Showing papers in "Philosophical Magazine in 1965"

A theory of the electrical properties of liquid metals

Abstract: The simple theory of resistivity originally applied to pure liquid metals (Ziman 1961, Bradley et al. 1962) is formally extended to liquid alloys. In dilute solutions, size difference between solute and solvent ions can be allowed for, approximately, by a modification of the pseudo-potential of the solute, in a manner reminiscent of the modification of solute valency suggested by Harrison and Blatt (1957) in dealing with the resistance of solid alloys. The theory explains qualitatively a number of interesting features of the behaviour of liquid binary alloys, e.g. the failure of liquid solutions in polyvalent metals to obey the familiar rules of Nordheim and Linde. Some speculations on the anomalous behaviour of mercury amalgams are included. A bibliography of experimental results for both resistivity and density in liquid alloys is collected in Appendix II.

The screened model potential for 25 elements

Abstract: The Fourier transform of the self-consistent screened model potential has been calculated for 25 elements. The results are presented in a form applicable to the potential in the solid or liquid metal or semiconductor, or to the electron-phonon interaction. They are reliable to about 0.01 ryd. The calculations are a continuation of the work by Heine and Abarenkov (1964), using the screening theory of Animalu (1965 a). The behaviour at high wave numbers and other points of detail are discussed much more thoroughly than before. Some average band effective masses are also calculated.

The model potential for positive ions

Abstract: Some experience is reported of numerical calculations with atomic pseudo-potentials in the sense of Austin et al. (1962). The pseudo-potentials and/or the pseudo-wave-functions are not sufficiently smooth to give really good convergence to an expansion of the model wave-function in a crystal in terms of plane waves. The pseudo-potential of an ion is therefore replaced by a simple model potential with two parameters. These are adjusted to reproduce exactly the observed spectroscopic energy levels of an electron in the field of the ion. In this way a model potential is set up to represent the ion, useful for calculations in solids and liquids. Some numerical results are given and more extensive tables are available from the authors.

Work hardening and deformation sub-structure in iron single crystals deformed in tension at 298°k

Abstract: Single crystal strips of Ferrovac E iron with various orientations were grown by a strain anneal method then strained various amounts in tension at room temperature. Slip lines on the polished surf...

Magnetic properties of dispersed magnetite powders

Abstract: Measurements have been made of coercive force, susceptibility, isothermal remanence, thermoremanence and thermal and alternating field demagnetization of annealed magnetite powders in nine size ranges from 1·5 μ to 120 μ. The samples were dispersed in plaster to occupy a few per cent by volume of the specimens and thus to simulate the magnetic properties of rocks. The results are compared with the theories of rock magnetism. Susceptibility agrees precisely with theory and saturation remanence has the same dependence on coercive force as the theory. The observed values of thermoremanence are smaller than expected but show a size dependence consistent with the predicted appearance of pseudo-single domain effects at small grain size.

Non-local dielectric screening in metals

Abstract: The potential seen by an electron in a metal is frequently set up by starting with the pseudo-potential or model potential of the bare ion, and dividing by a dielectric constant to take account of the screening of the conduction electrons. Such a procedure assumes that the pseudo-potential is a ‘local’ potential. But in reality the pseudo-potential is a non-local operator, and the theory is extended in the present paper to take this correctly into account. Numerical calculations have been made with the model potential of Heine and Abarenkov (1964). The results are in somewhat better agreement with experiments than those of the local screening approximation, but the difference between them is not as large as might have been expected.

On the formation of interstitial loops in b.c.c. metals

Abstract: Two types of interstitial loops with Burgers vector b = a/2 and b = a , respectively, have been observed experimentally in irradiated b.c.c. metals. The present paper discusses how these loops can form from a common loop nucleus consisting of a single platelet of interstitials on a {110} plane. Such a loop contains a stacking fault which is removed by one of two shears described by the following dislocation reactions: It is suggested that the latter higher energy, shear, requires thermal energy to assist it and therefore only occurs at elevated temperatures. Consideration is given to the elastic energy of the loops with particular reference to their shape and orientation. The calculations predict that the loops are initially rectilinear and as they grow they should change to the circular form and rotate into a pure edge orientation. The crystallographic symmetry of the a loops and their growth mechanism probably explains the observed retention of the rectilinear form.

Slip line and dislocation structures in fatigued copper

Abstract: Large grained long life push-pull fatigue specimens of 99-999% purity copper have been examined at 10% and 50% of the expected life. Carbon replicas of slip striations were examined, interference microscopy was used to measure the displacement across the striations and dislocation structure was examined by transmission electron microscopy. The total length of slip striation in a given area was measured as a function of fatigue life. After polishing the surface the pattern of slip striations which appeared when cycling was resumed was identical to that removed by polishing. The observations indicate that the slip striations are produced by a very large number of related dislocation sources situated in the planes of the striations. Intrusions and extrusions may be built up by the operation of two dislocation sources near the surface on closely adjacent parallel planes, one operated by a tensile stress and the other by the compressive stress. It is suggested that annealing a specimen after half the ...

The thermal equilibrium shape and size of holes in solids

Abstract: The generalized equations of state governing the attainment of equilibrium have been formulated for bubbles (holes containing gas under pressure) and for voids (holes containing no gas). The equilibrium shape of bubbles is shown to be that which produces the minimum surface energy (γ) for a specific volume, and is uniquely defined by the polar plot of surface energy with orientation (the γ plot). Examination of bubbles using the electron microscope has shown that their equilibrium shape is polyhedral, with faces formed from low index crystallographic planes. Measurements on such bubbles which should have relatively clean internal surfaces have provided information on the orientation dependence of surface energy: CU(600°c, γ100/γ110= 1·2, Al(550°c),γ100/γ110= 0·98;γ100/γ111= 1·13, MO(2000°c),γ100/γ110= 1·14. The equilibrium shape of bubbles in grain boundaries is discussed and shown to be dependent on the grain boundary energy. Holes not in complete equilibrium have been considered, and in this ca...

Optical absorption features associated with paramagnetic nitrogen in diamond

Abstract: A survey of the optical and E.P.R. absorption in a large number of diamonds revealed that most natural diamonds contain substitutional nitrogen exclusively in non-paramagnetic, platelet form. These diamonds are classified as Type I a. A small group of transparent natural diamonds contain dispersed, paramagnetic nitrogen. The optical properties of these diamonds are unlike those of other diamond types and were hitherto unknown. It is proposed that these diamonds be classified as Type I b. The absorption continuum in the ultra-violet and visible region and the infra-red bands in the region of single single-phonon absorption, characteristic of Type I b diamonds, are shown to be due to paramagnetic nitrogen in solid solution. Synthetic diamonds are found to be generally of Type I b, whereas natural Type I b diamonds are rare exceptions. This is attributed to the growth history of the specimens.

The deformation of magnesium single crystals

Abstract: Magnesium single crystals were deformed at room temperature. Dislocation distributions were determined by transmission electron microscopy of sections of crystals deformed by various amounts. In stage A the dislocations are mainly in the form of edge dipole bands. No dislocations with non-basal Burgers vectors were found. The density of dislocations varies linearly with strain and (stress)2. Slip line studies were carried out using the replica technique. The slip lines are long and terminate generally by cross-slip. The number of dislocations per slip line is approximately constant, the spacing of the slip lines varying inversely as the strain. Stage B is characterized by twinning and network formation. A model for stage A is proposed in which the dislocations from sources operating simultaneously trap one another and form dipole bands for edges and screws; the latter cross-slip and annihilate leaving the edges and an excess of screws of one sign. The flow stress is controlled by the internal str...

An investigation of thermal spikes by studying the high energy sputtering of metals at elevated temperatures

Abstract: The sputtering ratio of Au, Ag, Cu, Zn, Bi and Ge polycrystals bombarded with 45 kev Xe+, A+ and Ne+ ions has been studied as a function of temperature. In every case the curve of sputtering ratio as a function of temperature was initially constant, but at temperatures approaching the melting-point the curve exhibited an approximately exponential rise. The increase is explained in terms of evaporation from thermal spikes created by the individual ions, and analysis of the results has provided information as to the spikes' size, lifetime and temperature.

Grain boundary diffusion of iron, cobalt and nickel in alpha-iron and of iron in gamma-iron

Abstract: Grain boundary diffusion coefficients have been measured in the following systems: Iron in α-iron D gbδ = 11·2 × 10−7 exp (—41 500/RT) cm3/sec.Cobalt in α-iron D gbδ = 2·23 × 10−7 exp (—41 500/RT) cm3/sec.Nickel in α-iron D gbδ = 23·3 × 10−7 exp (—43 300/RT) cm3/sec.Iron in γ-iron D gbδ = 0·77 × 10−7 exp (—38 000/RT) cm3/sec. Standard radiotracer techniques were used. The results were derived using Fisher's solution for the grain boundary diffusion problem. The significance of the grain boundary diffusion constants, D gb and Q gb, is discussed.

Dislocation loops in irradiated iron

Abstract: Dislocation loops have been observed in iron foils irradiated with protons or Fe+ ions of energy 150 kev. Those produced by irradiation with Fe+ ions are shown to lie on {100} planes and to be pure edge in character. They are normally square, with the sides in the corresponding directions, and are formed by the condensation of interstitial atoms. It is believed that the loops are formed because of the excess concentration of interstitial atoms resulting from the direct insertion of atoms on bombardment. It is shown that either a homogeneous or a heterogeneous nucleation mechanism can account for the loop density observed. Finally, an estimate is made of the strain energy of various modes of condensation of point defects in iron. The most favoured are ½[111] (110), ½[111] (111), and [100] (100).

Slip character and the ductile to brittle transition of single-phase solids

Abstract: A transition from brittle to ductile behaviour of single-phase polycrystals is shown experimentally to be a consequence of a change in slip character rather than of a change in yield stress. In the case of a b.c.c. Fe-Co alloy, deformation by planar glide induces brittleness, whereas the alloy is ductile if it deforms exclusively by wavy glide. For a series of copper-base solid solutions loaded in mercury, the degree of embrittlement increases with planarity of glide. The Cottrell-Petch equation can account for such behaviour provided the appropriate parameter, k y, is re-considered in terms of the number of operable slip systems and the propensity for cross slip.

Thermal expansion of germanium and silicon at low temperatures

Abstract: The linear expansion coefficients (α) of silicon and germanium are reported at temperatures down to ca. 0/50. At temperatures below 16°K, the expansion coefficients are very small but are positive and hence the Gruneisen parameter γ=3α. VK/C (K is the bulk modulus and C/V is heat capacity per unit volume) is positive. At T∽0/15, γ has a minimum value of —0·1 for germanium and —0·42 for silicon. At higher temperatures α and γ become positive and agree with data of Gibbons. Individual γ-values for the transverse acoustic modes must be reduced considerably by dispersion to account for the deep minimum in γ(T).

An electron microscope study of neutron irradiation damage in alpha-iron

Abstract: The technique of transmission electron microscopy has been used to study neutron irradiation damage in α -iron. Foils were irradiated to a fission neutron dose of ∼ 2 × 1020 cm−2 at 60°c and the effects of annealing up to 500°c on the micro-hardness and structure were studied. The irradiated foils contained visible point defect clusters which remained unchanged on annealing up to 300°c. Above 300°c the size of clusters remained fairly constant but the number decreased steadily until they had all disappeared after annealing at 500°c. The micro-hardness also remained unchanged up to 300°c and then recovered steadily until it returned to the unirradiated value at 500°c. The results of a diffraction analysis carried out on loops in annealed foils indicated that their Burgers vector was in a ⟨111⟩ direction and that both interstitial and vacancy loops were present. It is suggested that the visible clusters in the irradiated iron are interstitial in nature, the vacancies being trapped by interstitial i...

A debris mechanism of cyclic strain hardening for F.C.C. metals

Abstract: Prismatic dislocation loops (dislocation debris), formed by fatigue or cyclic straining, have been observed in a number of metals. To account for these observations a mechanism of cyclic strain hardening is proposed which depends on the ease of loop formation and the behaviour of the loops after they are formed. In the initial rapid hardening stage, the rate of cyclic strain hardening is determined by the rate of formation of debris obstacles. Since the debris is thought to be formed by some variant of the double cross slip mecachism, changes of variable which lead to easier cross-slip result in higher hardening rates (because the rate of debris obstacle formation is increased). As the crystal fills up with debris, the motion of screw dislocations diminishes and the enforced strain is accommodated by the motion of the prismatic loops already present in the crystal. The basic motion of the prismatic dislocation loops is a flip-flop motion from one stable equilibrium position to another which is re...

Determination of the slip systems in single crystals of tungsten monocarbide

Abstract: Slip in single crystals of tungsten monocarbide has been observed around Vickers pyramid hardness indentations. The slip plane is {l100} at room temperature and the slip directions are postulated to be the and . The Vickers hardness value on a (0001) face is approximately twice the value determined for a (1100) face. This is attributed to the ease of motion of dislocations with Burgers vectors as opposed to the dislocations having Burgers vectors since the former can dissociate into partial dislocations. The occurrence of a {1100} slip plane is explained on the basis of the fact that a minimum number of only two covalent tungsten-carbon bonds need be broken for a unit of slip on this plane.

Crack propagation in single crystals of tungsten

Abstract: The effect of pre-existing micro-cracks on the fracture of [010] tungsten single crysta's has been investigated Cleavage cracks on (010) planes were introduced into the edge of the specimens by a spark-machining technique Tensile tests at 77°K showed that the fracture stress was dependent on the crack length according to the Griffith criterion At this temperature the specimens failed in a completely brittle manner A surface energy of 6300 ergs cm−2 was obtained in close agreement with the theoretical value for the true surface energy Metallographic examination of the fracture surfaces showed that the markings on the surface depend on the stress level at fracture One test was carried out to determine the effect of prestraining at 295°K on the subsequent fracture behaviour at 77°K The effect of temperature on the fracture of specimens containing a constant crack length was determined between 77 and 478°K As the temperature increased, the crack propagation mechanism changed from fast cleavag

Magnetic properties of hematite single crystals

Abstract: An experimental study of natural and synthetic single crystals of hematite containing small quantities of impurities was made to establish what effect impurities had on a variety of magnetic properties. In both types of crystals the average values for the weak magnetic moment were comparable regardless of the impurity. A triaxial anisotropy was observed in the basal plane of many natural crystals and for the samples with the largest anisotropy the major impurity was found to be Sn; this was also the only impurity which induced a triaxial anisotropy in the synthetic crystals. The coercive force was also much larger in the natural and synthetic crystals containing Sn. The transition temperature was found to be progressively lowered by increasing the amounts of certain impurities but the values for this temperature in synthetic and natural crystals containing the same impurities were often different. For the synthetic crystals, small additions of Sn depressed the transition temperature to a greater ...

Intersecting faults on basal and prismatic planes in aluminium nitride

Abstract: Intersecting faults on basal (0001) and prismatic (1210) planes in A1N (hexagonal, wurtzite structure) have been studied by diffraction contrast with the electron microscope. The faults on (1210) had fault vectors 3/2 p + 1/2 c=½[1011], as deduced from the observed vanishing conditions. The expected stair-rod dislocations at the intersections of the faults have been found, and they had screw components to their Burgers vectors. An atomic model for the faults on (1210) is proposed, and possible mechanisms of formation during crystal growth are discussed. The determination of the nature of faults in hexagonal crystals by means of diffraction contrast is considered in detail, since the information obtainable is different for h.c.p. than for f.c.c.

Stacking-fault tetrahedra in deformed face-centred cubic metals

Abstract: Stacking-fault tetrahedra have been observed in plastically deformed specimens of copper, silver, gold and several alloys. A dislocation reaction is proposed for the formation of these defects during deformation. From the size of the largest tetrahedra, estimates of stacking-fault energy are made which indicate that the stacking-fault energy of silver is less than that of gold and copper.

Conversion of crystalline germanium to amorphous germanium by ion bombardment

Abstract: Of particular interest in the theory of radiation damage are the size and structure of the damaged regions produced in crystalline solids by fast particle irradiation. In the present work crystalline germanium at room temperature or at 30°K has been bombarded in a Siemens electron microscope with 100 kev oxygen ions emanating from oxide-coated emission filaments. The size and structure of the damaged regions generated by the incident ions have been examined at both temperatures using transmission electron microscopy. Detailed evidence is presented which shows that the damaged regions had a size which was dependent on the temperature of the specimen during bombardment and a structure which was amorphous and less dense than the crystalline matrix in which they were located. Further, as the ion dose was increased to 2 × 1015 ions cm−2 the entire surface region to a depth of ∼600 A was observed to undergo a gradual transition from a crystalline to an amorphous state. In terms of radiation damage mode...

On the theory of ferroelectricity and anharmonic effects in crystals

Abstract: A microscopic theory of ferroelectricity is developed in which the temperature dependence of the normal modes of vibration arises from the anharmonic interactions between those normal modes. The conditions for the stability of the crystal are discussed and detailed expressions given for the physical properties of both the distorted and undistorted phases. The parameters of the thermodynamic theory of ferroelectricity are obtained in terms of those of the microscopic theory, and the approximations and validity of the thermodynamic theory are discussed. By using a model of strontium titanate which includes the anharmonic interactions between the normal modes of vibration in a very simple way, detailed calculations have been made of the infra-red reflectivity, linear expansion and dielectric properties of strontium titanate. The fairly quantitative agreement with experiment given by these calculations and the reproduction of the features of the ferroelectric transition in barium titanate suggest tha...

The mechanism of cavitation in magnesium during creep

Abstract: During the creep of magnesium, cavities form on some grain boundaries, limiting the ductility of the material and causing its density to decrease. It is shown that cavitation is eliminated if a hydrostatic pressure is superimposed which is equal to the creep stress. If the hydrostatic pressure is applied after cavities are already present in the material, continuing creep results in negligible density change and existing cavities do not continue to grow. When hydrostatic pressure is applied throughout the early stages of creep and is then removed, the subsequent cavitation develops at a rate which is unaffected by the prior deformation. The fractional change in density during creep is approximately proportional to the duration of creep to the power of 2·5 increasing to about 4 immediately prior to fracture. This rate of change of density is greater than can be accounted for by the growth of a constant number of cavities by vacancy condensation and suggests a progressive creation of nuclei. The nu...

Kinetics and mechanism of formation of forsterite (Mg2SiO4) by solid state reaction of MgO and SiO2

Abstract: The development of forsterite, Mg2SiO4, by solid state reaction in mixed powders of MgO and SiO2 in various mole ratios at temperatures 1100–1400°c and for times up to 144 hrs is followed by quantitative x-ray diffraction measurements. Powders are used consisting of coarse (15 μ) particles of one component in a fine-grained matrix of the other component. The results are consistent with a diffusion-controlled mechanism and up to about 60–70% reaction to agree with the Ginstling-Brounshtein equation. Reaction constants are evaluated and activation energies in the range 70–80 kcal/mole are obtained. Studies, using platinum markers, indicate that Mg is the main diffusing species. A cathode-luminescent image shows a thin product layer of enstatite between the principal product (forsterite) and the SiO2 phase, which is confirmed by electron microprobe measurements. It is considered that a thin zone of enstatite moves ahead of the main product forsterite as the reaction progresses into the silica component.

A theory of the electrical properties of liquid metals iv. quantitative calculations of resistivity and thermoelectric power

Abstract: Following the lines of the pseudopotential approach to the electronic transport properties of liquid metals as presented in I (Ziman 1961) and II (Bradley et al. 1962), systematic quantitative calculations have been made for the electrical resistivity and thermoelectric power of the liquid metals: Li, Na, K, Rb, Cs, Zn, Hg, Al, Tl, Pb and Bi. The pseudopotential used is that of Heine and Abarenkov (1964). The results are in fair agreement with experiment. Some deviations from the experimental figures may perhaps be rectified by refinements to the pseudopotential employed, and in other cases the correlation functions, as deduced from x-ray and neutron diffraction experiments, may be in error.

Faulted dislocation loops in quenched aluminium

Abstract: Dislocation loops containing stacking faults have been observed in quenched aluminium using the electron microscope. It is found that lowering the quenching rate relaxes the stringent purity conditions governing the retention of faulted loops, such that a high proportion of the loops in oil-quenched aluminium (99·97 % purity) contain faults. The results are discussed in terms of the influence of quenching stresses on the stability of the faulted loop. A low quenching rate also favours double-faulting, in which a Frank sessile dislocation loop is nucleated on an existing fault. The stability of this defect is attributed to the lower stacking-fault energy of a double fault compared with that of two intrinsic stacking faults. Under the action of stress the outer fault of the double defect can sometimes be removed to produce a faulted loop within a prismatic loop. The annealing behaviour of the four different types of dislocation loop, i.e. prismatic, single-faulted, double-faulted and faulted prisma...

The hardness of alkali halide crystals containing divalent ion impurities

Abstract: It was shown 30 years ago that the hardness of alkali halides increases considerably when low concentrations of divalent cations are incorporated in the crystal lattice. Recently, dielectric measurements have provided information about the manner in which the divalent cations are incorporated in the lattice. The measurements reported in this paper on NaCl: Ca++, NaCl: Mn++, KCl: Sr++, KCl: Ba++ and LiF: Mg++ were undertaken to observe the effect of the state of aggregation of the divalent impurities on the critical resolved shear stress. The principal results are: (1) the increases in critical shear stress is proportional to c2/3, where c is the concentration of divalent ion-vacancy pairs, (2) there is no increase in hardness as these divalent ion-vacancy pairs aggregate into groups of three (trimers), (3) in NaCl: Mn++, KCl: Sr++ and KCl: Ba++ there is no increase in hardness as these trimers grow into larger aggregates, (4) in LiF: Mg++ there is a large increase in hardness as the trimers grow ...