10 Physical properties of Materials

physical properties of solid waste materials and physical properties of natural and processed materials and what are physical properties of materials
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Published Date:25-10-2017
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Chapter 6 The physical properties of materials 3 6.1 Introduction that the mass of 1 m of copper, i.e. the density, is 28 24 3 8.56ð 10 ð 63.57ð 1.63ð 10 D 8900 kg m . The ways in which any material interacts and responds On alloying, the density of a metal changes. This to various forms of energy are of prime interest to sci- is because the mass of the solute atom differs from entists and, in the context of engineering, provide the that of the solvent, and also because the lattice essential base for design and innovation. The energy parameter usually changes on alloying. The parameter acting on a material may derive from force fields (grav- change may often be deduced from Vegard’s law, itational, electric, magnetic), electromagnetic radiation which assumes that the lattice parameter of a solid (heat, light, X-rays), high-energy particles, etc. The solution varies linearly with atomic concentration, but responses of a material, generally referred to as its numerous deviations from this ideal behaviour do physical properties, are governed by the structural exist. arrangement of atoms/ions/molecules in the material. The density clearly depends on the mass of the The theme of the structure/property relation which atoms, their size and the way they are packed. Metals has run through previous chapters is developed fur- are dense because they have heavy atoms and close ther. Special attention will be given to the diffusion of packing; ceramics have lower densities than metals atoms/ions within materials because of the importance because they contain light atoms, either C, N or O; of thermal behaviour during manufacture and service. polymers have low densities because they consist of In this brief examination, which will range from den- light atoms in chains. Figure 6.1 shows the spread in sity to superconductivity, the most important physical density values for the different material classes. Such properties of materials are considered. ‘Material Property Charts’, as developed by Ashby, are useful when selecting materials during engineer- ing design. 6.2 Density This property, defined as the mass per unit volume of 6.3 Thermal properties a material, increases regularly with increasing atomic 6.3.1 Thermal expansion numbers in each sub-group. The reciprocal of the density is the specific volume v, while the product of If we consider a crystal at absolute zero temperature, v and the relative atomic mass W is known as the the ions sit in a potential well of depth E below the r 0 atomic volume . The density may be determined by energy of a free atom (Figure 6.2). The effect of raising the usual ‘immersion’ method, but it is instructive to the temperature of the crystal is to cause the ions show how X-rays can be used. For example, a powder to oscillate in this asymmetrical potential well about photograph may give the lattice parameter of an fcc their mean positions. As a consequence, this motion 10 3 metal, say copper, as 0.36 nm. Then 1/3.6ð 10  causes the energy of the system to rise, increasing 28 or 2.14ð 10 cells of this size (0.36 nm) are found with increasing amplitude of vibration. The increasing in a cube 1 m edge length. The total number of amplitude of vibration also causes an expansion of the 3 28 28 crystal, since as a result of the sharp rise in energy atoms in 1 m is then 4ð 2.14ð 10 D 8.56ð 10 below r the ions as they vibrate to and fro do not since an fcc cell contains four atoms. Furthermore, 0 approach much closer than the equilibrium separation, the mass of a copper atom is 63.57 times the mass 24 r , but separate more widely when moving apart. When 0 of a hydrogen atom (which is 1.63ð 10 g) soThe physical properties of materials 169 Figure 6.1 Strength , plotted against density, (yield strength for metals and polymers, compressive strength for ceramics, 2/3 1/2 tear strength for elastomers and tensile strength for composites). The guide lines of constant / , / and / are used in minimum weight, yield-limited, design (Ashby, 1989, pp. 1273–93, with permission of Elsevier Science Ltd.). The change in dimensions with temperature is usually expressed in terms of the linear coefficient of expansion ˛,given by ˛D1/ldl/dT,where l is the original length of the specimen and T is the absolute temperature. Because of the anisotropic nature of crystals, the value of ˛ usually varies with the direction of measurement and even in a particular crystallographic direction the dimensional change with temperature may not always be uniform. Phase changes in the solid state are usually studied by dilatometry. The change in dimensions of a specimen can be transmitted to a sensitive dial gauge or electrical transducer by means of a fused silica rod. When a phase transformation takes place, because Figure 6.2 Variation in potential energy with interatomic the new phase usually occupies a different volume distance. to the old phase, discontinuities are observed in the coefficient of thermal expansion ˛ versus T curve. the distance r is such that the atoms are no longer Some of the ‘nuclear metals’ which exist in many interacting, the material is transformed to the gaseous allotropic forms, such as uranium and plutonium, show phase, and the energy to bring this about is the energy a negative coefficient along one of the crystallographic of evaporation. axes in certain of their allotropic modifications.170 Modern Physical Metallurgy and Materials Engineering The change in volume with temperature is important given by   in many metallurgical operations such as casting, dECPdV dH welding and heat treatment. Of particular importance C D p dT dT is the volume change associated with the melting p or, alternatively, the freezing phenomenon since this where HDECPV is known as the heat content or is responsible for many of the defects, both of enthalpy, C is greater than C by a few per cent p v a macroscopic and microscopic size, which exist because some work is done against interatomic forces in crystals. Most metals increase their volume by when the crystal expands, and it can be shown that about 3% on melting, although those metals which 2 have crystal structures of lower coordination, such as C C D 9˛ VT/ˇ p v bismuth, antimony or gallium, contract on melting. where ˛ is the coefficient of linear thermal expansion, This volume change is quite small, and while the V is the volume per gram-atom and ˇ is the liquid structure is more open than the solid structure, compressibility. it is clear that the liquid state resembles the solid Dulong and Petit were the first to point out that the state more closely than it does the gaseous phase. For specific heat of most materials, when determined at the simple metals the latent heat of melting, which sufficiently high temperatures and corrected to apply is merely the work done in separating the atoms from to constant volume, is approximately equal to 3R, the close-packed structure of the solid to the more open where R is the gas constant. However, deviations liquid structure, is only about one thirtieth of the latent from the ‘classical’ value of the atomic heat occur heat of evaporation, while the electrical and thermal at low temperatures, as shown in Figure 6.3a. This conductivities are reduced only to three-quarters to deviation is readily accounted for by the quantum one-half of the solid state values. theory, since the vibrational energy must then be quantized in multiples of h,where h is Planck’s 6.3.2 Specific heat capacity constant and  is the characteristic frequency of the normal mode of vibration. The specific heat is another thermal property important According to the quantum theory, the mean energy in the processing operations of casting or heat of a normal mode of the crystal is treatment, since it determines the amount of heat 1 required in the process. Thus, the specific heat (denoted ED hvCfh/ exph/kT 1g 2 byC , when dealing with the specific heat at constant p 1 pressure) controls the increase in temperature, dT, where h represents the energy a vibrator will have 2 produced by the addition of a given quantity of heat, at the absolute zero of temperature, i.e. the zero-point dQ, to one gram of matter so that dQDC dT. p energy. Using the assumption made by Einstein (1907) The specific heat of a metal is due almost entirely that all vibrations have the same frequency (i.e. all to the vibrational motion of the ions. However, a atoms vibrate independently), the heat capacity is small part of the specific heat is due to the motion C DdE/dT v v of the free electrons, which becomes important at 2 high temperatures, especially in transition metals with D 3Nkh/kT electrons in incomplete shells. 2 The classical theory of specific heat assumes that exph/kT/fexph/kT1g an atom can oscillate in any one of three directions, This equation is rarely written in such a form because and hence a crystal of N atoms can vibrate in 3N most materials have different values of .Itis more independent normal modes, each with its characteristic usual to express as an equivalent temperature defined frequency. Furthermore, the mean energy of each nor- by  D h/k,where  is known as the Einstein E E mal mode will be kT, so that the total vibrational characteristic temperature. Consequently, when C is v thermal energy of the metal is ED 3NkT. In solid plotted against T/ , the specific heat curves of all E and liquid metals, the volume changes on heating are pure metals coincide and the value approaches zero at very small and, consequently, it is customary to con- very low temperatures and rises to the classical value sider the specific heat at constant volume. If N,the of 3NkD 3R' 25.2 J/g at high temperatures. number of atoms in the crystal, is equal to the number Einstein’s formula for the specific heat is in good of atoms in a gram-atom (i.e. Avogadro number), the agreement with experiment for T , but is poor for E heat capacity per gram-atom, i.e. the atomic heat, at low temperatures where the practical curve falls off constant volume is given by less rapidly than that given by the Einstein relationship.   However, the discrepancy can be accounted for, as dQ dE 1 C D D 3NkD 24.95 J K shown by Debye, by taking account of the fact that the v dT dT v atomic vibrations are not independent of each other. This modification to the theory gives rise to a Debye In practice, of course, when the specific heat is exper- characteristic temperature  , which is defined by D imentally determined, it is the specific heat at constant pressure, C , which is measured, not C , and this is k D h p v D DThe physical properties of materials 171 where  is Debye’s maximum frequency. Figure 6.3b without a rise in temperature, so that the specific heat D shows the atomic heat curves of Figure 6.3a plotted dQ/dT at the transformation temperature is infinite. against T/ ; in most metals for low temperatures In some cases, known as transformations of the sec- D 3 ond order, the phase transition occurs over a range T/ − 1 a T law is obeyed, but at high temper- D atures the free electrons make a contribution to the of temperature (e.g. the order–disorder transformation atomic heat which is proportional to T and this causes in alloys), and is associated with a specific heat peak ariseof C above the classical value. of the form shown in Figure 6.4b. Obviously the nar- rower the temperature range T  T ,the sharperis 1 c the specific heat peak, and in the limit when the total 6.3.3 The specific heat curve and change occurs at a single temperature, i.e. T D T ,the 1 c transformations specific heat becomes infinite and equal to the latent The specific heat of a metal varies smoothly with tem- heat of transformation. A second-order transformation perature, as shown in Figure 6.3a, provided that no also occurs in iron (see Figure 6.4a), and in this case phase change occurs. On the other hand, if the metal is due to a change in ferromagnetic properties with temperature. undergoes a structural transformation the specific heat curve exhibits a discontinuity, as shown in Figure 6.4. If the phase change occurs at a fixed temperature, the 6.3.4 Free energy of transformation metal undergoes what is known as a first-order trans- formation; for example, the ˛ to , to υ and υ to liq- In Section it was shown that any structural uid phase changes in iron shown in Figure 6.4a. At the changes of a phase could be accounted for in terms transformation temperature the latent heat is absorbed of the variation of free energy with temperature. The Figure 6.3 The variation of atomic heat with temperature. Figure 6.4 The effect of solid state transformations on the specific heat–temperature curve.172 Modern Physical Metallurgy and Materials Engineering relative magnitude of the free energy value governs the phase changes occur the more close-packed structure stability of any phase, and from Figure 3.9a it can be usually exists at the low temperatures and the more seen that the free energy G at any temperature is in turn open structures at the high temperatures. From this governed by two factors: (1) the value of G at 0 K, viewpoint a liquid, which possesses no long-range G , and (2) the slope of the G versus T curve, i.e. the structure, has a higher entropy than any solid phase 0 temperature-dependence of free energy. Both of these so that ultimately all metals must melt at a sufficiently terms are influenced by the vibrational frequency, and high temperature, i.e. when the TS term outweighs the consequently the specific heat of the atoms, as can be H term in the free energy equation. shown mathematically. For example, if the temperature The sequence of phase changes in such metals as titanium, zirconium, etc. is in agreement with this pre- of the system is raised from T to TC dT the change diction and, moreover, the alkali metals, lithium and in free energy of the system dG is sodium, which are normally bcc at ordinary temper- dGD dH TdS SdT atures, can be transformed to fcc at sub-zero temper- atures. It is interesting to note that iron, being bcc D C dT TC dT/T SdT p p (˛-iron) even at low temperatures and fcc ( -iron) at DSdT high temperatures, is an exception to this rule. In this case, the stability of the bcc structure is thought to be so that the free energy of the system at a temperature associated with its ferromagnetic properties. By hav- T is ing a bcc structure the interatomic distances are of the  T correct value for the exchange interaction to allow the GD G  SdT 0 electrons to adopt parallel spins (this is a condition for 0 magnetism). While this state is one of low entropy it is also one of minimum internal energy, and in the lower At the absolute zero of temperature, the free energy temperature ranges this is the factor which governs the G is equal to H ,and then 0 0 phase stability, so that the bcc structure is preferred.  T Iron is also of interest because the bcc structure, GD H  SdT 0 which is replaced by the fcc structure at temperatures 0 ° ° above 910 C, reappears as the υ-phase above 1400 C.  T which if S is replaced by C /TdT becomes This behaviour is attributed to the large electronic spe- p 0 cific heat of iron which is a characteristic feature of    T T most transition metals. Thus, the Debye characteristic GD H  C /TdT dT (6.1) 0 p temperature of -iron is lower than that of ˛-iron and 0 0 this is mainly responsible for the ˛ to transformation. Equation (6.1) indicates that the free energy of a given However, the electronic specific heat of the ˛-phase phase decreases more rapidly with rise in tempera- becomes greater than that of the -phase above about ture the larger its specific heat. The intersection of the ° 300 C and eventually at higher temperatures becomes free energy–temperature curves, shown in Figure 3.9a, sufficient to bring about the return to the bcc structure therefore takes place because the low-temperature ° at 1400 C. phase has a smaller specific heat than the higher- temperature phase. At low temperatures the second term in equation 6.4 Diffusion (6.1) is relatively unimportant, and the phase that 6.4.1 Diffusion laws is stable is the one which has the lowest value of H , i.e. the most close-packed phase which is 0 Some knowledge of diffusion is essential in associated with a strong bonding of the atoms. understanding the behaviour of materials, particularly However, the more strongly bound the phase, the at elevated temperatures. A few examples include higher is its elastic constant, the higher the vibrational such commercially important processes as annealing, frequency, and consequently the smaller the specific heat-treatment, the age-hardening of alloys, sintering, heat (see Figure 6.3a). Thus, the more weakly bound surface-hardening, oxidation and creep. Apart from structure, i.e. the phase with the higher H at low 0 the specialized diffusion processes, such as grain temperature, is likely to appear as the stable phase boundary diffusion and diffusion down dislocation at higher temperatures. This is because the second channels, a distinction is frequently drawn between term in equation (6.1) now becomes important and G diffusion in pure metals, homogeneous alloys and decreases more rapidly with increasing temperature, inhomogeneous alloys. In a pure material self-diffusion  for the phase with the largest value of C /TdT. p can be observed by using radioactive tracer atoms.  From Figure 6.3b it is clear that a large C /TdT In a homogeneous alloy diffusion of each component p is associated with a low characteristic temperature can also be measured by a tracer method, but in an and hence, with a low vibrational frequency such as inhomogeneous alloy, diffusion can be determined by is displayed by a metal with a more open structure chemical analysis merely from the broadening of the and small elastic strength. In general, therefore, when interface between the two metals as a function of time.The physical properties of materials 173 Figure 6.5 Effect of diffusion on the distribution of solute in an alloy. Inhomogeneous alloys are common in metallurgical practice (e.g. cored solid solutions) and in such cases diffusion always occurs in such a way as to produce a macroscopic flow of solute atoms down the concentration gradient. Thus, if a bar of an alloy, along which there is a concentration gradient (Figure 6.5) is heated for a few hours at a temperature where atomic migration is fast, i.e. near the melting point, the solute Figure 6.6 Diffusion of atoms down a concentration atoms are redistributed until the bar becomes uniform gradient. in composition. This occurs even though the individual atomic movements are random, simply because there with J the flux of diffusing atoms. Setting c  c D x 1 2 are more solute atoms to move down the concentration bdc/dx this flux becomes gradient than there are to move up. This fact forms the 2 1 2 basis of Fick’s law of diffusion, which is J Dp v b dc/dxD vb dc/dx x x v 2 DDdc/dx 6.3 dn/dtDDdc/dx (6.2) In cubic lattices, diffusion is isotropic and hence all six Here the number of atoms diffusing in unit time 1 orthogonal directions are equally likely so that p D . x 6 across unit area through a unit concentration gradient For simple cubic structures bD a and thus 1 is known as the diffusivity or diffusion coefficient, D. 2 1 2 1 2 1 It is usually expressed as units of cm s or m s and D D D D D D va D D (6.4) x y z 6 depends on the concentration and temperature of the p 1 2 alloy. whereas in fcc structures bD a/ 2and DD va , 12 1 2 To illustrate, we may consider the flow of atoms and in bcc structures DD va . 24 in one direction x, by taking two atomic planes A Fick’s first law only applies if a steady state exists and B of unit area separated by a distance b,as in which the concentration at every point is invariant, shown in Figure 6.6. If c and c are the concentrations 1 2 i.e. dc/dtD0for all x. To deal with nonstationary of diffusing atoms in these two planes c c  the 1 2 flow in which the concentration at a point changes corresponding number of such atoms in the respective with time, we take two planes A and B, as before, planes is n D c b and n D c b. If the probability separated by unit distance and consider the rate of 1 1 2 2 that any one jump in the Cx direction is p ,then increase of the number of atoms dc/dt in a unit x the number of jumps per unit time made by one atom volume of the specimen; this is equal to the difference is p ,where  is the mean frequency with which between the flux into and that out of the volume x an atom leaves a site irrespective of directions. The element. The flux across one plane is J and across the x number of diffusing atoms leaving A and arriving at other J C 1 dJ/dx the difference being dJ/dx. x B in unit time is p c b and the number making the We thus obtain Fick’s second law of diffusion x 1   reverse transition is p c b so that the net gain of x 2 dc dJ d dc x D D D (6.5) atoms at B is x dt dx dx dx p bc  c D J When D is independent of concentration this reduces x 1 2 x to 1 2 The conduction of heat in a still medium also follows the dc d c x D D (6.6) x same laws as diffusion. 2 dt dx174 Modern Physical Metallurgy and Materials Engineering and in three dimensions becomes The flux through any shell of radius r is2rDdc/dr       or dc d dc d dc d dc D D C D C D x y z 2D dt dx dx dy dy dz dz JD c  c  (6.11) 1 0 lnr /r  1 0 An illustration of the use of the diffusion equations is the behaviour of a diffusion couple, where there Diffusion equations are of importance in many diverse is a sharp interface between pure metal and an alloy. problems and in Chapter 4 are applied to the diffusion Figure 6.5 can be used for this example and as the of vacancies from dislocation loops and the sintering solute moves from alloy to the pure metal the way in of voids. which the concentration varies is shown by the dotted lines. The solution to Fick’s second law is given by 6.4.2 Mechanisms of diffusion   p  x/2 Dt c 2 0 2 The transport of atoms through the lattice may conceiv- cD 1p expy  dy (6.7) 2  ably occur in many ways. The term ‘interstitial diffu- 0 sion’ describes the situation when the moving atom where c is the initial solute concentration in the alloy 0 does not lie on the crystal lattice, but instead occu- and c is the concentration at a time t at a distance pies an interstitial position. Such a process is likely x from the interface. The integral term is known as in interstitial alloys where the migrating atom is very the Gauss error function (erf (y)) and as y1, small (e.g. carbon, nitrogen or hydrogen in iron). In erf y 1. It will be noted that at the interface where this case, the diffusion process for the atoms to move xD 0, then cD c /2, and in those regions where the 0 from one interstitial position to the next in a perfect 2 2 curvature ∂ c/∂x is positive the concentration rises, lattice is not defect-controlled. A possible variant of in those regions where the curvature is negative the this type of diffusion has been suggested for substitu- concentration falls, and where the curvature is zero tional solutions in which the diffusing atoms are only the concentration remains constant. temporarily interstitial and are in dynamic equilibrium This particular example is important because it can with others in substitutional positions. However, the be used to model the depth of diffusion after time energy to form such an interstitial is many times that to t, e.g. in the case-hardening of steel, providing the produce a vacancy and, consequently, the most likely concentration profile of the carbon after a carburizing mechanism is that of the continual migration of vacan- time t, or dopant in silicon. Starting with a constant cies. With vacancy diffusion, the probability that an composition at the surface, the value of x where atom may jump to the next site will depend on: (1) the the concentration falls to half the initial value, i.e. probability that the site is vacant (which in turn is pro- p 1 1 erfyD ,is given by xD Dt. Thus knowing portional to the fraction of vacancies in the crystal), 2 D at a given temperature the time to produce a given and (2) the probability that it has the required activa- depth of diffusion can be estimated. tion energy to make the transition. For self-diffusion The diffusion equations developed above can also be where no complications exist, the diffusion coefficient transformed to apply to particular diffusion geometries. is therefore given by If the concentration gradient has spherical symmetry 1 2 about a point, c varies with the radial distance r and, DD a f exp S C S /k f m 6 for constant D, ð exp E /kTexpE /kT   f m 2 dc d c 2 dc D D C (6.8) D D exp E C E /kT 6.12 2 0 f m dt dr r dr The factor f appearing in D is known as a correla- When the diffusion field has radial symmetry about a 0 tion factor and arises from the fact that any particular cylindrical axis, the equation becomes diffusion jump is influenced by the direction of the   2 d c 1 dc dc previous jump. Thus when an atom and a vacancy D D C (6.9) 2 dt dr r dr exchange places in the lattice there is a greater prob- ability of the atom returning to its original site than and the steady-state condition dc/dtD0is given by moving to another site, because of the presence there 2 of a vacancy; f is 0.80 and 0.78 for fcc and bcc d c 1 dc C D 0 (6.10) lattices, respectively. Values for E and E are dis- 2 f m dr r dr cussed in Chapter 4, E is the energy of formation of f which has a solution cD AlnrC B. The constants A a vacancy, E the energy of migration, and the sum m and B may be found by introducing the appropriate of the two energies, QD E C E , is the activation f m 1 boundary conditions and for cD c at rD r and 0 0 energy for self-diffusion E . d cD c at rD r the solution becomes 1 1 1 c lnr /rC c lnr/r  0 1 1 0 The entropy factor exp S C S /k is usually taken to be f m cD unity. lnr /r  1 0The physical properties of materials 175 In alloys, the problem is not so simple and it is found that the self-diffusion energy is smaller than in pure metals. This observation has led to the sugges- tion that in alloys the vacancies associate preferentially with solute atoms in solution; the binding of vacancies to the impurity atoms increases the effective vacancy concentration near those atoms so that the mean jump rate of the solute atoms is much increased. This asso- ciation helps the solute atom on its way through the lattice, but, conversely, the speed of vacancy migration is reduced because it lingers in the neighbourhood of Figure 6.8 ˛-brass–copper couple for demonstrating the the solute atoms, as shown in Figure 6.7. The phe- Kirkendall effect. nomenon of association is of fundamental importance in all kinetic studies since the mobility of a vacancy some practical importance, especially in the fields of through the lattice to a vacancy sink will be governed metal-to-metal bonding, sintering and creep. by its ability to escape from the impurity atoms which trap it. This problem has been mentioned in Chapter 4. 6.4.3 Factors affecting diffusion When considering diffusion in alloys it is impor- tant to realize that in a binary solution of A and B The two most important factors affecting the diffu- the diffusion coefficients D and D are generally not A B sion coefficient D are temperature and composition. equal. This inequality of diffusion was first demon- Because of the activation energy term the rate of diffu- strated by Kirkendall using an ˛-brass/copper couple sion increases with temperature according to equation (Figure 6.8). He noted that if the position of the inter- (6.12), while each of the quantities D, D and Q 0 faces of the couple were marked (e.g. with fine W or varies with concentration; for a metal at high temper- 5 3 2 1 Mo wires), during diffusion the markers move towards atures Q³ 20RT , D is 10 to 10 m s ,and m 0 12 2 1 each other, showing that the zinc atoms diffuse out of D' 10 m s . Because of this variation of diffu- the alloy more rapidly than copper atoms diffuse in. sion coefficient with concentration, the most reliable This being the case, it is not surprising that several investigations into the effect of other variables neces- workers have shown that porosity develops in such sarily concern self-diffusion in pure metals. systems on that side of the interface from which there Diffusion is a structure-sensitive property and, therefore, D is expected to increase with increasing is a net loss of atoms. lattice irregularity. In general, this is found experi- The Kirkendall effect is of considerable theoretical mentally. In metals quenched from a high temper- importance since it confirms the vacancy mechanism 9 ature the excess vacancy concentration ³10 leads of diffusion. This is because the observations cannot to enhanced diffusion at low temperatures since DD easily be accounted for by any other postulated D c expE /kT. Grain boundaries and disloca- 0 v m mechanisms of diffusion, such as direct place- tions are particularly important in this respect and exchange, i.e. where neighbouring atoms merely produce enhanced diffusion. Diffusion is faster in the change place with each other. The Kirkendall effect cold-worked state than in the annealed state, although is readily explained in terms of vacancies since the recrystallization may take place and tend to mask the lattice defect may interchange places more frequently effect. The enhanced transport of material along dislo- with one atom than the other. The effect is also of cation channels has been demonstrated in aluminium where voids connected to a free surface by dislo- cations anneal out at appreciably higher rates than isolated voids. Measurements show that surface and grain boundary forms of diffusion also obey Arrhe- nius equations, with lower activation energies than for volume diffusion, i.e. Q ½ 2Q ½ 2Q .This vol g.b surface behaviour is understandable in view of the progres- sively more open atomic structure found at grain boundaries and external surfaces. It will be remem- bered, however, that the relative importance of the various forms of diffusion does not entirely depend on the relative activation energy or diffusion coefficient values. The amount of material transported by any dif- fusion process is given by Fick’s law and for a given composition gradient also depends on the effective area through which the atoms diffuse. Consequently, since Figure 6.7 Solute atom–vacancy association during diffusion. the surface area (or grain boundary area) to volume176 Modern Physical Metallurgy and Materials Engineering ratio of any polycrystalline solid is usually very small, it is only in particular phenomena (e.g. sintering, oxi- dation, etc.) that grain boundaries and surfaces become important. It is also apparent that grain boundary diffu- sion becomes more competitive, the finer the grain and the lower the temperature. The lattice feature follows from the lower activation energy which makes it less sensitive to temperature change. As the temperature is lowered, the diffusion rate along grain boundaries (and also surfaces) decreases less rapidly than the dif- fusion rate through the lattice. The importance of grain boundary diffusion and dislocation pipe diffusion is discussed again in Chapter 7 in relation to deformation Figure 6.9 Anelastic behaviour. at elevated temperatures, and is demonstrated con- vincingly on the deformation maps (see Figure 7.68), where the creep field is extended to lower temperatures is often used, whereω andω are the frequencies on 1 2 when grain boundary (Coble creep) rather than lattice the two sides of the resonant frequency ω at which 0 p diffusion (Herring–Nabarro creep) operates. the amplitude of oscillation is 1/ 2 of the resonant Because of the strong binding between atoms, pres- amplitude. Also used is the specific damping capacity sure has little or no effect but it is observed that with E/E,where E is the energy dissipated per cycle extremely high pressure on soft metals (e.g. sodium) of vibrational energy E, i.e. the area contained in a an increase in Q may result. The rate of diffusion stress–strain loop. Yet another method uses the phase also increases with decreasing density of atomic pack- angle˛ by which the strain lags behind the stress, and ing. For example, self-diffusion is slower in fcc iron if the damping is small it can be shown that or thallium than in bcc iron or thallium when the results are compared by extrapolation to the transfor- υ 1 E ω ω 2 1 1 tan˛D D D DQ (6.13) mation temperature. This is further emphasized by the  2 E ω 0 anisotropic nature of D in metals of open structure. Bismuth (rhombohedral) is an example of a metal in By analogy with damping in electrical systems tan ˛ 6 1 which D varies by 10 for different directions in the is often written equal toQ . lattice; in cubic crystalsD is isotropic. There are many causes of internal friction arising from the fact that the migration of atoms, lattice defects and thermal energy are all time-dependent 6.5 Anelasticity and internal friction processes. The latter gives rise to thermoelasticity and occurs when an elastic stress is applied to a specimen For an elastic solid it is generally assumed that stress too fast for the specimen to exchange heat with its and strain are directly proportional to one another, but surroundings and so cools slightly. As the sample in practice the elastic strain is usually dependent on warms back to the surrounding temperature it expands time as well as stress so that the strain lags behind the thermally, and hence the dilatation strain continues to stress; this is an anelastic effect. On applying a stress at increase after the stress has become constant. a level below the conventional elastic limit, a specimen The diffusion of atoms can also give rise to will show an initial elastic strain ε followed by a e anelastic effects in an analogous way to the diffusion gradual increase in strain until it reaches an essentially of thermal energy giving thermoelastic effects. A constant value,ε Cε as shown in Figure 6.9. When e an particular example is the stress-induced diffusion of the stress is removed the strain will decrease, but a carbon or nitrogen in iron. A carbon atom occupies small amount remains which decreases slowly with the interstitial site along one of the cell edges slightly time. At any time t the decreasing anelastic strain is distorting the lattice tetragonally. Thus when iron given by the relation εDε expt/ where  is an is stretched by a mechanical stress, the crystal axis known as the relaxation time, and is the time taken oriented in the direction of the stress develops favoured for the anelastic strain to decrease to 1/e' 36.79% of sites for the occupation of the interstitial atoms its initial value. Clearly, if is large, the strain relaxes relative to the other two axes. Then if the stress is very slowly, while if small the strain relaxes quickly. oscillated, such that first one axis and then another is In materials under cyclic loading this anelastic effect stretched, the carbon atoms will want to jump from leads to a decay in amplitude of vibration and therefore one favoured site to the other. Mechanical work is a dissipation of energy by internal friction. Internal therefore done repeatedly, dissipating the vibrational friction is defined in several different but related ways. energy and damping out the mechanical oscillations. Perhaps the most common uses the logarithmic decre- The maximum energy is dissipated when the time per ment υD lnA /A , the natural logarithm of suc- n nC1 cessive amplitudes of vibration. In a forced vibration cycle is of the same order as the time required for the experiment near a resonance, the factorω ω /ω diffusional jump of the carbon atom. 2 1 0The physical properties of materials 177 Figure 6.10 Schematic diagram of aKeO torsion pendulum. The simplest and most convenient way of studying Figure 6.11 Internal friction as a function of temperature this form of internal friction is by means of a KeO for Fe with C in solid solution at five different pendulum torsion pendulum, shown schematically in Figure 6.10. frequencies (from Wert and Zener, 1949; by permission of The specimen can be oscillated at a given frequency the American Institute of Physics). by adjusting the moment of inertia of the torsion bar. The energy loss per cycleE/E varies smoothly with the frequency according to the relation discussed above, and hence occurring in different fre-     quency and temperature regions. One important source E E ω of internal friction is that due to stress relaxation across D 2 2 E E 1Cω max grain boundaries. The occurrence of a strong internal friction peak due to grain boundary relaxation was first and has a maximum value when the angular frequency ° demonstrated on polycrystalline aluminium at 300 C of the pendulum equals the relaxation time of the by Ke ˆ and has since been found in numerous other process; at low temperatures around room temperature metals. It indicates that grain boundaries behave in this is interstitial diffusion. In practice, it is difficult to a somewhat viscous manner at elevated temperatures vary the angular frequency over a wide range and thus and grain boundary sliding can be detected at very low it is easier to keepω constant and vary the relaxation stresses by internal friction studies. The grain boundary time. Since the migration of atoms depends strongly on sliding velocity produced by a shear stress is given temperature according to an Arrhenius-type equation, byDd/ and its measurement gives values of the the relaxation time  D 1/ω and the peak occurs 1 1 viscosity  which extrapolate to that of the liquid at at a temperature T . For a different frequency value 1 the melting point, assuming the boundary thickness to ω the peak occurs at a different temperatureT ,and 2 2 bed' 0.5nm. so on (see Figure 6.11). It is thus possible to ascribe Movement of low-energy twin boundaries in crys- an activation energy H for the internal process tals, domain boundaries in ferromagnetic materials and producing the damping by plotting ln  versus 1/T, or from the relation dislocation bowing and unpinning all give rise to inter- nal friction and damping. lnω /ω  2 1 HDR 1/T  1/T 1 2 In the case of iron the activation energy is found to 6.6 Ordering in alloys coincide with that for the diffusion of carbon in iron. Similar studies have been made for other metals. In 6.6.1 Long-range and short-range order addition, if the relaxation time is  the mean time An ordered alloy may be regarded as being made up 3 an atom stays in an interstitial position is  ,and 2 of two or more interpenetrating sub-lattices, each con- 1 2 from the relation DD a v for bcc lattices derived taining different arrangements of atoms. Moreover, the 24 previously the diffusion coefficient may be calculated term ‘superlattice’ would imply that such a coher- directly from ent atomic scheme extends over large distances, i.e.   2 the crystal possesses long-range order. Such a perfect 1 a DD arrangement can exist only at low temperatures, since 36  the entropy of an ordered structure is much lower than Many other forms of internal friction exist in met- that of a disordered one, and with increasing tempera- als arising from different relaxation processes to those ture the degree of long-range order,S, decreases until178 Modern Physical Metallurgy and Materials Engineering at a critical temperatureT it becomes zero; the general c form of the curve is shown in Figure 6.12. Partially- ordered structures are achieved by the formation of small regions (domains) of order, each of which are separated from each other by domain or anti-phase domain boundaries, across which the order changes phase (Figure 6.13). However, even when long-range order is destroyed, the tendency for unlike atoms to be neighbours still exists, and short-range order results Figure 6.13 An antiphase domain boundary. above T . The transition from complete disorder to c complete order is a nucleation and growth process and may be likened to the annealing of a cold-worked has pointed out that the ease with which interlocking structure. At high temperatures well above T ,there c domains can absorb each other to develop a scheme are more than the random number of AB atom pairs, of long-range order will also depend on the number of and with the lowering of temperature small nuclei possible ordered schemes the alloy possesses. Thus, in of order continually form and disperse in an other- ˇ-brass only two different schemes of order are possi- wise disordered matrix. As the temperature, and hence ble, while in fcc lattices such as Cu Au four different 3 thermal agitation, is lowered these regions of order schemes are possible and the approach to complete become more extensive, until atT they begin to link c order is less rapid. together and the alloy consists of an interlocking mesh of small ordered regions. Below T these domains c 6.6.2 Detection of ordering absorb each other (cf. grain growth) as a result of antiphase domain boundary mobility until long-range The determination of an ordered superlattice is usu- order is established. ally done by means of the X-ray powder technique. In Some order–disorder alloys can be retained in a a disordered solution every plane of atoms is statisti- state of disorder by quenching to room temperature cally identical and, as discussed in Chapter 5, there are while in others (e.g. ˇ-brass) the ordering process reflections missing in the powder pattern of the mate- occurs almost instantaneously. Clearly, changes in the rial. In an ordered lattice, on the other hand, alternate degree of order will depend on atomic migration, so planes become A-rich and B-rich, respectively, so that that the rate of approach to the equilibrium configu- these ‘absent’ reflections are no longer missing but ration will be governed by an exponential factor of appear as extra superlattice lines. This can be seen Q/RT the usual form, i.e. RateDAe . However, Bragg from Figure 6.14: while the diffracted rays from the A planes are completely out of phase with those from theB planes their intensities are not identical, so that a weak reflection results. Application of the structure factor equation indicates that the intensity of the superlattice lines is 2 2 2 proportional to jFjDS f f  , from which A B it can be seen that in the fully-disordered alloy, where SD 0, the superlattice lines must vanish. In some alloys such as copper–gold, the scattering factor difference f f  is appreciable and the A B superlattice lines are, therefore, quite intense and easily detectable. In other alloys, however, such as iron–cobalt, nickel–manganese, copper–zinc, the term f f  is negligible for X-rays and the A B Figure 6.12 Influence of temperature on the degree of order. super-lattice lines are very weak; in copper–zinc, for Figure 6.14 Formation of a weak 100 reflection from an ordered lattice by the interference of diffracted rays of unequal amplitude.The physical properties of materials 179 example, the ratio of the intensity of the superlattice size can be obtained from a measurement of the line lines to that of the main lines is only about 1:3500. breadth, as discussed in Chapter 5. Figure 6.15 shows In some cases special X-ray techniques can enhance variation of order S and domain size as determined from the intensity and breadth of powder diffraction this intensity ratio; one method is to use an X- lines. The domain sizes determined from the Scherrer ray wavelength near to the absorption edge when line-broadening formula are in very good agreement an anomalous depression of the f-factor occurs with those observed by TEM. Short-range order is which is greater for one element than for the other. much more difficult to detect but nowadays direct As a result, the difference between f and f is A B measuring devices allow weak X-ray intensities to be increased. A more general technique, however, is to measured more accurately, and as a result considerable use neutron diffraction since the scattering factors information on the nature of short-range order has for neighbouring elements in the Periodic Table can been obtained by studying the intensity of the diffuse be substantially different. Conversely, as Table 5.4 background between the main lattice lines. indicates, neutron diffraction is unable to show the High-resolution transmission microscopy of thin existence of superlattice lines in Cu Au, because the 3 metal foils allows the structure of domains to be exam- scattering amplitudes of copper and gold for neutrons ined directly. The alloy CuAu is of particular interest, are approximately the same, although X-rays show since it has a face-centred tetragonal structure, often them up quite clearly. ° ° referred to as CuAu 1 below 380 C, but between 380 C Sharp superlattice lines are observed as long as ° and the disordering temperature of 410 Cit has the 3 order persists over lattice regions of about 10 mm, CuAu 11 structures shown in Figure 6.16. The002 large enough to give coherent X-ray reflections. When planes are again alternately gold and copper, but half- long-range order is not complete the superlattice lines way along thea-axis of the unit cell the copper atoms become broadened, and an estimate of the domain switch to gold planes and vice versa. The spacing between such periodic anti-phase domain boundaries is 5 unit cells or about 2 nm, so that the domains are easily resolvable in TEM, as seen in Figure 6.17a. The isolated domain boundaries in the simpler superlat- tice structures such as CuAu 1, although not in this case periodic, can also be revealed by electron micro- scope, and an example is shown in Figure 6.17b. Apart from static observations of these superlattice struc- tures, annealing experiments inside the microscope also allow the effect of temperature on the structure to be examined directly. Such observations have shown that the transition from CuAu 1 to CuAu 11 takes place, as predicted, by the nucleation and growth of anti-phase domains. 6.6.3 Influence of ordering on properties Specific heat The order–disorder transformation has a marked effect on the specific heat, since energy is necessary to change atoms from one configuration to another. However, because the change in lattice arrangement takes place over a range of temperature, the specific heat versus temperature curve will be of the form shown in Figure 6.4b. In practice the excess spe- cific heat, above that given by Dulong and Petit’s law, does not fall sharply to zero atT owing to the exis- c Figure 6.15 Degree of orderð and domain size (O) tence of short-range order, which also requires extra ° during isothermal annealing at 350 C after quenching from ° energy to destroy it as the temperature is increased 465 C (after Morris, Besag and Smallman, 1974; courtesy of Taylor and Francis). aboveT . c Figure 6.16 One unit cell of the orthorhombic superlattice of CuAu, i.e. CuAu 11 (from J. Inst. Metals, 1958–9, courtesy of the Institute of Metals).180 Modern Physical Metallurgy and Materials Engineering contribution to the electrical resistance. Accordingly, superlattices belowT have a low electrical resistance, c but on raising the temperature the resistivity increases, as shown in Figure 6.18a for ordered Cu Au. The 3 influence of order on resistivity is further demonstrated by the measurement of resistivity as a function of com- position in the copper–gold alloy system. As shown in Figure 6.18b, at composition near Cu Au and CuAu, 3 where ordering is most complete, the resistivity is extremely low, while away from these stoichiomet- ric compositions the resistivity increases; the quenched (disordered) alloys given by the dotted curve also have high resistivity values. Mechanical properties The mechanical properties are altered when ordering occurs. The change in yield 0.05µ (a) stress is not directly related to the degree of ordering, however, and in fact Cu Au crystals have a lower yield 3 stress when well-ordered than when only partially- ordered. Experiments show that such effects can be accounted for if the maximum strength as a result of ordering is associated with critical domain size. In the alloy Cu Au, the maximum yield strength is exhibited 3 by quenched samples after an annealing treatment of 5 ° min at 350 C which gives a domain size of 6 nm (see Figure 6.15). However, if the alloy is well-ordered and the domain size larger, the hardening is insignificant. In some alloys such as CuAu or CuPt, ordering produces a change of crystal structure and the resultant lattice strains can also lead to hardening. Thermal agitation is the most common means of destroying long-range order, but other methods (e.g. deformation) are equally effective. Figure 6.18c shows that cold work has a 0.05µ (b) negligible effect upon the resistivity of the quenched (disordered) alloy but considerable influence on the Figure 6.17 Electron micrographs of (a) CuAu 11 and well-annealed (ordered) alloy. Irradiation by neutrons (b) CuAu 1 (from Pashley and Presland, 1958–9; courtesy or electrons also markedly affects the ordering (see of the Institute of Metals). Chapter 4). Electrical resistivity As discussed in Chapter 4, any Magnetic properties The order–disorder pheno- form of disorder in a metallic structure (e.g. impuri- menon is of considerable importance in the application ties, dislocations or point defects) will make a large of magnetic materials. The kind and degree of order Figure 6.18 Effect of (a) temperature, (b) composition, and (c) deformation on the resistivity of copper–gold alloys (after Barrett, 1952; courtesy of McGraw-Hill).The physical properties of materials 181 affects the magnetic hardness, since small ordered bismuth a poor conductor in the solid state is destroyed regions in an otherwise disordered lattice induce on melting. strains which affect the mobility of magnetic domain In most metals the resistance approaches zero at boundaries (see Section 6.8.4). absolute zero, but in some (e.g. lead, tin and mer- cury) the resistance suddenly drops to zero at some finite critical temperature above 0 K. Such metals are called superconductors. The critical temperature is dif- 6.7 Electrical properties ferent for each metal but is always close to absolute zero; the highest critical temperature known for an ele- 6.7.1 Electrical conductivity ment is 8 K for niobium. Superconductivity is now One of the most important electronic properties of met- observed at much higher temperatures in some inter- als is the electrical conductivity, , and the reciprocal metallic compounds and in some ceramic oxides (see of the conductivity (known as the resistivity, )is Section 6.7.4). defined by the relationRDl/A,whereR is the resis- An explanation of electrical and magnetic properties tance of the specimen, l is the length and A is the requires a more detailed consideration of electronic cross-sectional area. structure than that briefly outlined in Chapter 1. There A characteristic feature of a metal is its high electri- the concept of band structure was introduced and the cal conductivity which arises from the ease with which electron can be thought of as moving continuously the electrons can migrate through the lattice. The high through the structure with an energy depending on the thermal conduction of metals also has a similar expla- energy level of the band it occupies. The wave-like nation, and the Wiedmann–Franz law shows that the properties of the electron were also mentioned. For the ratio of the electrical and thermal conductivities is electrons the regular array of atoms on the metallic lattice can behave as a three-dimensional diffraction nearly the same for all metals at the same temperature. grating since the atoms are positively-charged and Since conductivity arises from the motion of con- interact with moving electrons. At certain wavelengths, duction electrons through the lattice, resistance must be governed by the spacing of the atoms on the metallic caused by the scattering of electron waves by any kind lattice, the electrons will experience strong diffraction of irregularity in the lattice arrangement. Irregularities effects, the results of which are that electrons having can arise from any one of several sources, such as tem- energies corresponding to such wavelengths will be perature, alloying, deformation or nuclear irradiation, unable to move freely through the structure. As a since all will disturb, to some extent, the periodicity consequence, in the bands of electrons, certain energy of the lattice. The effect of temperature is particularly levels cannot be occupied and therefore there will be important and, as shown in Figure 6.19, the resistance energy gaps in the otherwise effectively continuous increases linearly with temperature above about 100 K energy spectrum within a band. up to the melting point. On melting, the resistance The interaction of moving electrons with the metal increases markedly because of the exceptional disor- ions distributed on a lattice depends on the wavelength der of the liquid state. However, for some metals such of the electrons and the spacing of the ions in the as bismuth, the resistance actually decreases, owing direction of movement of the electrons. Since the ionic to the fact that the special zone structure which makes spacing will depend on the direction in the lattice, the wavelength of the electrons suffering diffraction by the ions will depend on their direction. The kinetic energy of a moving electron is a function of the wavelength according to the relationship 2 2 ED h /2m (6.14) Since we are concerned with electron energies, it is more convenient to discuss interaction effects in terms of the reciprocal of the wavelength. This quantity is called the wave number and is denoted by k. In describing electron–lattice interactions it is usual to make use of a vector diagram in which the direction of the vector is the direction of motion of the moving electron and its magnitude is the wave number of the electron. The vectors representing electrons having energies which, because of diffraction effects, cannot penetrate the lattice, trace out a three-dimensional surface known as a Brillouin zone. Figure 6.20a shows such a zone for a face-centred cubic lattice. It is made Figure 6.19 Variation of resistivity with temperature. up of plane faces which are, in fact, parallel to the most182 Modern Physical Metallurgy and Materials Engineering Figure 6.20 Schematic representation of a Brillouin zone in a metal. widely-spaced planes in the lattice, i.e. in this case the f111g and f200g planes. This is a general feature of Brillouin zones in all lattices. For a given direction in the lattice, it is possible to consider the form of the electron energies as a function of wave number. The relationship between the two quantities as given from equation (6.14) is 2 2 ED h k /2m (6.15) which leads to the parabolic relationship shown as a broken line in Figure 6.20b. Because of the existence of a Brillouin zone at a certain value of k, depending on the lattice direction, there exists a range of energy values which the electrons cannot assume. This pro- duces a distortion in the form of the E-k curve in the neighbourhood of the critical value of k and leads to the existence of a series of energy gaps, which cannot be occupied by electrons. The E-k curve showing this Figure 6.21 Schematic representation of Brillouin zones. effect is given as a continuous line in Figure 6.20b. The existence of this distortion in the E-k curve, In Figure 6.21a the two zones are separated by an due to a Brillouin zone, is reflected in the density energy gap, but in real metals this is not necessarily of states versus energy curve for the free electrons. the case, and two zones can overlap in energy in the As previously stated, the density of states–energy N(E)-E curves so that no such energy gaps appear. curve is parabolic in shape, but it departs from this This overlap arises from the fact that the energy of form at energies for which Brillouin zone interactions the forbidden region varies with direction in the lattice occur. The result of such interactions is shown in and often the energy level at the top of the first zone Figure 6.21a in which the broken line represents the has a higher value in one direction than the lowest N(E)-E curve for free electrons in the absence of energy level at the bottom of the next zone in some zone effects and the full line is the curve where a other direction. The energy gap in the N(E)-E curves, zone exists. The total number of electrons needed to which represent the summation of electronic levels in fill the zone of electrons delineated by the full line all directions, is then closed (Figure 6.21b). in Figure 6.21a is 2N,where N is the total number For electrical conduction to occur, it is necessary of atoms in the metal. Thus, a Brillouin zone would that the electrons at the top of a band should be be filled if the metal atoms each contributed two able to increase their energy when an electric field is electrons to the band. If the metal atoms contribute applied to materials so that a net flow of electrons in more than two per atom, the excess electrons must be accommodated in the second or higher zones. the direction of the applied potential, which manifestsThe physical properties of materials 183 itself as an electric current, can take place. If an there are no unfilled levels at the top of the d-band energy gap between two zones of the type shown into which electrons can go, and consequently both in Figure 6.21a occurs, and if the lower zone is just the electronic specific heat and electrical resistance is filled with electrons, then it is impossible for any low. The conductivity also depends on the degree to electrons to increase their energy by jumping into which the electrons are scattered by the ions of the vacant levels under the influence of an applied electric metal which are thermally vibrating, and by impurity field, unless the field strength is sufficiently great to atoms or other defects present in the metal. supply the electrons at the top of the filled band with Insulators can also be modified either by the applica- enough energy to jump the energy gap. Thus metallic tion of high temperatures or by the addition of impu- conduction is due to the fact that in metals the number rities. Clearly, insulators may become conductors at of electrons per atom is insufficient to fill the band up elevated temperatures if the thermal agitation is suffi- to the point where an energy gap occurs. In copper, for cient to enable electrons to jump the energy gap into example, the 4s valency electrons fill only one half of the unfilled zone above. the outers-band. In other metals (e.g. Mg) the valency band overlaps a higher energy band and the electrons 6.7.2 Semiconductors near the Fermi level are thus free to move into the Some materials have an energy gap small enough empty states of a higher band. When the valency band to be surmounted by thermal excitation. In such is completely filled and the next higher band, separated intrinsic semiconductors, as they are called, the current by an energy gap, is completely empty, the material is carriers are electrons in the conduction band and either an insulator or a semiconductor. If the gap is holes in the valency band in equal numbers. The several electron volts wide, such as in diamond where relative position of the two bands is as shown in it is 7 eV, extremely high electric fields would be Figure 6.22. The motion of a hole in the valency necessary to raise electrons to the higher band and the band is equivalent to the motion of an electron in material is an insulator. If the gap is small enough, the opposite direction. Alternatively, conduction may such as 1–2 eV as in silicon, then thermal energy be produced by the presence of impurities which may be sufficient to excite some electrons into the either add a few electrons to an empty zone or higher band and also create vacancies in the valency remove a few from a full one. Materials which band, the material is a semiconductor. In general, the have their conductivity developed in this way are lowest energy band which is not completely filled with commonly known as semiconductors. Silicon and electrons is called a conduction band, and the band germanium containing small amounts of impurity have containing the valency electrons the valency band. For semiconducting properties at ambient temperatures a conductor the valency band is also the conduction and, as a consequence, they are frequently used in band. The electronic state of a selection of materials electronic transistor devices. Silicon normally has of different valencies is presented in Figure 6.21c. completely filled zones, but becomes conducting if Although all metals are relatively good conductors of some of the silicon atoms, which have four valency electricity, they exhibit among themselves a range electrons, are replaced by phosphorus, arsenic or of values for their resistivities. There are a number of antimony atoms which have five valency electrons. reasons for this variability. The resistivity of a metal The extra electrons go into empty zones, and as a depends on the density of states of the most energetic electrons at the top of the band, and the shape of the N(E)-E curve at this point. In the transition metals, for example, apart from pro- ducing the strong magnetic properties, great strength and high melting point, the d-band is also responsi- ble for the poor electrical conductivity and high elec- tronic specific heat. When an electron is scattered by a lattice irregularity it jumps into a different quan- tum state, and it will be evident that the more vacant quantum states there are available in the same energy range, the more likely will be the electron to deflect at the irregularity. The high resistivities of the transi- tion metals may, therefore, be explained by the ease with which electrons can be deflected into vacant d- states. Phonon-assisted s-d scattering gives rise to the non-linear variation of  with temperature observed at high temperatures. The high electronic specific heat is also due to the high density of states in the unfilled d- band, since this gives rise to a considerable number of Figure 6.22 Schematic diagram of an intrinsic electrons at the top of the Fermi distribution which can semiconductor showing the relative positions of the be excited by thermal activation. In copper, of course, conduction and valency bands.184 Modern Physical Metallurgy and Materials Engineering result silicon becomes an n-type semiconductor, since Trivalent impurities in Si or Ge show the opposite conduction occurs by negative carriers. On the other behaviour leaving an empty electron state, or hole, hand, the addition of elements of lower valency than in the valency band. If the hole separates from the so-called acceptor atom an electron is excited from silicon, such as aluminium, removes electrons from the valency band to an acceptor level E³ 0.01 eV. the filled zones leaving behind ‘holes’ in the valency Thus, with impurity elements such as Al, Ga or In band structure. In this case silicon becomes a p-type creating holes in the valency band in addition to those semiconductor, since the movement of electrons in one created thermally, the majority carriers are holes and direction of the zone is accompanied by a movement the semiconductor is of the p-type extrinsic form of ‘holes’ in the other, and consequently they act (see Figure 6.23b). For a semiconductor where both as if they were positive carriers. The conductivity electrons and holes carry current the conductivity is may be expressed as the product of (1) the number given by of charge carriers, n, (2) the charge carried by each 19 (i.e. eD 1.6ð 10 C) and (3) the mobility of the Dn e Cn e (6.16) e e h h carrier, . A pentavalent impurity which donates conduction where n and n are, respectively, the volume con- e h electrons without producing holes in the valency band centration of electrons and holes, and  and  the e h is called a donor. The spare electrons of the impurity mobilities of the carriers, i.e. electrons and holes. atoms are bound in the vicinity of the impurity atoms Semiconductor materials are extensively used in electronic devices such as the p–n rectifying junction, in energy levels known as the donor levels, which transistor (a double-junction device) and the tunnel are near the conduction band. If the impurity exists diode. Semiconductor regions of either p-or n-type in an otherwise intrinsic semiconductor the number of can be produced by carefully controlling the distribu- electrons in the conduction band become greater than tion and impurity content of Si or Ge single crystals, the number of holes in the valency band and, hence, and the boundary between p-and n-type extrinsic the electrons are the majority carriers and the holes the semiconductor materials is called a p–n junction. Such minority carriers. Such a material is ann-type extrinsic a junction conducts a large current when the voltage is semiconductor (see Figure 6.23a). applied in one direction, but only a very small cur- rent when the voltage is reversed. The action of a p–n junction as a rectifier is shown schematically in Figure 6.24. The junction presents no barrier to the flow of minority carriers from either side, but since the concentration of minority carriers is low, it is the flow of majority carriers which must be considered. When the junction is biased in the forward direction, i.e. n- type made negative and thep-type positive, the energy barrier opposing the flow of majority carriers from both sides of the junction is reduced. Excess majority car- riers enter the p and n regions, and these recombine continuously at or near the junction to allow large cur- rents to flow. When the junction is reverse-biased, the energy barrier opposing the flow of majority carriers is raised, few carriers move and little current flows. A transistor is essentially a single crystal with two p–n junctions arranged back to back to give either a p–n–p or n–p–n two-junction device. For a p–n–p device the main current flow is provided by the positive holes, while for a n–p–n device the electrons carry the current. Connections are made to the individual Figure 6.23 Schematic energy band structure of (a) n-type and (b) p-type semiconductor. regions of the p–n–p device, designated emitter, base Figure 6.24 Schematic illustration of p–n junction rectification with (a) forward bias and (b) reverse bias.The physical properties of materials 185 with high-voltage equipment and can protect it from transient voltage ‘spikes’ or overload. 6.7.3 Superconductivity At low temperatures (20 K) some metals have zero electrical resistivity and become superconductors. This superconductivity disappears if the temperature of the metal is raised above a critical temperature T , c if a sufficiently strong magnetic field is applied or when a high current density flows. The critical field Figure 6.25 Schematic diagram of a p–n–p transistor. strengthH , current densityJ and temperature T are c c c interdependent. Figure 6.26 shows the dependence of H on temperature for a number of metals; metals with c and collector respectively, as shown in Figure 6.25, high T and H values, which include the transition c c and the base is made slightly negative and the collector elements, are known as hard superconductors, those more negative relative to the emitter. The emitter- with low values such as Al, Zn, Cd, Hg, white-Sn are base junction is therefore forward-biased and a strong soft superconductors. The curves are roughly parabolic current of holes passes through the junction into the 2 and approximate to the relationH DH 1T/T  c 0 c 2 n-layer which, because it is thin (10 mm), largely where H is the critical field at 0 K; H is about 0 0 reach the collector base junction without recombining 5 1.6ð 10 A/m for Nb. with electrons. The collector-base junction is reverse- Superconductivity arises from conduction elec- biased and the junction is no barrier to the passage of tron–electron attraction resulting from a distortion of holes; the current through the second junction is thus the lattice through which the electrons are travelling; controlled by the current through the first junction. this is clearly a weak interaction since for most metals A small increase in voltage across the emitter-base it is destroyed by thermal activation at very low tem- junction produces a large injection of holes into the peratures. As the electron moves through the lattice base and a large increase in current in the collector, to it attracts nearby positive ions thereby locally caus- give the amplifying action of the transistor. ing a slightly higher positive charge density. A nearby Many varied semiconductor materials such as InSb electron may in turn be attracted by the net positive and GaAs have been developed apart from Si and Ge. charge, the magnitude of the attraction depending on However, in all cases very high purity and crystal the electron density, ionic charge and lattice vibrational perfection is necessary for efficient semiconducting frequencies such that under favourable conditions the operations and to produce the material, zone-refining effect is slightly stronger than the electrostatic repul- techniques are used. Semiconductor integrated circuits sion between electrons. The importance of the lattice are extensively used in micro-electronic equipment ions in superconductivity is supported by the obser- and these are produced by vapour deposition through vation that different isotopes of the same metal (e.g. masks on to a single Si-slice, followed by diffusion of Sn and Hg) have different T values proportional to c the deposits into the base crystal. 1/2 M ,where M is the atomic mass of the isotope. Doped ceramic materials are used in the construc- Since both the frequency of atomic vibrations and tion of thermistors, which are semiconductor devices 1/2 the velocity of elastic waves also varies as M , with a marked dependence of electrical resistivity upon the interaction between electrons and lattice vibrations temperature. The change in resistance can be quite significant at the critical temperature. Positive temper- ature coefficient (PTC) thermistors are used as switch- ing devices, operating when a control temperature is reached during a heating process. PTC thermistors are commonly based on barium titanate. Conversely, NTC thermistors are based on oxide ceramics and can be used to signal a desired temperature change during cooling; the change in resistance is much more gradual and does not have the step-characteristic of the PTC types. Doped zinc oxide does not exhibit the linear volt- age/current relation that one expects from Ohm’s Law. At low voltage, the resistivity is high and only a small current flows. When the voltage increases there is a sudden decrease in resistance, allowing a heavier cur- rent to flow. This principle is adopted in the varistor, Figure 6.26 Variation of critical field H as a function of c a voltage-sensitive on/off switch. It is wired in parallel temperature for several pure metal superconductors.186 Modern Physical Metallurgy and Materials Engineering (i.e. electron–phonon interaction) must be at least one change in density of states with e/a ratio. Supercon- cause of superconductivity. ductivity is thus favoured in compounds of polyvalent The theory of superconductivity indicates that the atoms with crystal structures having a high density of electron–electron attraction is strongest between elec- states at the Fermi surface. Compounds with high T c trons in pairs, such that the resultant momentum of values, such as Nb Sn (18.1 K), Nb Al (17.5 K), V Si 3 3 3 each pair is exactly the same and the individual elec- (17.0 K), V Ga (16.8 K), all crystallize with the ˇ- 3 trons of each pair have opposite spin. With this partic- tungsten structure and have an e/a ratio close to 4.7; ular form of ordering the total electron energy (i.e. T is very sensitive to the degree of order and to devi- c kinetic and interaction) is lowered and effectively ation from the stoichiometric ratio, so values probably introduces a finite energy gap between this organized correspond to the non-stoichiometric condition. state and the usual more excited state of motion. The The magnetic behaviour of superconductivity is as gap corresponds to a thin shell at the Fermi surface, remarkable as the corresponding electrical behaviour, but does not produce an insulator or semiconductor, as shown in Figure 6.28 by the Meissner effect for because the application of an electric field causes the an ideal (structurally perfect) superconductor. It is whole Fermi distribution, together with gap, to drift observed for a specimen placed in a magnetic field to an unsymmetrical position, so causing a current to HH , which is then cooled down below T ,that c c flow. This current remains even when the electric field magnetic lines of force are pushed out. The specimen is removed, since the scattering which is necessary to is a perfect diamagnetic material with zero inductance alter the displaced Fermi distribution is suppressed. as well as zero resistance. Such a material is termed At 0 K all the electrons are in paired states but as an ideal type I superconductor. An ideal type II super- the temperature is raised, pairs are broken by thermal conductor behaves similarly at low field strengths, with activation giving rise to a number of normal electrons HH H , but then allows a gradual penetration cl c in equilibrium with the superconducting pairs. With of the field returning to the normal state when pen- increasing temperature the number of broken pairs etration is complete atHH H . In detail, the c2 c increases until atT they are finally eliminated together c field actually penetrates to a small extent in type I with the energy gap; the superconducting state then superconductors when it is below H and in type II c reverts to the normal conducting state. The supercon- superconductors when H is below H , and decays cl ductivity transition is a second-order transformation away at a penetration depth³100–10 nm. 2 and a plot of C/T as a function of T deviates from The observation of the Meissner effect in type I the linear behaviour exhibited by normal conducting superconductors implies that the surface between the metals, the electronic contribution being zero at 0 K. normal and superconducting phases has an effective The main theory of superconductivity, due to Bardeen, positive energy. In the absence of this surface energy, Cooper and Schrieffer (BCS) attempts to relate T to c the specimen would break up into separate fine regions the strength of the interaction potential, the density of superconducting and normal material to reduce the of states at the Fermi surface and to the average fre- work done in the expulsion of the magnetic flux. A quency of lattice vibration involved in the scattering, negative surface energy exists between the normal and provides some explanation for the variation of T c and superconducting phases in a type II superconduc- with thee/a ratio for a wide range of alloys, as shown tor and hence the superconductor exists naturally in in Figure 6.27. The main effect is attributable to the a state of finely-separated superconducting and nor- mal regions. By adopting a ‘mixed state’ of normal and superconducting regions the volume of interface is maximized while at the same time keeping the volume Figure 6.27 The variation of T with position in the c periodic table (from Mathias, 1959, p. 138; courtesy of Figure 6.28 The Meissner effect; shown by the expulsion of North-Holland Publishing Co.). magnetic flux when the specimen becomes superconducting.The physical properties of materials 187 of normal conduction as small as possible. The struc- 6.7.4 Oxide superconductors ture of the mixed state is believed to consist of lines In 1986 a new class of ‘warm’ superconductors, based of normal phases parallel to the applied field through on mixed ceramic oxides, was discovered by J. G. which the field lines run, embedded in a supercon- Bednorz and K. A. Muller. ¨ These lanthanum–copper ducting matrix. The field falls off with distances from oxide superconductors had a T around 35 K, well c the centre of each line over the characteristic distance above liquid hydrogen temperature. Since then, three , and vortices or whirlpools of supercurrents flow mixed oxide families have been developed with much around each line; the flux line, together with its cur- higher T values, all around 100 K. Such materials c rent vortex, is called a fluxoid. AtH , fluxoids appear c1 give rise to optimism for superconductor technology; in the specimen and increase in number as the mag- first, in the use of liquid nitrogen rather than liquid netic field is raised. AtH , the fluxoids completely fill c2 hydrogen and second, in the prospect of producing a the cross-section of the sample and type II supercon- room temperature superconductor. ductivity disappears. Type II superconductors are of The first oxide family was developed by mixing particular interest because of their high critical fields and heating the three oxides Y O , BaO and CuO. 2 3 which makes them potentially useful for the construc- This gives rise to the mixed oxide YBa Cu O , 2 3 7x tion of high-field electromagnetics and solenoids. To sometimes referred to as 1–2–3 compound or YBCO. produce a magnetic field of ³10 T with a conven- The structure is shown in Figure 6.30 and is basically tional solenoid would cost more than ten times that of made by stacking three perovskite-type unit cells one a superconducting solenoid wound with Nb Sn wire. 3 above the other; the top and bottom cells have barium By embedding Nb wire in a bronze matrix it is pos- ions at the centre and copper ions at the corners, the sible to form channels of Nb Sn by interdiffusion. 3 middle cell has yttrium at the centre. Oxygen ions sit The conventional installation would require consid- half-way along the cell edges but planes, other than erable power, cooling water and space, whereas the those containing barium, have some missing oxygen superconducting solenoid occupies little space, has no ions (i.e. vacancies denoted byx in the oxide formula). steady-state power consumption and uses relatively This structure therefore has planes of copper and little liquid helium. It is necessary, however, for the oxygen ions containing vacancies, and copper–oxygen material to carry useful currents without resistance ion chains perpendicular to them. YBCO has a T c in such high fields, which is not usually the case in value of about 90 K which is virtually unchanged annealed homogeneous type II superconductors. For- when yttrium is replaced by other rare earth elements. tunately, the critical current density is extremely sen- The second family of oxides are Bi–Ca–Sr–Cu–O x sitive to microstructure and is markedly increased by materials with the metal ions in the ratio of 2111, precipitation-hardening, cold work, radiation damage, 2122 or 2223, respectively. The 2111 oxide has only etc., because the lattice defects introduced pin the flux- one copper–oxygen layer between the bismuth-oxygen oids and tend to immobilize them. Figure 6.29 shows layers, the 2122 two and the 2223 three giving rise to the influence of metallurgical treatment on the critical an increasing T up to about 105 K. The third family c current density. is based on Tl–Ca–Ba–Cu–O with a 2223 structure having three copper–oxygen layers and a T of about c 125 K. While these oxide superconductors have high T c values and high critical magnetic field (H )-values, c they unfortunately have very low values of J,the c critical current density. A high J is required if they c are to be used for powerful superconducting magnets. Electrical applications are therefore unlikely until the J value can be raised by several orders of magni- c tude comparable to those of conventional supercon- 6 2 ductors, i.e. 10 Acm . The reason for the low J c is thought to be largely due to the grain boundaries in polycrystalline materials, together with dislocations, voids and impurity particles. Single crystals show J c 5 2 values around 10 Acm and textured materials, pro- 4 2 duced by melt growth techniques, about 10 Acm , but both processes have limited commercial applica- tion. Electronic applications appear to be more promis- ing since it is in the area of thin (1 µ m) films that high J values have been obtained. By careful deposi- c Figure 6.29 The effect of processing on the J versus H c tion control, epitaxial and single-crystal films having curve of an Nb–25% Zr alloy wire which produces a fine 6 2 J × 10 Acm with low magnetic field dependence precipitate and raises J (from Rose, Shepard and Wulff, c c 1966; courtesy of John Wiley and Sons). have been produced.