Lecture Notes on Equilibrium Point Defects

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

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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

3. Methods for studying point defects

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

3. Methods for studyingpoint defects

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

3. Methods for studying point defects

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,.

3. Methods for studyingpoint defects

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.

3. Methods for studying point defects

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.

3. Methods for studying point defects

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

3. Methods for studying point defects

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

3. Methods for studying point defects

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

3. Methods for studyingpoint defects

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 << 1, and A$ is the change in the phase

3. Methods for studying point defects

41

of the temperature oscillations caused by the relaxation. The difference between the specific heats measured at a low and a high modulation frequency thus depends on the vacancy contribution under equilibrium, AC, and on X.The relaxation time decreases with increasing temperature because of an increase in the mobility of the vacancies. The mobility is proportional to e x p ( - H M / k , T ) , where H M is the enthalpy of vacancy migration. When the density of internal sources (sinks) for the vacancies does not depend on temperature, the quantity X can be written as

where To is a temperature for which X = 1. The relaxation is also observable through changes in the phase of the high-frequency temperature oscillations in the sample. Modulation calorimetry usually employs the so-called adiabatic regime. This means that the oscillations of the heat losses from the sample are much smaller than the oscillations of the heating power. In this regime, the phase shift between the oscillations of the heating power and the temperature oscillations in the sample is close to 90'. Due to the relaxation, this phase shift decreases. The phase measurements are made along with the measurements of the amplitude of the temperature oscillations in the sample.

42

3. Methods for studying point defects

Prediction of the relaxation phenomenon in tungsten The magnitude of the relaxation in specific heat is the ratio of the equilibrium specific heat, C,, to the specific heat measured at a sufficiently high frequency of the temperature oscillations, IC(X)I. At a given modulation frequency, this ratio first increases with increasing temperature, reaches a maximum and then fails due to the decrease of the relaxation time. Assuming a constant density of internal sources (sinks) for the vacancies, it is easy to evaluate the magnitude of the relaxation and the change in the phase of the high-frequency temperature oscillations (Fig. 3.3). A technique for calorimetric measurements at modulation frequencies of the order of l o 5 Hz was developed (Kraftmakher 1981) and employed to observe the relaxation in the specific heat of tungsten and platinum (Kraftmakher 1985, 1990).

43

3. Methods for studying point defects

1.04

1.02

3600

0.06

0.04

0.02

3600

Fig. 3.3. Expected relaxation in specific heat of W. Ratio of specific heats measured at a low and a high modulation frequency and change in the phase of high-frequency temperature oscillations. The parameters employed are as follows: HF = 3.15 eV, SF = 6.5kB (Kraftmakher and Strelkov 1962), HM = 3 eV. To is a temperature for which X = 1.

44

3. Methods for studyingpoint defects

3.4. Summary It is commonly agreed that studying of point defects under equilibrium provides the most reliable data on the defectformation parameters. Measurements under equilibrium are especially important for determinations of the equilibrium defect concentrations. High-temperature specific heat is the property most promising for determining equilibrium defect concentrations. The only obstacle for this approach is the separation of the defect contributions. 0

Careful quenching experiments provide plausible values of the enthalpies of vacancy formation. However, it is difficult to evaluate the equilibrium vacancy concentrations. Observations of the equilibration of point defects make it possible to unambiguously separate defect contributions to corresponding physical properties of metals, including specific heat. 0

Modulation calorimetry is the best tool to search for relaxation phenomena in specific heat. Under rapid temperature oscillations, the vacancy contribution to the specific heat may be almost completely excluded and the specific heat of a hypothetical defect-free crystal determined. The only obstacle to implement such measurements is the shortness of the equilibration time. The relaxation is therefore observable only with high frequencies of the temperature oscillations.

Chapter 4

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

46 51 59

Direct electric heating. Induction heating. Modulated-light heating. Electron bombardment. Separate heaters. Peltier heating.

4.4. Measurement of temperature oscillations

65

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

83

Principle of modulation dilatometry. Differential method. Bulk samples. Interferometric modulation dilatometer. Nonconducting materials. Measurement of extremely small periodic displacements.

4.6. Modulation measurements of electrical resistivity and thermopower

96

Temperature derivative of resistance. Direct measurement of thermopower.

100

4.7. Summary 45

46

4 . Modulation techniques

4.1. Introduction Modulation calorimetry and other modulation techniques turned out to be very useful tools for studying equilibrium point defects in metals. Modulation calorimetry is based on periodically modulating the power that heats the sample and creating thereby temperature oscillations in it around a mean temperature. The amplitude of these depends on the heat capacity of the sample. This principle was discovered at the beginning of the century but only since the 60s the modulation method has gotten a proper development based on modern experimental techniques. In modulation calorimetry, it is enough to measure the oscillations of the heating power and of the sample temperature. The use of periodic temperature changes provides important advantages. When the modulation frequency is sufficiently high, the correction for heat losses from the sample becomes negligible even at highest temperatures. In this respect, the method is comparable to adiabatic calorimetry. Against the pulse method, modulation calorimetry has the advantage that selective amplifiers and/or lock-in detectors measure the harmonic temperature oscillations. This feature becomes very important when good temperature resolution is required as, e.g., in studies of phase transitions. Modulation calorimetry provides a unique possibility to perform measurements with temperature oscillations in the range 1-10 mK and even smaller when necessarily. Modifications of the method differ by the ways to modulate the heating power (heating by an electric current, radiation or electron-bombardment heating, induction heating, use of separate heaters, Peltier heating) and by the methods to detect the temperature oscillations (through the resistance of the sample or radiation

4. Modulation techniques

47

from it, using thermocouples, resistance thermometers, or pyroelectric sensors). Modulation calorimetry permits measurements in wide temperature ranges, from fractions of a kelvin up to melting points of refractory metals. In many cases, it is possible to assemble compensation schemes whose balance does not depend on the amplitude of the power oscillations. The measurements may be controlled by a data-acquisition system and fully automated. All these features have made the modulation method attractive and widely used. In treating the data, the mean temperature and the amplitude of the temperature oscillations are assumed to be constant throughout the sample. In this respect, this technique differs from the method of temperature waves. As a rule, the measurements are made in the so-called adiabatic regime, when the amplitude of the temperature oscillations is inversely proportional to the heat capacity of the sample. Corbino (1910) has developed the theory of modulation calorimetry and carried out the first modulation measurements of specific heat. He measured the temperature oscillations in the sample through the oscillations in the sample's resistance. They were detected by passing through the sample a supplementary AC current of a frequency equal to that of the temperature oscillations or by the third-harmonic technique (Corbino 191 1). This work has been highly praised by Fermi (1937) who called Corbino his teacher. Considerable advances in modulation calorimetry have been made in the 1960s. At the first stage, the method was used exclusively at high temperatures, and its most important feature was the smallness of the correction for heat losses. The samples, in the form of a wire or a rod, were heated by an electric current passing through them. The temperature oscillations were determined from oscillations in the resistance of the sample or radiation from it. Using this method, the high-temperature

48

4. Modulation techniques

specific heat of refractory metals has been measured. Later, the method was employed in studies of phase transitions, when the main requirement is good temperature resolution. In these experiments, the samples also were heated by an electric current. At the second stage, modulation calorimetry was used at low and intermediate temperatures and for studying nonconducting materials. A good finding turned out to be the modulated-light heating. Even at low and intermediate temperatures, the traditional domain of adiabatic calorimetry, the modulation method ensures better temperature resolution and higher sensitivity. In addition, small dimensions of the samples often are of importance. Absolute values obtainable by the modulation technique are less accurate than those by adiabatic calorimetry. Even under very favorable conditions, the inaccuracy of modulation measurements amounts about 1-2%. Nevertheless, due to the excellent resolution, the method is widely used in studies of phase transitions (structural transitions, ferro- and antiferromagnets, ferroelectrics, superconductors, liquid crystals, biological materials). It is applicable also under high pressures. During a long time, there was no special term for the modulation technique. The term ‘modulation method for measuring specific heat’ was proposed in the paper describing the equivalent-impedance method for determining heat capacity of wire samples (Kraftmakher 1962). However, most investigators have acquainted with modulation calorimetry only owing to the famous papers by Sullivan and Seidel (1966, 1967, 1968). The authors considered thermal coupling in a system including a sample, a heater, and a thermometer, and performed measurements at low temperatures. Sullivan and Seidel (1968) stressed the essential advantages of this technique as follows. “(1) The sample may be coupled thermally to a bath. (2) The method is a steady-state measurement. (3) Changes in heat capacity with some experimentally variable parameter may be

4. Modulation techniques

49

recorded directly. (4) Extremely small heat capacities may be measured with accuracy. (5) The method possesses a precision an order of magnitude better than existing techniques." The term 'AC calorimetry' introduced by the authors is now generally accepted. Recently, a new term appeared, 'the temperaturemodulated calorimetry' (Gmelin 1997). Modulation calorimetry provides unique opportunities: temperatures down to one tenth of a kelvin; magnetic fields up to 30 T; pressures up to 3.5 GPa; samples as small as 1 pg; a resolution of the order of 0.01%. At liquid helium temperatures, temperature oscillations necessary to perform modulation measurements are of the order of K. The specific heat is measurable versus an external parameter, e.g., magnetic field or pressure. This approach has been proposed and confirmed by Sullivan and Seidel (1967, 1968). One of the recent achievements of the method is the noncontact calorimetry successfully employed in space during a mission of the shuttle 'Columbia' (Wunderlich et al. 1997). Modulation calorimetry has become such a necessary technique that a commercial instrument has already appeared (calorimeter ACC-1, ShinkuRiko, Inc.). The modulation principle is also powerful for studying some other thermophysical properties. Measurements of oscillations of the sample's length permit direct determinations of thermal expansivity. This technique arounds the main drawback of hightemperature dilatometry caused by the creep of the samples and significantly improves the temperature resolution. By registering oscillations of the sample's resistance caused by the temperature oscillations, the temperature derivative of resistance is available. Direct measurements of this quantity more reliably show the behavior of electrical resistivity, e.g., near phase transitions of the second order. Measuring temperature oscillations by two thermocouples provides a direct comparison of their

50

4. Modulation techniques

thermopowers. The modulation method was applied also to measurements of the spectral absorptance. High sensitivity and unique temperature resolution are peculiar to all the modulation methods. Both features are owing to the periodic character of the temperature changes. Employment of selective amplifiers and/or lock-in detectors therefore reduces any influence of a noise and interference. The theory of the measurements is simple and quite adequate to the experimental conditions. In contrast to pulse or dynamic calorimetry, modulation calorimetry is a steady-state method: the amplitude and the phase of the temperature oscillations in the sample do not depend on time. Electronic equipment for modulation measurements (oscillators, amplifiers, oscilloscopes, lock-in detectors) is widely used in fundamental and applied studies and is easily accessible. It is possible to perform the measurements using common scientific instruments, as well as data-acquisition systems and computers for controlling the measurements and processing the data. Many review papers and book chapters are devoted to modulation calorimetry (Filippov 1966, 1967, 1984; Gmelin 1997; Kraftmakher 1973a, 1984, 1988, 1992a). Specific items of modulation calorimetry were considered by Hatta and lkushima (1981), Garland (1985), Huang and Stoebe (1993), Finotello and lannacchione (1995), Birge et al. (1997), Finotello et al. (1997), Hatta (1997ab), Jeong (1997), Minakov (1997), Hatta and Nakayama (1998), Hatta and Minakov (1 999), Kraftmakher (1992b, 1994a, 1996a). Reviews of modulation dilatometry and other modulation techniques are also available (Kraftmakher 1973b, 1978a, 1989).

4. Modulation techniques

51

4.2. Basic theory of modulation calorimetry The basic theory of modulation calorimetry is very simple (Corbino 1910). The power heating the sample is modulated by a sine wave and thus equals p o + p s i n o t . The sample's temperature therefore oscillates around a mean value To. For a short time interval At, during which the quantities considered are assumed to be constant, the heat balance equation is given by

(Po + p s i n o t ) A t = mcAT

+ P(T)At.

(4.1)

Here m , c and T are the mass, specific heat and temperature of the sample, P ( T ) is the power of the heat losses from the sample, co = 2nf is the angular modulation frequency, and AT is the change of the sample's temperature during the time interval At. The equation has a very simple meaning: heat input = heat accumulated in the sample + heat losses. Denoting T = To + 0 and assuming 0 << To, P ( T ) = P(T,) + P ' 0 . Thus, one obtains

mc@' + P(T,)

+ P'O

=po

+ psinot,

(4.2)

where 0' = d@/dt. The steady-state solution to this equation is

0

=

0,

O,sin(ot - $),

=

tan4

@ / m c o ) s i n $ = (pcos4)/P' = = p l ( m2 c 2 o2 + P 12 ) 112 , =

mco/P1.

(4.3b)

(4.3c) (4.3d)

4. Modulation techniques

52

The quantity P ' = dP/dT is called the heat transfer coefficient. The above results (4.3) may be presented as frequency dependences of 0, and tan$, as well as a polar diagram @,($) (Fig. 4.1). The condition tan4 >> 1 (sin4 E 1) is a criterion of the so-called adiabatic regime in modulation calorimetry. More rigorously, it should be called quasiadiabatic. When the phase shift 4 is close to 90°, the correction for heat losses is negligible: the oscillations of the heating power are much larger than those of the heat losses due to the temperature oscillations in the sample. Although the heat transfer coefficient grows rapidly with temperature, the regime of the measurements can be kept adiabatic by increasing the modulation frequency. Under adiabaticity conditions,

mc

=

p/oO,.

(4.4)

This is the basic equation of modulation calorimetry. The temperature oscillations may be written in a complex form, showing the amplitude and the phase of the temperature oscillations. Under adiabaticity conditions (4 = goo),

0 = -ip/mco.

(4.5)

In a nonadiabatic regime, when tan$ < 10, the heat capacity is calculated from the relation

mc

=

(p/o@,)sin4.

(4.6)

When an electric current heats a wire sample, then it is necessary to take into account the temperature dependence of the sample's resistance, as it was done by Corbino (1910). Hence, R = R, + R'O, where R, and R' are the sample's resistance and its temperature derivative at the mean temperature.

53

4. Modulation techniques

20 8

c (ZI

-

+.'

15

h

cn + .-

5 .f!

lo

(ZI

- 5 0"

0

5

10

15

20

o/w,

90

Fig. 4.1. Frequency dependence of the amplitude and phase of temperature oscillations and polar diagram Oo($) (Rosenthal (1965). a,,is a frequency for which tan $ = 1.

4. Modulation techniques

54

When a DC current 1 , is superimposed with a small AC component isinot, the electric power dissipated in the sample equals

Since i << I , and R’O << R,, the last three terms are small. In addition, they correspond to power oscillations of higher frequencies, while selective amplifiers tuned to the fundamental frequency and/or lock-in amplifiers are usually employed in modulation measurements. The heat balance equation for the oscillating part of the heating power is

and the solution to it differs from the expressions (4.3) only by the phase shift:

0 0,

=

=

tancp

@,sin(ot - c p ) , (210iRo/mco)sincp, =

mco/(P’ - Io2R’).

(4.9a) (4.9b) (4.9c)

Here cp is the phase shift between the AC component of the heating current and the temperature oscillations. Usually, Io2R’ is several times smaller than P’. Under direct electrical heating, axial and radial temperature gradients may appear in the sample. However, for thin samples and under proper modulation frequencies the gradients are sufficiently small, as was shown by Holland and Smith (1966). The authors have given expressions for the magnitude and phase of all harmonic components of the temperature oscillations.

4. Modulation techniques

55

Radial and end effects were treated as corrections to the solutions for a long thin wire. It turned out that corrections to the usual expression for the heating power may be required only for a very large temperature coefficient of the sample’s resistance. The surface temperature oscillations may differ from those in the interior but this difference may become meaningful only in special, rather exotic cases. With separate heaters and thermometers, one has to take into account thermal coupling in the calorimetric cell, as well as a finite thermal conductivity of the sample. Sullivan and Seidel (1968) considered a heater (heat capacity c h , temperature Th), a sample (C, , T,), and a thermometer (C, , T,) interconnected by thermal conductances Kh and K, (Fig. 4.2). An AC power applied to the heater provides an oscillating heat input to the sample. The heat flows through the sample and out to the heat sink (bath) via a thermal link Kb. At first the thermal conductivity of the sample was assumed to be infinite. The heat balance equations for the system are as follows (T’ dT/dt): chTh’

=

p o + p s i n o t - Kh(Th - Ts),

T,) C,T,’

=

K,(T,

-

T,).

(4.1Oa)

-

(4.1OC)

The temperature oscillations are sufficiently small to consider the parameters C and K to be constant. The steadystate solution for the above equations consists of two terms, a constant term depending upon Kb and an oscillatory term inversely proportional to the heat capacity of the calorimetric cell: (4.1 1)

4. Modulation techniques

56

where C = C , + c h + C,, B is a complicated expression involving quantities from the equations (4.10), and $ is a phase shift close to 90' under conditions discussed below.

bath

thermometer

Fig. 4.2. Presentation of calorimetric cell consisting of a heater, a sample, and a thermometer. As a rule, the modulation period is much longer than the equilibration time inside the cell and much shorter than that between the cell and the bath (Sullivan and Seidel 1968).

When (i) the heat capacities of the heater and of the thermometer are much smaller than that of the sample, (ii) the sample, the heater, and the thermometer come to equilibrium in a time much shorter than the modulation period, and (iii) the modulation period is much shorter than the sample-to-bath relaxation time z,, then, to first order in l / 0 2 z s 2 and 02(Th2

B

=

+

z,2),

[ l + l/o2 zs2

+ W2(Th2 + T t 2 ) ] 4 ,

(4.12a)

4. Modulation techniques

cot4

= I/OT, -

a( Zh +

'T,),

57 (4.12b)

where the relaxation times are defined as follows: z, = C/K,, 'Th = ch/K,, and 'T+ = C{Kt. To take into account the finite thermal conductance of the sample, Sullivan and Seidel (1968) considered a sample in the form of a slab heated uniformly at one side by a sine heat flux. The other side of the slab is coupled to the bath through thermal conductance K,. The temperature oscillations in the sample can be presented as

0,

=

p/[oC(l + ~ / o ~ ' T + ,a* 2 ~+ 22Kb/3K,)%],

(4.13)

' T ~is the internal relaxation time of the where T~ = :2 + z; + ,z: sample depending on its thermal diffusivity and thickness, and K, is the thermal conductance of the sample in the direction of heat flow. As a rule, z, is by two to three orders of magnitude larger than z, and the condition m27: >> 1 >> a2z2 is achievable by a proper choice of the modulation frequency. The term 2K,,/3Ks is small owing to the small thickness of the sample. Modulation measurements are usually performed under conditions when the above assumptions are well satisfied, so that the expression for the heat capacity retains the simple form (4.4). When the conditions of the measurements are adequate to the theoretical model, the quantity 00,should not depend on the modulation frequency. A decrease of this quantity at low frequencies shows that the criterion of adiabaticity is not satisfied, while the decrease at high frequencies is due to the inertia of the temperature sensor, a thermocouple or a resistance thermometer. It is necessary to choose the operating frequency inside an interval where the quantity 00,is constant. Velichkov (1992) considered a construction consisting of a substrate carrying a sample, a heater, and a thermometer (Fig. 4.3). Such a construction was successfully employed, e.g.,

50

4. Modulation techniques

in studies of thin layers deposited onto a substrate. The author has presented the results as 00,and tan4 versus the modulation frequency. He stressed a possibility to perform modulation measurements in a nonadiabatic regime when 07 < 10. Earlier, Varchenko and Kraftmakher (1973) examined this regime.

in

Kh

nut

Fig. 4.3. Presentation and equivalent electric circuit of a calorimetric cell containing a substrate with a sample, a heater, and a thermometer (Velichkov 1992). R = 1/K.

4. Modulation techniques

59

4.3. Modulation of heating power The choice of a method to periodically heat the sample depends on its shape and electrical resistivity and on the temperature range of the measurements. At high temperatures, direct electric heating or electron bombardment seem to be preferable. In studies of nonconducting samples, separate heaters or modulated-light heating are employed. Modulated-light heating is usable when there is no need to accurately determine the absolute values of the heat capacity. No data on heating-power oscillations are necessary in modulation measurements of thermal expansivity, temperature derivative of resistivity, or thermopower.

Direct electric heating Samples of sufficient electrical conductivity may be heated by passing an electric current through them. Several methods are known to modulate the heating power. (1) Heating by an AC current. The modulation frequency is twice the frequency of the current. The amplitude of the power oscillations equals the effective power dissipated in the sample. When the current is the only means to heat the sample in a wide temperature range, the amplitude of the temperature oscillations strongly depends on the temperature. The third-harmonic technique is applicable to measure the temperature oscillations. (2) Heating by a DC current on which a small AC component is superimposed. The modulation frequency equals to that of the AC component. The mean temperature and the amplitude of the temperature oscillations in the sample are controlled independently. The equivalent-impedance method is applicable,

60

4. Modulation techniques

and one measures the heat capacity of the sample from parameters of a bridge or a potentiometer circuit. (3) Heating by a modulated high-frequency current. This method is useful when the sample's temperature and its oscillations are measured by a thermocouple electrically connected to the sample. In this case, the low-frequency signal caused by the temperature oscillations can be separated from a high-frequency voltage that may appear in the thermocouple circuit. This method of heating was employed in observations of relaxation phenomena in specific heat when temperature oscillations of two frequencies were created in the sample simultaneously. (4) For heating the samples, it is also possible to employ rectangular unipolar pulses. When the modulation period is sufficiently short, the temperature oscillations are of triangular shape and their period equals to that of the pulses. Pulses of specific shape allow one to avoid such a coincidence (Jin et al. 1984). In all the cases of direct electric heating, it is easy to precisely determine the AC component of the heating power.

Induction heating Filippov and Makarenko (1968) have introduced this method. The sample was placed in an induction furnace fed by a modulated high-frequency current. The mean temperature of the sample was measured by a micropyrometer, while the temperature oscillations were detected by a photomultiplier. A blackbody cavity in the sample was used to avoid corrections for the spectral emittance of the sample. A separate coil served to evaluate the AC magnetic field created inside the furnace. Electrical conductivity of the sample, which also must be known, was measured by the four-probe method. With the induction

4. Modulation techniques

61

heating, it is difficult to accurately measure the power dissipated in the sample. The method was employed in noncontact calorimetry, including modulation measurements in space (Wunderlich et al. 1997).

Modulated-light heating This elegant method invented to measure the specific heat of nickel near the Curie point (Handler et al. 1967) has become very popular. A sample in the form of a foil is placed in a furnace that controls the mean temperature (Fig. 4.4). A light passed through a chopper heats the sample. A thermocouple followed by a lockin amplifier detects the temperature oscillations created in the sample. A photocell provides the reference voltage for the lock-in amplifier. The output DC signal of the amplifier is proportional to the input AC voltage and thus is inversely proportional to the heat capacity of the sample. It is fed to the Y-input of a plotter. The X-input of the plotter is driven by a second thermocouple, with the hot junction inside the furnace. The measurements are performed when the temperature of the furnace gradually changes. The adequate modulation frequency, which ensures the adiabatic regime of the measurements, depends on the thickness and thermal diffusivity of the sample. Connelly et al. (1971) have described this technique in more detail. A nickel single crystal of 3 ~ 3 ~ 0mm . 1 size was placed into a copper block inside a furnace. Helium gas at pressure 50 kPa provided the necessary heat exchange. A strip thermocouple of cross-section 70x5 pm measured the temperature oscillations in the sample. An incandescent lamp powered by a stabilized source provided a radiation constant within 0.4% for several hours. A control of the radiation by a thermopile permitted introducing corrections if necessarily. The increase in the mean temperature of the sample above the furnace amounted to 0.5 K.

62

4. Modulation techniques

In the range 25-60 Hz, the quantity 00, remained constant within 1%. The rate of heating the furnace was adjustable from 0.06 to 0.4 K.min-'.

Fig. 4.4. Modulated-light heating (Handler et al. 1967). The amplitude of the temperature oscillations in the sample is recorded under slow changing the mean temperature.

In the most measurements using this method, the amplitude of the heating-power oscillations was not determined but precautions were taken to make it to be independent of the sample's temperature. For this purpose, the samples were covered with a thin layer of graphite or PbS whose spectral absorptances are high and do not depend on temperature. To evaluate the oscillations in the heating power caused by the modulated light, Salamon (1970) employed the relation between the mean power supplied to the sample and the temperature

4. Modulation techniques

63

difference between the sample and the bath: P = KAT, where K is the thermal conductance between them. lkeda and lshikawa (1979)also used this approach.

Electron bombardment An important advantage of electron-bombardment heating is the possibility to use samples of irregular shape. At the same time, it is easy to determine the AC component of the heating power. The power supplied to the sample can be modulated as follows: (i) in a saturation regime, the accelerating voltage is modulated, while the electron-beam current remains constant; (ii) the temperature of the cathode is modulated or a control electrode is used to modulate the electron current, while the accelerating voltage is constant; (iii) the accelerating voltage is periodically switched on and off, so that the heating-power pulses are rectangular. A drawback of this method is that the power oscillations increase along with the mean heating power. The power dissipated in the sample equals I U , where I is the electron-beam current, and U is the accelerating voltage. A thermocouple is either welded to the sample or placed into a thinwall insulating capillary passing through the sample. Welding ensures better thermal coupling but a part of the anode current may appear in the thermocouple circuit. The second method requires longer modulation periods. An additional radiative heater was used to independently control the mean temperature of the sample and the amplitude of the temperature oscillations. Using this technique, the specific heat of iron was measured in the range 600-1250 K (Varchenko et al. 1978).

64

4. Modulation techniques

Separate heaters Separate heaters for periodic heating are used mainly at low and middle temperatures and in studies of nonconducting samples. The main requirements for a heater are its small heat capacity and good thermal coupling to the sample. Microresistors or thin deposited films may serve for this purpose.

Peltier heating This method has an advantage that no DC temperature increment is introduced. Johansen (1987) used a Peltier element to periodically heat the sample in an AC dilatometer.

4. Modulation techniques

65

4.4. Measurement of temperature oscillations Various methods exist to detect temperature oscillations. First, they were determined from oscillations in the resistance of the sample, thermionic current or radiation from it. Later, separate temperature sensors, thermocouples and resistance thermometers, were employed. A choice of the method to measure the temperature oscillations depends on the temperature range, shape, and electrical conductivity of the sample, and on the modulation frequency.

Use of oscillations in the sample's resistance With this method, the measurements are possible at relatively low temperatures when the radiation from the sample is small. In addition, high modulation frequencies are applicable because the problem of thermal inertia of a temperature sensor is avoided. The only drawback of the method is the need to know the temperature dependence of the sample's resistance and of its temperature derivative. The method is usable when these quantities exhibit no anomalies. To measure temperature oscillations, three techniques are known: the supplementarycurrent method, the third-harmonic technique, and the equivalent-impedance method. Evidently, in all the cases the temperature derivative of the sample's resistance, R', enters the expressions for determinations of the specific heat.

Supplementary-current method. Corbino

( 1 910)

has invented this method long ago. He observed a DC voltage drop across the sample when a supplementary AC current of a frequency equal to the frequency of the temperature oscillations passed through it. The supplementary current is much smaller

4 . Modulation techniques

66

than the heating current. The oscillations in the sample’s resistance are R’Oosin20t. A supplementary current isin(2ot causes a DC voltage drop across the sample that equals (iR’Oo/2)cos~.It reaches a maximum when the supplementary current has the same phase as the temperature oscillations. Gerlich et al. (1965) employed a supplementary current of a frequency much higher than the frequency of the temperature oscillations in the sample (Fig. 4.5).

x)

sample

potentiometer

320 Hz filter -0

O t 1 2 H z

1

Fig. 4.5. Setup employing supplementary current of a frequency much higher than the frequency of the temperature oscillations (Gerlich et al. 1965).

A l - H z current passes through temperature oscillations at 2 Hz. resistance are detected by means of a component of the voltage drop

the sample, setting up The oscillations in the 320-Hz current. The main across the sample is

4. Modulation techniques

67

compensated by a potentiometer. A lock-in amplifier measures the component related to the temperature oscillations, with the frequency 320 Hz as a reference. A 2-Hz signal obtained from the lock-in amplifier is fed through a filter to the Y-input of an oscilloscope. The X-input of the oscilloscope is connected to the oscillator of the frequency 1 Hz. The Lissajous pattern is photographed and the specific heat is evaluated from its shape and the temperature derivative of the sample's resistance. A modification of the method consists in using a supplementary current of a frequency close to that of the temperature oscillations (Kraftmakher and Tonaevskii 1972). This causes a difference-frequency voltage across the sample, being proportional to the oscillations in its resistance and to the supplementary current. An advantage of the method is that the difference-frequency signal depends only on the oscillations of the resistance. The method was used to directly determine the temperature coefficient of specific heat. In this case, it was necessary to measure a weak second-harmonic component of the temperature oscillations in the presence of a much stronger fundamental signal.

Third-harmonic technique. Corbino (191 1) has found that when an AC voltage is applied to a sample, the current through it contains a third-harmonic component proportional to the temperature oscillations and to the temperature derivative of the sample's resistance. He measured this component by compensating the fundamental-frequency current and observing the Lissajous pattern by means of a cathode-ray tube. When an AC voltage Usinot is applied to a sample, the temperature oscillations in it, under adiabatic conditions, are 0 = -Oosin2cot. The current passing through the sample equals i = ( U s i n o t ) / ( R , - R10,sin2ot). The oscillations of the sample's resistance are small, so that the current obeys the relation

4. Modulation techniques

68

i

=

U ( s i n o t + POosinot s i n 2 0 t ) / R o = = U [ s i n o t + ( p O o / 2 ) c o s o t - ( ~ O , / 2 ) c o s 3 o t ] / R 0 , (4.14)

where p = R’/R,. When the sample is fed through a sufficiently high resistance, the amplitude of the third-harmonic voltage across the sample, under adiabaticity conditions, is

V,

=

13RoR’/8mcw = U3R’/8R02rnco,

(4.15)

where I and U are the amplitudes of the fundamental-frequency current through the sample and of the voltage across it.

I jigita1 I.

.

I

lock-in amplifier frequency tripler

Fig. 4.6. Calorimetric setup using third-harmonic technique in a wide frequency range (Jung et al. 1992).

4. Modulation techniques

69

Filippov (1960) measured the third-harmonic signal by a bridge circuit with the sample as one of its arms. The bridge was balanced at the fundamental frequency, and the thirdharmonic output signal was observed by means of a selective amplifier and an oscilloscope. A variable capacitor shunting one of the arms served to compensate the additional fundamentalfrequency voltage caused by the temperature oscillations. With this approach, Rosenthal (1961, 1965) studied the temperature oscillations in thin wires over a wide frequency range. He presented the data in a convenient form of polar diagrams. Holland (1963) employed the third-harmonic method in measurements of specific heat of titanium and Skelskey and Van den Sype (1970) in measurements on gold. Birge and Nagel (1985, 1987) and Birge (1986) used this method in measurements over the range 10 mHz-6 kHz. Jung et al. (1992) have described a fully automated calorimeter employing the third-harmonic technique up to 10 kHz (Fig. 4.6). Measurements with different modulation frequencies were aimed at search for relaxation phenomena in specific heat. The method, often referred to as 3o technique, was used in many other studies.

Eguivalent-impedance method. Long ago, radio engineers have understood that a temperature-sensitive resistor, through which a DC current is flowing, can be represented by an equivalent impedance depending, in particular, on its heat capacity (Griesheimer 1947; Jones 1953; Van der Ziel 1958). However, a long time elapsed before thermophysicists have realized the advantages of specific-heat measurements based on this principle. This technique appeared to be very convenient for measurements on wire samples at high temperatures (Kraftmakher 1962). When a wire sample is heated by a current 1 = I, + i s i n o t ( i << l o ) its , resistance follows the relation

4. Modulation techniques

70

R

R,+ R'O = R, + R'O,sin(ot - cp) = = R,+ R'Oocoscp sinot - R'O,sincp cosot.

=

(4.16)

The voltage drop across the sample is

ZR

( I , + isinot)x(R, + R'Oocoscp s h o t - R'O,sincp cosot) = = IoRo + I,R'Oocoscp sinat - Z,R'O,sincp cosot + iRosinot + + iR'O,coscp sin20t - iR'@,sincp sinat cosot. (4.17) =

Neglecting the small terms, the AC voltage across the sample is

V

=

iRosinot + Z,R'O,coscp s h o t - Z,R'O,sincp cosot. (4.18)

This voltage contains two components, one in phase with the AC component of the current (the first two terms), and the other quadrature lagging (the third term). The impedance of the sample Z depends on the amplitude and phase relations between the AC components of the current through and the voltage across the sample. It may be written as a complex quantity Z = R, + A - iB. The quantities A and B are obtainable from (4.18) and (4.9b) and by dividing the AC voltage across the sample by the AC component of the current:

Z

=

R,

+ (2102RoR'/mcw)sincpcoscp - i(2Z02RoR'/mcw)sin2cp. (4.19)

The ratio B/A equals tancp, so that at high frequencies the additional real part of the equivalent impedance, A , is much smaller than the imaginary part, B (Fig. 4.7). The equivalent electric circuit can be represented by a resistor and a capacitor connected in series: Z = R, + A - i/oC*. The resistance equals R, + A , and the capacitance is C* = mc/2Z02R,R'. In an adiabatic regime, A << R,, and Z = R, - i(2Z02RoR'/mcw).The

4. Modulation techniques

71

equivalent capacitance C* does not depend on the frequency of the temperature oscillations. However, the large equivalent capacitance makes this approach to be impracticable. To avoid this difficulty, the equivalent circuit was displayed by a resistor R and a capacitor C connected in parallel:

Z , = R / ( 1 + o2 R 2 C 2 )

0.1

-

i o R 2 C / ( 1 + o2 R 2 C 2 ).

1

o/o

(4.20)

10

0

Fig. 4.7. Real and imaginary parts of equivalent impedance, Z = R, + A - iB. w/wo = B/A = tan cp, where oo is a frequency for which cp = 45'. As a rule, measurements are performed under conditions when A << B << R, and

B=w ~ C .

When obtaining the above relations, the temperature derivative of the sample's resistance was supposed to be positive. Under a sufficiently high modulation frequency,

4. Modulation techniques

72

o2 R 2 C2 << 1, A << R,, and sin2q z 1. Hence, R

=

R,, B

=

oR2C,

and from (4.19) and (4.20) one obtains the main relation f o r the equivalent-impedance technique:

mc

=

2102R'/02RC.

(4.21)

.

selective amplifier V

lock-in detector

t Fig. 4.8. Bridge circuit for measurements by equivalent-impedance technique (Kraftmakher 1962). The heat ca acit of a wire sample relates to the bridge parameters as mc = 21, R / w RC, where I, is the DC current passing through the sample, R' is the temperature derivative of the sample's resistance, o is the angular frequency of the temperature oscillations, and values of R and C correspond to the balance of the bridge.

f ,Y

4. Modulation techniques

73

The heat capacity of the sample is thus fitted to the parameters of the equivalent impedance, R and C. It is measurable by means of a bridge circuit, one arm of which is shunted by a variable capacitor (Fig. 4.8). A selective amplifier tuned to the modulation frequency allows one to achieve high sensitivity even with small temperature oscillations in the sample. For very small temperature oscillations, it is necessary to employ the lock-in detection. Using a dual-channel lock-in amplifier, one of the detectors is set to be sensitive to the in-phase component of the input voltage and the other to the quadrature component. Signs of the output DC voltages of the detectors show whether increasing or decreasing of the resistor R and capacitor C are needed to complete the compensation. Using this technique, the specific heat of tungsten was determined at temperatures up to 3600 K (Kraftmakher and Strelkov 1962). The bridge circuit is unsuitable at relatively low temperatures when the cold-end effects become significant. In this case, one has to use very long wire samples or to heat the current leads to which the sample is welded. As a good alternative, a potentiometer circuit is usable to measure the heat capacity of a central portion of the sample (Kraftmakher 1966a). A DC current with a small AC component added heats the wire sample (Fig. 4.9). The potential probes are much thinner than the sample and cause no significant temperature changes at the points where they are welded to the sample. To satisfy this requirement more reliably, one may heat the probes by passing an additional current through them. The voltage drops across the resistors R2 and R3 are compared by means of a selective amplifier. The resistor R1 and capacitor C1 are adjusted to obtain a proper amplitude of the A C current in the compensation circuit and a phase strictly coinciding with that of the A C component of the heating current. The amplifier is then switched to measure the equivalent impedance of the central portion of the sample.

74

4. Modulation techniques

As in the case of the bridge circuit, the parameters of the equivalent impedance, R and C, do not depend on the AC component of the heating current. An oscilloscope indicates the balance, and a lock-in amplifier is also applicable. The resistance R1 is much larger than R. Therefore, the amplitude and the phase of the current in the compensation circuit are independent of R and C. The expression (4.21) is valid but now it relates to the central portion of the sample.

Fig. 4.9. Potentiometer circuit that eliminates cold-end effects (Kraftmakher 1966a). Only a central portion of the sample is involved in the measurements.

All the quantities to be measured, i.e., m , I,,, o,R , and C , are determined very accurately. The mass of the sample is measured with an error less than I%, whereas errors in the other

4. Modulation techniques

75

quantities are smaller than 0.1%. The total accuracy of the measurements of specific heat depends mainly on errors in the accepted values of R'. A case when the temperature dependence of the sample's resistance is nearly linear is most favorable. Hence, the temperature derivative R' weakly depends on temperature. For example, for tungsten and molybdenum the temperature dependence of R' is of about 1% per 100 K. On the other hand, the method is inapplicable in studies of phase transitions accompanied by anomalies in electrical resistivity.

Photoelectric detectors Many workers determined the temperature oscillations from the sample's radiation. Lowenthal (1963) measured the specific heat of tungsten, tantalum, molybdenum, and niobium in the 1200-2400 K range. An AC current heated the samples, and a photomultiplier served as the temperature sensor. The dependence of the photomultiplier's current on the temperature of the sample was supposed to have the form I = A T n , with n weakly dependent on temperature. In this case, the amplitude of the temperature oscillations obeys the relation

0, = TV/n V,,

(4.22)

where V, and V are the DC and AC components of the photomultiplier output voltage. Filippov et al. (1964) and Filippov and Yurchak (1965) considered the dependence of the photomultiplier's current on the sample's temperature to be I = Bexp(-A/T). Therefore, 0 , = T 2 V / A V,.

(4.23)

76

4. Modulation techniques

However, neither of these methods is satisfactory because the quantities A , B, and n are temperature dependent ( A and B depend on the effective wavelength and on the spectral emittance of the sample). These difficulties can be avoided and an additional advantage gained by employing samples with blackbody cavities. When such samples are maintained at equal mean temperatures, the relation between the oscillations of the radiation from the cavities and the temperature oscillations is the same for all the samples. This enables one to directly compare temperature oscillations in various samples, without any calibration of the photosensor. With a sample of a known specific heat, comparative measurements of specific heat are possible. Tungsten and platinum may serve as reference materials for such measurements. Their specific heat has been determined using all the calorimetric methods already known. Tungsten is a good reference material for temperatures up to 3000 K and platinum up to 1500 K. The high-temperature specific heat of other metals and alloys is not so well known, and comparative measurements of their specific heats would be useful. Akhmatova (1965, 1967) used comparative measurements in studies of molten metals. The samples were placed in a niobium capillary heated by an AC current. The oscillations of the radiation from the empty and filled capillary kept at the same mean temperature were compared. This approach provides the ratio of the heat capacities of the molten metal and the capillary. To perform such measurements more accurately, a compensation circuit was proposed (Kraftmakher 1992ab). The balance of the circuit is independent of the AC component of the heating power (Fig. 4.10). A sample with a blackbody cavity is heated by a DC current I , with a small AC component. To exclude the cold-end effects, only a central portion of the sample restricted by thin potential probes serves for the measurements. The mean temperature and the amplitude of the temperature

4. Modulation techniques

77

oscillations are constant throughout the central portion. In the adiabatic regime, the specific heat is given by

mc

=

21,U/000,

(4.24)

where m and U are the mass of the central portion of the sample and the AC voltage drop across it.

Fig. 4.10. Use of samples with blackbody cavities (Kraftmakher 1992ab). Under compensation, rnc = KIoRC, where K depends only on the sample’s temperature.

The blackbody cavity is projected onto a photodiode. The AC voltage across the load resistor of the photodiode is proportional to the temperature oscillations in the sample: V , = K,O,, where

4, Modulation techniques

78

K , is a proportionality factor. Owing to the blackbody cavity, this factor depends only upon the mean temperature of the sample. The potential probes are connected to the input of an integrating RC circuit. The output voltage of this circuit is amplified by a wide-band amplifier and fed to the load resistor of the photodiode. The AC voltage at the capacitor C equals U / o R C (o2 R 2 C 2 >> 1). The compensation voltage is V2 = K 2 U / o R C , where K , is a proportionality factor. Under compensation, V , = V2, i.e.,

mc

=

2K,I,RC/K2

=

KI,RC.

(4.25)

The coefficient K = 2 K , / K , is the same for all the samples of a given mean temperature. Only the variable capacitor C serves for the compensation. Therefore, for two samples one obtains (4.26) where I,, and Io2 are DC currents providing equal mean temperatures of the samples, and C, and C 2 are the values of the capacitance corresponding to the compensation. This method allows one to perform comparative measurements on any conducting sample with a blackbody cavity. In addition, the full radiation from the cavity can be utilized instead of the radiation in a narrow spectral band. This recompenses the decrease in the radiant flux due to the small size of the cavity. During the measurements, the radiation from the cavity and from a standard strip lamp is in turn projected, by the same optical system, onto a photodiode or another photosensor. The standard lamp serves to determine the mean temperature of the samples. Both samples and the standard lamp are located in a vacuum chamber and are replaced by means of

4. Modulation techniques

79

a turn-plate. No corrections are thus needed for the reflectance and absorption of the radiation in the chamber's window.

Pyroelectric sensors Many investigators employed pyroelectric sensors to detect the temperature oscillations (Fig. 4.1 1). Coufal (1983) described a pyroelectric thin-film transducer made out of polyvinylidene difluoride (PVDF). A 9-pm PVDF foil with nickel electrodes was brought in thermal contact with the sample. The sensitivity of the transducer was typically 0.1 V.K-', while the response was flat in the range from lo-' to l o 7 K.s-'. Mandelis and Zver (1985) have developed a theory of this approach known as the photopyroelectric technique.

modulated light

I I

I

pyroelectric transducer

Fig. 4.11. Measurement of temperature oscillations by means of a pyroelectric transducer (Marinelli et al. 1994).

80

4. Modulation techniques

Thermocouples and resistance thermometers At middle and low temperatures, thermocouples and resistance thermometers are the most reliable tools for measuring the mean temperature of the sample and the temperature oscillations. Thermocouples are widely used now with the modulated-light heating or separate heaters. In some cases, they were formed by deposited thin films, 10 to 100 nm thick. Thus a good thermal coupling and low thermal inertia are achievable allowing measurements to be made over a wide frequency range. With thermocouples, it is possible to measure temperature oscillations in the range 1-10 mK, the smallest value was 0.1 mK (Bonilla and Garland 1974). Even smaller temperature oscillations, of the order of 1 pK, are measurable at liquid helium temperatures by means of resistance thermometers. When using thermocouples, one has to ensure good thermal coupling of the junction to the sample and to reduce the additional heat capacity. Usually, the sample is mounted on a pair of thermocouples formed using thin spot-welded wires, flattened in the junction region. One of the junctions of the crossed thermocouples detects the temperature oscillations in the sample, while the other measures its mean temperature (Craven et al. 1974). Recently, Garfield and Patel (1998) have described a method of preparing such thermocouples. The simplest way to employ a resistance thermometer is to include it in a DC bridge or potentiometer circuit. Since the resistance of the thermometer follows the temperature oscillations, the AC output voltage of the bridge immediately provides the necessary data. Usually, this voltage is measured by means of a lock-in amplifier.

4. Modulation techniques

81

Lock-in detection of periodic signals Lock-in detection is widely used for measuring weak periodic signals in the presence of signals of other frequencies or a noise. The expected signal is measured by a detector controlled by a reference voltage taken from the source of the modulation. The reference frequency thus exactly coincides with that of the expected signal, and the phase shift between them, under given conditions, remains constant. Using the modulation techniques, the reference voltage is provided either by the source of the modulated power or by a special sensor, e.g., a photocell when using modulated-light heating. The output signal of the detector is averaged over a time sufficiently long to suppress signals of other frequencies and a noise. The effective bandwidth of a lockin detector is inversely proportional to the averaging time. The detector is thus always tuned to the frequency of the signal to be measured and has a readily adjustable bandwidth. The operation of a lock-in detector may be explained as follows. An electronic switch controlled by the reference voltage changes periodically the polarity of the signal fed to an integrating RC circuit (Fig. 4.12). A DC output voltage at the output of the circuit appears only when the signal contains a component of the reference frequency. The DC voltage is proportional to this component and to cosine of the phase shift between it and the reference. The detector incorporates an adjustable phase shifter to obtain the maximum output voltage. When the signal contains no component of the reference frequency, the averaged output voltage remains zero. The reason is that no fixed phase relation exists between the reference and signals of other frequencies or a noise.

4. Modulation techniques

82

I

reference

sly1la

R

Fig. 4.12. Principle of lock-in detection. Averaged output voltage is proportional to the AC input signal of a frequency coinciding with that of the reference and to cosine of the phase shift between them.

Amplifiers employing lock-in detection are called lock-in amplifiers. Owing to the narrow bandwidth, they provide high sensitivity and noise immunity. Some types of lock-in amplifiers employ two detectors governed by references with phases shifted by 90'. This enables one to measure the signal and its phase shift relatively to the reference. Dual-channel lock-in amplifiers are useful when the signal to be measured or balanced contains both in-phase and quadrature components, e.g., when using the equivalent-impedance technique. A lock-in detector is efficient to measure small phase changes. In this case, the phase shift between the signal and the reference is set at 90'. Under such conditions, the output voltage of the detector is zero but the sensitivity to phase changes is the best.

4. Modulation techniques

83

4.5. Modulat ion d iIatometry Principle of modulation dilatometry Modulation dilatometry is capable of measurements of 'true' expansivity, the thermal expansion coefficient within a narrow temperature interval. It involves oscillating the sample's temperature around a mean value and measuring corresponding oscillations in the sample's length. With this approach, the linear expansivity is measured directly. The technique is very attractive because it ignores irregular changes in the sample's length due to external disturbances. In modulation dilatometry, only those changes in the length are measured that reversibly follow the temperature oscillations. This is an excellent method to avoid main problems peculiar to the traditional dilatometry at high temperatures: one measures only what is necessary, while all the unwanted disturbances appear beyond the measurements. Modulation measurements are possible with temperature oscillations of the order of 0.1 K and less if necessarily. At present, even a resolution of 1 K at high temperatures is a great improvement over traditional methods. To measure the oscillations in the sample's length, one can employ various techniques, including the most sensitive. Modulation dilatometry has been proposed long ago (Kraftmakher and Cheremisina 1965). It was applied to studying thermal expansion of metals and alloys at high temperatures (Kraftmakher 1967a, 1972; Glazkov 1985, 1987, 1988; Glazkov and Kraftmakher 1986). However, at that time this technique gained no recognition. Since 20 years it was re-invented and applied to nonconducting materials (Johansen 1987). In the first modulation measurements, a wire sample was heated by an AC current or by a DC current containing a small AC component (Fig. 4.13).

4. Modulation techniques

84

sample

oscillator

M

selective

Fig. 4.1 3. Modulation measurements of thermal expansivity. Only oscillations of the length are measured whose frequency is equal to that of temperature oscillations in the sample (Kraftmakher and Cheremisina 1965).

The upper end of the sample is fixed, while the lower one is pulled by a load or a spring and is projected onto the entrance slit of a photomultiplier. The AC voltage at the photomultiplier's output is proportional to the amplitude of the oscillations in the sample's length. The temperature oscillations are determined from either electrical resistance of the sample or radiation from it. In addition, they are available from the heat capacity of the sample. When a DC current 1, with a small AC component heats

4. Modulation techniques

85

the sample, the linear expansivity a = ( l / l ) d l / d T obeys the evidential expression

a

= mcoV/21KI0U,

(4.27)

where m, c and 1 are the mass, specific heat, and length of the sample, U is the AC voltage across it, V is the AC component at the output of the photomultiplier, and K is the sensitivity of the photomultiplier to the elongation of the sample. The sensitivity is determined under static conditions: K = dV,/dl, where V, is the DC voltage at the photomultiplier's output. In this case, a circuit can be assembled whose balance is independent of the AC component of the heating current. The AC component at the photomultiplier's output is balanced by that of a variable mutual inductance M with the heating current passing through its primary winding. The voltage at the secondary winding of the mutual inductance balances the AC voltage at the output of the photomultiplier under condition

a

=

mcw2M/2101RoK.

(4.28)

The measuring system incorporates a selective amplifier tuned to the modulation frequency, so that the dilatometer becomes insensitive to irregular mechanical perturbations or to oscillations of other frequencies. When an AC current heats the sample, the expansivity is given by

a

= 2mcoV/lPK,

(4.29)

where P is the mean electric power suppli d to the sample. The modulation frequency is in the range 10-100 Hz. It is sufficiently high to satisfy the criterion of adiabaticity. Otherwise, the temperature oscillations obey formulas for a nonadiabatic regime.

06

4. Modulation techniques

Differential method In the differential method (Kraftmakher 1967d), a wire sample consists of two portions joined together: a sample under study and a reference sample of known linear expansivity (Fig. 4.14).

Fig. 4.14. Differential method: oscillations in the length of the sample under study AB are balanced by those of the reference sample BC. Direct comparison of the expansivities is thus feasible (Kraftmakher 1967d).

The two portions are heated by DC currents from separate sources and by AC currents from a common oscillator. The

4. Modulation techniques

87

temperature oscillations in the two portions are of opposite phase. By adjusting the two AC components, the oscillations in the length of the portion under study are completely balanced by those of the reference portion. The photomultiplier acts only as a null indicator. Any variations in its sensitivity, as well as in the intensity of the light, etc., have no influence. To calculate the expansivity, the temperature oscillations in the two portions are to be determined. When the corresponding specific heats are known, one may use a relation that holds when the oscillations balance each other: cx,Z,

U , 1, / m , c l

=

a,Io2U,l,/m,c,.

(4.30)

The subscripts 1 and 2 refer to the main and the reference portions of the sample, respectively. The reference sample is kept at a constant mean temperature, so that all the related quantities are constant. Hence,

a

=

BU,ClN0,U,,

(4.31)

where B is a coefficient of proportionality. The measurements are thus reduced to nullifying the oscillations in the length of the composite sample and to measuring the DC current in the main portion and the AC voltages across the two portions. This technique was employed in studies of the thermal expansion of platinum (1000-2000 K) and tungsten (2000-2900 K). The samples were 0.05 mm thick. Their mean temperature was determined from the electrical resistance, while the temperature oscillations, of about 1 K, were calculated from the specific heat. Similar wires kept at constant mean temperatures served as reference samples. The sensitivity of the setup was about 1 nm. The nonlinear increase of the expansivity was attributed to the vacancy formation in the crystal lattice (Kraftmakher 1967a, 1972).

4. Modulation techniques

88

Bulk samples Another version is suitable for comparatively bulk samples, such as rods and foils (Kraftmakher and Nezhentsev 1971). To eliminate the cold-end effects, the temperature oscillations are created only in a central portion of the sample (Fig. 4.15).

-TI

7

sample

Fig. 4.1 5. Modulation dilatometer for bulk samples (Kraftmakher and Nezhentsev 1971). An electromechanical transducer ET balances oscillations in the sample’s length.

The sample is heated by a mains current. A small current of the same frequency, but with its phase linearly varying with time, is added to the mains current in the central portion. The superposition of the two currents causes oscillations in the power

4. Modulation techniques

89

dissipated in the central portion, and its temperature oscillates around a mean value. The modulation period ranges from 1 to 10 s. A thermocouple measures the mean temperature and the temperature oscillations in the sample. An electromechanical transducer ET, a small earphone, serves for the compensation. A blade glued to the earphone's membrane is illuminated and projected onto a photodetector PD. The output voltage of the photodetector is partly compensated and then fed to an amplifier. The transducer is connected to the amplifier's output to balance the changes in the sample's length. Owing to the high gain of the amplifier, the system provides almost complete compensation, while the sensitivity is about 10 nm. The changes in the current of the transducer are proportional to the changes in the sample's length. The changes in the current are recorded, along with the temperature oscillations in the sample. A thin wire heated by an electric current also is usable as a compensator. One its end is attached to the sample, while the other is pulled by a spring. A blade is mounted at the upper end of the spring and is projected onto a photodiode. Since the temperature oscillations in the sample are small, only small changes in the temperature of the wire are necessary for the compensation. The mean temperature of the wire is constant and the changes in its length are proportional to the changes in the heating current. At 1500 K, the time response and linearity of a 50-pm tungsten wire turned out to be quite acceptable. To determine absolute values of the thermal expansivity, the compensator requires a calibration. Samples of known thermal expansivity are usable for this purpose. In many cases, it is enough to know the temperature dependence of the expansivity. In addition, the data can be normalized by using mean values of the expansivity in wide temperature ranges available from traditional measurements.

4. Modulation techniques

90

Interferometric modulation d ilatometer Oscillations in the sample's length may be measured by various methods, including those of highest sensitivity. The interferometric method is one of such techniques.

41'

4b

recorder

I

-

LF oscillator I

DC source (P

MI

M2

t lock-in detector

selective amplifier

Fig. 4.16. lnterferornetric modulation dilatometer (Glazkov and Kraftrnakher 1983). Temperature oscillations are created only in a central portion of the sample. A piezoelectric transducer balances oscillations in the sample's length.

In an interferometric modulation dilatometer (Glazkov and Kraftmakher 1983),the samples were taken in the form of a thick wire or a rod (Fig. 4.16). The upper end of the sample is fixed,

4. Modulation techniques

91

while a small flat mirror M I is attached to its lower end. A DC current from a stabilized source heats the sample. To create the temperature oscillations only in a central portion of the sample, a small AC current is fed to it through thin wires. To prevent an offshoot of the A C current to the upper and lower portions of the sample, a coil of a high A C resistance is connected in series with the sample (not shown in the figure). The beam of a He-Ne laser passes through a beam splitter BS and falls onto the mirror M I and a second mirror M2 attached to a piezoelectric transducer PT. The intensity of the interference pattern is detected by a photodiode P. The amplified output voltage of the photodiode is applied to the transducer, and the oscillations of both mirrors are in phase. Owing to the high gain of the amplifier, the displacements of the mirror M2 follow those of the mirror M I . The A C voltage applied to the transducer is therefore strictly proportional to the oscillations in the sample's length. A selective amplifier tuned to the modulation frequency amplifies this voltage. A lock-in detector measures the amplified signal. The reference voltage for the detector is taken from the oscillator providing the modulation. The output voltage of the lock-in detector is proportional to the amplitude of the oscillations in the sample's length. To check the validity of a modulation dilatometer, it is probably sufficient to verify that the results obtained do not depend on the modulation frequency. The uncertainty in the oscillations of the sample's length is about 1%. The errors in evaluating the linear thermal expansivity depend mainly on the uncertainty in measurements of the temperature oscillations.

92

4. Modulation techniques

Nonconducting materials Johansen (1987) employed a computerized capacitance dilatometer (Johansen et al. 1986) in a regime of periodic changes of the sample’s temperature (Fig. 4.17). A Peltier module provides AC heating of a copper block containing the sample. Capacitance measurements are made by means of a precision bridge and a lock-in amplifier as the imbalance detector. A 10-”-m resolution was achieved and maintained over a large dynamic range, of the order of lop6 m. Unlike Joule heating, the Peltier module introduces no DC temperature increment. The temperature of the copper block is controllable at a fixed temperature point or in a linear sweep mode, in the range -6OOC to 150°C.

Fig. 4.1 7. Simplified diagram of modulation dilatometer for nonconducting samples (Johansen 1987).

4. Modulation techniques

93

Measurement of extremely small periodic displacements Several methods are known to measure extremely small periodic displacements. The periodic character of the displacements allows one to employ selective amplifiers and/or lock-in detectors. Therefore, the sensitivity of the measurements is much higher than the sensitivity to irregular displacements. Laser interferometry provides the most convenient method to measure small periodic displacements. Bruins et al. (1975) have developed such an interferometric technique (Fig. 4.18).

u4

plate

laser

-%

temperature controlled can I

-------------

I

polarizer

oscillator

-

lock-in amplifier

Fig. 4.18. Measurement of small periodic displacements by means of a Fabry-Perot interferometer (Bruins et al. 1975).

94

4. Modulation techniques

A spherical Fabry-Perot interferometer was used to determine piezoelectric constants but the authors pointed out that the method could be adapted to other measurements. The interferometer has several advantages over other interferometric arrangements. First, it is inherently much more sensitive than any two-beam interferometer, because of high sharpness of rnultiple-beam interference fringes. Second, it is a compact instrument mounted in a temperature-controlled environment. A piezoelectric transducer serves as a reference. The vibrations of l ~are measured with 5% accuracy. The theoretical about 4 ~ 1 0 - m sensitivity in a well-isolated environment was estimated to be m. The method may be used as a null technique. A voltage of the same frequency should be applied to the transducer and its amplitude and phase adjusted to exactly cancel the displacements of the sample under study. The sensitivity of the dilatometer is sufficient for any modulation measurements of thermal expansion but the authors did not consider such a possibility . Fanton and Kin0 (1987) have designed a simple setup for measurements of small periodic displacements (Fig. 4.19). The laser beam passes through a polarizing beam splitter and a birefringent Woilaston prism with its optical axis oriented at 45' to the direction of polarization. The prism splits the beam into equal-amplitude beams angularly separated by 0.5', which are focused to two spots on the sample. The reflected beams recombine in the Wollaston prism, pass through the splitter and interfere on a photodetector. One beam focused onto an undisturbed portion of the sample serves as a reference. The sample is heated periodically at the focus of the other beam. Both beams pass through the same optical components making the system resistant to a drift and vibrations. A lock-in amplifier reduces the influence of a noise and disturbances of other frequencies. The background optical noise of the system was 3 . 4 ~ 1 0 - m. l~

4. Modulation techniques

95

The system was used to obtain a thermal image of a flawed conducting strip, a 26-nm nickel film on a silicon wafer. The strip was notched and then heated by passing an 8-kHz current through it. The temperature oscillations in the strip were of about 10 mK. The authors have pointed out that it is possible to modulate the laser at a frequency close to that of the periodic displacements and to obtain a signal at the difference frequency.

polarizing beam splitter laser

Wollaston prism _.

U

sample lock-in photodetector

Fig. 4.19. Measurements of small periodic displacements (Fanton and Kin0 1987).

The sensitivity achievable by the above techniques is much better than necessary for dilatometric measurements. The thermal expansivity can be determined along the sample's thickness. In this case, one completely avoids the problem of temperature gradients in the sample.

96

4. Modulation techniques

4.6. Modulation measurements of electrical resistivity and thermopower Temperature derivative of resistance The modulation technique allows one to directly measure the temperature derivative of electrical resistance. The method consists in oscillating the sample's temperature around a mean value and measuring the oscillations in the sample's resistance along with the temperature oscillations. This technique was developed to study the anomaly in the resistivity of nickel near its Curie point (Kraftmakher 1967b). The nickel sample was heated by a DC current from a stabilized source and by an AC current amplitude-modulated with a period of a few seconds. A thermocouple measured the temperature oscillations in the sample. Simultaneously, measurements were made of voltage oscillations that appeared because the DC current passed through the oscillating resistance. To eliminate the cold-end effects, only a central portion of the sample was involved in the measurements. A two-channel plotter recorded the AC components of the thermocouple's voltage and of the voltage drop across the central portion of the sample. This technique was used in some other studies (Kraftmakher and Sushakova 1972, 1974; Kraftmakher and Pinegina 1974, 1978; Glazkov 1985, 1987, 1988; Glazkov and Kraftmakher 1986). Salamon et al. (1969) used a similar approach in studies of the resistivity of chromium near its Nee1 point, by means of modulated-light heating. Lederman et at. (1 974) have described this technique in more detail. The specific heat and the temperature derivative of resistance of iron near its Curie point were measured simultaneously (Fig. 4.20). The temperature within the furnace was linearly swept at a rate of 1 K.min-' or less. The modulation frequency was 8 Hz. Lock-in amplifiers

4. Modulation techniques

97

measured the AC voltages corresponding to the oscillations in the temperature and in the resistance of the sample. Chaussy et al. (1992) have re-invented this technique.

source

oscillator

chopper

Fig. 4.20. Measurement of temperature derivative of resistance by means of modulated-light heating. Oscillations in the temperature and in the resistance of the sample are recorded simultaneously (Lederman et al. 1974).

4. Modulation techniques

98

Direct measurement of thermopower Among all the modulation techniques, measurements of thermopower (the Seebeck coefficient) are the simplest. The method consists in measuring the same temperature oscillations by a thermocouple under study and by a reference one. Three groups have independently introduced this technique (Freeman and Bass 1970; Hellenthal and Ostholt 1970; Kraftmakher and Pinegina 1970).

photodetector A

Th

light source

I1 detector lock-in 1 thermocouple junction

T

chopper

Fig. 4.21. Measurement of thermopower using modulated-light heating (Hellenthal and Ostholt 1970).

Freeman and Bass (1970) and Hellenthal and Ostholt (1970) have described very similar methods to measure the Seebeck

4. Modulation techniques

99

coefficient. A modulated light heats the thermocouple junction (Fig. 4.21). A lock-in amplifier measures the signals from the two thermocouples. It is possible to compensate the signal at the amplifier's input or to continuously record it. In measurements at high temperatures, a small low-inertia furnace created temperature oscillations measured by the two thermocouples (Kraftmakher and Pinegina 1970, 1971, 1978; Kraftmakher 1971a).

100

4. Modulation techniques

4.7. Summary Modulation calorimetry is a tool very useful for measuring high-temperature specific heat of metals and thus for studying formation of equilibrium point defects. It provides data sufficiently accurate to separate nonlinear contributions to specific heat. Various modifications of modulation calorimetry differ by the ways to modulate the heating power and by the methods to detect the temperature oscillations. For measurements on wire samples, the most convenient and accurate technique is the equivalentimpedance method. Modulation dilatometry is a powerful technique to study thermal expansion of solids. It provides 'true' expansivities with good temperature resolution even at very high temperatures. Modulation measurements enable one to directly determine the temperature derivative of electrical resistance and thermopower (the Seebeck coefficient). These properties depend on pointdefect formation. While modulation calorimetry is commonly recognized and widely used, the other modulation techniques are still waiting for more wide employment.

Chapter 5

Enthalpy and specific heat of metals 5.1. Point defects and specific heat

102

Why point defects affect hig h-temperature specific heat. What was said about calorimetric data, and the opposite viewpoint.

5.2. Methods of calorimetry

105

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

117 127 129

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

101

132 136

102

5. Enthalpy and specific heat

5.1. Point defects and specific heat Why point defects affect high-temperature specific heat Point-defect formation consumes certain energy, so that the enthalpy and specific heat of a crystal with point defects are larger than those of a defect-free crystal. The extra molar enthalpy and specific heat caused the formation of equilibrium vacancies are given by

A C = (NHF2A / k, T 2) ex p (-HF/k, T ).

(5.2)

Here A is the pre-exponential factor entering the expression for the equilibrium vacancy concentration, and N is the Avogadro number. It was assumed that the enthalpy and entropy of vacancy formation do not depend on temperature. The extra specific heat originates mainly from the temperature dependence of the vacancy concentration. The relative increase in the specific heat at a given temperature is about one order of magnitude larger than the vacancy concentration. In low-melting-point metals, the nonlinear increase in the specific heat is much smaller than in refractory metals. Owing to the higher accuracy of calorimetric measurements at lower temperatures, it was confirmed that it also follows the expression describing the vacancy contribution. On the other hand, the calculations may become ambiguous at very low vacancy concentrations. Calorimetric measurements provide the most natural determination of equilibrium concentrations of point defects.

5. Enthalpy and specijic heat

103

However, a problem difficult to solve is the separation of the defect contribution. The specific heat of a hypothetical defectfree crystal cannot be calculated precisely. Many authors, referring to anharmonicity, disregard all vacancy-formation data derived from calorimetric measurements. As a rule, thermophysicists studying properties of metals at high temperatures fit their experimental data by polynomials not taking into account vacancy formation. However, calorimetric determinations of the parameters of vacancy formation, being doubtful from the theoretical point of view, appeared to be quite acceptable in practice. Usually, vacancy contributions are much larger than the uncertainties in the extrapolation of the data from middle temperatures. Otherwise, this approach had no chance to yield plausible formation enthalpies. The vacancy contributions can be unambiguously determined through observations of the vacancy equilibration. The nonlinear increase in high-temperature specific heat was first observed in alkali metals by Carpenter et al. (1938). Later, this phenomenon has been attributed to vacancy formation (Carpenter 1953). Pochapsky (1953) has calculated the formation parameters from the specific heat of lead and aluminum. Rasor and McClelland (1960a) have found a strong nonlinear increase in the specific heat of molybdenum and tantalum. The authors have concluded that it is too large to be ascribed to point defects. A similar nonlinear increase in specific heat was observed in thorium and rhenium (Wallace 1960; Taylor and Finch 1964). The parameters of vacancy formation in refractory metals have been evaluated for the first time from modulation measurements of the specific heat of tungsten, tantalum, niobium, and molybdenum (Kraftmakher and Strelkov 1962; Kraftmakher 1963ab, 1964).

104

5. Enthalpy and specific heat

What was said about calorimetric data, and the opposite viewpoint Calorimetric determinations of the defect-formation parameters were considered by Seeger (1973a) as follows: “From a theoretical point of view, the next most powerful experimental technique for studying equilibrium point defects is the measurement of the specific heat as a function of temperature. In principle, calorimetric measurements are capable of giving absolute concentrations, since the contribution per vacancy is the vacancy formation enthalpy, a quantity that may be determined independently from the temperature variation of the vacancy contribution. The calorimetric techniques were discussed in great detail at the Julich Conference with the conclusion that for most metals the main part of the high temperature rise of the specific heat is due to the lattice anharmonicity and not to vacancies, so that specific heat measurements are not particularly suitable for studying vacancies.” In essence, all what was claimed (e.g., Hoch 1970) is the following: (i) the vacancy concentrations in high-melting-point metals deduced from the calorimetric data are much larger than the differential-dilatometry results for low-melting-point metals; (ii) it is possible to approximate the nonlinear increase in specific heat by polynomials not taking into account vacancy formation. Later, the data based on measurements of specific heat or electrical resistivity at high temperatures have been disregarded by Siege1 (1978): “Measurements of the specific heat or the electrical resistivity are in general plagued by the difficulty in extracting relatively small vacancy-related contributions from large signals dominated by background contributions from the lattice.” The opposite viewpoint is that for specific heat the situation is just opposite: the vacancy contributions are large in comparison with the nonlinear background.

5. Enthalpy and specijk heat

105

5.2. Methods of calorimetry Calorimetric measurements seem, at first glance, to be simple and straightforward. By definition, one has to supply some heat to a sample and to measure the corresponding increment in its temperature. However, no simple solution to this task exists in a wide temperature range. First, the accuracy of temperature measurements in various temperature ranges is very different. Second, when the sample’s temperature is far from room temperature, it is impossible to completely avoid uncontrollable heat exchange between the sample and its surroundings. This drastically restricts the accuracy of calorimetric measurements at high temperatures. After many years of development, several methods proposed to solve the problem are as follows. (1) All possible precautions are undertaken to reduce the heat exchange between the sample and its surroundings (adiabatic calorimetry). (2) The enthalpy of the sample is measured instead of the specific heat: the sample heated up to a high temperature drops into a calorimeter usually kept at room temperature, and the heat released from the sample is measured (drop method). (3) Shortening the time of the measurements minimizes the influence of uncontrollable heat exchanges (modulation, pulse, and dynamic techniques). (4) The heat exchange between the sample and its surroundings is taken into account and involved in the measurements of specific heat (relaxation method).

Ad ia bat ic ca Io ri metry Adiabatic calorimetry reduces to a minimum the heat exchange between the calorimeter and the environment. For this purpose, the calorimeter is surrounded by the so-called adiabatic shield,

106

5. Enthalpy and specific heat

whose temperature is kept equal to that of the calorimeter during all the experiment. The calorimeter and the shield are placed in a vacuum chamber, so only the radiative heat exchange occurs. To reduce the heat flow through the electrical leads, they are thermally anchored to the adiabatic shield. An electrical heater heats the calorimeter. A resistance thermometer or a thermocouple measures the temperature. The heat supplied to the calorimeter and the temperature increment are thus accurately determined. With a temperature difference between the calorimeter and the shield, AT, the power of the radiative heat exchange is proportional to T 3 A T . Adiabatic calorimetry, being an excellent technique at low and middle temperatures, fails therefore at high temperatures. An enhanced heating rate reduces the role of heat losses. Braun et al. (1968) have designed an adiabatic calorimeter of continuous heating for the range 300-1900 K. It is capable of measurements of specific heat of solids (with an error of 2%) and liquids (3%), and latent heats of phase transitions in solids (0.5%) and of melting (1.5%). Three modes of operation are feasible: (i) the applied power is constant, while the heating rate is inversely proportional to the heat capacity of the sample; (ii) the heating rate is kept constant, so that the applied power is proportional to the heat capacity; (iii) with the heater switched off, the temperature difference between the calorimeter and the thermal shield is monitored. Buckingham et al. (1973) have developed a high-precision calorimeter of continuous heating. It was designed for measurements near phase-transition points where the rate of change of specific heat and thermal relaxation time may become very large. Stewart (1983) and Gmelin (1 987) have reviewed calorimetric techniques for low temperatures.

107

5. Enthalpy and specific heat

Drop method This technique was developed for measurements at high temperatures. The sample is placed in a furnace and heated up to a selected temperature measured by a thermocouple or an optical pyrometer (Fig. 5.1).

vacuum chamber

graphite furnace

r I

I

Y

~

adiabatic shield copper block (calorimeter)

Fig. 5.1. Simplified diagram of setup for drop calorimetry (Glukhikh et al. 1966).

108

5. Enthalpy and spec@ heai

Then the sample drops into a calorimeter kept at a temperature convenient for measurements of the heat released from the sample, usually at room temperature. A resistance thermometer measures the increment in the temperature of the calorimeter, which is proportional to the enthalpy of the sample. The calorimetric measurements are thus performed under conditions most favorable to reduce the unwanted heat exchange. The price for this gain is that the result of the measurements is the enthalpy of the sample, instead of its specific heat. The specific heat is obtainable as the temperature derivative of the enthalpy. When the specific heat is weakly temperature dependent, the method is quite adequate. The situation becomes more complicated when the specific heat varies in narrow temperature intervals but the corresponding changes in the enthalpy are too small to be precisely determined. An additional problem arises when phase transitions of the first order occur at the intermediate temperatures. The thermodynamic equilibrium in the sample after cooling is therefore doubtful. Drop calorimetry has been developed when there was no alternative for high temperatures. Nevertheless, it remains useful until today, especially for studying nonconducting materials. A special case of the drop technique is the levitation calorimetry, when the sample is levitated and heated by a highfrequency electromagnetic field (for reviews see Chekhovskoi 1984, 1992). Therefore, no container is needed, and the temperature range of the measurements can be extended above melting points of refractory materials. For instance, Arpaci and Frohberg (1984) have measured the specific heat of tungsten up to 4000 K.

5. Enthalpy and specific heat

109

Pulse and dynamic techniques The influence of any heat exchange between the sample and its surroundings is proportional to the time of the measurements. A decrease of this time is an efficient approach even at very high temperatures. The method is applicable whenever it is possible to heat the sample and to measure its temperature rapidly. Conducting samples heated by passing through them an electric current or by electron bombardment are well suitable for such measurements. The temperature of the sample is measured through its resistance or radiation. When the heat losses cannot be completely neglected, they can be determined and taken into account. Pulse calorimetry employs small increments in the sample's temperature, and the result of one measurement is the specific heat at a single temperature. A furnace or electric heating provides the initial temperature of the sample. Rasor and McClelland (1960b) have designed an apparatus with a sample placed in a graphite furnace to achieve the initial temperature. The temperature increment caused by an electrical pulse was measured by a photomultiplier (Fig. 5.2). By means of a four-channel oscilloscope, the heating current I , the voltage drop across the sample U , the temperature of the sample T , and the heating rate T,' = ( d T / d t ) , were recorded simultaneously. With heat losses neglected, the heat capacity of the sample is

The specific heat of molybdenum, tantalum, and graphite was measured by Rasor and McClelland (1960a) at high temperatures, up to 3920 K for graphite. The strong nonlinear increase in the specific heat of refractory metals has been observed for the first time.

7i

P 0 0 0

0 0 0

w

0 0 0

Iu

0 0 0

2

0

0 UI

P 0

2

P UI

-.L

P

specific heat (ca1.g-'.K-')

5. Enthalpy and specific heat

111

Dynamic calorimetry consists in heating the sample over a wide temperature interval. The heating power and the sample's temperature are measured continuously during the run. If the heat losses from the sample are significant, they are measured during the cooling period and then taken into account. These data are sufficient to evaluate the specific heat in the whole temperature range. Cezairliyan and coworkers at the National Bureau of Standards have developed a convenient and accurate subsecond technique for temperatures up to 3600 K (for reviews see Cezairliyan 1984, 1988, 1992). In one run, the specific heat, the electrical resistivity, the normal spectral emittance, and the hemispherical total emittance are measured in a wide temperature interval. This technique was successfully employed in studies of many high-melting-point metals and alloys. The specific heat of tungsten was measured up to 3600 K (Cezairliyan and McClure 1971). The temperature of the samples with blackbody models was measured by an optical pyrometer (Fig. 5.3). The power-balance equations for the sample during the heating and cooling periods are given by

+P

=

IU,

(5.4a)

mcT,' + P

=

0.

(5.4b)

mcT,,'

Here P is the power of the heat losses from the sample, and T,' = (dT/dt), is the cooling rate after ending the heating current. From these relations, the quantity P can be excluded. The heat capacity of the sample is

mc

=

IU/(Th' - Tcf),

where Th' and T,' relate to the same temperature of the sample.

112

5. Enthalpy and specijk heat

Cezairliyan and Righini (1996) have reviewed the high-speed optical pyromet ry .

amplifier

pyrometer

L-^LL^-.

.

Fig. 5.3. Dynamic calorimetry developed by Cezairliyan and coworkers (for reviews see Cezairliyan 1984, 1988, 1992).

A setup described by DobrosavljeviC and Maglic (1989) is based on heating wire samples with the rate up to 1500 K.s-’. The setup consists of a vacuum chamber, an electric power circuit, measuring and control devices, and a computer. The wire sample, 2 mm in diameter, has a total length about 200 mm. To exclude the cold-end effects, only a central portion of the sample, 20 mm long, is employed in the measurements. Three

5. Enthalpy and specific heat

113

0.05 or 0.1 mm thermocouples are spot-welded at the center and symmetrically at 1O-mm separations on both sides. The thermocouple legs serve also as potential leads to measure the voltage drop across the central portion of the sample. The data are collected with a l-kHz sampling rate. Contact temperature measurements in the presence of an electric current pose a problem because it is impossible to position both thermoelectrodes along the same equipotential line. The necessary corrections are based on comparison of the last thermocouple reading in the heating period and the first reading after ending the current. In the 300-1900 K temperature range, the maximum uncertainties were estimated to be 3% in specific heat and 1% in electrical resistivity. For other results by this technique see MagliC (1979), MagliC and DobrosavljeviC (1992), MagliC et al. (1994, 1997), and VukoviC et al. (1996).

Relaxation method This technique is based on measurements of the cooling (or heating) rate of a sample whose temperature differs from that of the surroundings (Bachmann et al. 1972; Denlinger et al. 1994). This rate depends on the temperature difference, heat capacity of the sample and the heat transfer coefficient, i.e., the temperature derivative of the heat losses from the sample. One of the methods employs a calorimeter in which a sample under study and a reference one are put in turn. The temperature dependence of the heat losses from the calorimeter remains the same, and it is easy to calculate the ratio of the specific heats of the two samples. In the step method, the calorimeter is first brought to a temperature T = To + A T , slightly higher than that of the surroundings, To. Then the heating current is switched off and the temperature of the calorimeter decays exponentially:

114

T

=

5. Enthalpy and specijic heat

T o + AT exp(-t/z),

where T = C/K is the relaxation time, C is the sum of the heat capacities of the sample and of the calorimeter, and K is the heat transfer coefficient. The determination of the specific heat thus includes measurements of the relaxation time and of the heat transfer coefficient. Under steady-state conditions, P = KAT, where P is the power applied to the calorimeter that is necessary to increase its temperature by AT. The temperature increment is small, so that the linear relation is valid. The internal time constant, which relates to equilibration inside the sample and between the sample, the heater, and the thermometer, is much shorter than the relaxation time z. In the sweep method, a wide temperature range is covered in one run. For this purpose, the steady-state temperature of the sample is determined beforehand as a function of the heating power. Then the specific heat of the sample is available from the measured cooling curve. The relaxation method is applicable to samples as small as a few micrograms. Zinov’ev and Lebedev (1976) employed the relaxation technique to measure the specific heat of tungsten in the range 2400-3600 K. After heating the wire sample to the highest temperature, the cooling curve was observed by means of a photomultiplier and an oscilloscope. The specific heat obeys the relation

mc

=

-P/T,’.

(5.7)

Here P is the power of heat losses from the sample, and T,’ is the cooling rate at this temperature. The results obtained by the authors fairly coincide with those from the modulation measurements (Kraftmakher and Strelkov 1962).

5. Enthalpy and specijk heat

115

Hatta (1979) has designed a relaxation calorimeter employing light heating. Ema et al. (1993) and Yao et al. (1998) have described calorimeters operating in both relaxation and modulation modes.

Rapid-heating experiments Rapid-heating techniques allows one to measure thermophysical properties of metals over wide temperature intervals including liquid state (for reviews see Lebedev and Savvatimskii 1974; Gathers 1986; Cezairliyan et al. 1990). Pottlacher et al. (1991, 1993) have reported on experiments with heating rates more than l o 9 K.s-’. Starting at room temperature, the measurements are performed far into the liquid phase of the metal under study, up to 10000 K. The necessary energy is stored in a 5.4 pF capacitor, with a charging voltage 4 to 8 kV (Fig. 5.4). The wire samples are typically 40 mm long and 0.25 mm thick. Water serves as the ambient medium to avoid peripheral discharges. The pressure in the vessel is variable up ~ A three-electrode spark gap triggers the main to 2 ~ 1 0Pa. discharge. The quantities measured during the entire run are as follows: (i) the current through the sample, by means of an induction coil; (ii) the voltage drop across the sample, by using a coaxial voltage divider; (iii) the radiance temperature of the sample, by a fast pyrometer; (iv) the final volume of the sample, by employing a shadowgraph technique with a 30-1-1s exposure. The melting temperatures are used as the calibration points, and the emissivity of the sample’s surface is considered as being independent of temperature. A special care should be taken to avoid a superheat of the samples that is quite probable in rapidheating experiments. From the measurements, the enthalpy, the electrical resistivity, and the volume expansion are deduced. The authors

116

5. Enthalpy and specific heat

have pointed out that more accurate results for the solid phase are available from static measurements. However, rapid heating permits measurements far above the melting point. Moreover, this technique has a potentiality to monitor vacancy equilibration and hence to reliably reveal vacancy contributions to enthalpy and electrical resistivity of metals.

sample flashlamp

T

-, window

-

T

A

oscilloscopes in shielded room

Fig. 5.4. Rapid-heating technique using heating rates of the order of 10’ K.s-’ (after Pottlacher et al. 1991, 1993).

Kaschnitz et al. (1992) have designed a microsecondresolution system. Heating rates in the range from l o 7 to l o 8 K.s-’ provide data more accurate than by using faster systems. Wire or tube-shaped samples are resistively heated by

5. Enthalpy and specijk heat

117

means of a discharge circuit. Energy is stored in a capacitor, 240 to 500 pF, which may be charged up to 10 kV. Typically, the heating current is about 5000 A and the pulse is 80 ps long. To measure the temperature of the sample, a lens produces its magnified image at the rectangular entrance of an optical fiber. The light passes through the fiber and enters a photodiode detector. The detector is self-calibrated with the plateau of the melting transitions. The thermal expansion of the samples is determined photographically, by means of a Kerr cell. The uncertainty of the data was estimated to be 3% for the enthalpy and 3% for the electrical resistivity, without corrections for thermal expansion.

5.3. Formation parameters from calorimetric data In the first modulation measurements of specific heat, including those by Corbino (1910, 1911), wire samples were heated by an AC current passing through them. In this case, the amplitude of heating-power oscillations equals the mean applied power and thus is easy to measure. The temperature oscillations were determined through oscillations in the sample's resistance. For the measurements, Corbino has developed two techniques, the supplementary-current method and the third-harmonic method. A good finding turned out to be the equivalent-impedance technique. Here, a DC current with a small AC component added heats a wire sample. One thus independently varies the mean temperature of the sample and the temperature oscillations in it. A bridge or potentiometer circuit is employed whose balance is independent of the AC component of the heating current. The heat capacity of the sample relates to the resistance and capacitance corresponding to the balance. A disadvantage of this technique is the requirement to know the temperature

118

5. Enthalpy and speciflc heat

dependence of the sample's resistance and of its temperature derivative. This drawback is peculiar to all methods based on determinations of the temperature oscillations through the sample's resistance. The bridge circuit was used to measure the specific heat of tungsten in the range 1500-3600 K (Kraftmakher and Strelkov 1962). Tungsten has a high melting point and for a long period it served for fabrication filaments for incandescent lamps and cathodes for vacuum tubes. The temperature dependence of its resistivity is therefore well known. The measurements provided a good opportunity to verify the equivalent-impedance technique. In the range 1500-2500 K, the results obtained are in good agreement with existing data. At higher temperatures, a strong nonlinear increase in the specific heat was found and attributed to point-defect formation in the crystal lattice. The equivalentimpedance method was employed in studies of other highmelting-point metals: tantalum, niobium, molybdenum, and platinum (Table 5.1). The nonlinear increase in the specific heat of all the metals was treated as a result of point-defect formation (Kraftmakher 1966b). Equations that fit experimental data take into account the vacancy contributions (Table 5.2). The nonlinear increase in high-temperature specific heat of metals was observed by very different calorimetric techniques. Especially numerous are the measurements on tungsten. The data presented here (Fig. 5.5) show that the phenomenon was observed by all the calorimetric techniques already known. Surprisingly, the scatter of the data is quite moderate. The results of the modulation measurements were confirmed by pulse and relaxation measurements. One may compare the results of the modulation measurements and the data by dynamic calorimetry (Righini et al. 1993). The difference between the two curves is less than 1% in the range 1500-2100 K and less than 3.5% in the range

119

5. Enthalpy and specific heat

Table 5.1 Specific heat determined by equivalent-impedance method (smoothed K-I). values,

J.mol-' .

T (K)

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600

W

28.65 29.2 29.75 30.3 30.85 31.4 31.95 32.5 33.05 33.7 34.45 35.4 36.4 37.7 39.1 41.0 43.1 45.9 49.15 52.95 57.35 62.8

Ta

27.8 28.05 28.35 28.65 28.9 29.15 29.45 29.75 30.1 30.4 30.75 31.1 31.5 31.95 32.5 33.15 33.95 34.95

Mo

30.55 31.2 31.85 32.5 33.15 33.95 35.0 36.2 37.65 39.55 42.0 44.8 48.1

Nb

27.6 28.1 28.65 29.15 29.65 30.2 30.8 31.55 32.4 33.3 34.35 35.6 36.95 38.5 40.6

Pt

29.9 30.45 31.0 31.55 32.1 32.85 33.95 33.3 36.95 38.9 41.3

120

5. Enthalpy and specijk heat

21 00-3200 K. This range is sufficient to calculate the parameters of vacancy formation. At higher temperatures, the difference grows rapidly and amounts to about 15% at 3600 K. However, above 3200 K the other results are disposed between the data considered. For other metals, the situation is not so good because only a few measurements have been carried out. In this respect, tungsten is rather a fortunate exception.

Table 5.2 High-temperature specific heat of metals from the modulation measurements. The data are fitted by equation taking into account vacancy formation: C = u + b T + cT-2exp(-d/T) J.mol-l.K-l.

Metal

W Ta Mo Nb Zr

Pt Ti Ni cu AU

La

U

20.5 24.35 22.0 20.7 20.5 24.5 24.05 27.2 23.65 23.85 24.7

bx105

545 285 650 525 420 545 440 530 500 525 650

CXIO-*O

740 220 170 30 35 25 46 50 5 2.5 37

dx 1 0-3

36.5 33 26 23.7 20.3 18.6 18 16.2 12.2 11.6 11.6

Reference

KS 1962 K 1963a K 1964 K 196313 KK 1966 KL 1965 Sh 1965 G 1987 K 1967c KS 1966a AK 1970

KS - Kraftmakher and Strelkov, K - Kraftmakher, KK - Kanel’ and Kraftmakher, KL - Kraftmakher and Lanina, Sh - Shestopal, G - Glazkov, AK - Akimov and Kraftmakher.

5. Enthalpy and specific heat

2000

2500

3000

3500

121

4000 n

Fig. 5.5. High-temperature specific heat of W. 1 - curve presenting data from modulation method (Kraftmakher and Strelkov 1962), pulse calorimetry (Affortit and Lallement 1968), and relaxation measurements (Zinov'ev and Lebedev 1976); 2 - drop method (Arpaci and Frohberg 1984); 3, 4 - pulse calorimetry (Yakunkin 1983; Senchenko and Sheindlin 1987); 5 - dynamic measurements (Righini et al. 1993).

The pulse measurements on molybdenum and tantalum (Rasor and McClelland 1960a) and the modulation measurements on tungsten and other refractory metals were the first direct determinations of the specific heat of these metals at high temperatures. Earlier, only drop calorimetry was employed above 2000 K, and the nonlinear increase in the specific heat was not seen. It has been shown (Kraftmakher 1971b) that the reason for

122

5. Enthalpy and specific heat

this was an employment of fitting polynomials not taking into account vacancy formation. Regrettably, many authors until today do not attempt to check whether their experimental results exhibit vacancy contributions. For example, Cezairliyan and coworkers fitted the high-temperature specific heat of refractory metals by polynomials not including the vacancy-related term. It is interesting to fit these data by the equation containing such a term. For niobium, tantalum, and tungsten, this procedure leads to very plausible values of the formation enthalpies (Fig. 5.6). For molybdenum, the formation enthalpy appeared to be somewhat lower than the expected value. This is probably due to relatively narrow temperature interval of the measurements, so that the vacancy-free part of the specific heat could not be determined accurately. The narrow temperature interval caused also a very low error of the fit in comparison to other metals. Using pulse calorimetry, Cezairliyan and coworkers have determined specific heat of W, Ta, Mo, Nb, V, Zr, Ti, Pd, Ni, and of several alloys. Results for Zr and Ti are presented here (Fig. 5.7). The specific heat of vanadium was measured also by Stanimirovie et al. (1999). The nonlinear increase in the specific heat is clearly seen (Fig. 5.8). However, the authors did not try to fit the data supposing point-defect formation. Many data on equilibrium vacancies in metals are based on calorimetric measurements (Table 5.3). Only formation parameters evaluated by the authors themselves are presented here. In low-melting-point metals, the nonlinear increase in specific heat is much smaller than that in refractory metals. However, owing to the higher accuracy of the measurements, it was measured and interpreted by Kramer and Nolting (1972) in terms of vacancy formation. The derived vacancy concentrations at the melting points of indium, tin, lead, zinc, antimony and aluminum are in the range from 5 ~ 1 0 -to~ 2 . 3 ~ 1 0 - ~whereas , in

123

5. Enthalpy and specific heat

1500

-

0.4

u)

c .-

2000

2500

3000

'I

3500

4000 K

C

3

4

0.3-

m

Y

C

0

'E

.-m>

0.2-

a, -0

2

0.1

m -0 C 0 c m

Fig. 5.6. High-temperature specific heat of refractory metals (Cezairliyan et al. 1970, 1971; Cezairliyan and McClure 1971; Cezairliyan 1971). Plot of the standard deviation versus assumed formation enthalpy shows the validity of the approximation taking into account vacancy formation.

124

5. Enthalpy and specific heat

40

I

1400

1800

1600

2200 K

2000

Fig. 5.7. High-temperature specific heat of Zr and Ti (Cezairliyan and Righini 1974; Cezairliyan and Miiller 1977).

""I 45

I 0

I

I

I

I

500

1000

1500

2000

2500 K

Fig. 5.8. Specific heat of V. 1 - Cezairliyan et al. (1974), 2 Stanimirovie et al. (1999). Is the nonlinear increase caused by pointdefect formation?

125

5. Enthalpy and specific heat

Table 5.3 Formation enthalpies and equilibrium vacancy concentrations at the melting points evaluated from the nonlinear increase in specific heat. A - adiabatic calorimetry, D - drop method, M - modulation technique, P - pulse calorimetry.

HF

Method,

=mp

Metal ( e V )

cs Rb K Na

In Sn Pb Zn Sb Al

La

0.28 30 0.31 25 0.23 48 0.42 14 0.255 76 0.35 30 0.425 5 0.455 13 0.48 20 0.39 23 0.61 23 1.13 12 1.17 20 0.79 22 11 0.66 1.0 120

Refer.

A A A A

A A A A P

M M C M C

M KN

KN P

A

KN

A

KN KN P KN

A P A D M

S AK

1965 1965 1953 1965 1953 1967 1972 1972 1953 1972 1972 1972 1953 1972 1985 1970

5

Method,

=lap

Metal ( e V )

Au Cu Ni Ti Pt Zr Cr Rh Nb

Mo

Ta W

1.0 1.05 1.4 1.55 1.6 1.75 1.2 1.9 2.04 1.68 2.24 1.86 2.9 3.15 3.3

Refer.

40 50 190 170 100 70 600 100 120 270 430 290 80 340 210

M M

M M M

M D M M D M

D M M D

KS 1966a K 1967c G 1987 Sh 1965 KL 1965 KK 1966 Ch 1979 G 1988 K 196333 CZ 1966 K 1964 CP 1970 K 1963a KS 1962 Ch 1981

M - Martin, C - Carpenter, KN - Kramer and Nolting, P - Pochapsky, S - Shukla et al., AK - Akimov and Kraftmakher, KS - Kraftmakher and Strelkov, K - Kraftmakher, G - Glazkov, Sh - Shestopal, KL - Kraftmakher and Lanina, KK - Kanel’ and Kraftmakher, Ch - Chekhovskoi, CZ - Chekhovskoi and Zhukova, CP - Chekhovskoi and Petrov.

126

5. Enthalpy and specific heat

specific heat (J.mol-’.K-’)

35

30

25

900

600

1500 K

1200

45

40

0

Pt 35

__

30

25

.. : . = == ..:... 0

’

500

1000

.

*

1500

2000 K

Fig. 5.9. Specific heat of Cu (Kraftmakher 1967c; Brooks et al. 1968) and Pt (Kraftmakher and Lanina 1965; Vollmer and Kohlhaas 1969): 0 - modulation measurements, - adiabatic calorimetry.

5. Enthalpy and specific heat

127

high-melting-point metals the concentrations are of the order of lo-*. The formation enthalpies obtained are in agreement with data from other techniques or theoretical estimates. Data on copper and platinum from modulation measurements and adiabatic calorimetry are shown here (Fig. 5.9).

5.4. Extra enthalpy of quenched samples The enthalpy of vacancies frozen in the crystal lattice by quenching may be released during annealing. If all equilibrium vacancies survive in the lattice after a quench and retain the same structure, measurements of the stored enthalpy were the best determinations of the vacancy concentrations. However, the stored enthalpy reduces because of vacancy losses during the quench. Many frozen-in vacancies form clusters whose enthalpy is smaller than that of the original single vacancies. Moya and Coujou (1973) have shown that this decrease depends on the size of the secondary defects. The stored enthalpy thus shows a lower limit of the vacancy-related enthalpy. Regretfully, such data are scarce. DeSorbo (1958, 1960) has determined the formation enthalpy (0.97 eV) and the equilibrium vacancy concentration in gold ( 5 . 5 ~ 1 0 -at~ the melting point). Pervakov and Khotkevich ( 1960) have obtained the vacancy concentration in gold at the melting point to be 2 . 1 ~ 1 0 - ~ This . figure is quite comparable with the calorimetric value, 4 ~ 1 0 ~ ~ (Kraftmakher and Strelkov 1966a). An additional informative parameter appears when the stored enthalpy is measured along with the quenched-in electrical resistivity. The ratio of these quantities, AHlAp, characterizes the type of defects in a given metal and should be independent of the vacancy losses. In the experiments mentioned above, this ratio was estimated to be 0.5 and 3 kJ.pQ-l cm-’, respectively. From equilibrium measurements, it is about 1 kJ.yQ-l.cm-’.

128

5. Enthalpy and specific heat

However, it is difficult to calculate and take into account changes in AH and Ap caused by vacancy clustering during or immediately after quenching.

time Fig. 5.10. Determination of extra enthalpy of aluminum. Deviations from expected temperature trace show the extra heat absorbed or released by the sample after a rapid change of its temperature (after Guarini and Schiavini 1966).

Guarini and Schiavini (1966) measured the vacancy-related enthalpy in aluminum by means of a microcalorimeter. The temperature of the calorimetric chamber was stabilized and the sample, held at a different temperature, was lowered into the thermopile. The temperature of the sample was changed, in both directions, from 3OO0C to 35OoC, 350°C to 4OO0C, and so on, and the temperature trace was recorded. In the absence of thermal reactions in the sample, the output voltage of the thermopile varies exponentially. This was checked by means of a copper

5. Enthalpy and specific heat

129

sample in which the vacancy contribution at these temperatures is negligible. When an extra heat is absorbed or released by the sample, the output voltage no longer behaves exponentially (Fig. 5.10). The extra heat was attributed to changes of the vacancy concentrations in the sample. The vacancy concentration at the melting point has been estimated to be 6 ~ 1 0 - ~ The . authors considered this value as being a lower limit because the initial part of the calorimetric curves could not be followed. Unexpectedly, the equilibration times turned out to be much longer than those from quenching experiments.

5.5. Question to be answered by rapid-heating experi ments How to derive vacancy-related enthalpy and resistivity. Rapid-heating data for tungsten and molybdenum Under very rapid heating, vacancies have no time to appear, and the enthalpy of the sample at a given premelting temperature should be smaller than that under a moderate heating rate. For molybdenum and tungsten, the vacancy-related enthalpy at the melting point calculated from the nonlinear increase in the specific heat is about 10%. To check this concept, an examination of typical data now available has been made (Kraftmakher 1996ab, 1997). The sources of the data included equilibrium measurements of the enthalpy, modulation, pulse and dynamic measurements of the specific heat, and rapid-heating determinations of the enthalpy at the melting points. From estimated times of the vacancy equilibration, only experiments with heating rates of the order of l o 8 K.s-’ or more may be expected not to contain the vacancy contribution.

130

5 . Enthalpy and spec@ heat

It turned out that certainly different values of the enthalpy at the melting points have been obtained under equilibrium and in rapid-heating measurements. To make a quantitative comparison, parts related to the nonlinear increase in the specific heat were separated from the results of equilibrium measurements. For this purpose, the experimental data were fitted by equations supposing vacancy formation. To fit the specific heat measured only at high temperatures, the enthalpy at 1500 K was taken to be 32.6 kJ.mol-’ for tungsten (Chekhovskoi 1981) and 33.5 kJ.mol-’ for molybdenum (Chekhovskoi and Petrov 1970). We thus have three sets of the enthalpies at the melting points (Table 5.4): (i) equilibrium enthalpies after subtracting the assumed vacancy contributions, HI; (ii) total equilibrium enthalpies including the vacancy contributions, H2; (iii) enthalpies from the rapid-heating measurements, H 3 , which are expected to be close to H , rather than to H2. With heating rates in the range 108-109 K.s-’, the uncertainty in the enthalpy is within 3 4 % (Pottlacher et al. 1993). The difference between H , and H2 is therefore quite detectable. For tungsten, the results of the rapid-heating experiments, H 3 , are close to the evaluated H , values and thus support the above concept. For molybdenum, the results lie between the two values, H , and H2. Possible explanations of this may be as follows: (i) the heating is not fast enough to completely avoid the vacancy formation; (ii) a superheat of the samples under high heating rates leads to an enhancement of the apparent melting point and of the corresponding enthalpy; (iii) vacancy formation accounts for only a part of the nonlinear increase in the specific heat. A way to distinguish between these possibilities is to carry out the measurements with various heating rates and to determine the apparent melting temperature independently. One may measure the enthalpy also at a selected premelting

131

5. Enthalpy and specific heat

temperature. The rapid-heating technique is a very promising tool to reveal the vacancy-related enthalpy of metals.

Table 5.4 Enthalpy of solid tungsten and molybdenum (kJ.mol-’) at the melting points: H1 - enthalpy not including the assumed defect contribution, H, - total enthalpy, Hg - results of rapid-heating experiments. ~_____

H1

H2

H3

Reference

Tungsten 112 112 109

122 116 116 112 111

109

117

Kraftmakher and Strelkov Cezairliyan and McClure Chekhovskoi Hixson and Winkler Pottlacher et al. Righini et al.

1962 1971 1981 1990 1993 1993

Molybdenum 81 84 83

91 92 89

85 83

89 90

85

87 87

Rasor and McClelland Kraftmakher Chekhovskoi and Petrov Seydel and Fischer Cezairliyan Righini and Ross0 Hixson and Winkler Pottlacher et al.

1960a 1964 1970 1978 1983 1983 1992 1993

132

5. Enthalpy and specific heat

5.6. Specific heat of tungsten - a student experiment In this student experiment (Kraftmakher 1994c), a gas-filled incandescent lamp (12 V, 10 W) is included in a bridge circuit and heated by a DC current with a small AC component (Fig. 5.11).

selective

source

Fig. 5.11. Circuit for a student experiment to measure specific heat of tungsten.

A variable capacitor C shunts one arm of the bridge to compensate the quadrature component of the voltage caused by

5. Enthalpy and specific heat

133

the temperature oscillations in the sample. An oscilloscope serves as the null indicator, and the Lissajous pattern is seen on its screen. A selective amplifier is useful to reduce a noise and interference. It should be tuned to introduce no phase shift of the signal. The ammeter in the circuit measures the total DC current feeding the bridge, so it is necessary to take into account the distribution of the current between the arms. In our case, R2/R1 = 10, and an additional factor, (10/11)2, should be included into the expression for specific heat. The modulation frequency, 80 Hz, meets the adiabaticity conditions. The radial temperature differences in the sample are negligible owing to the small thickness and high thermal conductivity of the metal. Axial temperature gradients appear only at the ends of the sample. The mean temperature and the amplitude of the temperature oscillations are constant throughout the sample except short portions near the ends. The situation is more favorable at higher temperatures. For this reason, we start the measurements at 1500 K. At lower temperatures, the influence of the ends of the filament becomes significant. From the measurements, one obtains the quantity

where p = R7R273 is the temperature coefficient of the sample's resistance. In our case, the R273 value is 1.10 It was obtained by using the same bridge. The DC current was excluded and the AC component gradually reduced. The sample's resistance is available from extrapolation of the data to zero heating current. For tungsten, the ratio R293IR27, equals 1.095. One presents the ratio mc/B as a function of the sample's resistance. From the R/R273 ratio, it is easy to find points on the graph corresponding to temperatures 1500 K, 1600 K, and so on,

a.

134

5. Enthalpy and spec$c heat

up to 3400 K (Table 5.5),and to determine the corresponding mc/p values.

Table 5.5 Resistance ratio, RIR,,,, and temperature coefficient of resistance, p = RrlR273,for tungsten (Roeser and Wensel 1941).

1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

7.78 8.41 9.04 9.69 10.34 11.00 11.65 12.33 13.01 13.69 14.38

621 629 637 645 654 661 669 676 682 688 694

2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600

15.08 15.78 16.48 17.19 17.90 18.62 19.35 20.08 20.82 21.56 22.30

699 704 709 713 718 723 728 733 737 739 740

Another possibility is to calculate the sample's resistance at these temperatures. They are set at the bridge, and the balance is achieved by adjusting the heating current and the capacitor C. It should be remembered that R273 in (5.8) relates to the resistance of the sample, while R relates to the resistance set at the bridge. This resistance is ten times larger than that of the sample. To calculate the absolute values of the specific heat, the mass of the filament should be determined. In our case, this mass was 3.5 f 0.2 mg. The weighing caused an error larger than

5. Enthalpy and specific heat

135

other errors of the measurements. As an alternative, one can fit the data at a point where results obtained by various methods, including the equivalent-impedance technique, are in good agreement. At 2000 K, the specific heat of tungsten equals 31.5 f 0.5 J.mol-’.K-’. Accepting the vacancy origin of the nonlinear increase in the specific heat, it is easy to evaluate the parameters of vacancy formation. The plot of InT2AC versus 1/T is a straight line with a slope -H,/k,. After the formation enthalpy is determined, the equilibrium vacancy concentration becomes available. A more rigorous determination of these values consists in fitting the experimental data by the equation taking into account the vacancy formation. Various values of HF should be tried. A plot of the standard deviation versus the assumed value of H , shows the most probable formation enthalpy and its uncertainty.

136

5. Enthalpy and specific heat

5.7.Summary Formation of equilibrium point defects strongly affects high-temperature specific heat of metals. This phenomenon is caused by the temperature dependence of the equilibrium defect concentration. Point-defect contributions to specific heat strictly relate to equilibrium concentrations of the defects. The only difficulty is a separation of the defect contributions. These contributions are nonlinear according to the temperature dependence of defect concentrations. The nonlinear increase in high-temperature specific heat of metals has been observed in all cases when the specific heat was measured accurately. In all the cases, calorimetric data provide plausible enthalpies of vacancy formation. However, equilibrium vacancy concentrations from calorimetric measurements are higher than those from other techniques. Calorimetric data show especially high vacancy concentrations in refractory metals. The majority of the scientific community believes that equilibrium vacancy concentrations in metals are too small to make significant contributions to specific heat and that the nonlinear effects are caused by anharmonicity. The opposite viewpoint is that anharmonicity is a mainly linear effect that does not impede the correct separation of nonlinear defect contributions. 0

Chapter 6

Thermal expansion of metals 6.1. Point defects and thermal expansion 6.2. Methods of dilatometry

138 140

Optical methods. Capacitance dilatometers. Dynamic techniques.

6.3. Differential dilatometry

147

Revision of Simmons-Balluffi data. Nowick-Feder example.

6.4. Equilibrium vacancy concentrations 6.5. High vacancy concentrations in some alloys and intermetalks 6.6. Lattice parameter and volume of quenched samples 6.7. Summary

137

152 156 157 160

6. Thermal expansion

138

6.1. Point defects and thermal expansion At high temperatures, vacancy formation leads to an increase in the volume of the sample and in the thermal expansivity. The only difficulty is to separate the vacancy contribution. Gertsriken (1954) was the first to derive equilibrium vacancy concentrations in some metals from the thermal-expansion data. Creation of a vacancy means that one atom leaves its lattice site and occupies a position on the surface. The increase in the volume should be equal to one atomic volume, 0. Due to relaxation of atoms around the vacancy, the formation volume V, becomes smaller than the atomic volume. For rough estimations, the ratio y = V F / Omay be taken to be 0.6. When an interstitial is created, the volume of the sample decreases. Because of the relaxation, the decrease is also smaller than one atomic volume. Flinn and Maradudin (1962) have developed a Green’s function method for calculating the static distortion and the energy change due to a point defect in crystal. In an isotropic sample, the relative linear expansion due to the vacancy formation, A U , is three times smaller than the relative change of the volume, A V / K

AV/V

=

3A1/1

=

yAexp(-H,/k,T).

(6.1)

This term makes only a small contribution to the regular thermal expansion. The results therefore strongly depend on the extrapolation of the data from intermediate temperatures. In an isotropic case, the increase in the linear thermal expansivity a = ( l / l ) d l / d T caused by vacancy formation is

Aa

= (y H F A / 3k ,

T 2 ) exp(-H,/k, T ) .

6. Thermal expansion

139

The relative increase in the thermal expansivity is much larger than that in the volume of the sample. (Fig. 6.1).

1000

1200

1400

1600

1800

2000 K

2000

2200

2400

2600

2800

3000 K

Fig. 6.1. Linear thermal expansivity of Pt and W measured by modulation technique (Kraftmakher 1967a, 1972). The nonlinear increase was attributed to vacancy formation.

140

6. Thermal expansion

6.2. Methods of dilatometry Nowadays, methods for measuring dilatation of solids provide sensitivity of the order of lo-’’ m and even better. However, difficulties in studying thermal expansion at high temperatures are caused by poor stability of the samples rather than by the lack of sensitivity. This is why one had to accept data on thermal expansivity averaged within wide temperature intervals. Important improvements in this field have been made in the last decades. Along with significant progress in the traditional dilatometry, two new techniques have appeared, modulation dilatometry and dynamic techniques. Most sensitive dilatometers employ optical interferometers and capacitance sensors. lnterferometric measurements became much easier by using lasers, owing to the high temporal and spatial coherence of the laser beam.

Optical methods Feder and Charbnau (1966) have designed a sensitive laser dilatometer employing a Fizeau-type interferometer (Fig. 6.2). The fringe shifts depend only upon the changes in the sample’s length. A He-Ne laser serves as the light source, and a special system counts the fringes. The entire assembly, consisting of a prism, two adjustable slits, and two phototubes, is first rotated, so that the fringes are perpendicular to the apex of the prism. Each fringe is then split into two parts with each phototube seeing one half of it. By connecting the phototubes to a two-pen chart recorder, the fringe shifts were recorded as a series of sine waves. This double-recording system senses changes in the direction of the fringe movement. The temperature control was

6. Thermal expansion

141

achieved by immersing the container with the sample into a regulated oil bath.

flats

Fig. 6.2. Simplified diagram of interferometric dilatometer designed by Feder and Charbnau (1966).

Jacobs et al. (1970) have developed a very sensitive dilatometer based on the dependence of Fabry-Perot resonances on the mirror separation. The sample is formed into a spacer separating the mirrors (Fig. 6.3). An oscillator of variable frequency modulates a stabilized He-Ne laser. This makes the laser spectrum consisting of variable frequency sidebands. A change in the sample’s temperature causes a change in the resonance frequencies. The modulation frequency is adjusted until one of the sidebands coincides with a Fabry-Perot resonance. When the sample’s temperature changes, a new modulation frequency must be found to achieve the maximum

6. Thermal expansion

142

transmittance. The change in the necessary frequency thus shows the relative change in the sample's length.

L M

laser

P

1 1 oscillator and amplifier

w4 L

sample /

00 vacuum chamber

Fig. 6.3. lnterferometric dilatometer developed by Jacobs et al. (1970).

L - lens, M - modulator, P - polarizer, SF - spatial filter, PM photomultiplier.

The sample has the form of a hollow cylinder I 0 cm long. Its ends are polished flat and approximately parallel. The laser beam collimated by a lens L passes through a modulator M. A polarizer P and a quarter-wave plate serve as an isolator against reflections back. The polarizer also suppresses the carrier frequency, so that only the sidebands are transmitted. The precision of the dilatometer is limited only by the laser stability and amounts to lo-'. This technique is a good finding for the traditional dilatometry. Suska and Tschirnich (1999) have designed a sensitive Fizeau-type interferometer with two cube-corner reflectors. In measurements on I - m bars, the overall uncertainty of the measurements is less than 2 ~ 1 0 K-'. - ~ Ruffino (1989) has reviewed optical dilatometry.

6. Thermal expansion

143

Capacitance d i latometers Johansen et al. (1986) have designed a sensitive capacitance dilatometer. The dilatometer integrated in an oven is of the parallel-plate capacitor type. The fine regulation of the oven was achieved by means of a Peltier element. In the range from -6OOC to 150°C, the oven can be stabilized to better than K. For large dilatations, a HP 4192 low-frequency impedance analyzer measures the capacitance. For a capacitance near 15 pF, the sensitivity is pF. A computer serves to average the analyzer’s data. For submicron range, the detection system is based on a manual capacitance bridge and a lock-in amplifier. The amplifier’s output signal is calibrated versus the deviation in capacitance between a fixed reference and the dilatometer’s capacitor. In this case, the sensitivity is 7 ~ 1 0 - ’m. ~ A quartzcrystal thermometer measures the temperature of the dilatometric cell. In addition, this dilatometer was modified to operate in a modulation regime (Johansen 1987). Recently, Rotter et al. (1998) have developed a miniature capacitance dilatometer suitable for measuring thermal expansion and magnetostriction of small samples and samples of irregular shape. The active length of the sample can be less than 1 mm. An AC bridge measures the capacitance. The absolute resolution is about lo-’’ m. The dilatometer was tested in the range 0.3-200 K and in magnetic fields up to 15 T.

6. Thermal expansion

144

Dynamic techniques Miiller and Cezairliyan (1982) developed a dynamic technique that involves resistively heating the sample from room temperatures to above 1500 K in less than 1 s. The sample is mounted in a chamber providing measurements either in vacuum or in a gas atmosphere (Fig. 6.4).

high-speed pyrometer

detector laser

sample

Fig. 6.4. Simplified diagram of dynamic dilatometer developed by Miiller and Cezairliyan (1982). This technique was successfully employed in studies of refractory metals (Miiller and Cezairliyan 1982, 1985, 1988, 1990, 1991).

A photoelectric pyrometer capable of 1200 evaluations per second determines the sample's temperature. A small rectangular hole in the sample's wall, 0 . 5 ~ 1mm, serves as a blackbody

6. Thermal expansion

145

model. Simultaneously, the expansion of the sample is measured by the shift in the fringe pattern produced by a polarizedbeam laser interferometer. A Michelson-type interferometer is employed with the sample acting as a double reflector in the path of the light beam. The sample has the form of a tube with parallel optical flats on opposite sides. The distance between the flats, 6 mm, represents the length of the sample to be measured. The interferometer is thus insensitive to the translational motion of the sample. The rotational stability is monitored by reflecting the beam of an auxiliary laser from a third optical flat on the sample. During the pulse heating, a dual-beam oscilloscope displays the traces of the radiance from the sample and of the corresponding shift in the fringe pattern. This system has two very important advantages: (i) the measurements are related to the blackbody temperature, and (ii) only a central portion of the sample is involved in the measurements, so there is no need to take into account the temperature distribution along the sample. Using this technique, the authors have investigated thermal expansion of refractory metals at high temperatures (Miiller and Cezairliyan 1982, 1985, 1988, 1990, 1991). The dynamic dilatometer designed by Righini et al. (1986b) correlates the thermal expansion of the sample to its temperature profile. The longitudinal expansion of the sample is measured by an interferometer, while its temperature profile is determined by a scanning optical pyrometer (Fig. 6.5). Typical spacing between two consecutive measurements is 0.35 mm with a pyrometer viewing area 0.8 mm in diameter. Two massive brass clamps maintain the ends of the sample close to room temperatures and provide steep temperature gradients towards the ends. Two thermocouples spot-welded at the ends of the sample measure the temperature in the regions where pyrometric measurements are impossible. A corner cube retroreflector is attached to the lower (moving) clamp, while the beam bender is attached to the

6. Thermal expansion

146

switch

-I1 I1 I1 scanning

I

laser interferometer

I amplifiers, data acquisition system, cornputer

Fig. 6 . 5 . Simplified diagram of dynamic dilatometer developed by Righini et al. (1986b).

upper clamp. The resolution of the interferometer is about 0.15 pm, and 2000 measurements per second are feasible. All the results are sent to a data-acquisition system. Radiance temperatures measured by the pyrometer are transformed into true temperatures by either the electrical resistivity of the sample or data on the normal spectral emittance. The measurements on a niobium sample (Righini et al. 1986a) lasted from 0.3 s (fast) to 2.2 s (slow). For the profile measurements, fast experiments are preferable because the portion of the profile not known from the scanning pyrometer is below 6%. For slow experiments, this figure increases to 20%. On the other hand, in fast measurements the thermal-expansion polynomial is defined by few data points limited by the speed of

6. Thermal expansion

147

rotation of the mirror. A compromise must be found between a better knowledge of the temperature profile and a better definition of the thermal-expansion polynomial. The authors have stressed that the ideal experiment with this technique would either to bring the entire sample to the high temperature or to limit the measurements to the central portion of the sample.

6.3. Differential d iIato metry In usual measurements of thermal expansion, the background cannot be determined while the vacancies or interstitials are present in the crystal lattice. Differential dilatometry solves this problem. The history of this approach was given by Seeger (1973ab), Kluin (1992) and Hehenkamp (1994). Eshelby (1954) has demonstrated that the elastic relaxation around vacancies randomly distributed in a crystal affects the macroscopic volume of the sample and the volume of the unit cell equally. With an equilibrium vacancy concentration c,, the relative increase in the volume of the sample is A V / Y = 3A111 = yc, (y = V,/Q). The average volume corresponding to one lattice site decreases: Avlv = 3AaIa = - ( 1 - y)c,, where a is the lattice parameter. The difference between the two quantities equals the vacancy concentration c,, regardless of y: c, = AVIV - A v l v

=

3(A1/1 - A a l a ) .

(6.6)

More rigorously, this difference corresponds to the difference between the concentrations of the vacancies and interstitials, c, and ci: c, - ci = 3(A111 - Aala).

(6.7)

6. Thermal expansion

148

This relation is considered as being independent of the state of aggregation of the thermally generated defects and of any detailed model of the lattice dilatation produced by them. To achieve the necessary accuracy, the length of the sample 1 and the lattice parameter a are measured simultaneously. The equilibrium concentrations of interstitials are much smaller than of vacancies, so that this relation is quite adequate for determinations of the vacancy concentrations. For hexagonal and tetragonal crystals, the necessary relation is easily obtainable: c, - c i = 2 [ ( A l / l ) , - A d a ]

+ ( A l / l ) 2 - Ac/c,

(6.8)

where subscripts 1 and 2 relate to the bulk thermal expansion along the two crystallographic axes. Using this technique, Simmons and Balluffi (1960ac, 1962, 1963) have carried out the well-known measurements (Fig. 6.6).

1

8

Al

A

A

8

A 8

n v

A

800

A

.*

0 * *

1200

1600 K

Fig. 6.6. Differential-dilatometry data reported by Simmons and Balluffi (1960ac, 1962, 1963). During three decades, these data were regarded to be the most reliable.

6. Thermal expansion

149

Revision of Simmons-Balluffi data Hehenkamp and coworkers (Kluin and Hehenkamp 1991; Hehenkamp 1992; Hehenkamp et al. 1992; Mosig et al. 1992; Kluin 1992) have developed a new apparatus for differentialdilatometry measurements. For the bulk measurements, the sample was 20 mm long and 18 mm in diameter. The powder sample for the X-ray determinations was placed in the middle of the massive sample. The furnace with the sample was mounted in a vacuum chamber equipped with a beryllium window for X-rays and two quartz windows for a laser beam. A helium atmosphere at 5 ~ 1 Pa 0 ~was a good compromise between the X-ray absorption and the thermal coupling between the sample and the furnace. The bulk thermal expansion was measured by means of a single-slit diffraction pattern. The top of the sample formed a slit with the surrounding cage. When the thermal expansion of the cage is smaller than that of the sample, the slit width decreases with increasing temperature. The shift in the diffraction patterns formed by a laser beam was recorded by means of a high-resolution photodiode array and a multichannel analyzer. Due to the decrease in the width of the slit, the accuracy of the measurements increases at higher temperatures. The accuracy of the temperature control was better than 0.1 K. For each temperature, the diffraction pattern was registered several times to minimize the statistical error. A standard spectrometer measured the X-ray expansion. Only measurements when the lattice parameter determined from two reflections showed a difference less than m were utilized. The authors have determined vacancy concentrations in silver and copper and some alloys (Fig. 6.7). At elevated temperatures, openings in the heating and shielding elements allowing the passage of X-rays give rise to

6. Thermal expansion

1 50

800

900

1000

1100

1200

1300 K

Fig. 6.7. Equilibrium vacancy concentrations in A g and AgSn alloys obtained by differential dilatometry (Mosig et al. 1992).

temperature gradients inside the sample. To avoid this drawback, one may employ neutron diffraction instead of X-rays to measure the microscopic thermal expansion. Adlhart et al. (1975) used this method in studies of vacancy formation in sodium. The bulk expansion was measured by laser interferometry. Trost et al. (1986) reported on similar measurements on copper and gold. The authors considered this modification to be a promising one for studying metals with higher melting temperatures, such as nickel and platinum, or even niobium and molybdenum.

6. Thermal expansion

151

Nowi c k-Fed e r exa mpI e Differential dilatometry provides low vacancy concentrations in low-melting-point metals, less than IOW3 at the melting points. These data, along with low extra resistivities of quenched samples, caused the opinion about smallness of the vacancy concentrations in metals. However, the validity of this approach is doubtful. In real samples, vacancies appear from internal sources such as voids, grain boundaries, and vacancy clusters. The corresponding increase in the outer volume of the sample may therefore be smaller than under ideal conditions. The assumption about a uniform distribution of the sources (sinks) for the vacancies is generally invalid. Nowick and Feder (1972) have pointed out an ultimate example of the situation. For a perfect wire sample, only its surface serves as the source of the vacancies. In this case, vacancy formation results in an increase of the thickness of the sample, while the length remains unchanged. Hence, one cannot consider measurements of thermal expansion to be a very reliable method for studying equilibrium point defects. This relates to differential dilatometry as well. Theoretical calculations (e.g., Yamamoto et al. 1973; Matthai and Bacon 1985) show that the relaxation of atoms around a created vacancy is nonuniform. The nearest-neighbor atoms move inward the vacancy, so that the lattice parameter in this region decreases. At the same time, the next-neighbor atoms may move outward. An example exists when differential dilatometry provides vacancy concentrations larger than any other technique. For a strongly compressible crystal, X-ray measurements alone, made on a sample constrained at a constant volume, are sufficient to determine the vacancy concentrations. With this approach, Simmons (1994) studied vacancy formation in quantum solid

152

6. Thermal expansion

heliums. The author has pointed out that these measurements revealed vacancy concentrations larger than those from other properties, including the specific heat.

6.4. Equ iIibrium vacancy concentrations Many data on equilibrium vacancy concentrations in metals have been obtained from measurements of thermal expansion. Two main conclusions can be made: (i) the results derived by a linear extrapolation of the expansivity (Table 6.1) are several times larger than those from differential dilatometry (Table 6.2); (ii) usually, the linear extrapolation of the expansivity leads to vacancy concentrations smaller than those from the nonlinear increase in specific heat. The linear extrapolation of the expansivity from intermediate temperatures to separate the vacancy contributions can be justified as follows. (1) As a rule, a temperature range exists where the thermal expansivity increases linearly with increasing temperature. (2) Theoretical calculations of anharmonicity predict a linear temperature dependence of the expansivity of a defectfree crystal at high temperatures. (3)The nonlinear increase in the expansivity satisfies the expression describing vacancy formation and provides plausible formation enthalpies. The validity of such a procedure could be confirmed by observations of the relaxation in thermal expansion caused by the vacancy equilibration. Differential dilatometry is commonly regarded as being the most reliable method to determine equilibrium vacancy concentrations. It is even considered 'absolute technique'. However, the data sometimes strongly contradict each other. For instance, the new differential-dilatometry data on copper and silver (Kluin and Hehenkamp 1991; Hehenkamp et al. 1992; Mosig et al. 1992) revealed vacancy concentrations several

153

6. Thermal expansion

times larger than those reported by Simmons and Balluffi (1960c, 1963). Differential dilatometry has not yet been applied to metals with high melting points. Therefore, equilibrium vacancy concentrations in nickel, titanium, platinum, zirconium, chromium, rhodium, niobium, molybdenum, tantalum, and tungsten are now available only from calorimetric measurements.

Table 6.1 Vacancy concentrations in metals derived by means of a linear extrapolation of thermal expansivity. ~

Metal

Sn Cd Pb Zn

Al As Au cu Ni Pt

Mo W

c

mp

(10-4)

14 6 24 9 33 20 20 17.4 24 14 13.5 110

80 70 70 190 230

~ _ _ _ _

Reference

Gertsriken and Slyusar Current Gertsriken and Slyusar Gertsriken and Slyusar Gertsriken and Slyusar Gilder and Wallmark Gertsriken Gertsriken Gertsriken and Slyusar Gertsriken and Slyusar Gertsriken Glazkov Kraftmakher Glazkov and Kraftmakher Glazkov Chekhovskoi and Petukhov Kraftmakher

1958 1974 1958 1958 1958 1969 1954 1954 1958 1958 1954 1987 1967a 1986 1988 1987 1972

6. Thermal expansion

154

Table 6.2 Vacancy concentrations in metals obtained by differential dilatometry.

Metal

Na Li Sn Bi Cd Pb Zn

Ms Al

Au cu

c"p

7.5 7.8 4.4 10.3

6.2 6.6 4.5 1.5 1.7 3 4.9 7.2 3 11 9.4 1.7 5.2 7.2 1.9

2.1 7.6

Reference

Feder and Charbnau Adlhart et al. Feder Balzer and Sigvaldason Matsuno Feder and Nowick Janot and George Feder and Nowick Feder and Nowick Current and Gilder Balzer and Sigvaldason Janot et al. Feder and Nowick Nenno and Kauffman Simmons and Balluffi Simmons and Balluffi Mosig et al. Simmons and Balluffi Simmons and Balluffi Trost et al. Kluin and Hehenkamp

1966 1975 1970 1979a 1977 1972 1975 1958 1967 1977 197913 1970 1958 1959 1960a 1960c 1992 1962 1963 1986 1991

Recently, Wang and Reeber (1 998) considered the influence of equilibrium vacancies on thermal expansion of refractory BCC metals V, Nb, Mo, Ta, and W . Scarce measurements of the lattice parameter in several high-melting-point metals (Edwards et al. 1951; Waseda et al. 1975), together with the

6. Thermal expansion

155

bulk-expansion data now available, rather confirm high vacancy concentrations in these metals (Fig. 6 . 8 ) .

l-

id 4 0

15

Y

1000

1500

2000

2500

3000

3500 K

Fig. 6.8. Linear thermal expansivity of some high-melting-point metals: 0 - bulk thermal expansivity: Pt - Kraftmakher (1967a), Ta - Miiller and Cezairliyan (1982), W - recommended values (Swenson et al. 1985), Nb - Righini et al. (1986a), - X-ray data (Edwards et al. 1951; Waseda et al. 1975).

156

6. Thermal expansion

6.5. High vacancy concentrations in some alloys and intermetallics Kluin and Hehenkamp (1991) reported new differentialdilatometry data on vacancy formation in copper and CuGe alloys. In copper, the vacancy concentration at the melting point appeared to be 7 . 6 ~ 1 0 - ~i.e., , four times higher than the Simmons-Balluffi value. The concentrations in the alloys are even larger. For Cu-2.7at.YoGe alloy, the vacancy concentration at the melting point is 1 . 8 ~ 1 0 - Similar ~. results were obtained for AgSn alloys by Mosig et al. (1992). In Ag-8.6at.%Sn alloy, the vacancy concentration at the melting point is five times higher than in pure silver. In all the cases already known, the vacancy concentration increases along with the impurity concentration. This poses an intriguing question: how far such a behavior continues and what is the ultimate equilibrium concentration? Recently, Kerl et al. (1999) have found high concentrations of thermal vacancies in FeAl and FeSi by means of differential dilatometry. The concentrations correspond to formation entropies in the range 4kB to 5kB. The authors claim that differential dilatometry is "the only known direct and absolute technique for measuring equilibrium concentrations of vacancies which does not need any further assumptions." Wolff et al. (1999) have confirmed high vacancy concentrations in FeAl and FeSi alloys by means of positron-annihilation technique.

6. Thermal expansion

1 57

6.6. Lattice parameter and volume of quenched samples Extra concentrations of point defects after quenching cause changes in the volume and in the lattice parameter of the sample, which disappear during annealing. Balluffi et al. (1970) proposed an experiment that seemed to be very simple and informative. It consists in measurements of the length and of the lattice parameter of the sample immediately after quenching and during annealing. Results expected in such an experiment would show the anneal of quenched-in vacancies (Fig. 6.9). asquenched

Fig. 6.9. Expected changes in All1 and Aala in the experiment proposed by Balluffi et al. (1970).

Because of an extra vacancy population, the length of the sample 1 should increase, whereas the lattice parameter a should decrease. During annealing, both quantities return to the

158

6. Thermal expansion

equilibrium values. Eventually, the volume of the sample and the volume evaluated from the lattice parameter must become equal. It was surprising to the authors that such an experiment has never been performed. Still more surprising, it has not been performed until today. However, separate parts of this proposal were realized. Fraikor and Hirth (1967) measured a contraction of a quenched gold sample during annealing at 2OoC. With y = 0.525, a good agreement was achieved with the Simmons-Balluffi data. In similar study with platinum, Kopan' (1965) has found the vacancy concentration at the melting point to be 2 . 6 ~ 1 0 - ~Harrison . and Wilkes (1972) carried out a dilatometric study of an aluminum sample after quenching. To achieve agreement with the Simmons-Balluffi data, the authors had to accept y = 0.96. Using the so-called liquisol quenching (i.e., quenching from the liquid state), Laine (1972) observed changes in the lattice parameters of cadmium corresponding to a change in the microscopic volume of about 2 ~ l O - ~ After . annealing for 1 h at 2OO0C, the lattice parameters returned nearly to the values of a slow cooled sample. Taking y = 0.5, the vacancy concentration at the melting point was determined to be ~ x I O - ~several , times larger than the differential-dilatometry data. On the other hand, Suryanarayana (1973) has found no change in the lattice parameter in aluminum after a liquisol quenching. The lattice parameter of the samples quenched from temperatures in the range from 7OO0C to llOO°C was nearly the same as that of annealed samples. The author has concluded that the lattice parameter is not significantly affected by vacancies and hence calculations of the vacancy concentrations from such measurements are unreliable. In quenched samples of V,Ga, Waegemaekers et al. (1988) observed very large changes in the bulk density, up to 5% at 900°C, whereas the lattice parameter remained constant.

6. Thermal expansion

159

Hertz and Peisl (1975) reported measurements of the lattice parameter and electrical resistivity of quenched platinum foils. The accuracy of the measurements of Aala was better than lop6. The quenched-in resistivity, A p , was determined at liquid helium temperature. As expected, a linear relation holds between the relative change in the microscopic volume, Avlv, and the quenched-in resistivity:

( A v l v ) l A p = -(7.2 It: O.9)x1O4 C2-I.m-l.

(6.9)

The authors have pointed out that it is of great interest to study both quantities during thermal annealing and possibly to observe a change of the above ratio due to the divacancy and cluster formation. However, the achieved resolution was insufficient for this purpose. Earlier, the vacancy-formation volume in platinum was determined to be VF z 0.670, where 0 is the atomic volume (Emrick 1972). Hence, the relation between the relative change in the microscopic volume and the extra resistivity should be

( A v / v ) / A P = -0.33/~,,

(6.10)

where pv is the electrical resistivity caused by a unit vacancy concentration. The right side of this relation is available also from equilibrium measurements. In such measurements on platinum, pv was deduced to be 2 . 4 ~ 1 0 -R.m ~ (Kraftmakher and Lanina 1965). The above ratio thus equals

( A v l v ) l A p = - 1 . 4 ~ 1 0C2-I.m-l. ~

(6.11)

This figure is only two times larger than that from the quenching experiment, despite the estimated vacancy concentrations differ by one order of magnitude.

160

6. Thermal expansion

6.7.Summary Vacancy formation markedly affects thermal expansivity of metals. The strong increase in expansivity of high-melting-point metals is clearly seen by modern dilatometric methods. The scientific community believes that differential dilatometry is the best or even a unique method to determine equilibrium vacancy concentrations. Regretfully, it has not yet been applied to high-melting-point metals. For low-melting-point metals, it yields equilibrium vacancy concentrations less than 1O-3. New differential-dilatometry data on silver and copper by Hehenkamp and coworkers have shown the vacancy concentrations several times larger than those reported by Simmons and Balluffi. Differential dilatometry has not yet been applied to metals with high melting points. Therefore, equilibrium vacancy concentrations in nickel, titanium, platinum, zirconium, chromium, rhodium, niobium, molybdenum, tantalum, and tungsten are now available only from calorimetric measurements. Dilatometric data probably provide underestimated vacancy concentrations. The main sources and sinks for vacancies are internal imperfections in the crystal lattice, so that the increase in the outer volume of the sample may be much smaller than in an ideal case. On the other hand, very high concentrations of thermal vacancies have been found in some intermetallics. The scarce data on temperature dependence of the lattice parameter of high-melting-point metals, along with data on bulk thermal expansion, confirm rather than disprove high vacancy concentrations. Further measurements of the lattice parameter of these metals at high temperatures are very desirable.

Chapter 7

Electrical resistivity of metals 7.1. Influence of point defects on electrical resistivity

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

161

173 178

7. Electrical resistivity

162

7.1. Influence of point defects on electrical resistivity The formation of point defects results in an increase of electrical resistivity. Kauffman and Koehler (1952, 1955) were the first to observe an extra electrical resistivity of gold wires after quenching. Like impurities, vacancies give rise to an additional scattering of conduction electrons. The vacancy contribution is proportional to the vacancy concentration: Ap

=

pvc, = p , A e x p ( - H F / k , T ) .

(7.1)

Here p, = Ap/c, denotes the electrical resistivity induced by a unit vacancy concentration. Theoretical calculations of this parameter are in poor agreement (Table 7.1).

Deviations from Matthiessen's rule. Extra resistivity of vacancies and of vacancy clusters In a first approximation, the coefficient p, was assumed to be independent of temperature (Matthiessen's rule). Generally, this assumption is invalid (for a review see Bass 1972). Additional uncertainties therefore arise when results of equilibrium and quenching experiments are compared. Schule (1998ab) observed a strong temperature dependence of the extra resistivity caused by Frenkel pairs in nickel and platinum. The extra resistivity decreases with increasing temperature. On the other hand, Matthiessen's rule is well fulfilled for vacancies in copper. To experimentally determine the influence of vacancies on electrical resistivity, it is necessary to somehow find their concentrations. Although very different vacancy concentrations

163

7. Electrical resistivity

have been found in various experiments, the pv values often are in satisfactory agreement.

Table 7.1 Theoretical values of electrical resistivity induced by a unit vacancy concentration, pv = Ap/cv f2.m). 1 - Abeles (1953), 2 - Reale (1962), 3 - Manninen et al. (1977), 4 - Volkov (1980), 5 - Avte et at. (1993).

Mstal

cs

Rb K Na Li Sn

1

2

2.76 2.55 2.39 1.92 1.57

3.57 3.23 2.73 1.78 1.22 3.256 1.498 5.582 1.397

Cd

Pb Zn T1

w

2.782 3.350

Al Ca

w

Au

cu Be Ni FO W

3

0.89 0.77 0.71 0.87 0.94 0.62 1.39

4

5

1.03 0.73 1.79

0.25

1.08

0.57-0.75 0.87-1.64 0.61-0.78 0.55-0.69 0.45-0.64

0.84 1.25 0.98

1.45 1.45 1.28 0.652

0.45 5.41 7.40 8.20

When vacancies cluster in large units, their contribution to electrical resistivity decreases. Vacancy clusters in quenched

164

7. Electrical resistivity

samples may contain several thousands of vacancies. Martin and Paetsch (1973) evaluated the resistivity of clusters containing up to l o 2 vacancies. For such clusters, the decrease of the resistivity amounts to about 50% and it tends to be larger for clusters containing more vacancies.

7.2. Resistivity of metals at high temperatures How to derive formation parameters Vacancy contributions to electrical resistivity at high temperatures were first observed in alkali metals (MacDonald 1953) and in gold and copper (Meechan and Eggleston 1954). The extra resistivities at the melting points are of the order of Q.m. The main problem is to correctly separate the vacancy contributions. In some cases, such a procedure appeared to be quite reliable. In studies on aluminum, Simmons and Balluffi (1960b) used three different methods to extract the vacancyrelated resistivity. (1) The resistivity of a hypothetical defect-free crystal was approximated by a theoretical relation T = a + bX + c X 2 , where X = ln(p/T). The coefficients of this relation were found from the experimental data in the range 430-610 K, and the deviations from this dependence at higher temperatures were attributed to equi Iibrium vacancies. (2) The resistivity of a hypothetical defect-free crystal was taken to obey the relation p = A + BT + C T 2 . (3) The vacancy contribution was separated with an extrapolation by eye of the temperature dependence of the resistivity. All the methods provided the same extra resistivity at the melting point, 3.4~10-’ Q.m. This agreement is owing to the relatively large vacancy contribution (Fig. 7.1). Generally, the

165

7. Electrical resistivity

situation is not so good, and the extrapolation from intermediate temperatures becomes critical. Some authors therefore believe that this approach is unsuitable for studying the vacancy formation. For instance, Cook et al. (1979) have performed an analysis of the uncertainties of calculated and experimental values of the electrical resistivity of potassium. These uncertainties allow one to choose the formation enthalpy in the range 0.2 to 0.6 eV.

.. n

E

c 00

10

I

0 7 W

9

8 L

400

500

600

700O C

Fig. 7.1. Electrical resistivity of Al at high temperatures (Simmons and

Balluffi 1960b): 0 - experimental data, __ extrapolation from intermediate temperatures. Due to the relatively large vacancy contribution, various methods to extract it give very close results.

166

7.Electrical resistivity

Why measurements of temperature derivative of resistivity are preferable The situation is more favorable when the temperature derivative of the resistivity, dp/dT, is measured directly. The vacancyinduced increase in this derivative is

Direct measurements of dp/dT are feasible by the modulation technique. It consists in periodic oscillating the temperature of the sample and observing corresponding oscillations in its resistance. Rigorously speaking, the temperature derivative of the resistance (not of the resistivity) is available from such measurements. Modulation measurements on aluminum and platinum (Kraftmakher and Sushakova 1972, 1974) have clearly shown the nonlinear increase in the temperature coefficient of resistance, TCR = (l/R,,,)dR/dT (Fig. 7.2). This increase was attributed to vacancy formation. The two metals are the only exceptions, for which the results of equilibrium and careful quenching experiments are in reasonable agreement. Still more important, the vacancy concentrations in these metals estimated from the extra resistivity are consistent with the results based on the nonlinear increase in the specific heat. Formation enthalpies and extra resistivities at melting points obtained in equilibrium measurements are presented below (Table 7.2).

7. Electrical resistivity

TCR ( 1 o

167

- ~K-')

6.0

5.5

5.0

4.5

400

1000

600

1200

1400

1000 K

800

1600

1800

2000 K

Fig. 7.2. Temperature coefficient of resistance (TCR) of A l and Pt measured by modulation technique (Kraftmakher and Sushakova 1972, 1974). The nonlinear increase was attributed to vacancy formation.

7. Electrical resistivity

168

Table 7.2 Formation enthalpies and extra resistivities at melting points obtained in equilibrium measurements.

Metal

H,

Cd Pb

0.38 0.72 0.81 0.77 0.8 1.06 1.11 1.02 0.95 0.92 1.3 1.0 1.03 1.4 1.6 1.7 2.0

Ms Al

Au

cu

Ni Pt Rh

(ev)

~ p -

5 1.9 2.8 3.4 3.2 1.1 0.4 5.3

1.4 2.3 1 1.2 1.45 26 24

21 14

n.m)

Reference

Hillairet et al. Leadbetter et al. Mairy et al. Simmons and Balluffi Kraftmakher and Sushakova Ascoli et al. Schule and Scholz Ascoli et al. Schule and Scholz Ascoli et al. Hehenkamp and Sander Scholz and Schule Schiile Glazkov Kraftmakher and Lanina Kraftmakher and Sushakova Glazkov

1969 1966 1967 1960b 1972 1970 1982 1970 1982 1970 1979 1977 1998a 1987 1965 1974 1988

7. Electrical resistivity

169

7.3. Quenched-in resistivity The main objective of quenching experiments is to retain in the crystal lattice all the vacancies presented under equilibrium conditions. A determination of the formation enthalpy requires measurements of the extra resistivity corresponding to various temperatures. Serious obstacles were met when using this approach, namely: (i) many vacancies have time to annihilate owing to their high mobility and numerous internal sinks in the samples; (ii) many vacancies form clusters, and their contribution to the resistivity becomes therefore smaller; (iii) deformations during quench create new dislocations, which serve as sinks for the vacancies; (iv) the influence of the vacancies on the resistivity depends on temperature, so it is difficult to compare results of equilibrium and quenching experiments.

Quenching n superfluid helium An importanl achievement was the quenching into superfluid helium developed by Rinderer and Schultz (1964). The method consists in inserting a wire sample in liquid helium cooled down to the superfluid state. The sample is heated by passing through it an electric current. A thin, of about 0.1 mm, layer of helium gas appears around the wire. Owing to the high thermal conductivity of superfluid helium, the heat dissipated in the sample spreads all over the liquid preventing it from boiling. After interrupting the heating current, the gas layer disappears and the temperature of the sample rapidly decreases. The resistivity is measured at low temperatures, where the defect contribution prevails. The advantages of the method are evident: (i) the quenching rate is sufficiently high; (ii) liquid helium provides extremely pure quench medium comparable to ultra-high vacuum conditions;

170

7.Electrical resistivity

(iii) the sample's temperature after quenching is below 2 K, and the resistivity is measured under very favorable conditions immediately after quenching. Schultz (1964) employed this technique for quenching tungsten wires. Mundy and Ockers (1983)have built a quenching apparatus in which the gaseous environment could be controlled to an oxygen partial pressure of less than Pa. The authors examined the effect of oxygen on the quenched-in resistivity of high-purity copper wires. These measurements could not provide reliable data on quenched-in resistivity. Kin0 and Koehler (1967)have shown that dislocations in gold cannot accept vacancies until the supersaturation grows large enough to overcome the line tension. The vacancy losses during quenching turned out to be not sensitive to the dislocation density. It was therefore suggested that the major losses occur by the production of vacancy tetrahedra at impurities. At the same time, an important role of dislocations was demonstrated in other quenching experiments. From quenching experiments with gold, Wang et al. (1968) concluded the presence of tightly bound divacancies having the migration enthalpy close to 0.69eV. Levy et al. (1973)quenched high-purity aluminum samples from temperatures 350-820 K down to 4.2 K. The behavior of vacancies during the quench and further annealing was studied by measurements of the resistivity and by electron microscopy. In niobium, Schwirtlich and Schultz (1980b) have obtained very low quenched-in resistivity that was near the limit of detection. This low value, about 3~10-l~ R.m, was explained by high quenching losses.

7. Electrical resistivity

171

Quenching with reduced cooling rate The losses of vacancies during a quench decrease when using high cooling rates. However, thermal stresses in the sample increase in this case causing formation of new dislocations. On the other hand, pure samples having low dislocation densities allow one to reduce the vacancy losses even under moderate cooling rates. Such experiments were undertaken on platinum and palladium by Khellaf et al. (1988). With p, = 4 . 6 ~ 1 0 -R.m, ~ the vacancy concentration in platinum at the melting point was evaluated to be 9 . 4 ~ 1 0 - ~No . reliable data have been obtained for palladium because of very small quenched-in resistivity. A part of quenching experiments is presented here (Table 7.3).

Annealing experiments Many authors studied the thermal annealing of quenched-in vacancies. Siege1 (1966ab) investigated the vacancy annealing in gold by measuring changes in the quenched-in resistivity. The precipitate structure in the quenched samples was observed by a transmission electron microscope. The author has reported essential features of quenching and annealing phenomena, namely: (i) the resistivity-annealing rate drops continually for increased purity of the samples; (ii) a large rise in the density of the vacancy precipitates, with a corresponding decrease in the size, occurs with increased content of dissolved impurities; (iii) the measured migration energy is independent of the purity of the samples.

7. Electrical resistivity

172

Table 7.3 Formation enthalpies and extra resistivities at melting points obtained in quenching experiments.

Metal

Pb

Ms Al

Au CU

Ni Pt

V Mo

W

H,

(ev) ~ p - (lom9R.m)

0.45 0.79 0.77 0.69 0.97 0.97 0.70 1.27 1.3 1.6 1.6 1.5 1.51 1.3 1.3 2.7 3.24 3.2 3.3 3.6 3.1 3.3 3.67 3.6

Leadbetter et al. 1966 Tzanetakis et al. 1976 Bass 1967 Babic et al. 1970 Tzanetakis et al. 1976 Mori et al. 1962 Lengeler 1976 Furukawa et al. 1976 Bourassa and Lengeler 1976 Berger et al. 1979 Mamalui et al. 1968 Wycisk and Feller-Kniepmeier 1976 Jackson 1965 Heigl and Sizmann 1972 Zetts and Bass 1975 Khellaf et al. 1988 Arakelov et al. 1975 Mamalui et al. 1976 Suezawa and Kimura 1973 Schwirtlich and Schultz 1980 Schultz 1964 Gripshover et al. 1970 KUnZ 1971 Khoshnevisan et al. 1974 Mamalui et al. 1976 Rasch et al. 1980 Park et al. 1983

0.14 0.6 0.8 2.2-5.3 1.1 1.2 0.9 0.12 0.12 1 0.1 15 13 6 4.3 6 0.5 0.1 0.05 0.2 1 0.5 0.5 0.5 1 2

Reference

7. Electrical yes ist ivity

173

7.4. Comparison of data from two methods The vacancy contributions to the electrical resistivity of metals were observed in both equilibrium and quenching experiments, though the obtained values are very different. This difference is quite explainable by the drawbacks inherent to the two techniques and by deviations from Matthiessen’s rule. The only exceptions already mentioned are aluminum and platinum (Fig. 7.3). For other metals, the difference in the values from the two techniques amounts to one order of magnitude. Schule and Scholz (1979) have reviewed experimental data on vacancies and divacancies in FCC metals Cu, Ag, Au, Al, Ni, and Pt. No manifestation of point defects in tungsten was seen in measurements of its electrical resistivity at high temperatures. The resistivity at the premelting temperatures is about lop6 n.m, whereas the errors of the measurements are nearly 1%. The errors are caused mainly by uncertainties in the temperature measurements. Still more important, the reported data were based on the room-temperature shape of the samples. Cezairliyan and McClure (1971) used a linear approximation of the temperature dependence of the resistivity of tungsten in the range 2000-3600 K. This fit makes no hint about a vacancy contribution. However, this contribution becomes clear (Kraftmakher 199613) after introducing corrections for thermal expansion, which are necessary to calculate correct values of p and dp/dT:

dp/dT

=

(dp*/dT)

x

1/1298 + p * x a .

7. Electrical resistivity

174

extra resistivity ( I 0-l’ i2.m)

2

8

6

4

10

104n (K-’)

10

12

14

16

104/r (K-’) Fig. 7.3. Extra electrical resistivity of metals. W: 1 - Schultz 1964, 2 - Gripshover et al. 1970, 3 - Kunz 1971, 4 - Rasch et al. 1980, 5 - Park et al. 1983. Pt: 1 - Bacchella et at. 1959, 2 - Kraftmakher and Lanina 1965, 3 - Rattke et al. 1969, 4 - Heigl and Sizmann 1972, 5 - Kraftmakher and Sushakova 1974. Al: 1 - Bradshaw and Pearson 1956, 2 - Gertsriken and Slyusar 1958, 3 - Simmons and Balluffi 1960b, 4 - BabiC et at. 1970, 5 - Kraftmakher and Sushakova 1972.

7.Electrical resistivity

175

Here p* denotes the resistivity based on the roomtemperature shape of the sample, the ratio 1 / 1 2 9 8 corresponds to its linear thermal expansion, and a = ( l/1298)dudT is the linear thermal expansivity. The thermal expansivity of tungsten determined in the 2000-2900 K range by modulation dilatometry (Kraftmakher 1972) was extrapolated up to 3600 K and employed in the calculations:

a=3.5~10+ - ~1.4x1OV9T+ 2.74x106T2exp(-36540/T).

(7.5)

The thermal expansion in the range 298 to 2000 K was taken from the recommended data (Swenson et at. 1985). From the corrected values, the vacancy contribution is clearly seen (Fig. 7.4). 40 h

v

I

Y

E

c 7

7

30

0 r

v

t-

Y a -0

20

1500

2000

2500

3000

3500

4000 K

Fig. 7.4. Temperature derivative of electrical resistivity of W and Nb at high temperatures: - original data based on ambient-temperature shape of the samples (Cezairliyan and McClure 1971; Righini et al. 1985), 0 - values after introducing corrections for the thermal expansion .

176

7. Electrical resistivity

The extra resistivity of tungsten at the melting point is 5x10-’ R.m. This figure is quite comparable with the quenched-in resistivities. However, quantitative comparisons are difficult because of possible deviations from Matthiessen’s rule. In similar calculations for Nb, experimental data by Righini et al. (1985) were used. Their dp*/dT data show a positive deviation from a straight line. This deviation increases after introducing corrections for the thermal expansion according to Righini et al. (1986a). The estimated extra resistivity of niobium at the melting point is 7x10-’ R.m. Errors in measurements of electrical resistivity above 2500 K are of the order of 1%. They are caused mainly by uncertainties in temperature measurements and in thermal-expansion data. In the case of tungsten and niobium, the defect contributions to the resistivity became evident after introducing corrections for thermal expansion. For other refractory metals, the situation is probably similar. However, the above correction is insufficient to resolve the existing disagreement. With pv values of the order of lop5 C2.m that are commonly accepted for refractory metals, the estimated defect concentrations remain much smaller than those from the nonlinear increase in specific heat. More careful measurements of the resistivity are desirable, as well as calculations and experimental determinations of the influence of point defects on the resistivity. A probable reason for low extra resistivities in refractory metals may be deviations from Matthiessen’s rule (see Schule 1998ab). Recently, Stanimirovic et al. (1999) have determined the electrical resistivity of vanadium up to 1900 K. From these data, the temperature derivative of resistivity manifests a nonlinear contribution at high temperatures (Fig. 7.5). This contribution would be more pronounced if corrections for thermal expansion were introduced. Very similar behavior has been found in palladium by Khellaf et al. (1987). In both cases, the authors did

7. Electrical resistivity

177

not try to fit the data by the equation taking into account pointdefect formation. A straightforward method exists for determining the vacancy contributions to electrical resistivity. Under very rapid heating, vacancies have no time to arise, and the measured 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 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.

.. .. 6. b

.. ..

5-

4I

I

I

I

400

800

1200

1600

I

2000 K

Fig. 7.5. Temperature derivative of electrical resistivity of V (StanimiroviC et al. 1999). Does the nonlinear part manifest a pointdefect contribution?

170

7. Electrical resistivity

7.5.Summary Theoretical calculations provide very different estimates of influence of point defects on electrical resistivity of metals. 0 Vacancy contributions to electrical resistivity of metals are seen in equilibrium measurements and in quenching experiments. However, the data from the two techniques are in disagreement. This may be caused by three reasons. First, vacancy losses make the quenched-in vacancy concentrations smaller than the concentrations at high temperatures. Second, quenched-in vacancies form clusters whose resistivity becomes smaller than that of single vacancies. Finally, Matthiessen's rule is generally invalid, so that the extra resistivity caused by of the same vacancy concentration depends on temperature.

The defect contribution is clearly seen when the temperature derivative of resistance is measured directly. Modulation technique provides such a possibility. The low extra resistivity of quenched molybdenum samples may result from the well-known drawbacks inherent to all quenching experiments. The failure of the quenches on tantalum and niobium probably has the same reason. Only two exceptions are known, aluminum and platinum, when the vacancy contributions to the high-temperature electrical resistivity and the quenched-in resistivities are in reasonable agreement. Still more important, these contributions do not contradict the vacancy concentrations based on calorimetric measurements.

Chapter 8

Positron annihilation 8.1. Positron-annihilation techniques

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

179

192 194 195 196

180

8. Positron annihilation

8.1. Positron-annihilation techniques Why vacancies affect positron annihilation Positron annihilation is a tool very sensitive for detecting vacancy-type defects in metals (for reviews see Seeger 1973ab; Doyama and Hasiguti 1973; Siege1 1978; West 1979; Schaefer 1987; Hautojarvi 1987; Puska and Nieminen 1994). When a positron enters the sample, its kinetic energy becomes equal to the thermal energy very rapidly, in a time of the order of lo-’* s. Then the positron annihilates with an electron far away a nucleus, usually with a conduction electron. The mean lifetimes of positrons in metals are of the order of lo-” s. Two y-quanta appear at the annihilation. The informative parameters are the positron lifetime, the energy spectrum (the Doppler broadening of the annihilation line), and the angle correlation of the annihilation quanta (Fig. 8.1). Vacancies presented in the crystal lattice affect all these parameters. They have a negative charge and attract the positrons. However, the electron density in the vacancy is lower than in the regular lattice. The lifetime of positrons captured by vacancies therefore increases. The number of positrons in the sample is much smaller than that of vacancies. Under positron intensities currently employed in experiments, no more than one positron is presented in the sample at any time. The kinetic energy of a thermalized positron is therefore much smaller than that of electrons. The Doppler broadening, dE, and the angular distribution of the annihilation quanta, dO, hence relate only to electrons. Annihilations with higher-energy core electrons contribute more to the largest values of dE and dO than do annihilations with lower-energy valence or conduction electrons. The Doppler-broadening and

8. Positron annihilation

181

angular-correlation data give information about the kinetic energy of the annihilation electrons. In a vacancy, the local fraction of high-energy electron states is reduced. The shapes of the dE and dO distributions for the positron annihilations in vacancies become therefore narrower.

angular correlation 180’5 dO c

Doppler broadening

Fig. 8.1. Principles of positron-annihilation techniques.

Berko and Erskine (1967) were the first to point out a probable relation between vacancies and the positron annihilation in metals. They measured the angular distribution of y-quanta in aluminum. The distribution from a deformed sample exhibited a narrowing in comparison to that after annealing. The authors concluded that they “are inclined to believe rather that the positron tends to seek out the lower density regions typically obtained around dislocations and possibly at point defects (vacancies).” Two months later, MacKenzie et al. (1967) have presented measurements of positron lifetimes in some lowmelting-point metals (indium, cadmium, zinc, aluminum) over a

182

8.Positron annihilation

range from room temperature to the melting points or to 40OoC. Marked effects, amounting to a 30% increase in the lifetime, were considered to be caused by equilibrium vacancies. The authors ruled out dislocations as a prime cause of the phenomenon because of the lack of a hysteresis in the measurements. Bergersen and Stott (1969) have developed a model for trapping of positrons by vacancies. The authors have explained the observed temperature dependence of the positron lifetimes, including the saturation at high temperatures. They have evaluated the enthalpies of vacancy formation in metals studied by MacKenzie et al. (1967). Connors and West (1969) have presented a similar model a month later. Doyama (1972) considered thermal detrapping of positrons captured by lattice vacancies. He has noted that the lifetime of a trapped positron is often much longer than the time for a vacancy to spend at a lattice point. Kuribayashi and Doyama (1975) considered the effect of migration of vacancies on the positron trapping. At sufficiently high temperatures, the mean lifetime of positrons trapped by vacancies z1 becomes higher than the mean time z2,during which a vacancy stays at a certain lattice point. The authors have posed two important questions. (1) At high temperatures, when z1 > 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,

8. Positron annihilation

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

8. Positron annihilation

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-

8. Positron annihilation

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

8. Positron annihilation

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)

8. Positron annihilation

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.

8. Positron annihilation

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).

8. Positron annihilation

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

8. Positron annihilation

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. Positron annihilation

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.

8. Positron annihilation

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. Positron annihilation

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.

10. Equilibration of point defects

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

10. Equilibration of point defects

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

10. Equilibration ofpoint defects

(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.

10. Equilibration ofpoint defects

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

10. Equilibration ofpoint defects

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

10. Equilibration of point defects

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

10. Equilibration of point defects

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

10. Equilibration of point defects

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 << C or the relaxation time remains sufficiently long >> 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). Generally, this approach should indicate a lower limit of the relaxation. Only under very favorable conditions, when X 2 >> 1 before the quench and X 2 << 1 after it, the change in the amplitude of the high-frequency temperature oscillations should reveal the true magnitude of the relaxation (Fig. 10.5). Otherwise, this change corresponds to only a part of the phenomenon. Measurements under various modulation frequencies should make clear the temperature dependence of the relaxation. Along with providing data on the equilibration, this approach would verify the vacancy origin of the phenomenon. The measurements are performed as follows. At every mean temperature, the sample is annealed during 1 h. Then the heating current is interrupted and the sample quenched. After about 1 s, the sample returns to the initial temperature. During all the time, including the quench, the output signals are stored by the dataacquisition system. When they alter after the quench, the sample is kept at this temperature until the magnitude of the signal returns to its initial value. To exclude possible errors in the temperature scale, the measured relaxation can be directly compared with the nonlinear increase in the heat capacity of the sample. For this purpose, the heat capacity is measured by the equivalent-impedance technique. With this variant of modulation calorimetry, the ratio

(k

10. Equilibration ofpoint defects

228

of the heat capacity, mc, to the temperature derivative of the resistance of the sample, I?’, is measured. For tungsten, this derivative at high temperatures changes by less than 1% per 100 K, and this change is mainly linear. The nonlinear increase in the mclR’ ratio thus shows the nonlinear increase in the specific heat, with which the observed relaxation should be compared.

4

0.10

d

t

ns

CI

0.08

’c

I

-

0.06

z 0.04 5 >

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.

10. Equilibration ofpoint defects

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

10. Equilibration of point defects

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

10. Equilibration of point defects

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

10. Equilibration of point defects

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).

10. Equilibration of point defects

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

10. Equilibration of point defects

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.

10. Equilibration of point defects

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. Equilibration of point defects

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

11. Parameters of vacancyformation

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.

11. Parameters of vacancyformation

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

11. Parameters of vacancy formation

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.

11. Parameters of vacancy formation

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;

11. Parameters of vacancyformation

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

11. Parameters of vacancy formation

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

11. Parameters of vacancy formation

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.

11. Parameters of vacancyformation

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

11. Parameters of vacancy formation

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 is the spectral density of the square of the heating-power oscillations, and y = m C p d P r . The ratio of the two spectral densities can be written in the form

where A = 4 k B P ' / < p f 2 2 is constant at a given temperature, z = f/fo, fo is the modulation frequency at which the temperature oscillations become 62 times smaller than at zero frequency, and y = C,/C, is the ratio of the two specific heats.

0

1

2

3

5

4

temperature oscillations (1o

- K) ~

Fig. 12.4. Results of measurements of small temperature oscillations at 2200 K.

The above expression makes it possible to determine the C,IC, ratio by the least-squares method. Such a determination

12. Discussion

269

is feasible even without an evaluation of the temperature oscillations and of the mass and the heat transfer coefficient of the sample. The only requirement is to make the spectral density of the noise used for the modulation to be frequency independent.

Fig. 12.5. Spectra of temperature fluctuations (0) and of temperature oscillations caused by modulation of heating power (--.-).The spectra fitted at low frequencies show the C&, ratio (Kraftmakher and Krylov 1980b).

12. Discussion

270

The temperature fluctuations were observed in thin tungsten wires, 3.5 pm thick and 1 mm long. At 2200 and 2400 K, the spectra obtained are in good agreement with the theoretical prediction (Fig. 12.5). Below 20 Hz, the spectral density is constant, whereas at high frequencies it is inversely proportional to the frequency squared. The C,/C, ratio was found to be 1.4 f 0.1 (Fig. 12.6). The evaluated constant-volume specific heat of the crystal lattice is somewhat smaller than the classical limit 3 N k , = 24.9 J.mol-’.K-’. The experimental determination of the constant-volume specific heat of tungsten confirmed a weak or even negative anharmonic contribution to the specific heat. Further determinations of the C,IC, ratio unquestionably deserve efforts. Nowadays, a data-acquisition system and a computer could be employed for processing the data.

25 I

0‘

I

I

I

I

I

I

1.2

1.4

1.6

1.8

2.0

C,/C, Fig. 12.6. Determination of the C,/C, method.

ratio

ratio by the least-squares

12. Discussion

271

Chui et at. (1992) observed temperature fluctuations at low temperatures. The authors stated that the central point of the problem is the applicability of the relation between fluctuations of energy U and of temperature T . One viewpoint consists in accepting the relation AU = CAT, where C is the heat capacity of a subsystem thermally connected to a reservoir. Energy exchanges between them lead to the above formulas for the temperature fluctuations. Another viewpoint accepts the statement that temperature of a canonical ensemble does not fluctuate. Temperature fluctuations in a microcanonical ensemble are possible but in this case < A T 2 > = 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

294

13. Conclusions

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

13. Conclusions

295

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

13. Conclusions

296

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

13. Conclusions

297

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).

298

13. Conclusions

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.

13. Conclusions

299

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.

300

13. Conclusions

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

13. Conclusions

301

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