Equilibrium vacancies and thermophysical properties of metals

EQUILIBRIUM VACANCIES AND THERMOPHYSICAL PROPERTIES OF METALS

Yaakov KRAFTMAKHER The Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel. [email protected]

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 299 (1998) 79—188

Equilibrium vacancies and thermophysical properties of metals Yaakov Kraftmakher The Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel. [email protected] Received September 1997; editor: A.A. Maradudin Contents 1. Introduction 2. Basic theory of point-defect formation 3. Methods for studying point defects in metals 3.1. Measurements in equilibrium 3.2. Quenching experiments 3.3. Observation of the vacancy equilibration 4. Enthalpy and specific heat of metals at high temperatures 4.1. Point defects and specific heat of metals 4.2. High-temperature calorimetry 4.3. Formation enthalpies and equilibrium vacancy concentrations 4.4. Extra enthalpy of quenched samples 4.5. Question to be answered by rapid heating 5. Thermal expansion of metals at high temperatures 5.1. Point defects and thermal expansion 5.2. Methods of dilatometry 5.3. Differential dilatometry 5.4. Equilibrium vacancy concentrations 5.5. Lattice parameter and volume of quenched samples 6. Point defects and electrical resistivity of metals 6.1. Influence of point defects on resistivity 6.2. Resistivity of metals at high temperatures 6.3. Quenched-in resistivity 6.4. Extra resistivity caused by the vacancy formation 7. Method of positron annihilation 7.1. Positron-annihilation techniques 7.2. Experimental data 8. Hyperfine interactions 8.1. Perturbed angular correlation of c-quanta

82 88 91 91 95 96 99 99 100 107 109 110 112 112 113 119 121 123 124 124 124 126 129 130 130 135 137 137

8.2. Mo¨ssbauer spectroscopy and nuclear magnetic resonance 9. Influence of point defects on other physical properties 10. Microscopic observation of quenched-in defects 10.1. Electron microscopy 10.2. Field-ion microscopy 11. Relaxation phenomena caused by equilibration of point defects 11.1. Electrical resistivity 11.2. Specific heat 11.3. Positron annihilation 11.4. Equilibration times from relaxation measurements 12. Discussion 12.1. Proposals for determination of vacancy contributions to the enthalpy and specific heat of metals 12.2. Equilibrium vacancy concentrations 12.3. Comparison of methods for studying vacancy formation 12.4. Enthalpies and entropies of the vacancy formation 12.5. Anharmonic contributions to the specific heat and thermal expansivity 12.6. Thermal defects in alloys and intermetallic compounds 12.7. Self-diffusion in metals 12.8. Point defects and melting 13. Summary 14. Conclusions References

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Abstract Despite a significant progress in studies of point defects, some important problems have not received unambiguous solutions. 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. There exist two opposite viewpoints on equilibrium point defects in metals: (1) Defect contributions to the physical properties of metals at high temperatures are small and cannot be separated from the effects of anharmonicity. The only suitable methods are the positron-annihilation spectroscopy providing enthalpies of the vacancy formation and the differential dilatometry yielding equilibrium vacancy concentrations. The equilibrium vacancy concentrations at the melting points are in the range 10~4 to 10~3. Reasonable values of the formation enthalpies deduced from the nonlinear increase in the high-temperature specific heat of metals are accidental and the derived defect concentrations are improbably large, so that this approach is generally erroneous. (2) In many cases, defect contributions to the specific heat of metals are much larger than nonlinear effects of the anharmonicity and can be separated without crucial errors. This approach is quite adequate for determination of the defect parameters, especially, the equilibrium vacancy concentrations. The equilibrium concentrations at the melting points are of the order of 10~3 in low-melting-point metals and of 10~2 in high-melting-point metals. The strong nonlinear effects in the specific heat and thermal expansivity of metals at high temperatures can be explained by the formation of point defects. Examination of these effects rules out the anharmonicity as a possible origin of this phenomenon. Important arguments supporting this viewpoint have appeared in the last decade. It may turn out that just calorimetric determinations provide most reliable values of equilibrium vacancy concentrations in metals. The aim of the review is to discuss the experimental results and theoretical considerations favoring both claims. At present, the first point of view is shared by the majority of the scientific community. Regrettably, the data supporting the second viewpoint were never displayed and discussed together, and the criticism of this viewpoint never included a detailed analysis. In this review, the main attention is paid to equilibrium vacancies in metals and their relation to thermophysical properties of metals. Along with a discussion of experimental data and theoretical estimates now available, some approaches are proposed that seem to be most suitable to solve the question. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 61.72.Ji, 61.72.Cc, 65.40.#g Keywords: Point defects; Vacancy formation; Metals; High temperatures; Thermophysical properties

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1. Introduction The formation of point defects in solids was predicted by Frenkel [1]. At high temperatures, the thermal motion of atoms becomes more intensive and some of atoms obtain energies sufficient to leave their lattice sites and occupy interstitial positions. In this case, a vacancy and an interstitial atom, the so-called Frenkel pair, appear simultaneously. A way to create only vacancies has been shown later by Wagner and Schottky [2]: 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). This mechanism dominates in solids with close-packed lattices where the formation of vacancies requires considerably smaller energies than that of interstitials. In the last four decades, many theoretical and experimental studies of point defects in metals have been carried out. Many monographs [3—11], conference proceedings [12—23], and reviews [24—33] have been published. A brief history of the studies of point defects in metals is presented below (Table 1). 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 that increases rapidly with temperature. The equilibrium vacancy concentration, c , is 7 c "exp(!G /k ¹)"exp(S /k ) exp(!H /k ¹)"A exp (!H /k ¹) , (1) 7 F B F B F B F B where G denotes the Gibbs energy of the vacancy formation, H is the formation enthalpy, S the F F F formation entropy (not including the configurational entropy), k the Boltzmann constant, and B ¹ the absolute temperature. The enthalpy of formation is H "E #p» , where E is the formation energy and » is the F F F F F defect volume. The term p» becomes important when the pressure reaches a few kilobars, and F usually the enthalpy and energy of the defect formation are not significantly different. The formation entropy, S , results from vacancy-induced changes in the lattice vibration frequencies. F After a 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.

Fig. 1. Point defects in crystal lattice: V — vacancy, I — interstitial atom, FP — Frenkel pair, D — divacancy.

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Table 1 Brief history of studies of point defects in metals Item

Reference

Prediction of vacancy—interstitial pair formation Mechanism of vacancy formation Calculations of point-defect parameters Extra resistivity of quenched samples Defect contribution to resistivity at high temperatures Defect contribution to specific heat Vacancy parameters from thermal expansion Theory of defect concentrations Differential dilatometry

Frenkel, 1926 [1] Wagner and Schottky, 1930 [2] Huntington and Seitz, 1942 [34,35] Kauffman and Koehler, 1952—1955 [36,37] MacDonald, 1953 [38] Carpenter, 1953 [39], Pochapsky, 1953 [40] Gertsriken, 1954 [41] Vineyard and Dienes, 1954 [42] van Duijn and van Galen, 1957 [43] Feder and Nowick, 1958 [44] Nenno and Kauffman, 1959 [45] DeSorbo, 1958 [46] Hirsch et al., 1958 [47] Mu¨ller, 1959 [48] Jackson and Koehler, 1960 [49] Rasor and McClelland, 1960 [50] Simmons and Balluffi, 1960—1963 [51—55] Kraftmakher and Strelkov, 1962—1964 [56—59] Rinderer and Schultz, 1964 [60,61] Seidman and Balluffi, 1965 [62] Berko and Erskine, 1967 [63] MacKenzie et al., 1967 [64] Cezairliyan et al., 1970—1971 [65—68] Kramer and No¨lting, 1972 [69] Seeger, 1973 [26,27] Skelskey and Van den Sype, 1974 [70] Maier et al., 1979 [71] Miiller and Cezairliyan, 1982—1991 [72—76] Kraftmakher, 1985 [77] Schaefer and Schmid, 1987 [32,78] Varotsos, 1988 [79] Hehenkamp et al., 1992 [80—83]

Stored energy in quenched Au Electron microscopy of quenched samples Observation of point defects by a field-ion microscope Proposal to observe point-defect equilibration Nonlinear increase in specific heat of Mo and Ta Differential-dilatometry data on Al, Ag, Au and Cu Specific heat and vacancies in refractory metals Quenching in superfluid helium Equilibration of vacancies in Au Influence of vacancies on positron annihilation in metals Specific heat of refractory metals Specific heat and vacancies in low-melting-point metals Evidence of the priority of studies under equilibrium Relaxation phenomenon in specific heat of Au Positron-annihilation data on refractory metals Thermal expansion of refractory metals Relaxation phenomenon in specific heat of W Vacancy equilibration in Au from positron annihilation Theoretical bounds for formation entropies New differential-dilatometry data on Ag and Cu

The point-defect formation in metals is well established. As a rule, the enthalpies of the vacancy formation obtained by various experimental techniques are in a reasonable agreement. In many cases, however, dramatic differences were found in the equilibrium vacancy concentrations governed also by the formation entropies. This contradiction is especially strong in refractory metals. Point defects affect many physical properties of metals. Vacancies cause an increase of the volume and thermal expansivity (coefficient of thermal expansion) of a crystal. The scattering of conduction electrons by point defects contributes to the electrical resistivity. The crystal possesses an extra enthalpy and specific heat. Vacancies form traps for positrons, and this phenomenon is also utilized for studying the vacancy formation. The point-defect mechanism prevails in diffusion

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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 resistivity; their mechanical properties, thermopower, and parameters of the positron annihilation also alter. Quenched-in defects can be studied by an electron or field-ion microscope. Manifestations of point defects in the physical properties of metals were observed in the 1930s but the correct interpretation of these appeared much later. In the 1950s, the influence of the point defects on the electrical resistivity, specific heat and thermal expansion of metals was understood. The point-defect contribution to the high-temperature resistivity has been discovered at the same time as the extra resistivity of quenched samples. Nevertheless, the most of the further studies were performed using quenching. The drawbacks inherent to quenching experiments became clear in a short time. In the 1960s, many investigators employed the differential-dilatometry technique. 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 corresponds to the difference between equilibrium concentrations of the vacancies and interstitials. Equilibrium concentrations of interstitials are believed to be negligible, and this technique is now considered as the most proper one to determine the 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, they did not exceed 10~3 at the melting points of the metals. In the 1970s, a new method for studying point defects and defect clusters appeared, the perturbed angular correlation of c-quanta. Radioactive nuclei, introduced into a host material, decay to an excited state which decays to the ground state by emission of two successive c-quanta. This emission senses an interaction of the nucleus with extranuclear fields that occurs during the time between the two emissions. In the 1970s, studies of point defects under equilibrium conditions have been recognized as basically superior to any non-equilibrium experiments. This viewpoint has been clearly formulated by Seeger [26]: “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 2 The basic theory required for the analysis of equilibrium measurements is in general more straightforward and much simpler than that required for handling situations far from equilibrium.’’ Two new experimental methods have been developed at that time, the positron annihilation and the perturbed angular correlation of c-quanta. Positrons can be captured by vacancies, and their lifetime is therefore changed, as well as the parameters of the annihilation c-quanta. This approach seemed to be very promising. However, serious difficulties inherent to it are not overcome until today: the 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. The perturbed angular correlation of c-quanta senses the interaction between a defect-produced electric-field gradient and the nuclear quadrupole moment of a probe

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Fig. 2. Specific heat of metals. Pb, Al — adiabatic calorimetry (Kramer and No¨lting [69]); W, Pt — modulation calorimetry [56,84]; Cr — drop method (Kirillin et al. [85]); Mo, Nb — dynamic calorimetry (Cezairliyan et al. [65], Righini et al. [86]). Difference between low-melting-point and high-melting-point metals is clearly seen. Fig. 3. Linear thermal expansivity of high-melting-point metals. Pt — modulation method [87]; Ir — traditional dilatometry (Halvorson and Wimber [88]); Nb, Ta — dynamic technique (Righini et al. [89], Miiller and Cezairliyan [72]); W: 1 — modulation method [90], 2 — recommended values (Swenson et al. [91]).

atom. This technique may be capable of discriminating defects of different structure and has a potentiality to reveal equilibrium defect concentrations. However, no data on equilibrium defect concentrations in pure metals were reported until today. The nonlinear increase in the high-temperature specific heat of metals has been discovered long ago. It is especially strong in refractory metals (Fig. 2) but a time had elapsed before this fact has been accepted. The measurements on refractory metals were performed using the pulse and modulation techniques not recognized at that time. Earlier, the drop method was employed for such measurements, so that only the enthalpy of the samples was measured directly. The pointdefect contribution to the enthalpy is of about one order of magnitude smaller than that to the specific heat and is hardly to be seen using the drop method. A common opinion therefore appeared that the high-temperature specific heat of metals depends linearly on temperature and the specific heat at the melting point does not differ strongly from the value at room temperatures. After the nonlinear increase became evident from direct measurements of the specific heat, some authors began to take it into account in the approximation of the enthalpy data. The strong nonlinear increase in the thermal expansivity of metals is also obvious (Fig. 3). However, the origin of these phenomena remains under debate. The problem consists in a correct separation of the defect contributions because an unknown part of the nonlinear increase may originate from the anharmonicity. A reliable method to separate the defect contributions is well known: the specific heat should be measured under 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 W and Pt. In both cases, the difference between the specific heats measured using slow and rapid temperature changes appeared in an agreement with the nonlinear increase in the specific heat. This means that the nonlinear increase in the specific heat of the two metals is caused by the point-defect formation.

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A simple empirical rule has been established: the enthalpies of the defect formation are nearly proportional to the melting temperatures, as well as the enthalpies of the self-diffusion and of the vaporization. Despite a significant progress in studies of point defects, some important problems have not received unambiguous solutions. One of the most practically important questions relates to the equilibrium concentrations of point defects. It is indeed surprising that this fundamental problem is still under debate. For instance, equilibrium vacancy concentrations in refractory metals at the melting points based on the nonlinear increase in the specific heat are of the order of 10~2. On the other hand, low quenched-in electrical resistivity in these metals corresponds to defect concentrations two orders of magnitude smaller. There exist two opposite viewpoints on the equilibrium point defects in metals: (1) Defect contributions to the physical properties of metals at high temperatures are small and cannot be separated from the effects of anharmonicity. The only suitable methods are the positron-annihilation spectroscopy providing the enthalpies of the vacancy formation and the differential dilatometry yielding the equilibrium vacancy concentrations. The equilibrium vacancy concentrations at the melting points are in the range 10~4—10~3. Reasonable values of the formation enthalpies deduced from the nonlinear increase in the high-temperature specific heat of metals are accidental and the derived defect concentrations are improbably large, so that this approach is generally erroneous. (2) In many cases, defect contributions to the specific heat of metals are much larger than nonlinear effects of the anharmonicity and can be separated without crucial errors. This approach is quite adequate for determination of the defect parameters, especially, the equilibrium vacancy concentrations. The equilibrium concentrations at the melting points are of the order of 10~3 in low-melting-point metals and of 10~2 in high-melting-point metals (Fig. 4). The strong nonlinear effects in the specific heat and thermal expansivity of metals at high temperatures can be explained by the formation of point defects. Examination of these effects rules out the anharmonicity as a possible origin of this phenomenon. Important arguments supporting this viewpoint have appeared in the last decade. It may turn out that just calorimetric determinations provide most reliable values of equilibrium vacancy concentrations in metals. The aim of this review is to discuss the experimental results and theoretical considerations favoring both claims. The first point of view is shared by the majority of the scientific community. Regrettably, the data supporting the second viewpoint were never displayed and discussed together, and the criticism of this viewpoint never included a rigorous and detailed analysis. Important new arguments have appeared in the last decade. First, the relaxation phenomenon in the specific heat caused by the vacancy equilibration was observed. Such measurements were proposed long ago and considered as a crucial determination of the equilibrium vacancy concentrations. Second, recent differential-dilatometry measurements on Ag and Cu revealed vacancy concentrations in these metals three times larger than those commonly accepted during three decades. High concentrations of thermally generated vacancies were observed in many alloys and intermetallic compounds. Lastly, thermodynamic relations favoring high entropies of the vacancy formation in metals have been found. All these results support the second point of view. At the same time, the weakness of the first viewpoint is now clearly seen. In essence, there exist only two reasons for this opinion, namely: (1) the differential-dilatometry data on low-melting-point metals, and (2) low extra resistivities of quenched samples and small concentrations of quenched-in vacancies observed in high-melting-point metals by the electron and field-ion microscopy.

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Fig. 4. Equilibrium vacancy concentrations in metals calculated from nonlinear increase in specific heat. The concentrations at melting points are of the order of 10~3 in low-melting-point metals and of 10~2 in high-melting-point metals.

The main subject of the review is the equilibrium vacancy concentrations in metals and their correlation with the thermophysical properties of metals at high temperatures. An attempt will be made to answer two important questions: (1) what are the equilibrium vacancy concentrations in metals, and (2) what is the nature of the strong nonlinear effects in the specific heat of metals at high temperatures. The majority of the scientific community consider the two questions as to have no relation to each other. As a rule, physicists studying the point-defect formation in metals ignore data based on calorimetric measurements. At the same time, those which deal with the thermophysical properties of metals do not take into account expected point-defect contributions. This situation was caused by the opinion that equilibrium concentrations of point defects are too small to affect markedly the thermophysical properties. In this review, an attempt will be made to show that this well-established opinion needs a reconsideration. The issues reviewed below are related mainly to the equilibrium vacancy concentrations. Though the author always believed that the above items (1) and (2) are closely related, the commonly accepted viewpoint is also presented in the review. Along with a discussion of the experimental data and theoretical

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estimates now available, some approaches are proposed that seem to be most promising to solve the question.

2. Basic theory of point-defect formation Point defects are imperfections of the 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 those for interstitials, so that vacancies are the dominated point defects in equilibrium. Along with monovacancies, vacancy clusters of various sizes may exist in the crystal lattice under equilibrium and after quenching. The formation of a vacancy can be considered as removal of one interior atom from the crystal and replacement of the atom on the crystal surface. The Gibbs energy of the vacancy formation is defined as the corresponding change of the Gibbs 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 will be associated with an enthalpy of formation, H , and a vibrational entropy, S , F F resulting from a disturbance of the neighbors of the vacancy. In addition, a configurational entropy, S , appears that is equal to C S "k ln[(N#n)!/N!n!] . (2) C B Then Stirling’s approximation gives S "k N ln[(N#n)/N]#k n ln[(N#n)/n] . C B B The change in the Gibbs energy of the crystal due to the vacancy formation is given by

(3)

DG"nH !¹S !n¹S . (4) F C F The vacancy formation lowers the Gibbs energy of the crystal until an equilibrium vacancy concentration is reached. This equilibrium concentration, c , fulfills the requirement ­(DG)/­n"0, 7 i.e., H #k ln c !¹S "0 . (5) F B 7 F The quantity G "H !¹S denotes the Gibbs energy of the vacancy formation that governs F F F the equilibrium vacancy concentration. Vacancies are stable in the crystal at any temperature above the absolute zero, and their equilibrium concentration rapidly increases with temperature. The formation entropy reflects changes in the vibration frequencies of the atoms surrounding the vacancy. These frequencies become lower than before the vacancy was formed. Thus the vacancy formation entropy is positive. This entropy equals to DS "3nek ¹ ln(u@/u) , (6) F B where u and u@ are the unperturbed and perturbed vibration frequencies, and e is a quantity proportional to the volume perturbed by the vacancy.

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Since the surface of the crystal and internal defects act as sources and sinks for the vacancies, the equilibrium concentrations of the vacancies and of the interstitials are independent. In metals, the enthalpy of the interstitial formation is markedly larger than that of the vacancy formation. In addition, the entropy of the interstitial formation is rather negative. Vacancies are therefore considered as dominated equilibrium point defects in metals. At premelting temperatures, divacancies may measurably contribute with an equilibrium concentration given by c "(z/2) exp(DS /k ) exp(H /k ¹) c2 . 27 27 B B B 7

(7)

Here DS , H , and z denote the association entropy and the binding enthalpy of a divacancy, 27 B and the coordination number of a lattice site, respectively. At a fixed temperature, the enthalpies and entropies of the vacancy formation can be considered as independent quantities. However, their temperature derivatives (at constant pressure) are interrelated through the thermodynamic relation: (­H/­¹) "¹(­S/­¹) . P P

(8)

On increasing temperature, the interatomic distances increase and 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 [3] considered the formation enthalpy to linearly decrease with increasing temperature: H "H !a¹. F 0

(9)

This dependence gives a contribution of exp(a/k ) to the pre-exponential factor in the expression B for the equilibrium defect concentration. In such a case, the corresponding contribution to the entropy should be negative due to the interrelation between the temperature derivatives of the formation enthalpy and entropy. Later, Vineyard and Dienes [42] have shown that the entropy of the vacancy formation depends only on the lattice vibration frequencies before and after the vacancy formation and that no further contribution to the entropy arises even though the formation enthalpy remains temperature dependent. The theory of the lattice vibrations in a disordered crystal lattice was developed by Maradudin [92]. 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 [93]: (1) To choose a proper potential for the description of the metallic bond. (2) To find the atomic configuration after the lattice relaxation. (3) 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 calculations of the formation enthalpies and entropies, lattice relaxation, the enthalpies of migration and energies of the binding of defects. As a rule, calculations of the formation enthalpies are in agreement to each other and with experimental data. At the same time, a strong contradiction is seen in the formation entropies. Generally, calculations of the formation entropies are less accurate than those of the formation enthalpies. Such calculations for Cu in a wide temperature range have been made by Foiles [94] by Monte Carlo simulations and some approximate techniques. The author pointed out that the

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Fig. 5. Temperature dependence of Gibbs energy of vacancy formation in Cu (Foiles [94]). 1 — quasiharmonic method, 2 — local harmonic approximation, 3 — Monte Carlo simulations.

harmonic methods underestimate the temperature dependence of the Gibbs energy of the vacancy formation (Fig. 5). Significant temperature dependence of the vacancy formation entropy in Na has been found in molecular-dynamics simulations by Smargiassi and Madden [95]. The formation entropy is 1k to 2k at low temperatures and increases to about 4k —5k in the vicinity of the B B B B melting point. The Gibbs energy of the vacancy formation decreases nonlinearly from about 0.35 eV at low temperatures to about 0.17 eV close to the melting. The equilibrium vacancy concentration in Na at the melting point is thus predicted as 5]10~3. Also, an unexpected conclusion has been drawn from these molecular-dynamics simulations: it turned out that the role of interstitials at high temperatures cannot be dismissed. Najafabadi and Srolovitz [96] calculated the vacancy formation energy in Cu by Monte Carlo simulations and free-energy-minimization methods. From the simulations, the 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 to 0.25 eV. Other techniques employed in the calculations (quasi-harmonic and free-energy-minimization methods) showed much weaker temperature dependence, as well as the calculations by Rickman and Srolovitz for Au [97]. The increase in the crystal volume associated with the vacancy formation, i.e., the vacancy formation volume, » , satisfies the thermodynamic relation F » "(­G /­p) "!k ¹(­ ln c /­p) . (10) F F T B V T The vacancy formation volume is thus obtainable from the pressure dependence of the equilibrium vacancy concentration. The basic theoretical conclusions concerning equilibrium point defects in metals may be briefly summarized as follows: (1) Point defects, vacancies and interstitials, are thermodynamically stable since they lower the Gibbs energy of the crystal. The equilibrium concentrations of point defects rapidly increase with temperature. (2) In metals, vacancies are the predominant point defects in equilibrium. Their concentrations at high temperatures are much larger than that of interstitials. Divacancies are the only defects whose equilibrium concentrations at high temperatures may become comparable with those of monovacancies.

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Table 2 Theoretical values of the vacancy formation enthalpies. 1 — Kornblit [98,99], 2 — Kostromin et al. [100], 3 — Harder and Bacon [101], 4 — Ackland et al. [102], 5 — Krause et al. [103], 6 — Rosato et al. [104], 7 — Ghorai [105], 8 — Korhonen et al. [106] H (eV) F Metal

1

Rb K Na Li Pb Al Ag Au Cu Ni Pd Pt V Cr Rh Ir Nb Mo Ta W

0.335 0.321 0.379 0.458

2

3

4

5

6

7

8

0.573 0.85 1.14

0.66 1.021 0.962 1.191 1.458

1.20 1.57

2.5 2.2

1.83 1.91

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

(3) Generally, the enthalpies of the vacancy formation calculated by various theoretical methods are in a satisfactory agreement to each other (Table 2) and do not contradict the experimental values. Theoretical calculations of the vacancy formation volumes, » , are in agreement with F experimental data (Table 3). (4) Various approaches in calculations of the vacancy formation entropies, S , lead to very F different values. Theoretical calculations of the temperature dependence of the formation parameters, H , S and » , have recently appeared. These parameters increase with temperature. F F F According to some calculations, the Gibbs energy of the vacancy formation at the melting point may be 1.5—2 times smaller than at low temperatures. In conclusion, it is worthwhile to mention some recent papers [112—115]. 3. Methods for studying point defects in metals 3.1. Measurements in equilibrium At present, it is commonly agreed that equilibrium point defects are to be studied at high temperatures. In principle, any physical property influenced by the defects can be used to determine

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Table 3 Theoretical values of the vacancy formation volume, » , in units of atomic volume, X, and the experimental data. F 1 — estimates by Seeger [27] based on the activation volumes of self-diffusion, 2 — Jacucci and Taylor [107], 3 — Bauer et al. [108], 4 — Harder and Bacon [101], 5 — Ackland et al. [102], 6 — Rosato et al. [104], 7 — experimental data [109—111] » /X F Metal K Na Li In Sn Cd Pb Zn Al Ag Au Cu Ni Pd Pt V Cr Ir Rh Nb Mo Ta W

1

0.25 0.2—0.25 0.45 0.25 0.5 0.4 0.35—0.4 0.65 0.75 0.65 0.8

2

3

4

5

6

0.78 0.73 0.77 0.88

0.76 0.72 0.8 0.8 0.77 0.76

7

0.59—0.71! 0.65—0.72! 0.43—0.68!

0.62" 0.69 0.63 0.79

0.52#

0.7$

0.74 0.84 0.75 0.77 0.96 0.73 0.83 0.68

! Two values correspond to the formation volumes at the abosolute zero and melting temperatures. " Ref. [109]. # Ref. [110]. $ Ref. [111].

their equilibrium concentrations. The important advantages of equilibrium measurements over non-equilibrium ones have been well understood many years ago. These advantages were clearly formulated by Seeger [26,27]: (1) Temperature and pressure are the only parameters that govern the equilibrium concentrations of point defects. The number of different kind of the defects created in equilibrium is quite limited. In most cases, quantitative interpretation may be obtained by taking into account only one additional type of defect, 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 can be directly expressed in terms of the enthalpy and entropy of the defect formation.

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In equilibrium, point defects can be studied through various physical properties: the enthalpy and specific heat, thermal expansion, electrical resistivity, thermopower, positron annihilation, perturbed angular correlation of c-quanta. Such measurements should provide most reliable data on the equilibrium defects. However, properties of a hypothetical defect-free crystal are unknown and cannot be calculated precisely. It is therefore impossible to unambiguously separate the point-defect contributions. Also, one has to know the relation between the defect concentration and the contribution to the given physical property. The only exception is considered the differential dilatometry that provides itself the necessary background data. The criteria for the choice of a suitable physical property are quite clear: (1) The magnitude of the defect contribution and the reliability of separating it. (2) The accuracy of the measurements. (3) The knowledge of parameters entering the relations between the defect contributions and concentrations of the defects. The most adequate property seems the specific heat for which all the criteria listed above are well fulfilled: (1) It is very likely that the specific heat of a hypothetical defect-free crystal weakly depends on temperature. (2) There exists, at least in principle, a straightforward experimental approach to separate the defect contribution by observations of the defect equilibration. (3) In many cases, the defect contributions are much larger than the errors of the measurements. (4) The defect concentration can be evaluated immediately from the extra specific heat. The increase in the specific heat is caused only by defects whose concentration reversibly follows the temperature. In other words, it is due not to the presence of the defects in the crystal lattice (this influence is much smaller) but to the temperature dependence of the equilibrium defect concentration. Clearly, studies of point defects should also include measurements of other properties, e.g., the thermal expansion. A special approach, the differential dilatometry, has been developed to directly determine the equilibrium defect concentrations. It consists in simultaneous measurements of the macroscopic thermal expansion of the sample and the dilatation of its unit cell measured by means of X-ray. The difference between them shows the difference between the concentrations of vacancies and 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, there exist some reasons to consider data from the differential dilatometry as underestimated ones. Still more important, no data were obtained by this technique on high-melting-point metals. Data on macroscopic thermal expansion are now available for many metals including refractory metals. Earlier, such data were inapplicable because of large inherent errors. The temperature dependence of the thermal expansivity (the thermal expansion coefficient) usually was believed to be linear. Owing to improvements in the traditional dilatometry and to development of the modulation and dynamic techniques, the measurements became much more accurate. Now the nonlinear increase in the thermal expansivity is quite obvious. Another property depending on the concentrations of point defects is the electrical resistivity. It can be measured more accurately than the specific heat or thermal expansivity. However, the point-defect contribution to the resistivity at high temperatures is relatively small. Determinations of the vacancy contributions to various physical properties require measurements in wide temperature ranges. They should also include measurements in intervals where these contributions are still negligible. Such measurements provide data necessary for the approximation of the properties of a defect-free crystal. Otherwise, the uncertainties in the derived vacancy

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parameters may become unacceptably large. The vacancy contributions to the enthalpy, volume and electrical resistivity of metals at high temperatures are relatively small. They should be added to the regular temperature dependence of these properties, usually quadratic. These properties are thus to be approximated by equation X"a#b¹#c¹2#d exp(!H /k ¹) . (11) F B The situation is much more favorable when the specific heat, thermal expansivity, or the temperature derivative of resistivity are measured directly. The vacancies affect strongly these properties whereas a linear extrapolation from intermediate temperatures is sufficient to separate the vacancy contributions. Hence, the specific heat, thermal expansivity and the temperature derivative of resistivity obey the relation ½"A#B¹#C¹~2 exp(!H /k ¹) . (12) F B To deduce the formation enthalpy from the defect contribution, D½, one should plot ln(¹2D½) versus 1/¹. The plot is a straight line with a slope equal to !H /k (Fig. 6). This procedure was F B applied to the nonlinear increase in the specific heat of refractory metals and yielded quite reasonable values of the formation enthalpies [56—59]. No data on the point-defect formation in these metals were available 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 above equation should be evaluated by means of the least-squares method. Trying various formation enthalpies, H , one plots the standard deviation F versus the assumed value. The minimum in the curve indicates the most probable formation enthalpy whereas the width of the curve shows its uncertainty. Correct formation enthalpies strongly support the validity of the approximation employed. When close formation enthalpies are deduced from various physical properties then one can conclude that the nonlinear changes in them are of a common origin. Another technique widely used in determinations of the formation enthalpies is the positron annihilation. The vacancies form traps for positrons and influence the parameters of the positron

Fig. 6. Determination of formation enthalpies from nonlinear increase in specific heat. Slope of the plot ln(¹2DC) versus 1/¹ equals to !H /k . F B

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annihilation. This method is very sensitive to vacancy-type defects. It was employed for determinations of the vacancy formation enthalpies in many metals. Regretfully, vacancy concentrations are not available from this technique. 3.2. Quenching experiments Most studies of point defects in metals employed measurements of the quenched-in electrical resistivity. This approach was very attractive due to the apparent simplicity of the measurements and evaluations. However, the drawbacks inherent to quenching experiments became evident in a short time. This was well understood already at the Argonne Conference [12]. Later, at the Ju¨lich Conference, the situation was exhaustively reviewed by Balluffi et al. [116]. Quenching experiments include studies of the properties of samples with vacancies frozen in the crystal lattice by quenching. Such properties are the extra enthalpy, changes in the volume and lattice parameter, the electrical resistivity, thermopower, parameters of the positron annihilation and of the perturbed correlation of c-quanta. One can also study quenched samples by an electron or field-ion microscope. The main disadvantage of equilibrium measurements is completely excluded here because quenched samples are compared with well-annealed samples defect concentrations in which are negligible. Unfortunately, the concentrations of quenched-in vacancies may be much smaller than the equilibrium concentrations at high temperatures. During quenching, many vacancies annihilate or form clusters. The vacancy-induced changes in the properties become therefore smaller. The distribution of the defects in quenched samples is thus not a representative of the equilibrium distribution. The mobility of the vacancies increases rapidly with temperature, and this discrepancy grows when the temperature approaches the melting points. Due to interactions between the vacancies and other imperfections in the sample, the situation becomes even more complicated. The extra electrical resistivity after quenching, Do, is measured as a function of the quench temperature. The plot of ln Do versus 1/¹ is a straight line with a slope governed by the formation enthalpy. However, instead of a straight line one often obtains a curve with a slope decreasing at higher temperatures. Correct values may be obtained by quenching from low and intermediate temperatures or by introducing corrections for vacancy losses during quenching (Fig. 7). The extra 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 that amounts only 1 ppm of the resistivity at high temperatures. The measurements thus ensure high sensitivity allowing one to use the quench temperatures far below the melting point and hence to reduce the losses of the vacancies during quenching. The enthalpy stored by the quenched-in vacancies can be released during annealing. Measurements of the stored enthalpy may provide the formation enthalpies and the vacancy concentrations. Clearly, results of such measurements may be valuable only if the vacancy losses and formation of secondary defects in the sample do not alter markedly the enthalpy related to the equilibrium vacancies. Nevertheless, measurements of the extra enthalpy of quenched samples, combined with measurements of the extra resistivity, are very desirable. Regrettably, such measurements are scarce. An important information is available from measurements of the changes in the volume and in the lattice parameter of the sample after quenching and during subsequent annealing.

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Fig. 7. (a) Extra resistivity of quenched Au wires at various cooling rates: 1 — 104, 2 — 2]104, 3 — 5]104 K s~1, 4 — extrapolation to infinite cooling rate. (b) Quenched-in resistivity as a function of reciprocal cooling rate (after Mori et al. [117]).

3.3. Observation of the vacancy equilibration Observations of the vacancy equilibration after rapid changes in the temperature of the sample can unambiguously reveal the vacancy contributions to the related property. This approach is capable of solving the main problem of equilibrium measurements. The only difficulty inherent to it is caused by short relaxation times. They are generally short and decrease rapidly when the 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 resistivity is measured versus 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. As a rule, the extra 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 to be easily measured. (2) The sample is rapidly heated 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 an important advantage that both initial and final states of the sample are well 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 positron-annihilation technique. Owing to the high sensitivity of the method, the measurements were carried out far below the melting point, so that the equilibration times were sufficiently long. Using the resistivity

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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 pre-melting 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 the temperature around a mean value that the vacancy equilibration cannot follow them. Under rapid temperature oscillations, the influence of the vacancies is almost completely excluded. This statement relates only to the properties that depend on changes in the vacancy concentrations during the measurements: the specific heat, thermal expansivity, and the temperature derivative of resistivity. If the vacancy concentration does not follow the temperature oscillations and retains a mean value, then 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 at 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. The relaxation phenomenon in specific heat can be searched conveniently by modulation calorimetry. It consists in periodically modulating the power applied to the sample and registering the temperature oscillations in it around a mean temperature (for a review see Refs. [118—120]). This technique makes it possible to directly compare the specific heats measured at various frequencies of the temperature oscillations in the sample. Relaxation phenomena appear when the period of the modulation 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, it depends on the frequency and the relaxation time. From the complex expression for the specific heat, C(X)"C#DC/(1#iX), one obtains [121]: DC(X)D2"(C2#C2X2)/(1#X2) , 0

(13)

tan D/"XDC/(C #CX2) . 0

(14)

Here C and DC denote the specific heat of a vacancy-free crystal and the vacancy contribution, X"uq is the product of the angular frequency of the temperature oscillations and the relaxation time, C "C#DC is the equilibrium specific heat measured when X2;1, and D/ is the change in 0 the phase of the temperature oscillations. The observable difference between the specific heats measured at a low and a high modulation frequency thus depends on the vacancy contribution under equilibrium, DC, and on X. Its temperature behavior reflects the decrease of X when the temperature increases. If the density of internal sources (sinks) for the vacancies does not depend on temperature, the quantity X obeys the expression X"exp(H /k ¹!H /k ¹ ) , M B M B 0 where H is the enthalpy of vacancy migration, and ¹ is a temperature for which X"1. M 0

(15)

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Fig. 8. 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: H "3.15 eV, S "6.5k [56], H "3 eV. ¹ is a temperature for which X"1. F F B M 0

The relaxation can also be observed through changes in the phase of the high-frequency temperature oscillations in the sample. Usually, the modulation calorimetry employs the so called adiabatic regime of the measurements. 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 can be made along with the measurements of the amplitude of the temperature oscillations in the sample. At a given modulation frequency, the magnitude of the relaxation first increases with temperature, reaches a maximum and then falls because of the decrease in the relaxation time. Assuming a constant density of internal sources and sinks for the vacancies, it is easy to evaluate the temperature dependence of the ratio of the specific heats measured at a low and a high frequency of the temperature oscillations, C /DC(X)D, and of the change in the phase of the high-frequency 0 temperature oscillations (Fig. 8). A technique for measuring the specific heat at frequencies of the temperature oscillations of the order of 105 Hz was developed [122] and employed to observe the relaxation in the specific heat of W and Pt [77,123—125].

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4. Enthalpy and specific heat of metals at high temperatures 4.1. Point defects and specific heat of metals The extra molar enthalpy and specific heat related to the formation of equilibrium vacancies are as follows: DH"NH c , (16) F 7 DC"(NH2A/k ¹2) exp(!H /k ¹) . (17) F B F B Here A is the pre-exponential factor entering the expression for equilibrium vacancy concentration, and N the Avogadro number. It was assumed that the enthalpy and entropy of the 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 the calorimetric measurements at lower temperatures, it was confirmed that it also follows the expression describing the vacancy formation. However, the calculations may become ambiguous at very low vacancy concentrations since minor nonlinear contributions may originate from other reasons. Calorimetric measurements seem the most natural method for determining equilibrium concentrations of point defects. The defect formation consumes a certain energy, and the enthalpy and specific heat of a crystal with point defects is larger than those of a defect-free crystal. The only drawback of this approach appears from the ambiguity in separating the defect contribution. The specific heat of a defect-free crystal cannot be calculated precisely, so that the defect contribution is hardly to be separated. This conclusion, quite correct in principle, caused the wide-spread negative regard to the data from specific-heat measurements. Many authors, referring to the anharmonicity, disregard all vacancy-formation data based on measurements of the specific heat. As a rule, thermophysicists studying properties of metals at high temperatures fit their experimental data by polynomials not taking into account the vacancy formation. However, calorimetric measurements, being doubtful from the theoretical point of view, appeared quite acceptable in practice. It turned out that the vacancy contributions are much larger than the uncertainties in the extrapolation of the data from intermediate temperatures where these contributions are negligible. Otherwise, this approach had no chance to yield reasonable formation enthalpies. Still more important, the vacancy contributions can be reliably determined through observations of the relaxation in the specific heat caused by the vacancy equilibration. The nonlinear increase in the specific heat was first observed in alkali metals by Carpenter et al. [126]. Later, the phenomenon has been attributed to the vacancy formation, and the formation enthalpies and equilibrium vacancy concentrations have been evaluated [39]. The formation parameters were also calculated from the specific heat of Pb and Al [40]. A strong nonlinear increase in the specific heat of Mo and Ta has been found by Rasor and McClelland [50]. However, the authors concluded that it is too large to be caused by point defects. A similar nonlinear increase

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in the specific heat was observed in Th and Re [127,128]. The parameters of the vacancy formation in refractory metals were first evaluated from the modulation measurements of the specific heat of W, Ta, Nb and Mo [56—59]. In 1973, the calorimetric measurements were considered by Seeger [26] 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 which may be determined independently from the temperature variation of the vacancy contribution. The calorimetric techniques were discussed in great detail at the Ju¨lich 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 that was claimed (e.g., [129]) is the following: (1) The vacancy concentrations in refractory metals deduced from the calorimetric data are much larger than results of the differential dilatometry for low-melting-point metals. (2) The nonlinear increase in the specific heat at high temperatures can be approximated by polynomials not taking into account the vacancy formation. Later, measurements of the specific heat or the electrical resistivity at high temperatures were disregarded by Siegel [130]: “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 the specific heat the situation turned out just opposite: as a rule, the vacancy contributions are large in comparison to the nonlinear background. 4.2. High-temperature calorimetry Calorimetric measurements seem, at first glance, to be simple and straightforward. By definition, one has to supply some heat to the sample and to measure the corresponding increment in its temperature. However, no simple solution of this task exists in a wide temperature range. First, the accuracy of temperature measurements in various temperature ranges is very different. Second, it is impossible to completely avoid uncontrollable heat exchange between the sample and its surroundings when the temperature of the sample is far from room temperature. This restricts drastically the accuracy of the calorimetric measurements at high temperatures. During many years of development, several approaches have been proposed to solve the problem, namely: (1) All possible precautions are undertaken to reduce the unwanted heat exchange between the sample and its surroundings (the 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 (the drop method). (3) The influence of the uncontrollable heat exchange is minimized by shortening the time of the measurements (the 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 (the relaxation method).

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4.2.1. Adiabatic calorimetry The adiabatic calorimetry reduces to a minimum the heat exchange between the calorimeter and the environment. For this purpose, the calorimeter is surrounded by a shield whose temperature is kept equal to that of the calorimeter during all the experiment, the so called adiabatic shield. Clearly, an exact equality of the temperatures is unachievable. The calorimeter and the shield are placed in a vacuum chamber, so that only a radiative heat exchange occurs. An electrical heater serves to heat the calorimeter. The temperature is measured by a resistance thermometer or a thermocouple. The heat supplied to the calorimeter and the temperature increment are thus accurately known. To reduce the heat flow through the electrical leads, they are thermally coupled to the adiabatic shield. With a temperature difference between the calorimeter and the shield, D¹, the power of the radiative heat exchange is proportional to ¹3D¹. The adiabatic calorimetry, being an excellent technique at low and intermediate temperatures, fails therefore at high temperatures. In measurements on metals, an enhanced heating rate reduces the role of heat losses at high temperatures. An adiabatic calorimeter of continuous heating for the range 300—1900 K was developed by Braun et al. [131]. It is capable of measurements of the specific heat of solids (with an error of 2%) and liquids (3%), and the latent heats of phase transitions in solids (0.5%) and of the melting (1.5%). Three modes of operation are feasible: (1) The applied power is constant, and the heating rate is inversely proportional to the heat capacity of the sample. (2) The heating rate is kept constant, so that the applied power is proportional to the heat capacity. (3) With the heater switched off, the temperature difference between the calorimeter and the thermal shield is monitored. Many metals were investigated by this group. 4.2.2. Drop method This technique was developed for measurements at high temperatures. The sample is placed in a furnace and heated to a selected temperature measured by a thermocouple or an optical pyrometer. 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. The increment in the temperature of the calorimeter is proportional to the enthalpy of the sample. It is measured by a resistance thermometer. The calorimetric measurements are thus made 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, instead of the 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 determined precisely. An additional disadvantage appears when phase transitions of the first order occur at the intermediate temperatures and the thermodynamic equilibrium in the sample after cooling is doubtful. The drop calorimetry has been developed when there was no alternative for high temperatures. Nevertheless, it remains useful until today, especially for measurements on nonconducting materials. 4.2.3. Modulation calorimetry The modulation calorimetry is based on periodically modulating the power that heats the sample and thereby creating temperature oscillations in it around a mean temperature. Their

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amplitude depends on the heat capacity of the sample. This principle was discovered at the beginning of the century by Corbino [132,133]. The periodic changing the temperature provides important advantages. When the modulation frequency is sufficiently high, then corrections for heat losses become negligible even at extremely high temperatures. The harmonic temperature oscillations are determined using selective amplifiers and lock-in detectors, so that even very small temperature oscillations become measurable. This feature is very important when a good temperature resolution is required as in studies of phase transitions. The modifications of this technique differ by the ways to modulate the heating power (heating by an electrical current, radiation or electron-bombardment heating, induction heating, use of separate heaters) and by the methods to detect the temperature oscillations (through the resistance of the sample or radiation from it, using a thermocouple or a resistance thermometer). In treating the data, the mean temperature and the amplitude of the temperature oscillations are considered to be constant throughout the sample. As a rule, the measurements are made in the adiabatic regime when the amplitude of the temperature oscillations is inversely proportional to the heat capacity of the sample. The basic equation of the modulation calorimetry is mc"p/uH ,

(18)

where m and c are the mass and the specific heat of the sample, p and u are the amplitude and the angular frequency of the oscillations in the heating power, and H is the amplitude of the temperature oscillations in the sample. Even at low and intermediate temperatures, the traditional domain of the adiabatic calorimetry, the modulation method ensures better temperature resolution and higher sensitivity. Also, small dimensions of the samples are often of importance. On the other hand, absolute values obtainable by the modulation technique are less accurate than those by the adiabatic calorimetry. Even under very favorable conditions for modulation measurements, the inaccuracy of absolute values of the specific heat is of about 1%. Nevertheless, the method is widely used owing to its high resolution. The term “modulation method for measuring specific heat” was proposed in the paper [134] describing the bridge circuit for measurements on wire samples (Fig. 9). However, for many investigators the acquaintance with this technique began from the papers by Sullivan and Seidel [135,136]. The authors considered a calorimetric system including a sample, a heater, and a thermometer, and performed measurements at low temperatures. They introduced the term “AC calorimetry” now widely recognized. Sullivan and Seidel stressed the essential advantages of the method as follows [136]: “(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 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.” It was shown [137] that measurements in a non-adiabatic regime are possible with about the same accuracy as in the adiabatic regime. Though the heat losses under non-adiabatic conditions may be large, they can be taken into account. This approach requires measurements of the phase shift, /, between the oscillations of the power applied to the sample and the temperature oscillations. The rigorous expression is mc"(p/uH) sin / .

(19)

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Fig. 9. Bridge circuit for modulation measurements by equivalent-impedance technique [134]. The heat capacity of a wire sample relates to the bridge parameters as mc"2I2R@/u2RC, where I is the DC current passing through the 0 0 sample, R@ is the temperature derivative of its resistance, u is the angular frequency of the temperature oscillations, and values of R and C correspond to the balance of the bridge.

The modulation calorimetry is capable of direct measurements of the temperature derivative of specific heat [138]. The measurements are based on determinations of the second harmonic in the temperature oscillations caused by the temperature dependence of the specific heat. Employing the modulation technique, the high-temperature specific heats of W [56], Ta [57], Nb [58], Mo [59], Pt [84], Zr [139], Au [140], Cu [141] and La [142] have been measured. The nonlinear increase in the specific heat was attributed to the vacancy formation, and the formation parameters were evaluated. The results of the modulation measurements on Pt and Cu [84,141] can be compared with those from the adiabatic calorimetry [143,144] (Fig. 10). 4.2.4. Pulse and dynamic techniques The influence of the unwanted 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 electrical 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 avoided, they are small and can be taken into account. The pulse calorimetry employs small increments in the temperature, and the result of such measurements is the specific heat at a single temperature. The initial temperature of the sample is provided by a furnace or electrical heating. Rasor and McClelland [145] developed a system in which the sample was placed in a graphite furnace to obtain the initial temperature. The temperature increment caused by an electrical pulse was measured by a photomultiplier (Fig. 11). With a four-channel oscilloscope, the heating current, I, the voltage drop across the sample, º, the temperature of the sample, ¹, and the heating rate, ¹@ "(d¹/dt) , were recorded simultaneously. The heat capacity of the sample is given by ) ) mc"Iº/¹@ . (20) )

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Fig. 10. Specific heat of Pt [84,143] and Cu [141,144]: v — modulation measurements, j — adiabatic calorimetry. Fig. 11. Measurement of specific heat of Mo and Ta (Rasor and McClelland [50,145]). The authors concluded that the nonlinear increase in the specific heat is too large to be attributed to vacancy formation.

The specific heat of Mo, Ta and graphite was measured [50] at high temperatures, up to 3920 K for graphite. The strong nonlinear increase in the specific heat of Mo and Ta was observed for the first time. The authors considered that it is too high to be attributed to the vacancy formation. The dynamic technique consists in heating the sample over a wide temperature interval. The heating power and the temperature are measured continuously during all the run. These data are sufficient to evaluate the specific heat in the whole temperature range. A convenient and accurate subsecond technique for temperatures up to 3600 K has been developed by Cezairliyan and coworkers [65—68,146,147] at the National Bureau of Standards (Fig. 12). The temperature of the samples with blackbody models was measured by an optical pyrometer. The power-balance equations for the sample during the heating and cooling periods are given by mc¹@ #P"Iº, )

(21)

mc¹@ #P"0 . #

(22)

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Fig. 12. Dynamic calorimetry developed by Cezairliyan et al. [65—68,146,147]. In one run, specific heat, electrical resistivity, normal spectral emittance, and 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.

Here P is the power of the heat losses from the sample, and ¹@ "(d¹/dt) is the cooling rate after # # termination of the heating current. From these relations, the heat capacity of the sample is mc"Iº/(¹@ !¹@ ) , (23) ) # where ¹@ and ¹@ relate to the same temperature of the sample. ) # The setup described by Dobrosavljevic´ and Maglic´ [148] is based on heating wire samples with the rates up to 1500 K s~1. The calorimetric system 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 of about 200 mm. The central portion of the sample is about 20 mm long. Three 0.05 or 0.1 mm thermocouples are spot-welded at the center and symmetrically at 10 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 at a 1 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 terminating the current. In the 300—1900 K temperature range, the maximum uncertainties were estimated as 3% in the specific heat and 1% in the electrical resistivity. 4.2.5. 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 [149,150]. This rate depends on the temperature difference, heat capacity of the sample and 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 sample can be put in turn. The temperature dependence of the heat losses from the calorimeter remains the same, so that 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 ¹"¹ #D¹, slightly higher 0

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than that of the surroundings, ¹ . Then the heating current is switched off and the temperature of 0 the calorimeter decays exponentially: ¹"¹ #D¹ exp(!t/q) , (24) 0 where q"C/K is the relaxation time, C is the sum of the heat capacity of the sample and the calorimeter itself, and K is the heat transfer coefficient, i.e., the temperature derivative of the heat losses from the sample. The determination of the specific heat thus includes measurements of the relaxation time and of the heat transfer coefficient. In steady-state conditions, P"K D¹, where P is the power dissipated in the calorimeter that is necessary to increase its temperature by D¹. The temperature increment is small, so that the linear relation is valid. The internal time constant (through the sample and between the sample, the heater, and the thermometer) is much shorter than the external time constant, q. 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 even to samples as small as a few micrograms. The relaxation technique was employed by Zinov’ev and Lebedev [151] in measurements on W in the range 2400—3600 K. After heating the wire sample to a high temperature, the heating current was terminated and the cooling curve was measured by means of a photomultiplier and an oscilloscope. The specific heat obeys the relation mc"!P/¹@ . (25) # Here P is the power radiated from the sample, and ¹@ is the cooling rate at this temperature. The # results obtained by the authors coincide with those from the modulation measurements [56]. 4.2.6. Rapid-heating experiments The rapid-heating technique allows measurements of thermophysical properties of metals over wide temperature intervals including liquid state. Many new results were obtained using this method (for reviews see [152—154]). In rapid-heating experiments, the time of heating the sample up to the melting point can be made comparable with the time necessary for the vacancy equilibration. Experiments with heating rates more than 109 K s~1 have been reported by Pottlacher et al. [155,156]. Starting at room temperature, the measurements are performed far into the liquid phase of the metal under study, up to 10 000 K. In the setup developed (Fig. 13), the energy was stored in a 5.4 lF capacitor, with a charging voltage 4—8 kV. The wire samples were typically 40 mm long and 0.25 mm thick. Water served as the ambient medium to avoid peripheral discharges. The pressure in the vessel could be varied up to 2]108 Pa. An initial pulse triggered a flash light for background illumination of the sample and the main discharge was triggered with the aid of a three-electrode spark gap. The quantities measured during the run were as follows: (1) The current through the sample, by means of an induction coil. (2) The voltage drop across the sample, using a coaxial voltage divider. (3) The radiance temperature of the sample, by a fast pyrometer. (4) The final volume of the sample, employing a shadowgraph technique with a 30 ns exposure time. The temperature was calculated using the melting temperatures as calibration points, and the

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Fig. 13. Rapid-heating technique using heating rates of the order of 109 K s~1 (after Pottlacher et al. [155,156]).

emissivity of the sample surface was considered as being independent of temperature. A special care should be taken to avoid a superheat of the samples that is quite probable in rapid-heating experiments. From the measurements, the enthalpy, the electrical resistivity and the volume expansion were deduced. The authors pointed out that more accurate results for the solid phase are available from static measurements. However, rapid heating allows one to perform measurements far above the melting point. This technique has a potentiality to monitor the vacancy equilibration and hence to reveal the vacancy contributions to the enthalpy and the resistivity of metals. A microsecond-resolution system was reported by Kaschnitz et al. [157]. Heating rates of 107—108 K s~1 permit more accurate measurements of the thermophysical properties than using faster systems. Wire or tube-shaped samples are resistively self-heated with a RCL discharge circuit. Energy is stored in a capacitor, 240—500 lF, which may be charged up to 10 kV. Typically, the heating current is of about 5000 A and the pulse is 80 ls 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, with a Kerr cell providing an exposure time of 30 ns. The uncertainty of the obtained values was estimated as 3% for the enthalpy and 3% for the electrical resistivity, without corrections for the thermal expansion. 4.3. Formation enthalpies and equilibrium vacancy concentrations The nonlinear increase in the high-temperature specific heat of metals was observed by all known methods. Especially numerous are measurements on W. The data presented here (Fig. 14) show that the phenomenon was observed by very different calorimetric techniques [56,151,158—162]. Surprisingly, the scatter of the data is quite moderate. The results of the modulation measurements [56] were confirmed by the pulse calorimetry [158] and by the relaxation method [151]. Curve 1 represents the data from the three sources. Also, one can compare the results of the modulation measurements [56] and the recent data by the dynamic

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Fig. 14. High-temperature specific heat of W. 1 — Curve presenting data from modulation method [56], pulse calorimetry (Affortit and Lallement [158]), and relaxation measurements (Zinov’ev and Lebedev [151]); 2 — drop method (Arpaci and Frohberg [159]); 3, 4 — pulse calorimetry (Yakunkin [160], Senchenko and Sheindlin [161]); 5 — dynamic measurements (Righini et al. [162]).

method [162]. The difference between the two curves is less than 1% in the range 1500—2100 K and less than 3.5% in the range 2100—3200 K. This temperature range is quite sufficient to calculate the parameters of the vacancy formation. At higher temperatures, the difference grows rapidly and amounts to about 15% at 3600 K. However, above 3200 K most of the other results is 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, W is rather a fortunate exception. The measurements on Mo and Ta by the pulse method [50] and the modulation measurements on W and some other metals were the first direct determinations of the specific heat of these metals at high temperatures. Earlier, only the drop technique was employed above 2000 K. The nonlinear increase in the specific heat was not seen by the drop technique. It was shown [163] that the reason for this was an employment of fitting polynomials not taking into account the vacancy formation. Regrettably, until today many authors make no attempts to check whether their experimental results contain vacancy contributions. For example, Cezairliyan et al. [65—68] employed four-term polynomials to fit the high-temperature specific heat of refractory metals. It is interesting to fit these data by the equation containing the vacancy-related term. For Nb, Ta and W, this procedure leads to very reasonable values of the formation enthalpies (Fig. 15). For Mo, the formation enthalpy appeared somewhat lower than the expected one. 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 exactly. Calorimetric measurements provided many data on equilibrium vacancies in metals (Table 4). Only results evaluated by the authors themselves are given here. In low-melting-point metals, the nonlinear increase in the 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 No¨lting [69] in terms of the vacancy formation. The derived vacancy concentrations at the melting points of In, Sn, Pb, Zn, Sb and Al appeared in the range 5]10~4—3]10~3, whereas in high-melting-point metals they are of the order of 10~2. The formation enthalpies obtained are in an agreement with the data from other techniques or theoretical estimates.

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Fig. 15. High-temperature specific heat of refractory metals (Cezairliyan et al. [65—68]). Plot of the standard deviation versus assumed formation enthalpy shows the validity of the approximation taking into account vacancy contributions.

4.4. Extra enthalpy of quenched samples The enthalpy of the vacancies frozen by quenching can be released during annealing. If all equilibrium vacancies survive in the crystal 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 losses of the vacancies during the quench. Many frozen-in vacancies form clusters and secondary defects whose enthalpy is smaller than that of the original vacancies. As was shown by Moya and Coujou [174], this decrease depends on the size of the secondary defects. The stored enthalpy can be considered as a lower limit of the enthalpy related to the equilibrium vacancies. Regretfully, such data are not numerous. The formation enthalpy (0.97 eV) and the equilibrium vacancy concentration in Au (4.5]10~4 at the melting point) have been determined by DeSorbo [46,175]. Pervakov and Khotkevich [176] evaluated the vacancy concentration in Au at the melting point as 2.1]10~3. This figure is quite comparable with the calorimetric value, 4]10~3 [140]. An additional informative parameter appears when the stored enthalpy is measured along with the extra resistivity. The ratio of these quantities, DH/Do, 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 as 0.5 and 3 kJ l)~1 cm~1,

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Table 4 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 Metal

Cs Rb K Na In Sn Pb Zn Sb Al

La

H F (eV)

C .1 (10~4)

Method, Refer.

0.28 0.31 0.23 0.42 0.255 0.35 0.425 0.455 0.48 0.39 0.61 1.13 1.17 0.79 0.66 1.0

30 25 48 14 76 30 5 13 20 23 23 12 20 22 11 120

A A A A A A A A P A A A P A D M

[164] [164] [39] [164] [39] [165] [69] [69] [40] [69] [69] [69] [40] [69] [166] [142]

Metal Au Cu Ni Ti Pt Zr Cr Rh Nb Mo Ta W

H F (eV)

C .1 (10~4)

Method, Refer.

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

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

[140] [141] [167] [168] [84] [139] [169] [170] [58] [171] [59] [172] [57] [56] [173]

respectively. From equilibrium measurements, it is of about 1 kJ l)~1 cm~1. However, it is difficult to take into account changes in both quantities, DH and Do, caused by the clustering of the vacancies during or immediately after quenching. The vacancy-related enthalpy in Al was measured directly by Guarini and Schiavini [177] by means of a precision 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 300°C to 350°C, 350°C to 400°C, 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 using a Cu sample in which the vacancy contribution at these temperatures is negligible. If an extra heat is absorbed or released by the sample, the output voltage no longer behaves exponentially (Fig. 16). The extra heat was attributed to changes in the vacancy concentrations in the sample. The vacancy concentration at the melting point has been evaluated as 6]10~4. The authors considered this result as a lower limit because the initial part of the calorimetric curves could not be followed. However, the equilibration times appeared several orders of magnitude longer than could be expected from quenching experiments. 4.5. Question to be answered by rapid heating Under very rapid heating, vacancies have no time to appear, so that the enthalpy of the sample at a given premelting temperature should be smaller than that under a moderate heating rate. For Mo

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Fig. 16. Determination of the extra enthalpy of Al by a precision microcalorimeter. Deviations from expected temperature trace shows the extra heat absorbed or released by the sample after a rapid change of its temperature (after Guarini and Schiavini [177]).

and W, the vacancy-related enthalpy calculated from the nonlinear increase in the specific heat is of about 10%. To check this concept, an examination of typical data now available has been made. The sources of the data included equilibrium measurements of the enthalpy [172,173], modulation, pulse and dynamic measurements of the specific heat [50,56,59,67,162,178,179], and rapid-heating determinations of the enthalpy at the melting points [156,180—182]. From the estimated time of the vacancy equilibration, only experiments with the highest heating rates, of the order of 108 K s~1 or more, may be expected not to contain the vacancy contribution. From these data, 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 the equilibrium measurements. For this purpose, the experimental data were fit by the equations taking into account the vacancy formation. To fit the specific heat measured only at high temperatures, the enthalpy at 1500 K was taken as 32.6 kJ mol~1 for W [173] and 33.5 kJ mol~1 for Mo [172]. We thus obtained three sets of the enthalpies at the melting points (Table 5): (1) The equilibrium enthalpies after subtracting the assumed vacancy contributions, H . (2) The total 1 equilibrium enthalpies including the vacancy contributions, H . (3) The enthalpies from the 2 rapid-heating measurements, H , which are expected to be close to H rather than to H . With 3 1 2 heating rates 108—109 K s~1, an uncertainty in the enthalpy is of about 3—5% [156]. The difference between H and H is therefore quite detectable. For W, the results of the rapid-heating 1 2 experiments, H , are close to the evaluated H values and thus support the proposed concept. For 3 1 Mo, the results lie between the two values, H and H . Possible explanations of this may be as 1 2 follows: (1) The heating is not fast enough to completely avoid the vacancy formation. (2) A superheating of the samples under high heating rates leads to an enhancement of the apparent melting point and the corresponding enthalpy. (3) The 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 using various heating rates and to measure independently the apparent melting temperature. Also, one can measure the enthalpy at a selected premelting temperature.

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Table 5 Enthalpy of solid tungsten and molybdenum at the melting points. H — enthalpy not including the assumed defect 1 contribution, H — total enthalpy, H — results of rapid-heating experiments 2 3 H 1

H (kJ mol~1) H 2

Tungsten 112 112 109

122 116 116

H 3

112 111 109 Molybdenum 81 84 83

117

91 92 89 85

85 83

89 90 87 87

Reference

Kraftmakher and Strelkov [56] Cezairliyan and McClure [67] Chekhovskoi [173] Hixson and Winkler [180] Pottlacher et al. [156] Righini et al. [162]

Rasor and McClelland [50] Kraftmakher [59] Chekhovskoi and Petrov [172] Seydel and Fischer [181] Cezairliyan [178] Righini and Rosso [179] Hixson and Winkler [182] Pottlacher et al. [156]

The rapid-heating technique may be regarded as a very useful tool to reliably determine the vacancy-related enthalpy of metals.

5. Thermal expansion of metals at high temperatures 5.1. Point defects and thermal expansion Point defects cause changes in the volume of the sample. The vacancy formation leads to an increase in the volume of the sample and in the thermal expansivity at high temperatures. The only difficulty is to correctly separate the vacancy contribution. Gertsriken [41] was the first to deduce the equilibrium vacancy concentrations in some metals from the thermal-expansion data. A creation of a vacancy means that one atom leaves its lattice site and occupies a position on the sample surface. The increase of the volume should thus be equal to one atomic volume, X. Due to the relaxation of the atoms around the vacancy, the formation volume, » , becomes smaller than the F atomic volume. For rough estimations, the ratio c"» /X may be taken as 0.5. When an interstitial F is created, the volume of the sample decreases. Because of the relaxation, the decrease is also smaller than one atomic volume. Flinn and Maradudin [183] developed a Green’s function method for calculating the static distortion and the energy change due to a point defect in crystal.

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In an isotropic sample, the relative linear expansion due to the vacancy formation is three times smaller than the relative changes of the volume: Dl/l"cc /3"(cA/3) exp(!H /k ¹) . (26) 7 F B 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. The increase in the linear thermal expansivity, a"dl/ld¹, is given by Da"(cH A/3k ¹2) exp(!H /k ¹) . (27) F B F B The relative increase in the thermal expansivity is much larger than that in the volume of the sample. Direct measurements of the expansivity are feasible by the modulation technique. 5.2. Methods of dilatometry At present, methods for measuring dilatation of solids provide a sensitivity of the order of 10~8 cm and even better (e.g. [184,185]. However, difficulties in studying thermal expansion of solids at high temperatures are caused by a 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. Fortunately, important improvements in this field have been made in the last few decades. Along with a significant progress in traditional dilatometry, two new approaches have been developed, namely modulation dilatometry and dynamic technique. 5.2.1. Traditional dilatometry Most sensitive dilatometers employ capacitor sensors and optical interferometers. Interferometric measurements of thermal expansion become much easier with employment of lasers, owing to high temporal and spatial coherence of the laser beam. A sensitive laser dilatometer was reported by Feder and Charbnau [186]. Changes in the length of the sample are measured using a Fizeau-type interferometer (Fig. 17). The accuracy of the

Fig. 17. Simplified diagram of interferometric dilatometer developed by Feder and Charbnau [186].

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measured change in the length is dependent only on the precision with which the wavelength of the light source is known and the shift in fringe pattern is determined. The light beam is reflected from the bottom optical flat upon which the sample rests. The incident and reflected beams interfere with each other on the lower face of the upper optical flat to produce the visible interference fringes. The fringe shifts are thus dependent only upon the changes in the length of the sample. A He—Ne laser was employed as the light source, and a special system was designed for the automatic fringe counting. 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 the incident fringes. By connecting the phototubes to a two-pen chart recorder, the fringe shifts can be recorded as a series of sine waves. This double-recording system was employed to sense changes in the direction of the fringe movement. The temperature control was achieved by immersing the tube containing the sample into a regulated oil bath. An instrument using a polarization interferometer, of the sensitivity of about 10~7 cm, has been developed by Roberts [187]. 5.2.2. Modulation dilatometry Direct measurements of the “true” expansivity, i.e., the coefficient of thermal expansion in a narrow temperature interval, are feasible by the modulation technique. This method was proposed long ago [188,189] and applied to studying thermal expansion of metals and alloys at high temperatures. Here the changes in the length of a sample when its temperature oscillates around a mean value are measured (for a review, see Refs. [119,190,191]). The thermal expansivity is thus measured directly. The method allows one to ignore irregular external disturbances or creep of the samples at high temperatures. Only those changes in the length of the sample are measured that follow reversibly the temperature oscillations. Since two decades the method was re-invented by Johansen [192] and employed for studying nonconducting materials. However, this promising technique has not gained a proper recognition until today. In studies of wire samples, they are heated by an AC current or by a DC current with a small AC component. The upper end of the sample is fixed whereas the lower one is pulled by a load or a spring and projected onto the entrance slit of a photomultiplier (Fig. 18a). The AC voltage at the output of the photomultiplier follows the oscillations of the length of the sample. The oscillations of the temperature in the sample are determined from either its electrical resistance or radiation from it, or calculated from the specific heat. When the sample is heated by a DC current I with a small 0 AC component, then the linear expansivity obeys the expression a"mcu»/2lKI º, (28) 0 where m, c and l denote the mass, specific heat, and length of the sample, u is the angular frequency of the temperature oscillations, º is the AC voltage across the sample, » is the AC component at the output of the photomultiplier, and K is the sensitivity of the photomultiplier to the elongation of the sample that is available from static measurements. The modulation frequency ranges from 10 to 100 Hz. It is considered to be sufficiently high to satisfy the adiabaticity conditions. Otherwise, the amplitude of the temperature oscillations is available from the formulas for the non-adiabatic regime [137]. A simple circuit can be assembled whose balance does not depend on the AC component in the heating current. The AC component

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Fig. 18. Modulation dilatometry. (a) Only changes in the length of the sample are measured that follow reversibly temperature oscillations [188]. (b) Compensation 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 [189].

at the output of the photomultiplier is compensated by a variable mutual inductance with the heating current being passed through its primary coil. The setup incorporates a selective amplifier tuned to the modulation frequency, which makes the dilatometer to be insensitive to the creep of the sample or mechanical perturbations. A compensation method is also feasible (Fig. 18b). The sample consists of two portions joined together, the sample under study and the reference sample of a known thermal expansivity. The two portions are heated by DC currents from separate sources and by AC currents from a common low-frequency oscillator. The temperature oscillations in the two portions are of opposite phase. By adjusting the AC currents, the oscillations in the length of the sample under study are compensated by those of the reference sample. Now the photomultiplier serves only as a null-indicator, and the influence of variations in its sensitivity, as well as in the light intensity, etc., is completely eliminated. To calculate the expansivity, the temperature oscillations in the two portions are to be determined. If their specific heats are known, then one can use the obvious relation that holds when the oscillations balance each other: a I º l /m c "a I º l /m c . 1 01 1 1 1 1 2 02 2 2 2 2

(29)

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Fig. 19. Linear thermal expansivity of Pt and W directly measured by modulation technique [87,90]. The nonlinear increase was attributed to vacancy formation.

Subscripts 1 and 2 refer to the sample under study and the reference, respectively. The reference sample is kept at a constant mean temperature, and all the related quantities are constant. Hence, a "Bc /I º , (30) 1 1 01 1 where B is a coefficient of proportionality. The measurements are thus reduced to measuring the DC current in and the AC voltage across the sample under study. This technique has been employed for studying the thermal expansion of Pt and W at high temperatures (Fig. 19). The samples were 0.05 mm thick. Their mean temperature was deduced from the electrical resistance whereas the temperature oscillations were calculated from the specific heat. Similar wires kept at constant mean temperatures served as the reference samples. The sensitivity of the setup was of about 10~7 cm. The oscillations in the length of the sample can be detected in various ways, including those of higher sensitivity. In the interferometric modulation dilatometer [193], the samples are used in the

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Fig. 20. Interferometric modulation dilatometer [193]. The temperature oscillations are created only in the central portion of the sample. A feedback circuit and piezoelectric transducer serve to balance the oscillations in the length of the sample.

form of a wire or a rod (Fig. 20). The upper end of the sample is fixed and a flat mirror M1 is attached to its lower end. The sample is heated by a DC current, and the temperature oscillations are created only in its central portion. For this purpose, a small AC current is fed to the central portion through thin probes welded to the sample. A laser beam is projected through a beamsplitter onto the mirror M1 and a second mirror M2 attached to a piezoelectric transducer. The intensity of the interference pattern is detected by a photodiode whose output voltage is fed to an amplifier. The amplified voltage is applied to the transducer with such a polarity that the oscillations of the two mirrors are in phase. The displacements of the mirror M2 are practically equal to those of the mirror M1. The AC voltage applied to the transducer is therefore proportional to the oscillations in the length of the sample. This voltage is measured by a lock-in amplifier. 5.2.3. Dynamic technique The dynamic dilatometry has been developed by Miiller and Cezairliyan [72—76]. The method 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. The temperature of the sample is determined by a photoelectric pyrometer capable of 1200 evaluations per second. Simultaneously, the expansion of the sample is measured by the shift in the fringe pattern produced by a polarized-beam laser interferometer. 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. A Michelson-type interferometer is employed with the sample acting as a double reflector in the path of the light beam (Fig. 21). The interferometer is thus insensitive to the translational motion of the sample. The rotational stability of the sample is monitored by reflecting the beam of an auxiliary laser from a third optical flat on the sample. A small rectangular hole in the wall of the sample, 0.5]1 mm, serves as a blackbody model for the temperature measurements.

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Fig. 21. Simplified diagram of dynamic dilatometer developed by Miiller and Cezairliyan. This technique was successfully employed in studies of refractory metals [72—76].

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: (1) The measurements are related to the blackbody temperature. (2) Only the central portion of the sample is involved in the measurements, so that there is no need to take into account the temperature distribution along the sample. Using this technique, the authors investigated the thermal expansion of refractory metals at high temperatures. The dynamic technique developed by Righini et al. [194] 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. Typical spacing between two consecutive measurements is 0.35 mm with a pyrometer viewing area of 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 upper clamp. The resolution of the interferometer is of about 0.15 lm, 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 resistivity of the sample or data on the normal spectral emittance. The measurements on a Nb sample [89] 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 rotation of the mirror. A compromise must therefore be found between a better knowledge of the temperature profile and a better definition of the thermal expansion polynomial. The authors 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. However, both methods are technically difficult.

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5.3. Differential dilatometry In measurements of the thermal expansion, the background cannot be determined while the vacancies or interstitials are present in the crystal lattice. To solve the problem, the differential dilatometry was proposed. The history of this approach was given by Seeger [26], Kluin [82] and Hehenkamp [83]. In 1954, Eshelby [195] demonstrated that the elastic relaxation around vacancies distributed randomly 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 7 in the volume of the sample is D»/»"3Dl/l"cc (c"» /X). The average volume corresponding 7 F to one lattice site decreases: Dl/l"3Da/a"!(1!c)c , where a is the lattice parameter. The 7 difference between the two quantities equals to the vacancy concentration c , regardless of c: 7 (31) c "D»/»!Dl/l"3(Dl/l!Da/a) . 7 More rigorously, this difference should correspond to the difference between the concentrations of the vacancies and interstitials, c and c : 7 * c !c "3(Dl/l!Da/a) . (32) 7 * This relation is considered to be 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, l, and the lattice parameter, a, are measured simultaneously. The equilibrium concentrations of interstitials are believed to be much smaller than those 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, namely: c !c "2[(Dl/l ) !Da/a]#(Dl/l ) !Dc/c , (33) 7 * 1 2 where subscripts 1 and 2 relate to the macroscopic thermal expansion along the two crystallographic axes. Using this technique, Simmons and Balluffi carried out the well-known measurements on Al, Ag, Au and Cu (Fig. 22). Hehenkamp et al. [80—83] developed a new differential-dilatometry apparatus. The sample for the macroscopic measurements 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 the X-rays and two quartz windows for the laser beam. A helium atmosphere at 5]104 Pa was a good compromise between the X-ray absorption and the thermal coupling between the sample and the furnace. The macroscopic thermal expansion was measured as follows. The top of the sample formed a slit with the surrounding cage. When the thermal expansion of the cage is much 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 temperature was increased in steps of about 15 K between the room temperature and the melting point of the sample. The accuracy of the

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Fig. 22. Differential-dilatometry data reported by Simmons and Balluffi [52—55]. During three decades, these data were regarded as the most reliable ones.

temperature control was better than 0.1 K. For each temperature, the diffraction pattern was registered several times to minimize the statistical error. The X-ray expansion was measured by a standard spectrometer. Only measurements where the lattice parameter determined from two reflections showed a difference of less than 10~12 cm were utilized. This technique provided data on equilibrium vacancy concentrations in Ag and Cu and some alloys (Fig. 23). At elevated temperatures, openings in the heating and shielding elements allowing the passage of X-rays give rise to temperature gradients. To avoid this drawback, the neutron diffraction can be employed instead of X-rays to measure the microscopic thermal expansion. Adlhart et al. [196] used this method in studies of the vacancy formation in Na. The macroscopic expansion was measured by the laser interferometry. Similar measurements on Cu and Au have been reported by Trost et al. [197]. The authors considered this modification as a promising one for studying metals with higher melting temperatures, such as Ni and Pt, or even Nb and Mo. The differential dilatometry showed low vacancy concentrations in many metals, less than 10~3 at the melting points. These data, along with the low extra resistivities of quenched samples, caused the opinion about smallness of the vacancy concentrations in metals. However, the validity of this approach seems doubtful. In real samples, vacancies may partly appear from internal defects such as voids, grain boundaries, vacancy clusters. The corresponding increase in the outer volume of the sample may therefore be smaller than under ideal conditions. The assumption about uniform distribution of the sources (sinks) for the vacancies is generally invalid. An ultimate example of such a situation was pointed out by Nowick and Feder [198]. For a perfect wire sample, only its surface serves as the source of the vacancies. In this case, the vacancy formation results in an increase of the thickness of the sample whereas the length remains unchanged. Hence, measurements of the thermal expansion cannot be considered as a very reliable method for studying equilibrium point defects. Theoretical calculations show (e.g., Refs. [199,200]) that the relaxation of atoms around a created vacancy is inhomogeneous. 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. It is worthwhile to recall an example when the differential dilatometry yielded vacancy concentrations larger than any other technique. For a strongly compressible crystal, X-ray measurements

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Fig. 23. Temperature dependence of equilibrium vacancy concentrations in Ag and AgSn alloys obtained by differential dilatometry (Mosig et al. [81]).

alone, made on a sample constrained at a constant volume, are sufficient to determine the vacancy concentrations. With this approach, Simmons [201] studied vacancy formation in quantum solid heliums. The author pointed out that the measurements revealed larger vacancy concentrations than can be deduced from the deviations of other properties, including the specific heat, from the postulated properties of a hypothetical defect-free crystal. 5.4. Equilibrium vacancy concentrations Many data have been obtained from the measurements of the thermal expansion (Table 6). Two main conclusions can be made: (1) The results derived by a linear extrapolation of the expansivity are several times larger than those from differential dilatometry. (2) The linear extrapolation of the expansivity leads to the vacancy concentrations close to or somewhat smaller than the values from the nonlinear increase in the 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, there exist temperature ranges where the thermal expansivity increases linearly with temperature. (2) Theoretical calculations of the anharmonicity predict a linear temperature dependence of the expansivity of a defect-free crystal at high temperatures. (3) The nonlinear increase in the thermal expansivity satisfies the expression describing the vacancy formation and provides reasonable values of the formation enthalpy. However, the validity of such a procedure could be confirmed only by observations of the relaxation in the thermal expansion caused by the vacancy equilibration. Differential dilatometry is commonly regarded as the most reliable method to determine equilibrium vacancy concentrations. It is even considered as the “absolute technique.” Regretfully, the data sometimes strongly contradict each other. For instance, the recent differential-dilatometry data on Cu and Ag [80,81] revealed vacancy concentrations three times larger than those reported by Simmons and Balluffi [53,55]. Differential dilatometry has not yet been applied to metals with high melting points. Therefore, equilibrium vacancy concentrations in Ni, Ti, Pt, Zr, Cr, Rh, Nb,

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Table 6 Vacancy concentrations in metals deduced from thermal-expansion data. L — linear extrapolation of the thermal expansivity, D — differential dilatometry, M — modulation dilatometry, Q — lattice parameter and volume of quenched samples Metal

Na Li Sn

Bi Cd

Pb

Zn

Mg

C .1 (10~4) 7.5 7.8 4.4 14 6 40.3 6.2 24 5.6 4.5 40 9 1.5 1.7 33 20 3 4.9 7.2

Method, Refer. D D D L L D D L D D Q L D D L L D D D

[186] [196] [202] [203] [204] [205] [206] [203] [207] [208] [209] [203] [44] [210] [203] [211] [212] [213] [214]

Metal Al

Ag

Au Cu

Ni Pt

Rh Mo W

C .1 (10~4)

Method, Refer.

20 3 11 9.4 17.4 24 1.7 5.2 14 7.2 13.5 2 7.6 110 26 80 70 70 190 230

L D D D L L D D L D L D D M Q M M M L M

[41] [44] [45] [52] [41] [203] [53] [81] [203] [54] [41] [55] [80] [167] [215] [87] [216] [170] [217] [90]

Fig. 24. Linear thermal expansivity of some high-melting-point metals: v — macroscopic thermal expansivity [73,87,89,91], ——— X-ray data (Edwards et al. [218], Waseda et al. [219]).

Mo, Ta and W are available only from the calorimetric data. Scarce measurements of the lattice parameter in several high-melting-point metals [218,219], together with the macroscopic-expansion data now available, rather confirm than disprove high vacancy concentrations in these metals (Fig. 24).

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5.5. 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 that disappear during annealing. Balluffi et al. [116] 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. The length was supposed to decrease during annealing and the lattice parameter to increase. Eventually, the macroscopic volume of the sample and the volume deduced 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 [220] measured the contraction in the length of a quenched Au sample during annealing at 20°C. With c"0.525, a good agreement was achieved with the Simmons-Balluffi data [54]. Harrison and Wilkes [221] carried out a dilatometric study of an Al sample after quenching. The sensitivity of the employed micrometer was better than 10~8 cm. To obtain an agreement with the Simmons-Balluffi data [52], the authors had to accept c"0.95. Using so called liquisol quenching (i.e., quenching from the liquid state), Laine [209] observed changes in the lattice parameters of Cd corresponding to a change in the microscopic volume of about 2]10~3. After annealing for 1 h at 200°C, the lattice parameters returned nearly to the values of a slow cooled sample. Taking c"0.5, the vacancy concentration at the melting point was determined as 4]10~3. This value is several times larger than the equilibrium differentialdilatometry data. On the other hand, Suryanarayana [222] has found no change in the lattice parameter in Al after liquisol quenching. The lattice parameter of the samples quenched from temperatures 700°C to 1100°C was nearly the same as that of annealed samples. The author concluded that this may suggest that the lattice parameter is not significantly affected by vacancies and hence calculations of the vacancy concentrations from such measurements may not be reliable. In quenched samples of V Ga , Waegemaekers et al. [223] observed very large changes in the 2 5 macroscopic density, up to 5% at 900°C, whereas the lattice parameter remained constant. Measurements of the lattice parameter and the electrical resistivity of quenched Pt foils were reported by Hertz and Peisl [224]. The accuracy of the measurements of Da/a was better than 10~5. The quenched-in resistivity, Do, was determined at liquid helium temperature, and the sensitivity was better than 10~11 ) cm. As expected, a linear relation was found between the relative change in the microscopic volume, Dl/l, and the Do values: (Dl/l)/Do"!720$90 )~1 cm~1.

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The authors pointed out that it is of great interest to study both quantities during thermal annealing and possibly 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 Pt was determined as » +0.67 ) [111], where X is the atomic volume. F Hence, the relation between the relative change in the microscopic volume and the extra resistivity should be (Dl/l)/Do"!0.33/o , 7 where o is the extra resistivity of a unit vacancy concentration. 7

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The right-hand side of this relation is available also from equilibrium measurements. In such measurements on Pt [84], o was deduced as 2.4]10~4 ) cm. This leads to the ratio 7 (Dl/l)/Do"!1400 )~1 cm~1.

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

6. Point defects and electrical resistivity of metals 6.1. Influence of point defects on resistivity Kauffman and Koehler [36,37] were the first to observe the extra electrical resistivity after quenching. This was the first reliable observation of point defects in metals. The formation of point defects results in an increase of the electrical resistivity. Like impurities, vacancies cause an additional scattering of conduction electrons. The vacancy contribution is proportional to the concentration: Do"o c "o A exp(!H /k ¹) . 7 7 7 F B

(37)

Here, o "Do/c denotes the resistivity induced by a unit vacancy concentration. Theoretical 7 7 calculations of this parameter coincide only within an order of magnitude (Table 7). As a first approach, the coefficient o was assumed to be independent of temperature (Matthiessen’s rule). 7 Generally, this assumption is invalid (for a review see [230]). Additional uncertainties therefore arise when results of equilibrium and quenching experiments are compared. To determine experimentally the influence of the vacancies on the resistivity, it is necessary to somehow determine their concentrations. Despite different vacancy concentrations found in various experiments, the o values often are in a satisfactory agreement. 7 When vacancies form clusters, their contribution to the electrical resistivity decreases. Vacancy clusters in quenched samples may contain several thousand of vacancies. Martin and Paetsch [231] evaluated the resistivity of clusters containing up to 102 vacancies. For such clusters, the decrease of the resistivity amounts to about 50% and it tends to be larger for clusters containing more vacancies. 6.2. Resistivity of metals at high temperatures The vacancy contributions to the electrical resistivity at high temperatures were first observed in alkali metals [38], Au and Cu [232]. The extra resistivities at the melting points are of the order of 10~6 ) cm. The main problem is to correctly separate the vacancy contributions. In some cases, such a procedure appeared to be quite reliable. In studies of Al, Simmons and Balluffi [233] used three different methods to extract the vacancy-related resistivity, namely: (1) The resistivity of a supposed defect-free crystal was approximated by a theoretical relation ¹"a#bX#cX2, where X"ln(o/¹). The coefficients of this relation were found from the

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Table 7 Theoretical values of the electrical resistivity induced by a unit vacancy concentration. 1 — Abele`s [225], 2 — Reale [226], 3 — Manninen et al. [227], 4 — Volkov [228], 5 — Avte et al. [229] o "Do/c (10~4 ) cm) 7 7 Metal

1

Cs Rb K Na Li Sn Cd Pb Zn Tl Mg Al Ca Ag Au Cu Be Ni Fe W

2.76 2.55 2.39 1.92 1.57

2

3.256 1.498 5.582 1.397 2.782 3.350

3

4

3.57 3.23 2.73 1.78 1.22

1.03 0.73 1.79

0.89 0.77 0.71 0.87 0.94 0.62 1.39

1.45 1.45 1.28

0.25

1.08

5

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 0.652

0.45 5.41 6.40 8.20

experimental data in the range 430—610 K, and the deviations from this dependence at higher temperatures were attributed to the equilibrium vacancies. (2) The resistivity of a defect-free crystal was taken as o"A#B¹#C¹2. (3) The vacancy contribution was separated with an extrapolation by eye of the temperature dependence of the resistivity. All the methods gave the same extra resistivity at the melting point, 0.34 l) cm. This agreement is owing to the relatively large vacancy contribution (Fig. 25). Generally, the situation is not so good, and the extrapolation from intermediate temperatures becomes critical. Some authors therefore consider this approach as an unsuitable one for studying the vacancy formation. For instance, an analysis of the uncertainties inherent to calculated and experimental values of the resistivity of K has been performed by Cook et al. [234]. These uncertainties allow one to choose the formation enthalpy in the range 0.2—0.6 eV. The situation is more favorable when the temperature derivative of the resistivity, do/d¹, is measured directly. The vacancy-induced increase in this derivative is D(do/d¹)"(o H A/k ¹2) exp(!H /k ¹) . 7 F B F B

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Fig. 25. Electrical resistivity of Al at high temperatures (Simmons and Balluffi [233]): v — experimental data, ——— extrapolation from intermediate temperatures. Due to relatively large vacancy contribution, various methods to extract it give very close results.

Direct measurements of do/d¹ are feasible by the modulation technique. It consists in periodically oscillating the temperature of the sample and observing the corresponding oscillations in its resistance. This technique was developed for studying anomalies in the resistivity of ferromagnets near the Curie points [235]. Since 25 years, the method was re-invented by Chaussy et al. [236]. Rigorously speaking, the temperature derivative of the resistance (not of the resistivity) is available from such measurements. The measurements on Al and Pt [237,238] have clearly shown a nonlinear increase in the temperature derivative of the resistivity (Fig. 26). This increase was considered as the vacancy contribution. The two metals are the only exceptions for which the results of equilibrium and careful quenching experiments are in a 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. 6.3. Quenched-in resistivity The main objective of quenching experiments is to retain in the crystal lattice all the vacancies created in equilibrium. The determination of the formation enthalpy requires measurements of the extra resistivity corresponding to various temperatures. However, serious obstacles were met in this approach, namely: (1) Many vacancies have time to annihilate owing to their high mobility and numerous internal sinks in the samples. (2) Many vacancies form clusters, and their contribution to the resistivity becomes therefore smaller. (3) Deformations during quench create new dislocations which serve as sinks for the vacancies. (4) The influence of the vacancies on the resistivity depends on temperature, so that it is difficult to compare results of equilibrium and quenching experiments. An important achievement was the superfluid-helium quenching technique developed by Rinderer and Schultz [60]. 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 layer of helium gas, of about 0.1 mm, appears around the wire. Owing to the high thermal conductivity of the superfluid helium, the heat dissipated in the sample spreads all over the liquid

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Fig. 26. Temperature coefficient of electrical resistance of Al and Pt measured by modulation technique [237,238]. The nonlinear increase was attributed to vacancy formation.

preventing it from boiling. After terminating 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 quite evident: (1) The quenching rate is sufficiently high. (2) Liquid helium provides extremely pure quench medium comparable with ultra-high vacuum conditions. (3) The temperature of the sample after quenching is below 2 K, and the resistivity is measured under very favorable conditions immediately after quenching. Schultz [61] employed this technique for quenching W wires. Mundy and Ockers [239] built a quenching apparatus in which the gaseous environment could be controlled to an oxygen partial pressure of less than 10~14 Pa. They employed the setup to examine the effect of oxygen on the quenched-in resistivity of thin Cu wires of high purity. These measurements could not provide reliable quenched-in resistivity data. The losses of vacancies during a quench can be reduced using high cooling rates. However, thermal stresses in the samples also increase in this case causing a formation of new dislocations. On the other hand, samples with low dislocation densities allow one to reduce the vacancy losses even under moderate cooling rates. Such experiments were undertaken on Pt and Pd by Khellaf et al. [240]. With o "4.6]10~4 ) cm, the vacancy concentration in Pt at the melting point was 7

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Table 8 Extra resistivities at the melting points and deduced formation enthalpies. E — equilibrium measurements, M — modulation measurements of the temperature derivative of resistance, Q — quenching experiments Metal

Cd Pb Mg Al

Ag Au

Cu

H F (eV)

Do .1 (n) cm)

Method, Refer.

0.38 0.72 0.45 0.81 0.79 0.77 0.77

500 190 14 280 60 340 80 220—530 320

E E Q E Q E Q Q M Q E E Q E Q E E Q Q E E Q E

0.8 0.69 1.06 1.11 0.97 1.02 0.97 0.95 0.92 0.70 1.27 1.25 1.3 1.3 1.0

110 40 110 530 120 140 230 90 12 40 100 12 120

[245] [246] [246] [247] [248] [233] [249] [250] [237] [248] [251] [252] [117] [251] [253] [252] [251] [254] [255] [252] [257] [256] [258]

Metal Ni

Pt

V Rh Mo

W

H F (eV)

Do .1 (n) cm)

Method, Refer.

1.6 1.6 1.4 1.6 1.5 1.51 1.7 1.3 1.3

100 10 2600 2400 1500 1300 2100 600 430 600 1400 50 10 5 20 100 50 50 50 100 200

Q Q M E Q Q M Q Q Q M Q Q Q Q Q Q Q Q Q Q

2.0 2.7 3.24 3.2 3.3 3.6 3.1 3.3 3.67 3.6

[259] [260] [167] [84] [261] [262] [238] [263] [240] [264] [170] [265] [266] [267] [61] [268] [269] [270] [265] [271] [272]

evaluated as 9.4]10~4. No reliable data have been obtained for Pd because of very small quenched-in resistivity. The thermal annealing of quenched-in vacancies was studied by many authors. Siegel [241] investigated the vacancy annealing in Au by measuring the changes in the quenched-in resistivity. The precipitate structure in the quenched samples was observed by a transmission electron microscope. Essential features of quenching and annealing phenomena have been reported, namely: (1) The resistivity annealing rate drops continually for increased purity of the samples. (2) A large rise in the density of the vacancy precipitates, with a corresponding decrease in the size, occurs with increased content of dissolved impurities. (3) The measured migration energy is independent of the purity of the samples. Kino and Koehler [242] have shown that dislocations in Au 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 many other quenching experiments.

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129

Fig. 27. Extra resistivity of metals. W: 1 — Schultz [61], 2 — Gripshover et al. [268], 3 — Kunz [269], 4 — Rasch et al. [271], 5 — Park et al. [272]. Pt: 1 — Bacchella et al. [273], 2 — Kraftmakher and Lanina [84], 3 — Rattke et al. [274], 4 — Heigl and Sizmann [262], 5 — modulation measurements [238], 6 — Khellaf et al. [240]. Al: 1 — Bradshaw and Pearson [275], 2 — Gertsriken and Slyusar [203], 3 — Simmons and Balluffi [233], 4 — Babic´ et al. [250], 5 — modulation measurements [237].

From quenching experiments with Au, Wang et al. [243] concluded the presence of tightly bound divacancies having the migration enthalpy close to 0.69 eV. Levy et al. [244] quenched high-purity Al samples from temperatures 350—820 K down to 4.2 K. The behavior of vacancies during the quench and further annealing has been studied by resistivity measurements and the electron microscopy. 6.4. Extra resistivity caused by vacancy formation The vacancy contributions to the electrical resistivity of metals were observed in both equilibrium and quenching experiments, though the obtained values are very different (Table 8). 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 Al and Pt (Fig. 27).

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For other metals, the difference in the values from the two techniques amounts to one order of magnitude.

7. Method of positron annihilation 7.1. Positron-annihilation techniques The positron annihilation is a very sensitive tool for detecting vacancy-type defects in metals (for reviews see Refs. [20,27—30,32,33]). When a positron enters the sample, its kinetic energy becomes equal to the thermal energy very rapidly, for a time of the order of 10~12 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 10~10 s. Two c-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. 28). 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 the positrons in the sample is much smaller than that of the 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 much smaller than that of electrons because the exclusion principle does not prevent it going to the lowest energy state of the system. The Doppler broadening, dE, and the angular distribution of the annihilation quanta, dh, hence relate only to electrons. Annihilations with higher-energy core electrons contribute more to the largest values of dE and dh than do annihilations with lower-energy valence or conduction electrons. The Doppler-broadening and angular-correlation data give an 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 dh distributions for the positron annihilations in vacancies become therefore narrower.

Fig. 28. Principles of positron-annihilation techniques. The vector diagram shows momentum conservation in positron—electron annihilations. Subscripts refer to longitudinal (L) and transverse (T) components.

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Berko and Erskine were the first to point out a probable relation between vacancies and the positron annihilation in metals. In 1967, they reported angular-correlation data on Al [63]. The angular 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. [64] presented measurements of positron lifetimes in some low-melting-point metals (In, Cd, Zn, Al) over a range from room temperature to the melting point or to 400°C. Marked effects, amounting to as much as a 30% increase in the lifetime, were considered as likely 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. In 1969, Bergersen and Stott [276] developed a model for trapping of positrons by vacancies in metals. The authors explained the observed temperature dependence of the positron lifetimes, including the saturation at high temperatures, and evaluated the enthalpies of the vacancy formation in metals studied by MacKenzie et al. [64]. A month later, a similar model was presented by Connors and West [277]. The positron annihilation is now considered as the most powerful technique for determining the enthalpies of the vacancy formation. For instance, it was stated by Schaefer [32] 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 annihilation parameters in a vacancy-free crystal weakly depend on temperature, so that the vacancy contribution can be separated without serious errors. In contrast to the Doppler broadening and the angular correlation, the lifetime spectroscopy supplies simultaneously the positron lifetime characterizing the type of trap and the trapping rate, which 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 from the delocalized to the localized states. The trapping rate is proportional to the concentration of the defects which capture positrons. The proportionality factor, i.e., the trapping rate per unit vacancy concentration, is called the specific trapping rate. The positron-annihilation technique is applicable to studies of the vacancy formation in equilibrium and to measurements on quenched samples [278]. This method was employed in observations of the vacancy equilibration at high temperatures [78,279,280]. Experimental methods for positron-annihilation studies of defects in metals are described in many papers (e.g., [28,281,282]). The theory of positron in solids is well developed. In a recent review [30], Puska and Nieminen stated that “...the theory underlying positron annihilation has developed from simple models describing the positron—solid interaction to ‘first-principles’ 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.” 7.1.1. Lifetime spectroscopy In its initial stage, the lifetime spectroscopy was based upon several simple assumptions: (1) The lifetime of positrons trapped in vacancy-type defects increases. (2) The specific trapping rate does

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not depend on temperature. (3) No detrapping of positrons occurs until annihilation. (4) The positron lifetimes in the regular lattice and in the vacancy may be obtained by fitting data from low and high temperatures. If n and n denote the concentrations of free and trapped positrons, the following equations are & 7 valid for their time rates of change: dn /dt"!n j !k c n , (39) & & & 7 7 & dn /dt"!n j #k c n . (40) 7 7 7 7 7 & Here j and j are the annihilation rates for a free and a trapped positron, respectively, c is the & 7 7 vacancy concentration, and k is the specific trapping rate. If some positrons annihilate in trapping 7 states and others while free, then any characteristic of the positron annihilation, F, will have the value of the weighted mean of this characteristic in both states, F and F : & 7 F"F P #F P . (41) & & 7 7 Here, P and P represent the probabilities of the annihilation in the free and trapped states. The & 7 fraction of positrons annihilating in the free state is given by P "j /(j #k c ) , & & & 7 7 whereas the fraction of positrons which annihilate when trapped in a vacancy is P "1!P "k c /(j #k c ) . 7 & 7 7 & 7 7 The mean positron lifetime obeys the relation:

(42)

(43)

q "q (1#q k c )/(1#q k c ) , (44) . & 7 7 7 & 7 7 where q "1/j and q "1/j are the annihilation lifetimes for a free and a trapped positron, & & 7 7 respectively. At low temperatures, when k c is small compared to j and j , the mean lifetime approaches q . 7 7 7 & & At high temperatures, when k c is large, the mean lifetime approaches q (Fig. 29). From the above 7 7 7 equation, the equilibrium vacancy concentration is c "(q !q )/q k (q !q ) . (45) 7 . & & 7 7 . Cotterill et al. [283] determined the positron lifetimes and the trapping probabilities in Al 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 pointed out that such measurements are probably inferior to equilibrium ones because of the inherent complexity of the quenching process. A detailed description of the lifetime spectroscopy and the data analysis was given by Hall et al. [284]. Small drops of 22NaCl in a neutral solution were dried onto two pieces of the sample. Then the pieces were clamped together and electron-beam welded around the edges. The decay of 22Na produces a positron and simultaneously a c-quantum of 1.28 MeV. This c-quantum signals the creation of the positron. The positron enters the sample, is thermalized and either drifts through the

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Fig. 29. Lifetimes of positrons in metals: free positrons, positrons captured by vacancies, and temperature dependence of mean lifetime.

crystal lattice or becomes trapped in a vacancy. It eventually annihilates with an electron, producing two c-quanta of 511 keV. The time delay between the 1.28 MeV and the 511 keV quanta is measured by a multichannel analyzer. The time resolution is usually better than 10~11 s. The data obtained provide a histogram representing the number of events as a function of the time delay. In this investigation, the vacancy formation was studied in Al, Au and in an Al—Zn 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 Al and Au, the authors concluded that the specific trapping rate increases with temperature. High-resolution positron-lifetime studies in Ag and Cu were carried out by Hehenkamp et al. [285,286]. Two lifetimes were seen in the lifetime spectra. The shorter lifetime, q , is given by 1 q "1/(j #k c ) . 1 & 7 7

(46)

When the vacancy concentration becomes significant, q decreases and a second lifetime, q "q 1 2 7 appears. The mean lifetime, q , was calculated as q "q I #q I , where I and I are the . . 1 1 2 2 1 2 corresponding intensities in the lifetime spectra. Two methods were employed to calculate the vacancy-formation enthalpies. The first method is based on measurements of the I /I ratio at various temperatures, whereas the second one employs 2 1 the I (j !j ) value. These two approaches gave markedly different values of the formation 2 1 2 enthalpies. The authors concluded that the most probable explanation of this result is trapping of the positrons during thermalization. 7.1.2. Doppler broadening The Doppler broadening of the c-line was used in many investigations. The annihilation radiation is detected by a lithium-drifted or intrinsic Ge crystal. The method possesses a high efficiency since the detector is placed close to the sample. A disadvantage of this technique is a low resolution. The samples under study are usually characterized by one of the so-called

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Fig. 30. Doppler broadening of the annihilation line: 1 — annealed samples, 2 — quenched samples. W-parameter shows the difference in the line shape at high and low temperatures.

“shape parameters.” The shape parameters S and ¼ are defined as S"R N(E)/R N(E) , M G

(47)

¼"[R N(E)#R N(E)]/R N(E) . L R G

(48)

Here N(E) is the number of counts per channel corresponding to the energy E, R N(E) is the G total number of counts within the annihilation line, and the intervals M, ¸ and R are chosen to include the maxima in the difference curve of the spectra at low and high temperatures (Fig. 30). Due to the vacancy formation, the S-parameter increases. The ¼-parameter is sensitive to the probability that positrons annihilate with high-energy core electrons and decreases when positrons are captured by vacancies. Maier et al. [71] employed this technique for studying the vacancy formation in V, Nb, Mo, Ta and W. 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. They were annealed at high temperatures in a vacuum of 10~6 Pa. Then 10—20 lCi of 22NaCl, vacuum-evaporated onto foils of the sample material, were placed into the tubes. The tubes were sealed by the electron-beam welding. Determinations of the 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 vapour pressure of V, Mo and W. The samples placed in a vacuum system were heated by an electron beam

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Fig. 31. Measurement of Doppler broadening in refractory metals (Maier et al. [71]).

(Fig. 31). 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 of 1.2 kV at 104 counts per second. The temperature dependence of the ¼-parameter was used to evaluate the formation enthalpy. In the case of V, the parameter D"S!¼ was employed because it exhibited less scatter of the data points than did the ¼-parameter. 7.1.3. Angular correlation of c-quanta In measurements of the angular correlation of annihilation c-quanta, the quanta are detected in coincidence by counters which are 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 simply 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. Measurements of the angular correlation of annihilation quanta was employed, e.g., by McKee et al. [287] to determine the vacancy-formation enthalpies in In, Cd, Pb, Zn and Al. 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 point, and then re-etched. For each sample, the data were accumulated during two days. The apparatus was set at a zero angle, and the coincidence counting rate was measured as a function of the temperature of the sample. The increase in the counting rates in the whole temperature range amounted to about 10%. The high-temperature saturation was clearly seen in Zn and Al. 7.2. 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

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Table 9 Vacancy formation enthalpies from positron-annihilation measurements (from review by Schaefer [32]). A — angular correlation, D — Doppler broadening, L — lifetime spectroscopy, M — mean lifetime Metal

H (eV) F

Method

Metal

H (eV) F

Method

In

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

L A D D L A A M M L A L M L M L M

Ni Co Pd Pt

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

D A D L D D D D L M D M D L M D

Sn Cd Pb Zn Tl Mg Al Ag Au Cu

V Cr Nb Mo

Ta W

temperature dependence. The annihilation parameters of quenched samples and changes of these during a thermal annealing are also informative. The positron-annihilation technique seemed very promising, and many investigations were carried out with it (Table 9). The method is regarded as the best one for determinations of enthalpies of the vacancy formation. However, serious drawbacks inherent to this technique became also evident, namely: (1) Vacancy concentrations are not available. (2) In some cases, a saturation in the annihilation parameters occurs far below the melting point. (3) The unknown temperature dependence of the specific trapping rate may markedly affect the results. (4) Detrapping of captured positrons may lead to ambiguous results. (5) The vacancy contribution has not been found in some metals. Seeger and Banhart [288] 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]10~10 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. In many cases, the formation enthalpies deduced from the positron-lifetime spectroscopy are markedly higher than those from the mean lifetime and the Doppler broadening. As was recently shown by Dryzek [289], the calculated specific trapping rate is extremely sensitive to the parameters of the potential used and may strongly depend on temperature. Trumpy and Petersen [290] have found the trapping rate to be a linear function of ¹~1@2. Puska and Manninen [291] calculated the trapping rate of positrons into small vacancy clusters and light substitutional impurities in metals. The trapping rates in materials may be very different. In order to extract

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a reliable information from the measurements, detailed data on the thermalization, diffusion, and capture of the positrons are necessary. Relatively low enthalpies of the vacancy formation were found by the positron-annihilation techniques in some intermetallic compounds. These low enthalpies lead to high vacancy concentrations. High concentrations of thermal vacancies in Fe Al [279,292,293] and Fe Si [280] were 3 3 deduced from the formation enthalpies. Clearly, any prediction of equilibrium vacancy concentration requires a knowledge of the formation entropy. In such cases, it would be useful to check the prediction by other methods of studying vacancy formation, including measurements of the specific heat.

8. Hyperfine interactions These techniques employ the hyperfine interaction 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 the hyperfine interactions include the perturbed angular correlation of c-quanta, the Mo¨ssbauer effect, and the nuclear magnetic resonance [294]. 8.1. Perturbed angular correlation of c-quanta In the 1970s, a new method for studying point defects and defect clusters appeared, the perturbed angular correlation of c-quanta. This technique allows 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 c-quanta. This emission senses an interaction of the nucleus in its intermediate state with extranuclear fields that occurs during the time between the two emissions. Due to the precession of the nuclear moment, the directional distribution of the emitted c-quanta becomes time dependent. In the time-differential perturbed angular correlation, the precessions of the nuclear spin of the excited nuclear state are monitored over its lifetime. The theory of the method was given by Steffen and Frauenfelder [295—297]. Details can be found in reviews devoted to this technique [298—302]. Usually, the experiments employ the isotope 111Cd. The 247-keV state in 111Cd is fed from the ground state of 111In (Fig. 32). 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 atoms cause perturbation factors that differ distinctly from those of a defect-free lattice. In cubic nonmagnetic metals, 111Cd, with a near-neighbor defect shows a discrete electric quadrupole interaction. The interaction is zero for undisturbed lattice sites, so that the near-neighbor states 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 for determinations of equilibrium defect concentrations at high temperatures.

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Fig. 32. Energy levels and parameters of decay of 111In employed in perturbed-angular-correlation measurements.

Hinman et al. [303] were probably the first to attribute changes in the angular correlation to vacancies created in Au by irradiation or cold work. More definite results were reported by Behar and Steffen [304,305]. The authors studied polycrystalline Ag foils bombarded with a-particles of various energies. The tracer nuclei were created in the samples by the Ag(a, 2n)In reaction. After the 111In ions come to rest in the lattice, they decay by electron capture to 111Cd. The time-differential perturbed-angular-correlation measurements revealed a quadrupole interaction with electric-field gradients. After annealing at 600°C for 12 h, the quadrupole interaction completely disappeared. It was therefore concluded that the most probable cause of the perturbation are vacancies and/or Ag interstitials. A similar phenomenon was observed in Cd [306]. Hohenemser et al. [307] observed the vacancy trapping and detrapping in Ni during isochronal annealing in the range 20—400°C. Mu¨ller et al. [308] implanted 111In ions into Cu, Al and Pt at 24 K and at room temperatures. In the case of Pt, the measurements have shown the trapping of a single defect interpreted as a monovacancy. Wichert et al. [309] studied the annealing of defects introduced in Cu by quenching and electron and proton irradiation. The stage-III recovery in irradiated samples was attributed to mobile vacancy-type defects. Irradiated Mo samples were studied by Weidinger et al. [310]. Three different defect configurations were identified, namely: a monovacancy, a divacancy, and a tetrahedral configuration of four vacancies with 111In atom in its center. Similar results were obtained in irradiated W [311]. Studies of defects in Ag, Al, Au, Cu, Ni, Pd and Pt were summarized and interpreted by Pleiter and Hohenemser [312]. The defects introduced by irradiation, quenching and ion implantation were divided 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 M1 1 1N plane, and tetrahedral vacancy clusters seen only in Ni. However, some of the observed states remained with undetermined structure. Collins et al. [313] studied Au samples, heavily deformed at 77 K, after annealing at temperatures up to 500 K. Lattice defects in Rh and Ir produced either by the implantation of 111In ions or by an irradiation with protons were examined by Hoffmann et al. [314]. The defect in Rh was

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interpreted as a monovacancy. In Ir, two defects were found, a monovacancy and a relaxed divacancy. The vacancy migration and clustering in Ni were observed after proton irradiation and deformation [315]. Collins et al. [302] studied vacancy defects in Au, Cu, Ni and Pt created by plastic deformation. The authors identified the structures of various multivacancy complexes by three methods: (1) Quadrupole interaction parameters were compared with calculations of electric-field gradients for 20 structures containing 1—4 vacancies. (2) Decoration of vacancy complexes with hydrogen atoms was determined and compared with calculations. (3) Annihilation of vacancies by mobile interstitials served to test the consistency of the identification. Mu¨ller and Hahn [316] examined the formation of constitutional and thermal defects in the intermetallic compound PdIn. Thermal defects were created by quenching the samples from high temperatures. An increasing concentration of randomly distributed monovacancies in the Pd sublattice was observed with increasing the quenching temperature. It was concluded that vacancies in the In sublattice are invisible because the electric-field gradients caused by them on a next-nearest-neighbor site of the In probe are too low. Unusually high concentrations of quenched-in vacancies, of about 15% at the melting point, were found in intermetallic compounds NiAl, CoAl and TiAl [317—319]. Very low vacancy concentration was observed in Ni Al. Diffusion processes in Cd and in dilute Cd—In alloys were 3 examined by Hanada [320]. An irreversible loss of the defect-related signal was observed in Cd below 500 K. This phenomenon was explained by migration of the probe atoms to unknown trap sites. Recently, the perturbed-angular-correlation technique was employed in studies of equilibrium point defects in NiO [321] and CoO [322], at temperatures up to nearly 1500 K. An enhancement of the unperturbed fraction in the angular-correlation spectra was observed at high temperatures. At the same time, the spectra of quenched samples showed well pronounced perturbed fractions. The authors explained this contradiction by a decrease in the vacancytrapping probability at high temperatures. The vacancy trapping becomes thus not effective which leads to rapid fluctuations of the electric-field gradients near the probe nuclei. This conclusion 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 al. [323] developed for this purpose a computer-controlled spectrometer (Fig. 33). The spectrometer records the time-dependent spectrum of events in which the first c-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 are tuned to the first c-quantum and start the time-to-amplitude converter (TAC), while the other two are tuned to the second c-quantum and stop the converter. The analog-to-digital converter (A/D) provides data for the computer. After an experimental run, the time spectrum of the perturbed 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 to be studied. 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 leads to significant enhancement of the

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Fig. 33. Simplified diagram of experimental setup for perturbed-angular-correlation studies (Jaeger et al. [323]).

local defect concentration which is necessary to obtain a measurable defect-induced signal and reveal the defects presented in the crystal lattice. (3) From the parameters measured, defects of different structure can be discriminated. (4) Various defects tend to anneal out and trap at probe atoms at different temperatures, which 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 temperatures. (5) Taking properly into account the defect-probe binding, absolute defect concentrations could be evaluated. However, the vacancy-trapping probability at high temperatures may become too low which will lead to loss of the defect-induced signal. This approach thus may become unsuccessful. For the present review, the last point is most important. Until today, no data on equilibrium point defects in pure metals were obtained by the perturbed-angular-correlation method. Measurements at high temperatures should meet no serious obstacle but an unavoidable difficulty is the temperature dependence of the vacancy-trapping probability. Such investigations, even far below the melting points, would be very useful in high-melting-point metals. 8.2. Mo¨ ssbauer spectroscopy and nuclear magnetic resonance The Mo( ssbauer spectroscopy is a useful technique for detecting interactions between point defects and neighboring atoms. Czjzek and Berger [324] used the method in studies of Fe—Al alloys. The Mo¨ssbauer c-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. Qualitative estimates of the effect of vacancies and interstitials were made. Using the Mo¨ssbauer effect in 197Au, Mansel et al. [325] measured changes in the Debye—Waller factor of Au in Pt after a low-temperature neutron irradiation. A reduction of up to 10% in the Debye—Waller factor was

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attributed to irradiation-produced defects. The influence of neutron irradiation was also studied on Al doped with 57Co [326]. A new component appeared in the spectrum after the irradiation was attributed to interstitials trapped in the immediate proximity of the Co atoms. Reintsema et al. [327] used 133Xe implanted in Mo, Ta and W as a source of the Mo¨ssbauer spectra. In Mo and W, four different Xe sites were identified: the substitutional Xe atoms and Xe atoms associated with one, two and three vacancies. For Ta, the results were less clear due to a considerable overlap of lines in the spectrum. Tanaka et al. [328] studied the vacancy-Sb complexes in Au. Their behaviour during heat treatments was investigated by 119Sb emission spectra. The spectra were measured immediately after quenching from 1073 K and after subsequent isochronal annealing at various temperatures. Wahl et al. [329] studied vacancy trapping at impurities in quenched and irradiated W. Minier et al. [330,331] employed the nuclear-magnetic-resonance technique to measure electricfield gradients caused by point defects in electron-irradiated Al and Cu. 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-III annealing. Recently, Konzelmann et al. [332] detected vacancy-type defects in quenched Cu samples by the nuclear quadrupole double resonance (NQDOR). The samples were rapidly quenched from the melt. The electric-field gradients caused by point defects cause a splitting of the energy levels of nuclei. The splitting is proportional to the field gradients and is different at crystallographically nonequivalent sites. Each atomic defect is thus characterized by a set of discrete level splittings. They were observed by NQDOR immediately after quenches and during subsequent annealing. By comparing the results obtained with those on irradiated samples, the conclusions were made as follows: (1) The irradiation-induced lines at frequencies above 100 kHz are due to self-interstitials. (2) Vacancy-type defects give rise to a broad distribution of NQDOR frequencies below 100 kHz. (3) 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. Influence of point defects on other physical properties Point defects affect many other physical properties of metals. A change in the thermoelectric power (the Seebeck coefficient) after quenching was observed long ago [333,334]. The vacancy contribution is of the order of 10~4 V per unit vacancy concentration and depends on temperature. Huebener measured the thermopower of quenched samples of Au [335] and Pt [336]. The vacancy-induced thermopower in Al was investigated by Rybka and Bourassa [337]. Bourassa et al. [338] studied the effect of pressure on the thermopower at high temperatures. The data for Al were interpreted as an influence of three types of thermally activated defects: the monovacancy, the divacancy, and the impurity-vacancy pair. The thermopower of Cu at high temperatures was measured [339] using the modulation technique. The method consists in creating temperature oscillations around a mean temperature and measuring them simultaneously by the thermocouple under study and by a reference one. The thermopower is thus measured directly. This technique was independently reported by three groups [340—342]. Vacancies influence the thermal conductivity and thermal diffusivity of metals. Temperature gradients in a sample cause gradients in the vacancy concentrations, and the diffusion of the

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Fig. 34. Thermal diffusivity of refractory metals at high temperatures (Kraev and Stel’makh [347,348]). The nonlinear decrease is caused by increase in specific heat.

vacancies contributes to the thermal conductivity [343,344]. Zinov’ev et al. observed this phenomenon in Pt [345] and Pd [346]. Nonlinear decrease in the thermal diffusivity of refractory metals (Fig. 34) was found by Kraev and Stel’makh [347,348]. It is due to the nonlinear increase in the specific heat. Similar behavior was observed in La [349] and Ru [350]. Mechanical properties of metals also depend on point defects. In computer simulations, Stabell and Townsend [351] have shown the effect of vacancies and interstitials upon the bulk modulus of W. The vacancies result in a decrease of the bulk modulus whereas interstitials cause an opposite change. The elastic moduli of Al over the temperature range 300—930 K were reported by Gerlich and Fisher [352]. 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 nonlinear decrease may be due to equilibrium vacancies. A decrease in the spontaneous magnetization of a ferromagnet was observed in equilibrium and quenching experiments [353,354]. Wuttig and Birnbaum [355] studied the resistivity and the magnetic after-effect in quenched samples of Ni. 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 were found, monovacancies mobile at 400 K and divacancies which become mobile at 320 K. An interesting phenomenon was observed by Celasco et al. [356] when measuring the current noise in thin Al films. The noise caused by fluctuations of the electrical resistance becomes visible when a 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 435°C and 475°C were estimated as 4.7 and 2.8 ms, respectively. The authors also evaluated the migration enthalpy and the vacancy contribution to the resistivity. Using the equilibrium vacancy concentrations from the differential dilatometry [52], the contribution of a unit vacancy concentration to the resistivity has been evaluated as o "1.8]10~4 ) cm. 7 Petz and Clarke [357] studied the electrical 1/f-noise in electron-irradiated Cu films. The sample was maintained at 90 K and irradiated by the beam of an electron microscope. The observed difference in the recovery of the noise and of the induced resistivity was explained by the authors as

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Table 10 Vacancy formation enthalpies evaluated from thermopower, thermal conductivity, and thermal diffusivity of metals Property

Metal

H (eV) F

Reference

Thermopower

Ag Pt

1.0 1.45

[333] [333]

Thermal conductivity

Pt Pd

1.45 1.7

[345] [346]

Thermal diffusivity

La Pd Ru

0.98 1.5 1.75

[349] [346] [350]

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. [358] investigated irradiation-induced defects in thin Al foils by measurements of the 1/f noise and 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. Formation enthalpies deduced from the measurements of various physical properties of metals do not contradict data obtained with basic techniques employed in studies of point defects (Table 10).

10. Microscopic observations of quenched-in defects The frozen-in defects 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. 10.1. Electron microscopy During annealing, vacancies quenched-in by a rapid cooling of the samples form precipitates observable by an electron microscope. Siegel [241] investigated pure Au samples, in the form of narrow foils, quenched from 700°C or 900°C. The vacancy precipitate structure in the samples annealed at temperatures 40°C and 60°C was observed by a transmission electron microscope (TEM). It was found 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 after quench from 900°C was estimated as (1.1$0.2)]10~4. This figure corresponds to the vacancy concentration at the melting point of about 3]10~4, which is lower than the

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differential-dilatometry value. The resistivity of a unit vacancy concentration was deduced as o "(1.8$0.4)]10~4 ) cm. In TEM investigations of an Au sample quenched from 1300 K, 7 Fraikor and Hirth [220] estimated the vacancy concentration at this temperature as 1.6]10~4. This value is also lower than the differential-dilatometry value. Rasch et al. [271] carried out an important investigation on W samples. Thin wires, 40—60 lm in diameter, were used to measure the resistivity, whereas foils of the same thickness served for TEM observations. The samples were prepared from single crystals of high purity. The residual resistivity of the crystals indicated the concentration of dissolved impurities of the order of 1 ppm. However, carbon impurities may form precipitates and thus could not be monitored by the residual resistivity. After preparation, the wire samples had a “bamboo structure” with an average grain size of 0.1 mm. The superfluid-helium quenching technique [60] was employed in the experiment. Some quenches were done in a vacuum chamber. Another cooling rate was used to estimate the quenching losses and to prove that helium did not enter the samples during the quench. Visible vacancy clusters (voids) were observed in samples quenched from the melting point, 3695 K, and subsequently annealed in the range 800—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 W of commercial purity. After quenches from the melting point, the concentration of vacancies stored in the voids was (2$1)]10~4. The authors accepted the lower limit of this value as the equilibrium vacancy concentration. The vacancy parameters were determined as H "3.67$0.2 eV, H "1.78$0.1 eV, S "2.3k , c "10~4, and F M F B .1 o "6.3]10~4 ) cm. A serious difficulty appeared in obtaining an agreement between these 7 parameters and self-diffusion data. Considering the results, the authors supposed that divacancies or interstitials were probably created in equilibrium but could not be quenched because of their high mobility. Kojima et al. [359] have shown that stacking fault tetrahedra formed in quenched samples are due to lattice vacancies, whereas interstitials form faulted dislocation loops. 10.2. Field-ion microscopy The field-ion microscopy (FIM) involves the atom by atom dissection of samples by pulse field evaporation and recording images produced at each stage of the process on a cine´ film. Until today, this technique is applicable only to quenched samples of high-melting-point metals. The observable defect concentrations may thus be smaller than in equilibrium. This approach is capable of a determination of the defect contribution to the electrical resistivity. 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 [360] reported on such an observation on Pt wires quenched from 1700°C. The quenched-in resistivity was measured at 4.2 K. The samples were prepared using electrical etching. Among about 9]105 lattice sites, 233 vacancies have been found (Fig. 35). The concentration of the frozen-in vacancies, of about 3]10~4 at the melting point, appeared much smaller than that from the specific-heat data. The author pointed out that the quenched-in concentration must be lower than the equilibrium one due to the vacancy losses which occur in any real quench. For the two samples, o was estimated as 4.8]10~4 and 7

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Fig. 35. Vacancy concentrations in two Pt samples quenched from 1700°C and observed by FIM on various crystallographic planes (Seidman [360]): j — samples were given a brief anneal in a molten salt solution at 325—369°C; h — samples were polished totally at room temperature.

5.75]10~4 ) cm. The author considered these values as upper limits of o . Hence, they do not 7 contradict the equilibrium value, 2.4]10~4 ) cm [84]. Park et al. [272] reported a similar study of quenched W samples. High-purity samples were quenched in ultra-high vacuum, and the quenched-in defects were observed as they were uncovered by the pulse-field evaporation. The extra resistivity was measured on the same quenched wires. A cooling rate of 3]104 K s~1 was achieved near the beginning of the quench where defect losses to pre-existing sinks are of major concern. No appreciable stresses 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 two samples investigated, the quenched-in resistivity extrapolated to the melting point appeared very different, of about 0.01 and 0.1 l) cm. 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 resulting film was then scanned to identify and count the quenched-in defects. Control observations on well-annealed samples served to determine the background not associated with the quenched-in defects. About 2]106 atomic sites were observed in the quenched samples and 106 in control samples. The authors have estimated the vacancy parameters as follows: H "3.6 eV, S "3.2k , F F B c "3]10~4, o "7]10~4 ) cm. Some important conclusions have been summarized by the .1 7 authors, namely: (1) Tungsten has a strong tendency to retain interstitial impurities, which interact strongly with point defects. (2) Due to the relatively low migration enthalpy, the vacancies migrate rather rapidly and tend to annihilate during quenching. (3) The vacancies form their own sinks (voids) during quenching by heterogeneous precipitation with impurities. (4) Vacancy losses during quenching have complicated to a greater or lesser degree nearly all the investigations of quenched W that have been carried out.

11. Relaxation phenomena caused by equilibration of point defects Observations of the relaxation phenomena caused by the vacancy equilibration, i.e., the approach to the equilibrium after a rapid change of temperature, have been proposed long ago

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[49,361,362]. The method was considered as the most reliable one to separate vacancy contributions to physical properties of metals. The characteristic time of setting up the equilibrium vacancy concentration, q, depends not only on the parameters of migration and temperature but also on the density of internal sources (sinks) for the vacancies. The relaxation time is proportional to the squared mean distance ¸ between the sources (sinks) and inversely proportional to the coefficient of self-diffusion of the vacancies, D : q"A¸2/D , where A is a numerical coefficient depending on 7 7 the geometry of the sources (sinks). The relaxation times in various samples thus may be very different. Quenching experiments have shown that this difference may amount even 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 the specific heat and in the thermal expansivity or the electrical resistivity. Pure and well-prepared samples having low dislocation densities and long relaxation times would enable observations of the phenomena at higher temperatures where the vacancy contributions become larger. The accuracy and reliability of the results thus could be improved. 11.1. Electrical resistivity In studies of the vacancy equilibration through the electrical resistivity, the samples were rapidly heated up to a selected temperature, kept at this temperature for a certain time and then quenched. The quenched-in resistivity was determined as a function of the exposure at the high temperature. This technique is very sensitive allowing 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 )/(c !c )+1!exp(!t/q) . (49) 0 7 0 Here, c is the initial vacancy concentration at the start temperature, c is the equilibrium 0 7 vacancy concentration at the final temperature, and q is the characteristic time of the equilibration. When c is negligible, then 0 c+c [1!exp(!t/q)] . (50) 7 Seidman and Balluffi [62] studied the vacancy equilibration in Au using two methods of measurements. First, a thin single-crystal slab was rapidly heated by a jet of compressed hot air into 875—920°C range, held for a short period of time and then quenched. The quenched-in vacancies were detected by precipitating them as vacancy tetrahedra observable by a transmission electron microscope. Second, polycrystalline Au foils were electrically heated to a high temperature and then quenched. The extra resistivity was measured versus the exposure of the sample at the high temperature. The results indicated that free dislocations were the predominant sources of thermally generated vacancies. The half-times of the vacancy equilibration were estimated as 80 ms at 653°C and 10 ms at 878°C. Heigl and Sizmann [262] carried out a similar investigation on Pt. The wire sample was raised to a quench temperature by a discharge of a capacitor at the heating rate of about 106 K s~1, kept at this temperature for an adjustable time and then quenched. Relatively long equilibration times, of the order of 10~2 s, were obtained at temperatures 800—950°C, far below the melting point. At these

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temperatures, a high efficiency of quenching should be expected. The estimated extra resistivity at the melting point was 1.3 l) cm. This figure is much larger than that in many investigations employing quenching from higher temperatures and quite comparable with the equilibrium value, 2.4 l) cm [84]. In studies on Al [363], the temperature of the sample was raised rapidly from an initial temperature, 160°C, to a selected higher temperature, from 310°C to 400°C. 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. 11.2. Specific heat The first attempts to observe the relaxation in specific heat were undertaken using the modulation technique at frequencies up to 103 Hz. No relaxation phenomenon has been found in Pt [364]. Modulation measurements on Au wire by Skelskey and Van den Sype [70] have shown a frequency dependence of the quantity c/R@, the ratio of the heat capacity to the temperature derivative of the resistance (Fig. 36). This ratio appears when temperature oscillations in the sample are measured through oscillations of its resistance. The increase in this quantity at higher frequencies is explainable if the relative vacancy contribution to the temperature derivative of the resistance is larger than that to the specific heat. The measurements were carried out only at a single temperature, 1164 K, so that the result could not be confirmed by a temperature dependence of the phenomenon. To search for the relaxation phenomena in specific heat, a method of the measurements at frequencies of the order of 105 Hz has been developed [122]. A wire sample is heated by a high-frequency current slightly modulated by a low-frequency voltage (Fig. 37). The high- and low-frequency temperature oscillations hence occur in the sample simultaneously. They are detected by a photomultiplier. The low-frequency component of its output signal proceeds to a lock-in amplifier. The high-frequency component selected by a resonant circuit is measured using

Fig. 36. Frequency dependence of the ratio of specific heat, c, to temperature derivative of resistance, R@, for Au at 1164 K (Skelskey and Van den Sype [70]). This ratio should increase at high frequencies if the relative defect contribution to R@ is larger than that to c.

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Fig. 37. Setup to observe relaxation in high-temperature specific heat of metals [122]. 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.

a 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 temperatures where the nonlinear increase in the specific heat is negligible and no relaxation is expected. At these temperatures, the recorded signal is adjusted to be zero. At a given mean temperature, the difference between the specific heats corresponding to the two frequencies is thus directly measured. The measurements were carried out on commercial W wires 8 lm thick and on vacuum incandescent lamps with W filaments 10—20 lm in diameter. The high frequency of the temperature oscillations was 3]105 Hz [77]. The character of the temperature dependence of the effect was always within the expectation (Fig. 38). This observation gained no recognition. For instance, Trost et al. [197] 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 Pt, the high frequency was 5]104 Hz [123]. The samples were cut from a foil 10 lm thick. Due to the lower melting temperature, the power heating

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Fig. 38. Ratio of specific heats measured at a low and a high frequency of the temperature oscillations in W [77] and Pt [123]. The high frequency was 3]105 and 5]104 Hz, respectively.

the sample and the amplitude of its oscillations becomes much smaller, which results in decrease of the temperature oscillations and, consequently, of the applicable modulation frequency. The observed relaxation effect appeared in an agreement with the nonlinear increase in the specific heat. As in the case of W, the scatter of the experimental points increases at temperatures where X"uq is close to unity since only in this range the effect depends strongly on X. The enthalpies of the vacancy formation in metals are nearly proportional to the melting temperatures, ¹ . This means that the ratio of the vacancy contribution to the specific heat at M a given temperature, DC, to its value at the melting point, DC , can be considered as a common M function of the ratio t"¹ /¹ for all metals: M DC/DC "t2 exp[K(1!t)] , (51) M where K"H /k ¹ . F B M This relation makes it possible to compare the phenomena observed in both metals [124,125]. The effect in Pt, even with a lower modulation frequency, was observed much closer to the melting point than in W. This is probably due to a lower dislocation density in the Pt samples. However, there exists no direct evidence that the phenomenon originates from the vacancy equilibration. A question therefore arises how to check this conjecture. A simple experimental approach can be proposed for this purpose. 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 increases drastically 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 to the initial temperature, the relaxation phenomenon may disappear. It should appear again after a proper annealing of the sample and recovery of its structure. Such an experiment would show clearly the nature of the relaxation. The experimental setup now can be much simpler. The sample is heated by a DC current with two AC components added, of a low and a high frequency (Fig. 39). Temperature oscillations of both frequencies are thus created in the sample simultaneously. The sine voltage of the reference

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Fig. 39. New setup for observing relaxation in specific heat. Measurements are performed during a period including quench and subsequent anneal of the sample. Fig. 40. Expected relaxation in specific heat as a function of X"uq, for DC/C"0.1. 1 — change in specific heat, 2 — shift in the phase of temperature oscillations.

frequency created by a lock-in amplifier is used for the high-frequency modulation. The radiation from the sample falls onto a silicon p-i-n photodiode type FND-100 (time constant 1 ns). 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 is fed to a selective amplifier and then to an AC/DC converter. At various mean temperatures of the sample, the irradiation of the photodiode is adjusted to maintain a constant magnitude of the low-frequency AC voltage from the photodiode. The output voltages from both channels are stored by a data-acquisition system. The measured quantities are monitored during a period of time 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 DC;C or the relaxation time remains sufficiently long (X2<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 relaxation (X2;1). Generally, this approach should indicate a lower limit of the relaxation. Only under very favourable conditions, when X2<1 before the quench and X2@1 after it, the change in the amplitude of the high-frequency temperature oscillations should reveal the true value of the relaxation (Fig. 40). Otherwise, this change corresponds to only a part of the relaxation phenomenon. Measurements under various modulation frequencies should make clear the temperature dependence of the phenomenon. Along with providing data on the equilibration, this approach would verify the vacancy origin of the relaxation. The phase shift in the temperature oscillations depends nonmonotonically on X. However, the phase measurements are very important because they would confirm the relaxation in the specific heat.

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Now the measurements are performed as follows. At every mean temperature, the sample is annealed during 1 h before the measurements. Then the heating current is interrupted and the sample quenched. After about 1 s, the sample is brought to the initial temperature. During all the time, including the quench, the output signals are stored by the data-acquisition system. When they alter after the quench, the sample is kept at this temperature until the magnitude of the signal is returned 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 specific heat of the sample. For this purpose, the specific heat was measured by the equivalent-impedance technique. With this variant of the modulation calorimetry, the ratio of the heat capacity, mc, to the temperature derivative of the resistance of the sample, R@, is measured. For W, this derivative at high temperatures changes by less than 1% per 100 K, and this change is mainly linear. The nonlinear increase in the mc/R@ ratio thus shows the nonlinear increase in the specific heat, and the observed relaxation can be compared with the nonlinear increase in this ratio. 11.3. Positron annihilation Using the positron-annihilation technique, Schaefer and Schmid [78] studied the vacancy equilibration in Au. The authors explained the advantages of this approach as follows: (1) The positron annihilation is sensitive to only vacancy-type defects. (2) It is applicable at high temperatures. (3) 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 diffusioncontrolled 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 the sample rapidly, for 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 c-line were measured (Fig. 41). After cooling the sample, the cycle repeated. The data were accumulated during 106 cycles. The temperature of the sample was raised 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 at these temperatures the vacancy concentrations are negligible. At 800 and 900 K, the relaxation times were determined as 11.7 and 3.6 ms, respectively. With a transmission electron microscope, the dislocation density in the sample after the measurements was estimated as 8]108 cm~2. The authors pointed out that the statistical precision of the measurements can be improved by optimizing the efficiency of c-detection and using a positron source of higher activity. The available temperature range could be thus extended to higher and lower temperatures. Using the positron-lifetime technique, Wu¨rschum et al. [279] studied vacancy formation and migration in intermetallic compounds Fe Al and Fe Al . Very slow equilibration was found 63 37 61 39 after rapid cooling the sample from 770 K to temperatures in the range 623—673 K. The vacancy equilibration was observed also in Fe Si [280]. 3 11.4. Equilibration times from relaxation measurements The extra resistivity of quenched samples and changes in the positron-annihilation parameters are certainly caused by vacancies. At the same time, the relation between the vacancy formation

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Fig. 41. Observation of vacancy equilibration in Au by positron-annihilation technique (after Schaefer and Schmid [78]). Fig. 42. Temperature dependence of equilibration times from various measurements: 1 — Au, resistivity (Seidman and Balluffi [62]); 2 — Al, current noise (Celasco et al. [356]); 3 — Au, positron annihilation (Schaefer and Schmid [78]); 4 — Pt, specific heat [123]; 5 — W, specific heat [77]. The straight lines correspond to H "7k ¹ and constant densities of M B M sources (sinks) for vacancies.

and the nonlinear increase in specific heat gained no recognition. It seems therefore very useful to compare results of all the relaxation experiments, even for various metals (Fig. 42). The relaxation time depends on the density of sources (sinks) for vacancies. The difference between the relaxation times in Au obtained by measurements of the resistivity [62] and by the positron annihilation [78], which amounts to 50 times, is thus quite understandable. Various densities of internal defects are probably responsible for such a difference in Pt samples. The migration enthalpies were considered to be proportional to the melting temperatures (H /k ¹ "7). The straight lines in the graph M B M correspond to q"B exp(H /k ¹), where q is the relaxation time, H is the migration enthalpy, and M B M B is a factor different for various samples. Assuming a temperature-independent density of sources (sinks) for vacancies, the temperature dependence of the relaxation time is available from a single measured value. In W, the relaxation times were deduced as 0.5 ls at 2600 K and 0.2 ls at 2700 K. The short relaxation times in W are consistent with the well-known fact that dislocation densities in refractory metals are much higher than those in metals such as Au or Pt. The comparison of the

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data is thus favourable for the conclusion that all the relaxation phenomena are of a common nature. The directions of further investigations of the phenomenon are evident: (1) A use of pure and well-prepared samples, which would enable the observations over a wider temperature range. (2) Measurements on metals in which low dislocation densities are obtainable. (3) Observations of the relaxation during a quench and consequent anneal of the samples. The very short relaxation times found in W may pose a question whether they are consistent with the vacancy origin of the relaxation. It is worthwhile to recall that the time necessary to create interstitials is much shorter. For example, it was recently shown by Fujiwara et al. [365] that Frenkel pairs in KCl and RbCl are formed in a stage which terminates within a few picoseconds after excitation.

12. Discussion 12.1. Proposals for determination of vacancy contributions to the enthalpy and specific heat of metals A straightforward approach was recently proposed [366] to reveal vacancy contributions to the 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 heating, in the time interval of interest, remains nearly constant. When the vacancies appear after the heating, then the shape of the initial cooling curve should depend on the vacancy contribution to the enthalpy and the relaxation time. Both quantities strongly depend on temperature. The absorbed heat can be determined 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 the decrease in the relaxation time (Fig. 43).

Fig. 43. Temperature traces expected after rapid heating of a sample to a premelting temperature [366]. At the highest temperature, vacancies have arisen during heating.

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Fig. 44. Calculation of the temperature drop, D¹, caused by vacancy formation after a rapid heating of tungsten sample to various premelting temperatures.

To evaluate the expected temperature drop, D¹, one has to take into account the equilibrium vacancy concentration at the final temperature of the sample after the equilibration. The vacancyrelated enthalpy equals to DH"H c "H exp(!G /k ¹). For W, the formation parameters F 7 F F B were taken from the calorimetric data. They predict the vacancy-related enthalpy of about 10 kJ mol~1 at the melting point. The heat-balance requirement is CD¹"DH, where C is the specific heat not including the vacancy contribution (C+40 J mol~1 K~1). From the graph (Fig. 44), the temperature drop due to the vacancy formation can be evaluated for any temperature achieved after a rapid heating. For W, the maximum temperature drop, after heating the sample to the melting point, amounts to about 160 K. This figure reduces to 105 K if the sample was heated to 3500 K and to 50 K after heating to 3200 K. For Mo, the situation is very similar. Hence, the phenomenon should be clearly seen if the equilibrium vacancy concentrations are of the order of 10~2 but become nonobservable if the concentrations were less than 10~3. From the measurements, the temperature dependence of the vacancy-related enthalpy could be evaluated. The rapid-heating experiment to be made is similar to those reported earlier [155,156]. However, owing to the proposed approach, a setup for the measurements can be much simpler. Now there is no need to measure the heating current and the voltage drop across the sample. All what one needs is to rapidly heat up the sample, within 10~8 or 10~7 s, to a premelting temperature and to observe the initial part of the cooling curve. An important point is to completely terminate the heating at

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the selected premelting temperature. Any uncontrollable heating will cause difficulties in the determination of the vacancy-related enthalpy. The proposed experiment allows one to reduce uncertainties usually inherent to rapid-heating experiments. The expected phenomenon should be clearly seen and amenable to a quantitative treatment. Still more important, the nature of the phenomenon would be quite evident. This technique may therefore offer the best 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 a dilatation of the sample immediately after a rapid heating. It is easy to show that the temperature drop in the sample due to the vacancy formation should be accompanied by an increase in its volume. Such unusual behaviour would be the best confirmation of the vacancy origin of the phenomenon. The same approach is probably suitable for determinations of the vacancy contributions to the electrical resistivity of metals. To be more informative, measurements of the extra resistivity should be accompanied by determinations of the vacancy-related enthalpy. However, the relative vacancy contribution to the electrical resistivity of refractory metals is much smaller than that to the enthalpy. There exists an additional possibility to check the origin of the nonlinear increase in hightemperature specific heat of metals. Under high pressures, the equilibrium vacancy concentrations decrease because of increase in the Gibbs energy of the vacancy formation. The vacancy contribution to the specific heat thus decreases under high pressures. This decrease should be related to the vacancy formation through the formation volume. Experimental data on the pressure dependence of the vacancy-induced resistivity (e.g., [109—111]) show that the pressures to be employed are quite moderate, of the order of 109 Pa. Such a pressure is probably insufficient to change markedly other contributions to the specific heat. The dynamic technique or the 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 directly on pressure. The best approach would be an employment of a blackbody model in the sample. 12.2. Equilibrium vacancy concentrations 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. The data obtained by various techniques (Table 11) clearly show that the situation is far from the optimistic expectations. Such expectations appeared every time when a new experimental approach was proposed, e.g., the differential dilatometry or the positron-annihilation technique. All quenching experiments may result in an underestimation of the vacancy concentrations, so that the data presented in the last three columns should be considered as only lower limits of the equilibrium concentrations. 12.2.1. Low-melting-point metals In low-melting-point metals, the situation is much more favourable than in refractory metals. Among FCC metals, Al and Au are the most extensively studied ones. In Al, the differential dilatometry shows the vacancy concentration at the melting point of about 10~3. The calorimetric data range from 1.1]10~3 to 2.2]10~3. The lower limit of the vacancy concentration determined from the changes in the enthalpy during the vacancy equilibration was estimated as 6]10~4.

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Table 11 Point-defect concentrations at the melting points from various techniques. L — linear extrapolation of thermal expansivity, D — differential dilatometry, C — specific heat, Q — volume and lattice parameter of quenched samples, E — quenched-in enthalpy, M — microscopic observations of quenched samples C Metal

L

Na Zn Sn Cd Pb Al Ag Au Cu Pt Mo W

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

D

33 14

24

80

6—9 3; 5 40.3 5 1.7 3—11 1.7; 5.2 7 2; 7.6

.1

C

(10~4) 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 5 1; 3

The extra resistivity at the melting point can be accepted as 0.3 l) cm, which does not contradict the vacancy concentrations given above. The specific vacancy resistivity, o , is thus in the range 7 1.5]10~4—3]10~4 ) cm. Au is the most favourable metal for studies of the vacancy formation. All experimental techniques now available were employed in studies of this metal. The determined equilibrium vacancy concentration at the melting point ranges from 7]10~4 (differential dilatometry) to 4]10~3 (specific heat). The extra resistivity at the melting point is of about 0.1 l) cm from quenching experiments, whereas the values from equilibrium measurements are 0.14 and 0.53 l) cm. A similar situation is seen in Cu but the quenched-in resistivity is of the order of 0.01 l) cm. In Pb, the difference between the data from the differential dilatometry and the specific heat amounts to one order of magnitude. A similar difference in the extra resistivity was found in equilibrium and quenching experiments. In Na, a BCC metal, there exists a significant discrepancy in the vacancy concentrations from the differential dilatometry and the calorimetric measurements. The situation in Li may be more favourable because the observed nonlinear increase in the specific heat [126] is much smaller than that in Na and K. The vacancy formation in HCP metals, Cd and Zn, was reviewed by Seeger [367]. The equilibrium concentrations at the melting points were determined by the differential dilatometry as 3]10~4 in Zn and 5]10~4 in Cd. In both metals, measurements of the thermal expansivity with a linear extrapolation from intermediate temperatures lead to concentrations several times larger. High extra resistivity, 0.5 l) cm at the melting point, was observed in Cd in equilibrium measurements [245]. The equilibrium vacancy concentration in Zn from the calorimetric measurements [69] is 2.3]10~3 at the melting point. The situation is thus unclear.

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12.2.2. High-melting-point metals For Pt, as well as for other high-melting-point metals, there exist no differential-dilatometry data. However, all other data now available are in a reasonable agreement to each other. The extra resistivity at the melting point determined in quenches from relatively low temperatures, 1.5 and 1.3 l) cm [261,262], are consistent with the equilibrium value, 2.4 l) cm [84]. The equilibrium vacancy concentration in Pt is probably much larger than those determined by the differential dilatometry in low-melting-point metals. The situation in refractory BCC metals is most difficult to understand. In 1962, the enthalpy of the vacancy formation in W (3.15 eV) and the equilibrium vacancy concentration (3.4]10~2 at the melting point) have been evaluated from the calorimetric measurements [56]. The measurements employed the method of equivalent impedance [133,368], a modification of the modulation calorimetry. These results have not been accepted as realistic ones, as well as the strong nonlinear increase in the specific heat of Mo and Ta reported earlier by Rasor and McClelland [50]. Modulation measurements of the specific heat were also carried out on Ta, Nb and Mo [57—59]. For all the metals, the vacancy-formation parameters have been evaluated. The strong nonlinear increase in the specific heat of refractory metals has been commonly recognized only after measurements by Cezairliyan and coworkers [65—68]. This recognition relates only to the behaviour of the specific heat whereas the origin of this phenomenon remains under debate. In 1964, Schultz [61] reported on the first successful quenching experiment on W. The extra resistivity was determined as 0.02 l) cm at the melting point. This figure became larger in further quenching experiments, and now the largest quenched-in resistivity in W amounts to 0.2 l) cm. Despite very different values of the extra resistivity, the deduced formation enthalpies show quite moderate scatter, in the range 3.1—3.67 eV. Quenching experiments appeared successful also on Mo but the obtained extra resistivity is several times smaller than in W. In 1972, the equilibrium vacancy concentration in W was evaluated from its thermal expansivity [90]. A linear extrapolation from intermediate temperatures was used to separate the vacancy contribution. Taking the formation volume as » "0.5 ), the vacancy concentration appeared F about 1.5 times smaller than that from the calorimetric data. In 1979, Maier et al. [71] determined the enthalpies of the vacancy formation in refractory metals by the positron-annihilation technique. The formation entropies were postulated as S "2k . With this value, the equilibrium vacancy concentrations at the melting points were F B calculated as 10~4 (V, Nb), 3.5]10~4 (Ta), 0.4]10~4 (Mo), and 0.25]10~4 (W). Later, Trost et al. [197] pointed out that the detrapping from monovacancies in these metals is so pronounced that the reported formation enthalpies may pertain mainly to divacancies. In 1985, the relaxation phenomenon in the specific heat caused by the vacancy equilibration was observed in W [77]. The relaxation appeared in a good agreement with the nonlinear increase in the specific heat. The phenomenon was observed also in Pt [123]. The results of the rapid-heating determinations of the enthalpy of W [156,180] can be considered as a confirmation of the vacancy origin of the nonlinear increase in the specific heat. At the same time, no manifestation of point defects in W was obtained in measurements of its electrical resistivity at high temperatures. The resistivity at the premelting temperatures is about 10~4 ) cm whereas the errors of the measurements are of about 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 [67] approximated

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Fig. 45. 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 [67], Righini et al. [86]), v — values after introducing corrections for thermal expansion.

the temperature dependence of the resistivity, in the range 2000—3600 K, as a linear function. This fit makes no hint about a vacancy contribution. However, this contribution becomes clear after introducing corrections for the thermal expansion of the sample, which are necessary to calculate true values of o and do/d¹: o(¹)"o*(¹) l(¹)/l , 298

(52)

do/d¹"(do*/d¹) l(¹)/l #o*(¹) a . 298

(53)

Here o*(¹) denotes the resistivity based on the room-temperature shape of the sample, the ratio l(¹)/l corresponds to its linear thermal expansion, and a"(1/l ) dl/d¹ is the linear thermal 298 298 expansivity. The thermal expansivity of W measured directly by the modulation technique [90] was employed in the calculations. These data were extrapolated up to 3600 K as follows: a"3.5]10~6#1.4]10~9¹#(2.74]106/¹2) exp(!36540/¹) .

(54)

The thermal expansion in the range 298—2000 K was taken from the recommended data [91]. From the do/d¹ values, the vacancy contribution is clearly seen (Fig. 45). The extra resistivity of W at the melting point was estimated as 0.5 l) cm [369]. This value 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. [86] were used. Their do*/d¹ data show a positive deviation from a straight line. This nonlinear deviation increases after introducing corrections for the thermal expansion according to Righini et al. [89]. The estimated extra resistivity of Nb at the melting point appeared of about 0.7 l) cm. The present situation looks as follows: (1) The low extra resistivity of quenched Mo samples may result from the well-known drawbacks inherent to all quenching experiments. The failure of the quenches on Ta and Nb probably has the same reason. Further efforts are thus necessary to improve the experimental technique. The positron annihilation may help to determine what a fraction of the equilibrium vacancies survive in

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the sample after quenching. It would be also useful to reveal how the vacancy-induced resistivity depends on temperature. (2) The errors in measurements of the 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 W and Nb, the defect contributions to the resistivity became evident after introducing corrections for the thermal expansion, which are necessary to calculate true values. The situation in other refractory metals is probably similar. However, the above correction is insufficient to resolve the existing disagreement. With q values of the order of 10~3 ) cm, that are 7 commonly accepted for refractory metals, the defect concentrations remain much smaller than those from the nonlinear increase in the specific heat or thermal expansivity. 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 simple and straightforward method can be tried to determine the vacancy contributions. Under very rapid heating, vacancies have no time to appear, and the measured resistance should correspond to a vacancy-free crystal. By 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. The electrical resistivity should be measured at a selected premelting temperature rather than at the melting point. (3) The vacancy contributions to specific heat become visible at temperatures above about two thirds of the melting temperature. The formation parameters deduced from the 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 the vacancy formation and migration. At the same time, the vacancy formation parameters are consistent with the enthalpies of the self-diffusion at lower temperatures. To understand this result, one may assume that the vacancies predominate 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 the equilibrium vacancy concentrations based on the calorimetric measurements, only a few data are available for high-melting-point metals. The quenched-in vacancy concentrations were determined by FIM only in Pt and W. In Pt, the estimated vacancy concentration at the melting point was c "3]10~4 [360]. The microscopic observations were accompanied by .1 measurements of the quenched-in resistivity. The extra resistivity at the melting point, Do , was .1 estimated as 0.2 l) cm, i.e., several times smaller than those in many other quenching experiments. The highest values of Do , 1.5 and 1.3 l) cm, have been obtained by Jackson [261] and Heigl and .1 Sizmann [262], respectively. With these values, the vacancy concentration in Pt at the melting point should be much larger, of about 2]10~3. From the equilibrium measurements [84], values of c "10~2 and Do "2.4 l) cm have been obtained. In W, the vacancy concentration at the .1 .1 melting point has been estimated as 10~4 through the observations of quenched samples by an electron microscope [271] and as 3]10~4 by a field-ion microscope [272]. The q values obtained 7 in these observations, 6.3]10~4 and 7]10~4 ) cm, appeared in a good agreement. The situation in refractory metals remains thus mysterious. Ways to resolve the problem could be formulated as follows: (1) Measurements of the lattice parameter of refractory metals at high temperatures are very desirable. Such measurements are known for a long time. The data now available rather confirm than disprove high vacancy concentrations in refractory metals.

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(2) Further observations of the relaxation in specific heat caused by the vacancy equilibration unquestionably deserve effort. Regretfully, this promising approach did not attract a proper attention. (3) Rapid-heating measurements proposed for a straightforward determination of the vacancyrelated enthalpy of metals (and, probably, of the contributions to the thermal expansion and resistivity) would be the best solution of the question.

12.3. Comparison of methods for studying vacancy formation The advantages of calorimetric determinations of equilibrium vacancy concentrations are clearly seen from a comparison of various techniques now available. The vacancy formation strongly affects the specific heat. An equilibrium vacancy concentration of 10~2 causes an increase in the specific heat by about 10%. The same concentration leads to the Dl/l!Da/a value of about 0.3%. Simultaneous measurements of the macroscopic dilatation and the lattice parameter allow one to reduce errors caused by uncertainties in the temperature of the sample. However, such measurements cannot guarantee a 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, which leads to an underestimation of equilibrium vacancy concentrations. To exclude errors in the temperature measurements and to compare various data, one probably can use independently measured Dl/l and Da/a values extrapolated to the melting point. In comparison with the positron annihilation, calorimetric measurements are much simpler and more straightforward. Furthermore, positron-annihilation data do not provide equilibrium vacancy concentrations. In addition, it turned out that even determinations of the formation enthalpies by the positron annihilation sometimes cause doubts. The reason for this may be the positron trapping during the thermalization, the detrapping of captured positrons, and the temperature dependence of the specific trapping rate. The only drawback of the calorimetric measurements is the unknown specific heat of a defectfree crystal. A separation of the vacancy contribution to specific heat seems therefore to be impossible (see, e.g., Refs. [370,371]). Hayes [372] claimed that “the distinction between pointdefect 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 presented in Refs. [373—377]. Fortunately, the separation of the vacancy contributions may be based not only on theoretical speculations but also 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 can be considered more optimistically. It is worthwhile to recall that specific-heat data yielded high concentrations of equilibrium point defects in some ionic crystals and molecular solids. Kanzaki [378] found a strong nonlinear increase in the specific heat of AgBr and attributed it to the point-defect formation even before the vacancy formation was observed in metals. Beaumont et al. [379] determined high vacancy concentrations in Ar and Kr, 1.3]10~2 at the triple points. These results did not cause as negative regards as similar data obtained in metals.

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12.4. Enthalpies and entropies of the vacancy formation Calorimetric measurements cannot answer what a type of point defects prevails at high temperatures. The increase in the thermal expansivity confirms the vacancy (or divacancy) formation. Quenching experiments, including microscopic observations, and the positron annihilation experiments lead to the same conclusion. Equilibration times from the relaxation measurements are compatible with the vacancy (or divacancy) nature of the defects. Generally, the formation enthalpies obtained by various techniques are in reasonable agreement with each other and with theoretical calculations. The only exceptions are some data on refractory metals. As the first approximation, the formation enthalpies are of about one half of the enthalpies of self-diffusion and one third of the enthalpies of vapourization. Disagreements in the equilibrium vacancy concentrations from various experimental techniques amount to one order of magnitude. The largest vacancy concentrations were obtained in refractory metals from nonlinear increase in their specific heat. 12.4.1. Comparison with critical vacancy concentrations New theoretical calculations of entropies of vacancy formation in metals have appeared in the last decade. For example, Wautelet [380] postulated that vacancies perturb the delocalized phonon spectrum via a relation, which seems quite reasonable: u@"u(1!ac ) . 7

(55)

Here u@ and u are the perturbed and unperturbed vibration frequencies, and a is a constant. With this conjecture, one obtains c "exp[3a/(1!ac )] exp(!H /k ¹) . 7 7 F B

(56)

From this approach, a critical temperature appears above which the crystal becomes unstable. This temperature can be related to the melting point. The critical vacancy concentration, c*, satisfies the requirement d¹/dc "0, which leads to 7 a2c*2!(3a2#2a)c*#1"0 .

(57)

For a given a, there exist definite values of c* and H /¹ . Hence, a and c* values can be deduced F M from experimental H /¹ quantities [380]. It is easy to see that the vacancy concentrations based F M on calorimetric data do not exceed the calculated critical values, c* (Fig. 46). However, the critical concentrations depend also on the defect-defect, phonon-temperature and local defect-phonon couplings [381]. 12.4.2. Thermodynamic bounds for formation entropies Another estimation of the formation entropies was proposed by Varotsos [79]. This approach is based on the concept of vacancy-formation parameters under constant volume and constant pressure. It is easy to obtain a thermodynamic relation S "S*#» bB . F F F

(58)

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Fig. 46. Line of critical vacancy concentrations ( ——— ) according to Wautelet [380] and vacancy concentrations at melting points from nonlinear increase in specific heat (v).

Here S "!(­G /­¹) is the vacancy-formation entropy under constant pressure, F F P S*"!(­G /­¹) is the formation entropy under constant volume, » "(­G /­p) is the formaF F V F F T tion volume, b is the volume thermal expansivity, and B is the isothermal bulk modulus. As has been shown [79], it is very likely that DS D'DS*D and S*40 . (59) F F F In most cases, S and » are positive, so that S 4» bB. Also, S '1» bB. The formation F F F F F 2 F entropies are thus restricted as follows: 1» bB(S 4» bB . (60) 2 F F F This inequality can be used to check vacancy-formation entropies obtained by various techniques. Let us apply it to the formation entropies based on the nonlinear increase in specific heat [369]. This increase was observed above about 0.8¹ (¹ is the melting point), so that the deduced M M parameters, H* and S*, can be related to temperatures of about 0.9¹ . Now we can calculate upper F F M limits for the formation entropies at temperatures ¹"0.9¹ (Table 12). Generally, bulk moduli at M high temperatures are not available, and the values at ¹"0.9¹ were taken as 0.7B(300 K). To M determine » values, a simple approximation » "0.6) was employed, where ) is the atomic F F volume at room temperatures. The upper limits for the vacancy-formation entropies were thus calculated as 0.42]b(0.9¹ )])(300 K)]B(300 K). The melting points and atomic volumes at M room temperatures were taken from a handbook [382], and the bulk moduli at room temperatures from Refs. [383,384]. Taking into account possible errors in the quantities involved, uncertainties in the calculated limits can be estimated as 30—50%. The lower limits for the formation entropies are two times smaller than the upper ones. This approach also leads to high entropies of the vacancy formation, in agreement with those from specific-heat data. It should be remembered that experimentally obtained formation enthalpies and entropies are interrelated and uncertainties in the entropies are in the range 0.5—1k . The check is also useful for metals in which vacancy B concentrations are of the order of 10~3. In these cases, the lower calculated limits of the formation entropies may become more important.

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Table 12 Upper limits for the vacancy formation entropies in metals at ¹"0.9¹ calculated according to Varotsos [79] M Metal

b (0.9¹ ) M (10~5 K~1)

B (300 K) (1011 N m~2)

X (300 K) (10~29 m3)

S/k (0.9¹ ) B M Upper limit

Sn Pb Al Ag Cu Ni Pt Ir Nb Mo Ta W

7.6 10.4 10.5 8.2 7.5 8.0 5.0 4.4 3.6 3.45 4.1 3.2

0.55 0.42 0.79 1.11 1.51 1.84 2.83 3.55 1.70 2.64 1.93 3.10

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

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

12.4.3. Temperature dependence of the formation parameters The temperature derivatives of the enthalpy and entropy of the vacancy formation must satisfy the thermodynamic relation (­H /­¹) "¹(­S /­¹) . F P F P

(61)

Two approaches are thus possible: (1) The formation enthalpy and entropy do not depend on temperature. (2) 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 that the relaxation of the atoms around a vacancy increases. The formation entropy must therefore increase with temperature, as well as the formation enthalpy. Gilder and Lazarus [385] and deVries [386] used this concept to explain curvatures in the Arrhenius plots of self-diffusion. Experimental data on the vacancy concentrations are not sufficiently accurate to reveal the temperature dependence of the Gibbs energies of the vacancy formation. As a simple approach, one can suppose a linear temperature dependence of the formation entropy: S "a¹. This relation seems to be quite acceptable for temperatures far above the Debye F temperature. Then H "H #a¹2/2 , F 0

G "H !a¹2/2 , F 0

(62)

where H is the formation enthalpy at the absolute zero temperature. 0 Earlier, the enthalpy and entropy of the vacancy formation were considered to be constant over the whole temperature range where the vacancy contributions to the specific heat were measured, above about 0.8¹ . The derived parameters, H* and S*, should be related to temperatures of about M F F 0.9¹ . A simple relation thus appears: M a"S*/0.9¹ . F M

(63)

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Table 13 Parameters of the vacancy formation calculated from the nonlinear increase in specific heat assuming a linear temperature dependence of the formation entropies Metal

¹ M (K)

H* F (eV)

S*/k F B

H 0 (eV)

A (eV)

H (¹ ) F M (eV)

Cs Rb K Na In Sn Pb 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

The enthalpies and Gibbs energies of the vacancy formation can be written in a form more convenient for practical use: H "H #A(¹/¹ )2 , F 0 M

G "H !A(¹/¹ )2 , F 0 M

(64)

where A"a¹2 /2. M It will be more convenient to consider parameters H /k ¹ and A/k ¹ which can be derived 0 B M B M from calorimetric data (Table 13). Considering the parameters as functions of the melting temperature, one can try to check how the difference appears between low-melting-point and highmelting-point metals. The first parameter is almost the same for all metals, regardless of the crystal structure and the melting temperature, and equals to 7$1 (Fig. 47). In contrast, the A/k ¹ ratio B M can be taken as linearly depending on the melting temperature: A/k ¹ "1.5#5.5]10~4¹ . B M M Earlier, the formation enthalpies were believed to be proportional to the melting temperatures [387—391]. Now we see that this is correct only for H whereas the ratio (H #A)/k ¹ increases 0 0 B M with the melting temperature. Equilibrium vacancy concentrations at the melting points are given by exp[!(H !A)/k ¹ ]. The quantity (H !A)/k ¹ decreases from 5.3 for ¹ "300 K to 3.5 0 B M 0 B M M for ¹ "3700 K. These two values correspond to the vacancy concentrations 5]10~3 and M

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Fig. 47. Parameters of vacancy formation, H /k ¹ (j) and A/k ¹ (v), based on calorimetric data and a linear 0 B M B M temperature dependence of formation entropies.

3]10~2, respectively. The above rule may be modified by individual properties of metals but generally the equilibrium vacancy concentrations at the melting point increase with the melting temperature. The temperature dependence of the Gibbs energy of the vacancy formation in Cu has been calculated by Foiles [94]. A very weak temperature dependence was obtained using quasiharmonic and local harmonic approximations. On the contrary, the Monte Carlo simulations have shown a strong temperature dependence of the formation enthalpies, as well as the moleculardynamics simulations in Na [95]. The temperature dependence of the Gibbs energy of the vacancy formation in Cu evaluated from the calorimetric data qualitatively agrees with the Monte Carlo simulations. Hence, the formation enthalpies should be presented as functions of temperature or, at least, one has to indicate the corresponding temperature. 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. A modified relation for the extra specific heat takes into account this temperature dependence: DC"NA[(H2/k ¹2)#dH /d¹] exp(!H /k ¹) . F B F F B

(65)

From this expression, the vacancy concentrations become smaller. However, the second term in the brackets is of about one order of magnitude smaller than the first one, so that only minor corrections are necessary.

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12.5. Anharmonic contributions to the specific heat and thermal expansivity The anharmonicity is generally considered as a probable reason for the nonlinear increase in high-temperature specific heat and thermal expansivity of metals. However, almost all theoretical calculations of the anharmonicity predict these contributions to be mainly linear, so that they cannot explain large nonlinear effects. For instance, Maradudin et al. [392] calculated the anharmonic contributions to the thermodynamic properties of solids. The constant-volume specific heat, C , was found to follow the equation 7 C "3Nk #A¹!B/¹2 . 7 B

(66)

The coefficient A may be positive or negative, as the authors have shown by numerical calculations for Pb. Considering the lattice dynamics of an anharmonic crystal, Cowley [393] concluded that the anharmonicity leads to a weak linear temperature dependence of the specific heat and thermal expansivity. Similar conclusions were made in many other theoretical studies (e.g., [394—399]). A linear extrapolation of the data from intermediate temperatures takes into account all linear contributions to the specific heat, including the electronic term. An experimental determination of the constant-volume specific heat appeared feasible by measurements of the temperature fluctuations in a sample under equilibrium conditions. The measurements have been carried out on thin W wires [400,401,124]. At temperatures 2200 and 2400 K, the lattice constant-volume specific heat appeared somewhat smaller than the classical limit 3R"24.9 J mol~1 K~1. This result supports the conclusion made in many theoretical calculations. MacDonald and Shukla [402] evaluated thermodynamic properties of the high-melting-point BCC metals V, Nb, Mo, Ta and W. The calculations failed to reproduce the rapid upward trend in the specific heat and thermal expansion above about 1800 K (Fig. 48). The authors pointed out that the vacancies are most likely to be responsible for the high-temperature behavior in these metals. However, Ferna´ndez Guillermet and Grimvall [403] interpreted the strong nonlinear

Fig. 48. Theoretical estimate of specific heat of W (MacDonald and Shukla [402]). 1 — lattice constant-volume specific heat, 2 — lattice plus electronic contribution, 3 — total constant-pressure specific heat, including vacancy contribution from positron-annihilation data [71], 4 — experimental data by Cezairliyan and McClure [67]. Accepting low vacancy concentrations, the theory failed to explain the experimental data.

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increase in the specific heat of Mo and W as an anharmonic effect. The thermodynamic properties of Cu and Al were computed by Zoli and Bortolani [404]. 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, there exist predictions of a strong nonlinear increase in specific heat caused by the anharmonicity and related instability of the crystal lattice. Plakida [405] considered the stability conditions for an anharmonic crystal lattice. In the case of fairly high temperatures or a fairly high energy of the zero vibrations, the lattice loses stability. Close to the critical temperature, the specific heat and thermal expansivity rapidly increase without limit. However, the observed temperature dependences are far away from the predictions of this theory, as well as of the theory developed by Ida [406]. Many authors believe that the vacancy contributions cannot be separated from the anharmonic terms. For instance, Grimvall [407] has shown that in a narrow temperature range the dependence X"A¹#B¹2, being specific to the anharmonicity, is similar to the relation X"C¹#D exp(!H /k ¹) when H /k ¹"3.4. However, the vacancy contributions are deterF B F B mined at temperatures where this ratio ranges from 10 to 15. To explain the strong nonlinear increase in the specific heat of some transition metals, White [408] attributed this phenomenon to electronic effects. He stated that vacancies make no measurable contribution to the specific heat at temperatures below 0.9¹ and the anharmonic contribuM tion linearly varies with temperature. The first statement is based on low vacancy concentrations found by the differential dilatometry in low-melting-point metals. The second statement, ruling out the 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. 12.6. Thermal defects in alloys and intermetallic compounds Equilibrium vacancy concentrations found by the differential dilatometry in alloys are much larger than those in pure metals. For example, the vacancy concentration at the melting point in Ag-8.6 at.%Sn alloy appeared five times larger than in pure Ag [81]. It would be very useful to study the vacancy formation in alloys with other techniques now available, including measurements of the specific heat. Such measurements on low-melting-point metals could be carried out by the adiabatic calorimetry. High concentrations of thermal vacancies were observed in many intermetallic compounds. Wasilewski et al. [409] studied the 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 as an indication of quenched-in thermal vacancies, in addition to defects due to the nonstoichiometry. The fraction of thermal vacancies, after quenching from 850°C, was of the order of 1%. Berner et al. [410] 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 [411] studied the vacancy formation in CoGa using the dilatometric technique. The relaxation in the length of the sample after changing its temperature was observed. Very long relaxation times, in the range 102—104 s, were determined at temperatures 870—1050 K. Van Ommen [412] observed the vacancy formation in NiGa by the differential dilatometry. Very low

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Fig. 49. Difference between macroscopic (v) and X-ray (L) densities revealed high concentration of equilibrium vacancies in V Ga (Waegemaekers et al. [223]). 2 5

formation enthalpy, 0.2 eV, was found. The equilibrium vacancy concentration at 1135 K thus amounted to 4.9%. Differential scanning calorimetry revealed high concentrations of quenched-in vacancies in intermetallic compounds NiAl [413], NiSb [414] and CoGa [415]. Waegemaekers et al. [223] found high vacancy content in quenched samples of V Ga . The 2 5 macroscopic and X-ray densities were measured after quenching the samples from temperatures 800—900°C. The macroscopic density decreases significantly with increase of the quench temperature. The X-ray density does not depend on the quench temperature, though the theory of the differential dilatometry predicts such a dependence. The difference between the two densities amounts to about 2% at 800°C and 5% at 900°C (Fig. 49). The authors concluded that thermal vacancies in this compound are created in both sublattices. Measurements of the specific heat of this compound are desirable, as well as of other materials with high concentrations of equilibrium vacancies. Low formation enthalpies and hence high equilibrium vacancy concentrations were determined by the positron-annihilation techniques in Fe Si [280]. Schaefer et al. [292,293] determined the 3 formation enthalpy in Fe Al as 1.18 eV and the vacancy concentration at the melting point as 3 6.6]10~2. In contrast, the vacancy concentration in Ni Al was estimated as 6]10~4. It should be 3 remembered, however, that the vacancy concentrations from the positron-annihilation techniques are based on certain assumptions about the positron trapping rate. Very low vacancy concentrations were observed in TiAl by Brossmann et al. [416], of the order of 10~4 at the melting point. At the same time, high vacancy concentrations were found in TiAl by the perturbed-angularcorrelation technique, as well as in NiAl and CoAl [313—315]. Kim [417,418] proposed a theory capable of describing satisfactorily the observed 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 10%.” 12.7. Self-diffusion in metals Generally, different mechanisms of self-diffusion in metals are possible. Many data show that the most probable mechanisms involve point defects (e.g., [5,419]). With the vacancy mechanism, the

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Fig. 50. Curvatures in Arrhenius plots of self-diffusion in metals according to fits by Neumann and To¨lle [421,422].

coefficient of self-diffusion is given by D"(a2fvg) exp(S /k #S /k ) exp[!(H /k ¹#H /k ¹)]"D exp(!Q/k ¹) . (67) F B M B F B M B 0 B Here a is the lattice parameter, f is the correlation factor of monovacancies, l is the attempt frequency for jumps of atoms into adjacent vacancies, g is a geometrical factor, H and H are the F M enthalpies of vacancy formation and migration, S and S are the corresponding entropies, Q is the F M enthalpy of self-diffusion, and D is the corresponding pre-exponential factor. 0 Experimental data on self-diffusion allow 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. [420] measured self-diffusion in W over the temperature range 1700—3400 K, encompassing a range of nine orders of magnitude in the coefficient of diffusion. An analysis of the existing data was given [421,422] by Neumann and To¨lle (Fig. 50). There exist two explanations of the observed curvature. The two-defect model assumes two competitive mechanisms of self-diffusion. With this approach, the coefficient of self-diffusion obeys the equation D"D exp(!Q /k ¹)#D exp(!Q /k ¹) . (68) 01 1 B 02 2 B Here the activation enthalpies for the two mechanisms, Q and Q , are independent of temper1 2 ature, as well as the corresponding pre-exponential factors, D and D . The self-diffusion in FCC 01 02 metals was considered by Mundy [423]. If divacancies are present in the crystal lattice at high

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Table 14 Parameters of two-component fits of self-diffusion calculated by Neumann and To¨lle [421, 422] Metal

D 01 (cm2 s~1)

Q 1 (eV)

D 02 (cm2 s~1)

Q 2 (eV)

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

temperatures, their contribution to self-diffusion is likely since divacancies are more mobile than monovacancies. The one-defect model assumes a strong temperature dependence of the activation enthalpy and entropy. At present, both models describe experimental data with nearly the same accuracy, so that one cannot favour 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 14). The ratio of them corresponds to difference in the activation entropies from 3k to 9k . The largest B B difference is seen in refractory metals. High values of D agree with the high formation entropies. 0 The quantity a2flg is usually of the order of 10~2 cm2 s~1, so that the factor exp(S /k #S /k ) at F B M B high temperatures should range from 102 to 105. Mundy et al. [424] measured the migration enthalpy in W at high temperatures. Its temperature dependence was accepted as a linear one. 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 to 2k . The B authors concluded that the observed curvature in the Arrhenius plot for self-diffusion in W could be explained if the formation enthalpy had a similar temperature dependence. Sabochik [425] calculated the free enthalpy of the vacancy migration in W using atomistic simulation. Below 1000 K the migration enthalpy is constant, whereas at higher temperatures it depends on temperature, increasing by 0.7 eV in the range 1500—2500 K. An important question arises in treatment of the self-diffusion in refractory metals. For W, the activation enthalpy at high temperatures differs from the sum of the enthalpies of the vacancy formation and migration now accepted. It was therefore supposed that divacancies or interstitials are responsible for the self-diffusion in tungsten at high temperatures. However, this assumption also meets difficulties, and further investigations are necessary to elucidate the situation.

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12.8. Point defects and melting Frenkel considered the melting as a result of softening of the crystal lattice by point defects. Many authors studied the vacancy mechanism of melting. Aksenov [426] considered stability of an anharmonic crystal with vacancies. The instability of a lattice with vacancies is governed by the balance of the energy of thermal vibrations of atoms and their binding energy in the self-consistent phonon field, and also 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 governing the interaction between atoms. The instability occurs when the vacancy concentration reaches 5—8%. Following this work, Moleko and Glyde [427] considered the stability of rare-gas solids. High vacancy concentrations were predicted at the instability points. Chudinov and Protasov [428] carried out computer simulations of melting in Cu by the molecular-dynamics technique. The temperature of the crystal and the number of various point defects were calculated as a function of the total energy per atom (Fig. 51). The results support

Fig. 51. Results of computer simulations by Chudinov and Protasov for Cu [428]). Caloric equation of state and defect concentrations: L — vacancies at the surface, v — unstable Frenkel pairs, j — vacancies in the bulk. Total number of atoms in the calculations was 1554.

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Frenkel’s concept that melting is due to the instability of the crystal to the defect formation. For the present discussion, the main point of these calculations is the conclusion that high defect concentrations appear in a solid before melting. Zhukov [429] has shown that an interaction between equilibrium vacancies can result in a thermodynamic instability of the crystal. When taking this interaction into account, a critical temperature appears above which the Gibbs energy of the crystal has no minimum and decreases at all vacancy concentrations. This means that above the critical point vacancies are generated spontaneously. The critical temperature can be identified with the melting point. From the equation derived, vacancy concentration at the critical point appeared e"2.7 times larger than the value calculated without taking into account the vacancy interaction. The entropy of the vacancy formation may increase along with the vacancy concentration [380]. In this case, a rapid increase of the vacancy concentration leads to melting. The stability limit of a crystal has been calculated by Fecht and Johnson [430]. Their approach was based on the equality of the enthalpies and entropies of the solid and liquid phases. The authors concluded that vacancy concentrations at the stability-limit point should be of the order of 10~1. 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: starting at a temperature where the vacancy concentrations are still negligible, and starting at a temperature where they are sufficiently large.

13. Summary 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 strongly affect the high-temperature specific heat, the thermal expansivity, and the temperature derivative of resistivity. From the vacancy contribution to the specific heat, the formation enthalpy and equilibrium vacancy concentration can be determined. The only difficulty is to correctly separate the vacancy contribution. Such an opportunity is provided by observations of the vacancy equilibration after a rapid temperature change. (2) The nonlinear increase in the high-temperature specific heat of metals was observed in numerous investigations by all the calorimetric techniques now available. The phenomenon is especially strong in refractory metals and indicates high equilibrium vacancy concentrations. This concept gained no recognition, and the phenomenon is commonly attributed to the anharmonicity. As a rule, thermophysicists treat their experimental data completely ignoring the vacancy formation. (3) In almost all cases, calorimetric measurements yield reasonable values of the enthalpies of vacancy formation. In low-melting-point metals, the obtained vacancy concentrations at the melting points are of the order of 10~3. These low values have been derived by the usual procedure of linearly extrapolating the specific-heat data from intermediate temperatures. The vacancy concentrations in high-melting-point metals appeared to be of the order of 10~2 (Fig. 52). The only

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Fig. 52. Vacancy concentrations at melting point from differential dilatometry (L) and specific-heat measurements (j). Note new differential-dilatometry data on Ag and Cu (v) by Hehenkamp et al. [80,81].

serious contradiction to the high vacancy concentrations in refractory metals is the low vacancyinduced resistivity. (4) Measurements of the stored enthalpy of quenched samples show lower limits of equilibrium vacancy concentrations. Such data have been obtained only for Au. Similar measurements on refractory metals would be much more important. (5) Under very rapid heating of 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 Mo and W, the expected difference based on the vacancy origin of the nonlinear increase in the specific heat amounts to about 10%. In contrast, vacancy concentrations less than 10~3 should make this difference too small to be visible. The rapid-heating data on W strongly support the high vacancy concentration. Further rapid-heating determinations of the enthalpy of metals at premelting temperatures are very desirable. (6) Vacancy concentrations deduced from the nonlinear increase in the thermal expansivity are somewhat smaller than those from the nonlinear increase in the specific heat. Vacancy formation partly involves internal sources in the crystal lattice (voids, grain boundaries, dislocations, vacancy clusters), so that changes in the outer volume of the sample may appear smaller than those under ideal conditions. Underestimated vacancy concentrations thus should be expected. (7) The differential dilatometry is regarded as the best or even the absolute method for determining the equilibrium vacancy concentrations. In pure metals, this technique yields vacancy concentrations less than 10~3 at the melting points. Recent differential-dilatometry measurements revealed vacancy concentrations in Ag and Cu three times larger than those commonly accepted during the last 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 10~4. However, a much easier aim is now topical, namely, to check whether 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 measure simultaneously the bulk and X-ray expansion, as necessarily in the case of low vacancy

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concentrations. Only a few determinations of the lattice parameter of high-melting-point metals have been made at high temperatures. With macroscopic-expansion data now available, they rather confirm than disprove high vacancy concentrations. High concentrations of thermal vacancies have been observed by the differential dilatometry in some alloys and intermetallic compounds. (8) Very different values of the extra electrical resistivity of metals have been found under equilibrium and after quenching. The relative vacancy contributions at high temperatures are generally small and cannot be separated unambiguously. On the other hand, only a fraction of equilibrium vacancies can survive in a sample after quenching. Further, many vacancies form clusters during quench, and their contribution to the resistivity becomes much smaller. Low values of the quenched-in resistivity are thus quite explainable. The vacancy-related resistivity may also depend on temperature. Only for Al and Pt the results from the two techniques are in a reasonable agreement. Still more important, 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 appeared successful only on Mo and W. The largest quenched-in resistivity in W is 0.2 l) cm 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) In equilibrium measurements, the vacancy contribution to the resistivity of refractory metals has not been found. The reason is that the reported values were based on the room-temperature shape of the samples. The vacancy contribution becomes clear after introducing corrections for the thermal expansion, which are necessary to calculate true resistivities. This procedure was applied to W and Nb, and the vacancy contributions at the melting points were estimated as 0.5 and 0.7 l) cm, respectively. However, with the commonly accepted values of o for refractory metals, of 7 the order of 10~3 ) cm, 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 the resistivity. A straightforward determination of the vacancy contribution to the 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 should be available. (10) Positron annihilation is now regarded as the best tool to determine the enthalpies of the vacancy formation. Regretfully, this technique does not provide the vacancy concentrations and is inapplicable to some metals. Moreover, 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. The positron annihilation may help in determinations of vacancy losses during quenching. An important application of this technique is an observation of the vacancy equilibration at high temperatures. (11) The perturbed-angular-correlation technique is an important tool 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 pure metals

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were reported until today. Nevertheless, this approach deserves attention because reliable determinations of equilibrium vacancy concentrations even far below melting points would be useful. Such measurements are very important in high-melting-point metals. (12) Only quenched samples of high-melting-point metals are suitable for observations by the field-ion microscopy. The method thus involves the drawbacks inherent to all quenching experiments and yields 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. Vacancy concentrations smaller than 10~3 at the melting points were found by the electron and field-ion microscopy. (13) Observations of the vacancy equilibration have been proposed long ago to reliably separate vacancy contributions to physical properties of metals. Such observations were considered as a crucial determination of equilibrium vacancy concentrations. Relaxation phenomena caused by the vacancy equilibration were observed through measurements of the resistivity (Al, Au, Pt) and the specific heat (Pt, W), and by the positron-annihilation technique (Au). The results confirmed the vacancy origin of the nonlinear increase in the specific heat of Pt and W. Despite very different relaxation times, all the phenomena observed are probably of one nature. The relaxation observed in the specific heat of W and Pt gained no recognition. Further development and application of this technique unquestionably deserves attention. (14) An important reserve has not yet been employed in the observations of the relaxation in the 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 should provide data at higher temperatures and thus reveal larger differences between the specific heats measured under slow and fast temperature changes. The vacancy origin of the relaxation in the specific heat could be checked by measurements including quench and subsequent annealing a sample. The relaxation time should significantly decrease after quenching and gradually increase during annealing at the high temperature. (15) Recently, a straightforward determination of vacancy contributions to the 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 form immediately after the heating. The initial cooling curve after the heating should show the heat absorbed by the vacancy formation. This method seems to be the simplest one to unambiguously determine the equilibrium vacancy concentrations. The same approach is probably suitable to determine the expansion of the sample caused by the vacancy formation. (16) An additional check of the origin of the nonlinear increase in specific heat may be 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 energy of the vacancy formation. The necessary pressures are quite moderate, and suitable calorimetric techniques are now available. (17) In the last decade, new relations concerning entropies of the vacancy formation have been found. First, it was shown that a perturbation in the delocalized phonon spectrum by vacancies may cause an 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 energy of the vacancy formation at the melting

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Fig. 53. Enthalpies of vacancy formation based on calorimetric data and a linear temperature dependence of the formation entropies.

points may be 1.5—2 times smaller than that at low temperatures. The temperature dependence of formation enthalpies based on specific-heat data and a linear temperature dependence of the formation entropies appeared in qualitative agreement with these calculations (Fig. 53). (18) There exists no evidence that the strong nonlinear increase in the specific heat of metals is caused by the anharmonicity. Theoretical calculations of the anharmonicity indicate mainly linear contributions, which may be even negative. It seems unlikely that a nonlinear anharmonicity contribution to the specific heat might be much larger than the linear term. The determination of the constant-volume specific heat of W also supports small or negative contribution of the anharmonicity. Even when the nonlinear increase in the specific heat is small, linear extrapolations lead to reasonable values of the formation enthalpies. This should be impossible if the nonlinear contributions of the anharmonicity were essential. (19) High vacancy concentrations in refractory metals do not contradict self-diffusion data. The parameters of vacancy formation deduced from the nonlinear increase in the specific heat should be compared with the self-diffusion data at high temperatures, above two thirds of the melting temperature. Such a comparison poses important questions which are difficult to answer unambiguously. The activation enthalpies of the self-diffusion at high temperatures are larger than the sum of the enthalpies of the vacancy formation and migration. On the other hand, the vacancy parameters are consistent with the self-diffusion enthalpies at lower temperatures. A possible explanation of this may be an assumption that the vacancies dominate also at high temperatures but their contribution to the self-diffusion becomes smaller than that of other defects, divacancies or interstitials, possessing smaller concentrations but higher mobilities. Curvatures in Arrhenius’ plots of the self-diffusion can be explained by the two-defect mechanism or by a strong temperature dependence of the enthalpies of formation and migration. In both cases, high entropies of the vacancy formation at high temperatures are favoured. (20) Large vacancy concentrations in high-melting-point metals correlate with equilibrium vapour pressures. The highest vapour pressures at the melting point are known in Cr, Mo and W exhibiting the largest nonlinear increase in the specific heat. This is in accordance with Frenkel’s concept of similarity of both phenomena, the evaporation and point-defect formation. (21) Until today, data on equilibrium point defects were obtained only for about a half of metals (Fig. 54).

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Fig. 54. Until today, equilibrium defect concentrations are determined only for half of the metals. Techniques providing such data are indicated as follows: C — calorimetry, D — differential dilatometry, R — resistometry.

14. Conclusions In 1973, Seeger formulated the situation as follows [26]: ‘‘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 a complete understanding. In this review, an attempt was made to show that the viewpoint shared by the majority of the scientific community needs a reconsideration. Recently, the subject was discussed at the International Conference on Diffusion in Materials (DIMAT-96, Nordkirchen). The author’s viewpoint was given in a paper entitled ‘‘An opposite view on equilibrium vacancies in metals’’ [431]. Some new ideas concerning equilibrium point defects and self-diffusion mechanisms in metals were presented by Seeger [432].

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Experimental approaches most suitable to reliably determine equilibrium defect concentrations in metals are as follows: (1) Measurements of the enthalpy of metals at the melting point or at a selected premelting temperature under various heating rates could show directly the vacancy-related enthalpy and, consequently, the equilibrium vacancy concentration. (2) A straightforward determination of the vacancy-related enthalpy could be based on observations of the initial cooling curve of a sample immediately after it was rapidly heated to a premelting temperature. Such an experiment seems to be the simplest one to reliably determine equilibrium vacancy concentration. (3) To check the origin of the nonlinear increase in high-temperature specific heat of metals, calorimetric measurements under high pressures seem to be very useful. (4) Pure and well-prepared samples of low dislocation densities and, consequently, long relaxation times should be employed in observations of the vacancy equilibration. The origin of the relaxation phenomenon in specific heat could be checked by measurements including quenching and subsequent annealing the samples. (5) Differential-dilatometry studies or only determinations of the lattice parameter of highmelting-point metals at high temperatures are very desirable. The aim of such measurements should be rather to check whether high vacancy concentrations in these metals can be ruled out than to measure small vacancy concentrations. (6) The 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 thermal-expansivity data are now available. The vacancy-related resistivity may be also determined in rapid-heating experiments. (7) To check the validity of calorimetric data, it would be useful to measure the specific heat of alloys and intermetallic compounds in which high vacancy concentrations were found by the differential dilatometry and the perturbed-angular-correlation technique or predicted by the positron annihilation. (8) Despite difficulties that are already evident, an attempt to determine equilibrium point defects in pure metals by perturbed-angular-correlation technique, especially in high-melting-point metals, unquestionably deserves attention.

Acknowledgements The author gratefully acknowledges the support by the Ministry of Science and Technology of Israel and by the Dr. Irving and Cherna Moskowitz Program for the Absorption of Scientists. Many thanks to the referee for useful comments.

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