Influence of ordering on properties

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,

Figure 6.15 Degree of order ⊲ð⊳ and domain size (O) does not fall sharply to zero at T c owing to the exis- during isothermal annealing at 350 °

tence of short-range order, which also requires extra 465 °

C after quenching from

C (after Morris, Besag and Smallman, 1974; courtesy energy to destroy it as the temperature is increased of Taylor and Francis) .

above T 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 below T c have a low electrical resistance, but on raising the temperature the resistivity increases, as shown in Figure 6.18a for ordered Cu 3 Au. The 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 3 Au and CuAu, 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

(a)

are altered when ordering occurs. The change in yield stress is not directly related to the degree of ordering,

however, and in fact Cu 3 Au crystals have a lower yield 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 3 Au, the maximum yield strength is exhibited 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

(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

of the Institute of Metals) or electrons also markedly affects the ordering (see

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

6.7.1 Electrical conductivity

zero; the highest critical temperature known for an ele- 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- metallic compounds and in some ceramic oxides (see Section 6.7.4).

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 nearly the same for all metals at the same temperature.

lattice can behave as a three-dimensional diffraction Since conductivity arises from the motion of con-

grating since the atoms are positively-charged and duction electrons through the lattice, resistance must be

interact with moving electrons. At certain wavelengths, caused by the scattering of electron waves by any kind

governed by the spacing of the atoms on the metallic of irregularity in the lattice arrangement. Irregularities

lattice, the electrons will experience strong diffraction can arise from any one of several sources, such as tem-

effects, the results of which are that electrons having perature, alloying, deformation or nuclear irradiation,

energies corresponding to such wavelengths will be since all will disturb, to some extent, the periodicity

unable to move freely through the structure. As a of the lattice. The effect of temperature is particularly

consequence, in the bands of electrons, certain energy important and, as shown in Figure 6.19, the resistance

levels cannot be occupied and therefore there will be increases linearly with temperature above about 100 K

energy gaps in the otherwise effectively continuous up to the melting point. On melting, the resistance

energy spectrum within a band. increases markedly because of the exceptional disor-

The interaction of moving electrons with the metal der of the liquid state. However, for some metals such

ions distributed on a lattice depends on the wavelength as bismuth, the resistance actually decreases, owing

of the electrons and the spacing of the ions in the to the fact that the special zone structure which makes

direction of movement of the electrons. Since the ionic 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 (6.14) Since we are concerned with electron energies, it is

EDh 2 /

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 most

182 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

f 1 1 1g and f2 0 0g 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

EDh 2 k 2 / 2m

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, due to a Brillouin zone, is reflected in the density

In Figure 6.21a the two zones are separated by an of states versus energy curve for the free electrons.

energy gap, but in real metals this is not necessarily As previously stated, the density of states–energy

the case, and two zones can overlap in energy in the curve is parabolic in shape, but it departs from this

N(E)-E curves so that no such energy gaps appear. form at energies for which Brillouin zone interactions

This overlap arises from the fact that the energy of occur. The result of such interactions is shown in

the forbidden region varies with direction in the lattice Figure 6.21a in which the broken line represents the

and often the energy level at the top of the first zone N(E)-E curve for free electrons in the absence of

has a higher value in one direction than the lowest zone effects and the full line is the curve where a

energy level at the bottom of the next zone in some zone exists. The total number of electrons needed to

other direction. The energy gap in the N(E)-E curves, fill the zone of electrons delineated by the full line

which represent the summation of electronic levels in in Figure 6.21a is 2N, where N is the total number

all directions, is then closed (Figure 6.21b). of atoms in the metal. Thus, a Brillouin zone would

For electrical conduction to occur, it is necessary

be filled if the metal atoms each contributed two that the electrons at the top of a band should be electrons to the band. If the metal atoms contribute

able to increase their energy when an electric field is more than two per atom, the excess electrons must be

applied to materials so that a net flow of electrons in accommodated in the second or higher zones.

the direction of the applied potential, which manifests

The physical properties of materials 183 itself as an electric current, can take place. If an

energy gap between two zones of the type shown in Figure 6.21a occurs, and if the lower zone is just filled with electrons, then it is impossible for any electrons to increase their energy by jumping into vacant levels under the influence of an applied electric field, unless the field strength is sufficiently great to supply the electrons at the top of the filled band with enough energy to jump the energy gap. Thus metallic conduction is due to the fact that in metals the number of electrons per atom is insufficient to fill the band up to the point where an energy gap occurs. In copper, for example, the 4s valency electrons fill only one half of the outer s-band. In other metals (e.g. Mg) the valency band overlaps a higher energy band and the electrons near the Fermi level are thus free to move into the empty states of a higher band. When the valency band is completely filled and the next higher band, separated by an energy gap, is completely empty, the material is either an insulator or a semiconductor. If the gap is several electron volts wide, such as in diamond where it is 7 eV, extremely high electric fields would be necessary to raise electrons to the higher band and the material is an insulator. If the gap is small enough, such as 1–2 eV as in silicon, then thermal energy may be sufficient to excite some electrons into the higher band and also create vacancies in the valency band, the material is a semiconductor. In general, the lowest energy band which is not completely filled with electrons is called a conduction band, and the band containing the valency electrons the valency band. For

a conductor the valency band is also the conduction band. The electronic state of a selection of materials of different valencies is presented in Figure 6.21c. Although all metals are relatively good conductors of electricity, they exhibit among themselves a range of values for their resistivities. There are a number of reasons for this variability. The resistivity of a metal 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

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 electrons at the top of the Fermi distribution which can

be excited by thermal activation. In copper, of course,

there are no unfilled levels at the top of the d-band into which electrons can go, and consequently both the electronic specific heat and electrical resistance is low. The conductivity also depends on the degree to which the electrons are scattered by the ions of the metal which are thermally vibrating, and by impurity atoms or other defects present in the metal.

Insulators can also be modified either by the applica- tion of high temperatures or by the addition of impu- rities. Clearly, insulators may become conductors at elevated temperatures if the thermal agitation is suffi- cient to enable electrons to jump the energy gap into the unfilled zone above.