concentrations decrease because of increase in the Gibbs free energy of vacancy formation. Vacancy contributions to specific heat thus decrease under high pressures. Experimental data on the

12. Discussion

285

pressure dependence of the vacancy-induced resistivity show that the pressures to be employed are quite moderate, up to l o 9 Pa. Such a pressure is probably insufficient to markedly change other contributions to the specific heat. Dynamic calorimetry and modulation calorimetry seem to be most suitable for measurements under high pressures. The temperature changes should be measured through the radiation from the sample that does not depend on pressure.

286

12. Discussion

Summary 0 Formation enthalpies and equilibrium vacancy concentrations obtained by calorimetric measurements do not contradict theoretical predictions. Recent computer simulations and some theoretical calculations predict high vacancy concentrations in metals at premelting temperatures.

Theoretical calculations of anharmonicity failed to explain the strong nonlinear increase in specific heat and thermal expansivity of metals. The experimental determination of the constantvolume specific heat of tungsten has confirmed a small or even negative anharmonic contribution to the specific heat. High concentrations of thermal defects have been found in some alloys and intermetallic compounds by means of differential dilatometry and positron annihilation. 0

Both mechanisms proposed to explain curvatures in Arrhenius plots for self-diffusion support high vacancy concentrations in metals. A straightforward method exists to unambiguously determine vacancy contributions to high-temperature enthalpy of metals and hence equilibrium vacancy concentrations. Calorimetric measurements under high pressures may be useful to reveal vacancy contributions to specific heat.

Chapter 13

Conclusions 13.1. Current knowledge of equilibrium point defects in metals 13.2. Actuality of Seeger's formulation 13.3. What could be done to reliably determine equilibrium defect concentrations

287

288 298

300

288

13. Conclusions

13.1. Current knowledge of equilibrium point defects in metals The current knowledge of equilibrium point defects in metals may be summarized as follows. (1) It is commonly agreed now that studies of point defects under equilibrium are basically superior to any non-equilibrium experiments. Vacancies are the defects prevailing in equilibrium. They strongly affect high-temperature specific heat and thermal expansivity of metals. From vacancy contributions to specific heat, the formation enthalpies and equilibrium vacancy concentrations are available. The only difficulty is to correctly separate the vacancy contribution. Observations of vacancy equilibration after a rapid temperature change provide such an opportunity. (2) The nonlinear increase in high-temperature specific heat of metals was observed by all calorimetric techniques now available. The phenomenon is especially strong in refractory metals thus indicating high vacancy concentrations. This concept gained no recognition, and the nonlinear increase in specific heat is commonly attributed to anharmonicity. As a rule, thermophysicists treat their experimental data ignoring vacancy formation. (3) Calorimetric measurements provide plausible enthalpies of vacancy formation. In low-melting-point metals, the vacancy concentrations at the melting points are of the order of These low values have been obtained by the usual procedure of linearly extrapolating the specific-heat data from intermediate temperatures. In high-melting-point metals, the equilibrium vacancy concentrations at the melting points are of the order of lo-* (Fig. 13.1).

13. Conclusions

1000,

289

I

70 .c

--E

W

c

.g

Niw

100

,

Ti

cu

w w

c

Q)

Au

0

c 0

10

%

0

im> I

1000

2000 3000 4000 melting temperature

K

Fig. 13.1. Vacancy concentrations at melting points from differential dilatometry (0) and specific-heat measurements (M). Note new differential-dilatometry data ( 0 ) on Ag and Cu (Kluin and Hehenkamp 1991; Hehenkamp et at. 1992; Mosig et al. 1992).

(4) Measurements of stored enthalpy of quenched samples show lower limits of equilibrium vacancy concentrations. Such data have been obtained only for gold. Similar measurements on refractory metals would be very important. (5) Under very rapid heating a sample to a premelting temperature, vacancies have no time to arise. Therefore, the enthalpy of the sample should correspond to a defect-free crystal and be smaller than that measured under moderate heating rates. For molybdenum and tungsten, the expected difference based on

13. Conclusions

290

the vacancy origin of the nonlinear increase in the specific heat amounts to about 10% (Fig. 13.2). In contrast, vacancy concentrations less than should make this difference too small to be measurable. The rapid-heating data on tungsten strongly support high vacancy concentration. Further rapidheating experiments are very desirable.

120

110

100

90

3000

3200

3400

3600

3800 K

Fig. 13.2. Enthalpy of tungsten measured under slow heating ( H . ) and expected under fast heating ( H I ) .

(6) Vacancy concentrations deduced from the nonlinear increase in thermal expansivity are somewhat smaller than those from the nonlinear increase in specific heat. Vacancy formation partly involves internal sources in the crystal lattice (voids, grain

13. Conclusions

291

boundaries, dislocations, vacancy clusters), so that the changes in the outer volume of the sample may be smaller than under ideal conditions. Underestimated vacancy concentrations thus are to be expected from thermal-expansion data. (7) The majority of the scientific community believes that differential dilatometry is the best or even the absolute method for determining equilibrium vacancy concentrations. In pure metals, this technique provides vacancy concentrations less than 10-3 at melting points. New differential-dilatometry measurements revealed vacancy concentrations in silver and copper several times larger than those commonly accepted during three decades. The method has not yet been applied to high-melting-point metals. At high temperatures, it is very difficult to measure vacancy concentrations of the order of However, a much easier aim is now topical, namely, to check either high vacancy concentrations can be ruled out or not. The necessary accuracy of such measurements is quite moderate, so there is no serious obstacle for them. Moreover, there is no need to simultaneously measure the bulk and X-ray expansion, as it is necessary in the case of low vacancy concentrations. Only a few determinations of the lattice parameter of high-melting-point metals have been made at high temperatures. With bulkexpansion data now available, they confirm rather than disprove high vacancy concentrations. High concentrations of thermal vacancies have been observed by differential dilatometry in some alloys and intermetallics. (8) Very different values of extra electrical resistivity have been found under equilibrium and after quenching. Only an unknown fraction of equilibrium vacancies survives in a sample after quenching. Further, many vacancies form clusters, and their contribution to the resistivity becomes smaller. Low values of quenched-in resistivity are thus quite explainable. The vacancy-related resistivity may also depend on temperature. Only for aluminum and platinum, results from measurements in

292

13. Conclusions

equilibrium and from quenching experiments are in reasonable agreement. Still more important, the vacancy concentrations in these metals estimated from the extra resistivity are consistent with the calorimetric data. The most serious problem arises in refractory metals. Quenching experiments turned out to be successful only on molybdenum and tungsten. The largest quenched-in resistivity in tungsten is 2XlO-’ R.m at the melting point. This low value caused a conjecture that defects of another type, divacancies or interstitials, are created at high temperatures but cannot be quenched-in because of their high mobility. (9) Earlier, vacancy contributions to electrical resistivity of refractory metals have not been found in equilibrium measurements. The reason is that the data were based on the room-temperature shape of the samples. The vacancy contribution becomes clear after introducing corrections for thermal expansion, which are necessary to correctly calculate the electrical resistivity. This procedure was applied to tungsten and niobium, and the estimated vacancy contributions at the melting points appeared to be 5x10-’ R.m and 7x10-’ R.m, respectively. However, with the commonly accepted values of p, for refractory metals, of the order of lop5 R.m, the extra resistivity remains too small to be consistent with the calorimetric data. More careful measurements of the resistivity are therefore desirable, as well as calculations and experimental determinations of the influence of point defects on resistivity. Such measurements and calculations should consider deviations from Matthiessen’s rule. A straightforward determination of the vacancy contribution to electrical resistivity could be based on rapid heating (up-quench) of a sample. Under very rapid heating, the measured resistance should correspond to a vacancy-free crystal. By varying the heating rate, the vacancy contribution and the equilibration time would be available.

13. Conclusions

293

(10) The majority of the scientific community believes that positron annihilation is the best tool to determine the enthalpies of vacancy formation. Regretfully, this technique does not provide vacancy concentrations. In addition, it is inapplicable to some metals. The detrapping of positrons from monovacancies may be so pronounced that the reported values of the formation enthalpies in some metals pertain rather to divacancies. Positron annihilation may help in determinations of vacancy losses during quenching. An important application of this technique is further observation of the vacancy equilibration at high temperatures. (1 1) The perturbed-angular-correlation technique is an important method capable of discrimination of various point defects and their clusters. When properly taking into account the temperature dependence of the defect-trapping probability, this technique has a potentiality to reveal equilibrium defect concentrations. However, the vacancy trapping becomes less effective at high temperatures. This may lead to very rapid fluctuations of the electric-field gradients near the probe nuclei and to loss of the defect-related signal. No data on equilibrium defect concentrations in metals have been obtained by this technique. Nevertheless, this approach deserves attention because reliable determinations of equilibrium vacancy concentrations even far below melting points would be useful. (12) Only quenched samples of high-melting-point metals are suitable for observations by field ion microscopy. The method involves the drawbacks peculiar to all quenching experiments and provides small defect concentrations. This technique is inapplicable to studies in equilibrium. Electron microscopy deals with samples in which precipitates formed by quenched vacancies become observable. In gold, platinum, and tungsten, vacancy concentrations smaller than at the melting points were found by electron and field ion microscopy. (13) Observations of the vacancy equilibration were proposed long ago to reliably separate vacancy contributions

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to physical properties of metals. Such observations were considered to be a crucial determination of equilibrium vacancy concentrations. Relaxation phenomena caused by the vacancy equilibration were observed through measurements of electrical resistivity (aluminum, gold, platinum) and specific heat (platinum, tungsten), and by positron annihilation (gold, some intermetallics). Despite very different relaxation times, all the phenomena observed are probably of one nature. The results have confirmed the vacancy origin of the nonlinear increase in the specific heat of platinum and tungsten. However, the relaxation observed in the specific heat gained no recognition. Further developments and applications of this technique are very desirable. (14) An important reserve has not yet been employed in observations of the relaxation in specific heat. Quenching and annealing experiments have shown that equilibration times in pure and well-prepared samples may be several orders of magnitude longer than in commercial wires. This means that well-prepared samples could provide data at higher temperatures and thus reveal larger differences between the specific heats measured with slow and fast temperature changes. The vacancy origin of the relaxation in specific heat could be checked by measurements including quenching and subsequent annealing of the samples. The relaxation time should significantly decrease after quenching and return to its initial value after annealing at the high temperature. (15) A straightforward determination of vacancy contributions to high-temperature enthalpy of metals has been proposed. If a sample is heated up to a premelting temperature so rapidly that vacancies have no time to arise, they will appear immediately after the heating. The initial cooling curve after the heating will show the heat absorbed by the vacancy formation. This method seems to be the simplest one to unambiguously determine equilibrium vacancy concentrations. The same approach is

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probably suitable to determine changes in the electrical resistivity and in the volume of the sample caused by the vacancy formation. (16) The origin of the nonlinear increase in specific heat may be checked by calorimetric measurements under high pressures. The equilibrium vacancy concentration and the related contribution to the specific heat should decrease according to the increase in the Gibbs free energy of vacancy formation. The necessary pressures are quite moderate, and appropriate calorimetric techniques are now available. (17) In the last decades, new relations concerning the entropies of vacancy formation have been found. First, it was shown that a perturbation in the delocalized phonon spectrum by vacancies may cause instability of the crystal above a certain critical temperature. The equation obtained fits this temperature, the formation enthalpy, and the critical vacancy concentration. This approach supports high vacancy concentrations. Second, theoretical calculations show that the formation entropies increase with temperature. The Gibbs free energy of vacancy formation at the melting points may become 1.5-2 times smaller than at low temperatures. (18) The temperature dependence of the formation enthalpies and Gibbs free energies of vacancy formation based on specific-heat data and a linear temperature dependence of the formation entropies are in qualitative agreement with theoretical calculations (Fig. 13.3). (19) No evidence exists that the strong nonlinear increase in high-temperature specific heat of metals is caused by anharmonicity. Theoretical calculations of anharmonicity indicate mainly linear contributions, which may be even negative. It seems improbable that a nonlinear anharmonic contribution to specific heat might be much larger than the linear term. Determinations of the constant-volume specific heat of tungsten also support a small contribution of anharmonicity. Even when

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the nonlinear increase in specific heat is small, a linear extrapolation leads to plausible values of the formation enthalpies. This should be impossible if nonlinear anharmonic contributions were essential.

0

1000

2000

3000

4000 K

Fig. 13.3. Parameters of vacancy formation based on calorimetric data and a linear temperature dependence of the formation entropies: formation enthalpies, .... Gibbs free energies of formation.

(20) High vacancy concentrations in refractory metals do not contradict self-diffusion data. The parameters of vacancy formation deduced from the nonlinear increase in specific heat should be compared with the self-diffusion data at high temperatures, above two thirds of the melting temperature. Such

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a comparison poses important questions, which are difficult to answer unambiguously. The activation enthalpies of selfdiffusion at high temperatures are larger than the sum of the enthalpies of vacancy formation and migration. On the other hand, the vacancy-formation parameters are in better agreement with the self-diffusion enthalpies at lower temperatures. A possible explanation of this may be an assumption that vacancies dominate also at high temperatures but their contribution to self-diffusion becomes smaller than that of other defects, divacancies or interstitials. These defects possess smaller concentrations but higher mobilities. Curvatures in the Arrhenius plots for self-diffusion can be explained by the twodefect mechanism or by a strong temperature dependence of the enthalpies of vacancy formation and migration. In both cases, high entropies of the vacancy formation at high temperatures are favored. (21) Large vacancy concentrations in high-melting-point metals correlate with equilibrium vapor pressures. The highest vapor pressures at the melting point are known in chromium, molybdenum, and tungsten, the metals that manifest the largest nonlinear increase in the specific heat. This is in accordance with Frenkel’s concept of similarity of evaporation and point-defect formation. (22) Until today, data on equilibrium point defects have been obtained only for about a half of metals (Fig. 13.4).

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13.2.Actuality of Seeger’s formulation About three decades ago, Seeger (1973a) described the situation as follows: “Unfortunately, it must be said that, at least in the case of metals, the secure and generally agreed-upon knowledge on the nature and the properties of point defects ... has fallen far short of the expectations held when this line of research was started about 20 years ago. Looking back, the reason for the lack of final success appears to be that the emphasis was laid on investigations in which the point defects were studied under conditions far from thermal equilibrium, and that this brought with it so many difficulties and uncertainties that more often than not alternative interpretations of the experiments were possible.” Despite intensive investigations, even today the situation is far from complete understanding. In this book, an attempt has been made to show that the viewpoint shared by the majority of the scientific community needs revision. Moreover, it may turn out that just calorimetric measurements provide the most reliable data on equilibrium point-defect concentrations in metals. The subject was discussed at the International Conference on Diffusion in Materials (DIMAT-96, Nordkirchen). The author’s viewpoint was given in a paper “An opposite view on equilibrium vacancies in metals” (Kraftmakher 1997). Seeger (1997) has presented some new ideas concerning equilibrium point defects and self-diffusion mechanisms in metals.

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Fig. 13.4. Equilibrium defect concentrations were determined only for about a half of metals. Techniques providing such data are indicated as follows: C - calorimetry, D - differential dilatometry, R resistometry.

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13.3. What could be done to reliably determine equilibrium defect concentrations Experimental methods most promising to reliably determine equilibrium defect concentrations in metals are as follows. Measurements of the enthalpies of metals at the melting points or at selected premelting temperatures under various heating rates could unambiguously show the vacancy-related enthalpies and, consequently, the equilibrium vacancy concentrations. Observations of the initial cooling curve immediately after a sample was rapidly heated up to a premelting temperature may become a straightforward determination of the vacancy-related enthalpy. Such an experiment seems to be the simplest one for this purpose. To test the origin of the nonlinear increase in high-temperature specific heat of metals, calorimetric measurements under high pressures would be very useful. Pure and well-prepared samples having low dislocation densities and long relaxation times should be employed in observations of the vacancy equilibration. The origin of the relaxation in specific heat could be tested by measurements including quenching and subsequent annealing of the samples. Further measurements of stored enthalpy of quenched samples are very desirable. Such measurements on refractory metals would be very important. Differential-dilatometry studies or only determinations of the lattice parameter of high-melting-point metals at high

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temperatures are very desirable. The aim of such measurements is to verify whether high vacancy concentrations in these metals can be ruled out. Electrical resistivity of refractory metals at high temperatures should be measured more carefully, and the thermal expansion of the samples must be taken into account. Necessary thermalexpansion data are now available. The vacancy-related resistivity may be also determined in rapid-heating experiments. To check the validity of calorimetric data, it would be useful to measure the specific heat of alloys and intermetallics, high vacancy concentrations in which have been found by differential dilatometry or predicted by positron-annihilation data. Despite difficulties that are already evident, attempts to determine equilibrium point defects in pure metals by the perturbed-angular-correlation technique, especially in highmelting-point metals, deserve attention.

Acknowledgments I would like to thank Dr. K.K. Phua for the offer to prepare this book for World Scientific and Ms. Lakshmi Narayan for the editing. The first version of this book has been published as a review paper in Physics Reports 299, 79-188 (1998). Many thanks to the editor, Professor A.A. Maradudin, and to the referee, for useful comments and suggestions. The support of the Ministry of Science and Technology of Israel and of the Dr. Irving and Cherna Moskowitz Program for the Absorption of Scientists is gratefully acknowledged.

302

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Index

AC calorimetry 49 activation enthalpy 276,277,297 activation entropy 277 adiabatic regime 41,47,61,70,77 adiabaticity 52,57,68,85, 133 Ag 5, 12, 13,23-26,149,150, 152-154,156, 160, 163,168, 173,182,

187,193,200,201,209,238, 239,241, 260, 262,272,278, 289, 29 1 AgBr 257 AgSn 150,156,272 A1 5, 7, 13,23-26,103, 122, 125, 128, 153, 154, 158, 163-168,170, 172-174,178, 181, 182, 186, 187, 192, 193, 200, 201, 205-207, 209-212,220, 234, 238, 239, 241, 250, 254, 260, 262, 291, 294 AlZn 187 anharmonic contribution 261,264,270,286,295,296 anharmonicity 8-10,103,104,136, 152,257,261,262,264,285,288, 295 annealing 37,38, 127,157-159,170,171,181,186,192,200,201, 205,206,21 1, 212,294,300 Ar 192,257 Arrhenius plot 248,252,275-277,286,297 atomic volume 24,26, 138,159,261 AU 5, 13, 23,25,26, 36,69, 120, 125, 127, 150, 153, 154, 158, 162164, 168, 170-173,182, 187, 193,200, 201, 205, 207, 212, 216, 219-221,229, 230, 233, 234, 238, 239, 242, 250, 278, 289, 293, 294 Be 149 Bi 154 bulk modulus 209,259,261

Index

324

Ca 163,262 Cd 26, 153, 154, 158, 163, 168, 181, 191, 193, 199, 200, 202, 239,

240 Cdln 202 Co 23, 193,205 CoAl 202,274 CoGa 272,273 constant-volume specific heat 261-263,265,267,270,286,295 Cr 7,24,25,26,96,125,153,160,193,243,244,250,297 Cs 125,163,240,250,262 CU 5, 12, 13, 21-26,61, 92, 120, 125-128,149, 150, 152-154,160,

162-164,168, 170, 172, 173, 182, 187, 192, 193,200, 201, 206, 207, 210, 238, 239, 242, 250, 252, 260, 262, 264, 271, 278-280, 289,291 CuGe 156 Debye temperature 248 Debye-Waller factor 205 defect equilibration 31,236,253 defect-formation parameters 44, 104 defect-trapping probability 293 deformation 4,169,201 detrapping of positrons 182,184,293 dislocation 2, 5, 169, 170, 171, 181, 182, 186, 187, 213, 215, 218,

219,225,229,233. 235,291,300 divacancy 3, 20, 24, 25, 27, 30, 159, 170, 173, 182, 201, 206, 207, 210,213,238,246,247,253,277,292,293,297 elastic modulus 209 equilibration time 39, 44,56, 129, 177,220, 234, 236, 247, 292, 294 equivalent-impedance method 48, 59, 65,72, 82, 100, 117-119,135,

227 faulted dislocation loops 213 Fe 23,24,96,163,182,205,231,242 Fe3AI 195,274 Fe3Si 195,231,274 Frenkel pairs 2,3,24, 162,235,238,280

Index

325

Gibbs free energy 2, 18, 19, 21-23, 27, 28, 248, 252, 254, 279, 284, 295, 296 grain boundaries 2, 151, 213, 290 heat balance 51, 54, 55 heat transfer coefficient 52, 113, 114, 265, 269 high pressures 48, 284-286, 295, 300

In 26, 122, 125, 181, 191, 193, 200, 202, 240, 250 instability point 278, 279 intermetallics 12, 160, 195, 196, 202, 236, 272, 273, 286, 291, 294, 300 interstitial 2, 3, 5 , 18-20, 22, 24, 27, 32, 138, 147, 148, 200, 202, 205, 206, 209, 213, 215, 235, 238, 247, 253, 277, 292, 297 interstitialcy model 238 ionic crystal 257 Ir 8, 25, 26, 201, 260 irradiation 4, 200, 201, 205, 206, 211, 212, 226 K 25, 26, 125, 163, 165, 240, 250, 262, 278 KCI 235 Kr 238, 257

La 120, 125, 208, 209 lattice parameter 5, 35, 38, 147, 148, 149, 151, 154, 157-160, 239, 256, 275, 291, 300 lattice relaxation 21 levitation calorimetry 108 Li 5 , 23-26, 154, 163, 240, 262, 278 liquisol quenching 158 magnetic after-effect 21 0 Matthiessen’s rule 162, 173, 176, 178, 292 Mg 154, 163, 168, 172, 192, 193 migration enthalpy 23, 170, 188, 210, 215, 233, 277 migration entropy 242, 277 MO 7, 13, 24-26, 75, 103, 109, 110, 118-122, 125, 129-131, 150, 153, 154, 160,172, 178, 190, 191, 193, 201, 205, 211, 239, 243-246, 250, 260, 263, 264, 278, 282, 289, 292, 297

326

Index

molecular solids 257 Monte Carlo simulations 21-23, 252 Na 5 , 22, 24-26, 125, 150, 154, 163, 187, 239, 240, 250, 252, 278 Nb 7, 8, 25, 26, 75, 76, 103, 118, 119, 120, 122, 125, 146, 150, 154, 155, 160, 170, 175, 176, 178, 182, 190, 191, 193, 211, 246, 250, 260, 263, 278, 292 Ni 23-26, 61, 96, 120, 122, 125, 150, 153, 160, 162, 163, 168, 173, 192, 193, 200, 201, 210, 242, 250, 260, 262, 278 Ni,AI 188, 195, 202, 274 NiAl 202, 232, 273, 274 NiGa 272 NiSb 273 nonadiabatic regime 52, 58, 85 noncontact calorimetry 49, 61 nuclear quadrupole double resonance 206

262, 153, 243172,

one-defect model 277 optical pyrometry 112 Pb 5 , 7, 25, 26, 103, 122, 125, 153, 154, 163, 168, 172, 191-193, 239, 241, 260, 262 Pd 23, 25, 26, 122, 171, 176, 193, 201, 202, 207, 209, 243, 244, 262 Pdln 202 positron-annihilation parameters 34, 192, 196, 233 Pt 7, 8, 9, 13, 25, 26, 42, 76, 87, 118-120, 125-127, 139, 150, 153, 155, 158, 159, 160, 162, 166-168, 171-174, 178, 193, 200, 201, 205, 207, 209, 214, 216, 220, 223-225, 233-235, 239, 243, 247, 250, 260, 278, 291, 293, 294 rare-gas solids 279 Rb 25, 125, 163, 240, 250, 262 Re 103, 244 relaxation time 9, 38-42, 56, 57, 106, 114, 218, 219, 227, 229, 231, 233, 235, 253, 271, 272, 281, 294, 300 R h 23-26, 125, 153, 160, 168, 201, 243, 250 Ru 23, 208, 209

Index

327

Sb 122, 125, 205, 241, 250 Sc 23 secondary defects 37, 127 Sn 26, 122, 125, 153, 154, 163, 193, 239, 240, 250, 260 sources (sinks) for vacancies 233, 234, 256 specific trapping rate 183-185, 187, 194, 256 Sr 262 stacking fault tetrahedra 213 stored enthalpy 37, 127, 239, 242, 289, 300 superfluid helium 13, 169 superheat 115, 130, 177, 247, 279 supplementary-current method 65, 117 Ta

8, 13, 25, 26, 109, 110, 118-122, 125, 153-155, 160, 178, 190, 191, 193, 205, 244-246, 250, 260, 263, 278 Tc 23 temperature fluctuations 265-267, 269-271 Th 103, 243, 244 thermal conductivity 55, 133, 169, 207, 209, 216 thermal diffusivity 57, 61, 207, 209, 216 thermodynamic instability 279 third-harmonic technique 47, 59, 65, 67-69, 117 Ti 23, 24, 120, 122, 124, 125, 243, 250 TiAl 195, 202, 274 TI 163, 193 two-defect model 275, 277 unstable Frenkel pairs 238, 280 up-quenching 292 26, 122, 124, 154, 172, 176, 177, 190, 191, 193, 243, 244, 246, 263, 278 V2Ga5 158, 273 vacancy clusters 18, 128, 151, 163, 188, 201, 213, 225, 235, 256, 291 vacancy equilibration 10, 13, 38, 39, 103, 116, 129, 152, 183, 218220, 225, 229, 230, 231, 236, 241, 246, 257, 288, 293, 294, 300 vacancy-free crystal 39, 40, 177, 247, 292 vacancy loss 37, 127, 170, 171, 178, 214, 215, 293 vacancy trapping 192, 200, 202, 205, 293

V

328

Index

vapor pressure 191, 297 vibration frequency 3, 19, 21, 257 voids 2, 151, 213, 215, 256, 290

W 7-9, 13, 24-26, 42, 43, 73, 75, 76, 87, 89, 103, 108, 111, 114, 122, 125, 129-135, 139, 153-155, 160, 163, 170, 172-176, 191, 193, 201, 205, 209, 213-216, 223-225, 228, 233-235, 244-248, 250, 260, 263, 264, 270, 275, 277, 278, 281, 283, 289, 290, 292-295, 297

118190, 239, 286,

Xe 205 X-rays 5, 32, 149-151, 155, 272-274, 291 Y 23

Zn 5, 26, 122, 125, 153, 154, 163, 181, 188, 191-193, 239, 241, 250 Zr 23, 24, 120, 122, 124, 125, 153, 160, 243, 250

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