Table Of ContentLECTURE NOTES ON
EQUILIBRIUM POINT DEFECTS
AND
THERMOPHYSICAL PROPERTIES
OF METALS
Yaakov Kraftmakher
Bar-/Ian Universitx Israel
orld Scientific
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LECTURNEO TESO N
EQUILIBRIUM POINT DEFECTS AND
THERMOPHYSICAL PROPERTIES OF METALS
Copyright 0 2000 by World Scientific Publishing Co. Re. Ltd
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Preface
Formation of point defects in solids has been predicted by
Frenkel (1926). At high temperatures, thermal motion of atoms
becomes more vigorous and some atoms acquire energies
sufficient to leave their lattice sites and occupy interstitial
positions. In this case, a vacancy and an interstitial atom (the so-
called Frenkel pair) appear simultaneously. Wagner and Schottky
(1930) have shown a way to create only vacancies: atoms leave
their lattice sites and occupy free positions on the surface or at
internal imperfections of the crystal (voids, grain boundaries,
dislocations). Such vacancies are often called Schottky
defects. This mechanism dominates in solids with close-packed
lattices, where formation of vacancies requires considerably
smaller energies than that of interstitials. Point defects are
thermodynamically stable because they enhance the entropy of a
crystal. The Gibbs free energy of the crystal reaches a
minimum at a certain defect concentration. From thermodynamic
considerations, point defects are present in a crystal at any
temperature. The equilibrium concentration of point defects
rapidly increases with increasing temperature.
Point-defect formation in metals is a well-documented
phenomenon. As a rule, the enthalpies of vacancy formation
obtained by various experimental techniques are in reasonable
agreement. In many cases, however, dramatic differences have
been found in equilibrium vacancy concentrations. These are
also governed by the formation entropies. This contradiction is
especially strong for refractory metals.
V
vi Preface
Despite the significant progress in the study of point defects
in metals, some important problems still do not have
unambiguous solutions. One of the most practically important
questions relates to equilibrium defect concentrations. It is
indeed surprising that this fundamental problem is still under
debate. Nowadays, two opposite viewpoints exist on equilibrium
point defects in metals.
(1) Defect contributions to physical properties of metals at
high temperatures are small and cannot be separated from the
effects of anharmonicity. The only methods appropriate for
studying point defects are positron-annihilation spectroscopy,
which provides the enthalpies of vacancy formation, and
differential dilatometry, which probes the equilibrium vacancy
concentrations. Equilibrium defect concentrations at the melting
points range from lo4 to lop3. That the formation enthalpies
deduced from the nonlinear increase in high-temperature specific
heat of metals are reasonable is just accidental, while the
derived defect concentrations are improbably large. Therefore,
this approach is generally erroneous.
(2) In many cases, defect contributions to the specific heat
of metals are much larger than the nonlinear effects of
anharmonicity. Thus, their separation does not introduce crucial
errors. This approach is quite appropriate for determining point-
defect parameters, especially, equilibrium defect concentrations.
Equilibrium defect concentrations at melting points are of the
order of in low-melting-point metals and of in high-
melting-point metals. Strong nonlinear effects in the high-
temperature specific heat and thermal expansivity of metals are
caused by the formation of equilibrium point defects.
Examination of these effects rules out anharmonicity as the
possible origin of this phenomenon. It may turn out that
calorimetric determinations provide the most reliable values of
equilibrium vacancy concentrations in metals.
Preface vii
This book discusses experimental results and theoretical
considerations favoring each claim. At present, the majority of
the scientific community holds the first viewpoint. Regrettably,
the data supporting the second viewpoint were never displayed
and discussed together, and the criticism of this viewpoint never
included a detailed analysis.
Important new arguments have appeared in the last decades.
First, the relaxation phenomenon in specific heat, caused by
vacancy equilibration, has been observed. Such measurements
were proposed long ago and considered to be crucial for
the determination of equilibrium vacancy concentrations. Second,
new differential-dilatometry measurements on silver and
copper revealed vacancy concentrations several times larger
than values commonly accepted for three decades. High
concentrations of thermally generated vacancies were observed
in some alloys and intermetallics. Finally, thermodynamic
relations favoring high entropies of vacancy formation in metals
have been found. All of these results support the second
viewpoint. At the same time, the weakness of the first viewpoint
is now clearly seen. In essence, only two results support this
opinion, namely: (i) differential-dilatometry data on low-melting-
point metals, and (ii) low extra resistivities of quenched samples
and small concentrations of quenched-in vacancies observed in
high-melting-point metals by electron and field ion microscopy.
In this book, the focus is on equilibrium point defects in
metals and their relation with the thermophysical properties of
metals at high temperatures. An attempt will be made to answer
two important questions: (i) what are the equilibrium vacancy
concentrations in metals, and (ii) what is the nature of the strong
nonlinear increase in the specific heat of metals at high
temperatures. The majority of the scientific community considers
these two questions to be unrelated. As a rule, physicists
studying point-defect formation in metals ignore calorimetric and
other thermophysical data from high-temperature measurements.
viii Preface
On the other hand, physicists studying thermophysical properties
of metals do not take into account the expected point-defect
contributions. This situation is caused by the opinion that
equilibrium concentrations of point defects are too small to
markedly affect the thermophysical properties.
The author's intention is to show that this well-established
opinion needs reconsideration. Though the author always
believed that the questions (i) and (ii) above are closely related,
the opposite viewpoint is also presented in this book. Along with
a discussion of the experimental data and the theoretical
estimates now available, some approaches are proposed that
seem to be most suitable for settling the questions discussed
above.
I gratefully remember my teacher, the late Professor
P.G. Strelkov (1899-1968), and my students and collaborators,
A.I. Akimov, I.M. Cheremisina, S.Y. Glazkov, O.M. Kanel',
S.D. Krylov, E.B. Lanina, V.P. Nezhentsev, T.Y. Pinegina,
G.G. Sushakova, V.L. Tonaevskii, and the late A.A. Varchenko.
I would like to thank in particular my following colleagues for
useful discussions: the late Dr. A. Cezairliyan, Professor
Th. Hehenkamp, Professor V.M. Koshkin, the late Professor
I.M. Lifshits, Dr. K.D. Maglic, Professor A.A. Maradudin,
Professor E.V. Matizen, Professor 1.1. Novikov, the late Professor
A.N. Orlov, Dr. F. Righini, the late Dr. G. Ruffino, Professor
H.-E. Schaefer, Professor A.V. Voronel. I am greatly indebted to
Professor A. Seeger for his constructive criticism.
Y. K.
Ramat-Gan,
February 2000.
Contents
1 .Introduction 1
1.1. Point defects in solids. Formation parameters 2
1.2. Influence of point defects on physical properties 4
1.3. Strong nonlinear increase in specific heat
and thermal expansivity of metals 6
1.4. Two viewpoints on equilibrium point defects in metals 9
2. Basic theory of point-defect formation 17
2.1. Thermodynamics of point-defect formation 18
2.2. Origin of the formation entropy 19
2.3. Temperature dependence of formation parameters 20
2.4. Results of theoretical calculations 23
2.5. Summary 27
3. Methods for studying point defects 29
3.1. Measurements in equilibrium 30
Advantages of equilibrium measurements
according to Seeger. Criteria for choice of a suitable
physical property. Equations to fit experimental data.
Determination of formation enthalpies.
3.2. Quenching experiments 35
Extra electrical resistivity of quenched
samples. Stored enthalpy.
3.3. Observation of vacancy equilibration 38
How to observe vacancy equilibration. Modulation
calorimetry as a tool to study vacancy equilibration.
Formulas for relaxation in specific heat. Prediction
of the relaxation phenomenon in tungsten.
3.4. Summary 44
4. Modulation calorimetry and related techniques 45
4.1. Introduction 46
4.2. Basic theory of modulation calorimetry 51
4.3. Modulation of heating power 59
Direct electric heating. Induction heating.
ix
Contents
X
Modulated-light heating. Electron bombardment.
Separate heaters. Peltier heating.
4.4. Measurement of temperature oscillations 65
Use of oscillations in the sample's resistance.
Photoelectric detectors. Pyroelectric sensors.
Thermocouples and resistance thermometers.
Lock-in detection of periodic signals.
4.5. Modulation dilatometry 83
Principle of modulation dilatometry. Differential
method. Bulk samples. lnterferometric modulation
dilatometer. Nonconducting materials. Measurement
of extremely small periodic displacements.
4.6. Modulation measurements of electrical
resistivity and thermopower 96
Temperature derivative of resistance.
Direct measurement of thermopower.
4.7. Summary 100
5. Enthalpy and specific heat of metals 101
5.1. Point defects and specific heat 102
Why point defects affect high-temperature
specific heat. What was said about calorimetric
data, and the opposite viewpoint.
5.2. Methods of calorimetry 105
Adiabatic calorimetry. Drop method. Pulse and dynamic
techniques. Relaxation method. Rapid-heating experiments.
5.3. Formation parameters from calorimetric data 117
5.4. Extra enthalpy of quenched samples 127
5.5. Question to be answered by rapid-heating experiments 129
How to derive vacancy-related enthalpy and resistivity.
Rapid-heating data for tungsten and molybdenum.
5.6. Specific heat of tungsten - a student experiment 132
5.7. Summary 136
6. Thermal expansion of metals 137
6.1. Point defects and thermal expansion 138
6.2. Methods of dilatometry 140
Optical methods. Capacitance
dilatometers. Dynamic techniques.
6.3. Differential dilatometry 147
Revision of Simmons-Balluffi data. Nowick-Feder example.
Contents xi
6.4. Equilibrium vacancy concentrations 152
6.5. High vacancy concentrations in some
alloys and intermetallics 156
6.6. Lattice parameter and volume of quenched samples 157
6.7. Summary 160
7. Electrical resistivity of metals 161
7.1. Influence of point defects on electrical resistivity 162
Deviations from Matthiessen's rule. Extra
resistivity of vacancies and of vacancy clusters.
7.2. Resistivity of metals at high temperatures 164
How to derive formation parameters. Why measurements
of temperature derivative of resistivity are preferable.
7.3. Quenched-in resistivity 169
Quenching in superfluid helium. Quenching with
reduced cooling rate. Annealing experiments.
7.4. Comparison of data from two methods 173
7.5. Summary 178
8. Positron annihilation 179
8.1. Positron-annihilation techniques 180
Why vacancies affect positron annihilation. Lifetime
spectroscopy. Mean positron lifetime. Doppler broadening.
S-, W-, and D-parameters. Angular correlation of y-quanta.
8.2. Experimental data 192
8.3. Drawbacks of positron-annihilation techniques 194
8.4. High vacancy concentrations in some intermetallics 195
8.5. Summary 196
9. Other methods 197
9.1. Hyperfine interactions 198
Perturbed angular correlation of y-quanta.
M6ssbauer spectroscopy. Nuclear magnetic resonance.
9.2. Other physical properties 207
Thermoelectric power. Thermal conductivity and thermal
diffusivity. Mechanical properties. Spontaneous magnetization
Current noise. Properties of superconductors.
9.3. Microscopic observation of quenched-in defects 212
Electron microscopy. Field ion microscopy.
9.4. Summary 216
10. Equilibration of point defects 21 7
10.1. Role of internal sources (sinks) for point defects 218
