Frank Theory Research Articles

Frank's kinematic theory of crystal dissolution applied to the prediction of apex angles of conical ion-bombardment structures

The construction process associated with the kinematic theorems of crystal dissolution due to Frank has been applied to the problem of determining the characteristics of surface topography caused by ion bombardment. Such an approach is shown to be valid by comparison between experimental results on and dissolution constructions applied to ion-etched spheres. Emphasis has been placed on determining the apex angle of equilibrium cone structures on crystalline materials. The values predicted for the apex angle by Frank's method have been found in good agreement with those measured experimentally and values derived from the peak of sputter yield curves or from ion channelling calculations. Using different starting shapes, the importance is stressed of ensuring the presence of adequate orientation directions without which only non-equilibrium dome type structures or pseudo-cones will develop.

Prediction of ion-bombarded surface topographies using Frank's kinematic theory of crystal dissolution

The two kinematic theorems of crystal dissolution by Frank are applied to surfaces undergoing ion-bombardment in order to determine the topographies evolved in any given time. The predicted shapes show good agreement with experimental observations.

Evaporation of Zinc Single Crystals

AbstractThe dislocation nature of defects leading to local evaporation of the {0001} face of zinc single crystals is proved. Two kinds of pits appear on the emergence points of dislocations, upon heating the crystals in vacuum, that is, hexagonal ones on screw dislocations and conical ones on edge dislocations (and probably on impurities). Hexagonal pits appeared at the growth points of whiskers upon evaporation. This is in excellent agreement with the assumption that whiskers grow by a screw dislocation mechanism.Deepening and broadening of the pits were studied by interferometry. It was found that the change in profile with time can be described by the equations of Frank's kinematic theory. The same theory was used to calculate the step flux along the basal face of zinc single crystals upon evaporation at 250, 270, and 290°C. The rate of advance of unit steps was determined from the flux and compared with the theoretically expected one. The temperature dependence of this rate made it possible to estimate the vibrational frequency of the atoms in the crystal lattice as well as the evaporation heat of zinc. The reasonable values obtained demonstrate the correctness of the interpretation of the experimental data.

The Effect of Non-Hydrostatic Stress on Intergranular Water Veins and Lenses in Ice

AbstractPolycrystalline ice at the melting point has been observed in the laboratory to contain veins of water at the places where three grains meet. Under uniaxial compression lenticular water inclusions appeared at grain boundaries perpendicular to the stress, while the nearby vein began to freeze. A similar effect occurred in tension on grain boundaries parallel to the stress. When the stress on the plane of the boundary was a pure shear stress, no effect was observed. The water lenses produced by stress increased in size and decreased in number after the stress was removed. The effect under compression is explained quantitatively by the combined effects of curvature and pressure on the melting point of an ice–water interface. The rate of formation of the lenses and of their coarsening is greatly reduced by the internal pressures set up in the lenses as a result of expansion on freezing and contraction on melting; transient creep to accommodate volume changes is an essential part of the process. The effect in a grain boundary under tension may arise from pressure caused by sliding on other grain boundaries; it was absent in a bicrystal.It is concluded that internal melting and freezing at grain boundaries and veins will occur in temperate glacier ice, with some effect, not discussed here, on its permeability to water. Any pure solid at its melting point which has a dihedral angle for the liquid phase in contact with a grain boundary between 0° and 60° should show similar behaviour, in that non-hydrostatic stress should cause liquid to move away from triple junctions between grains and into grain boundaries. There may be implications for the Frank theory of the upwelling of melt fluid in the Earth’s upper mantle.

The Effect of Non-Hydrostatic Stress on Intergranular Water Veins and Lenses in Ice

AbstractPolycrystalline ice at the melting point has been observed in the laboratory to contain veins of water at the places where three grains meet. Under uniaxial compression lenticular water inclusions appeared at grain boundaries perpendicular to the stress, while the nearby vein began to freeze. A similar effect occurred in tension on grain boundaries parallel to the stress. When the stress on the plane of the boundary was a pure shear stress, no effect was observed. The water lenses produced by stress increased in size and decreased in number after the stress was removed. The effect under compression is explained quantitatively by the combined effects of curvature and pressure on the melting point of an ice–water interface. The rate of formation of the lenses and of their coarsening is greatly reduced by the internal pressures set up in the lenses as a result of expansion on freezing and contraction on melting; transient creep to accommodate volume changes is an essential part of the process. The effect in a grain boundary under tension may arise from pressure caused by sliding on other grain boundaries; it was absent in a bicrystal.It is concluded that internal melting and freezing at grain boundaries and veins will occur in temperate glacier ice, with some effect, not discussed here, on its permeability to water. Any pure solid at its melting point which has a dihedral angle for the liquid phase in contact with a grain boundary between 0° and 60° should show similar behaviour, in that non-hydrostatic stress should cause liquid to move away from triple junctions between grains and into grain boundaries. There may be implications for the Frank theory of the upwelling of melt fluid in the Earth’s upper mantle.

The fracture of titanium carbide under static and sliding contact

Under static loading, crack formation in titanium carbide is shown to obey Auerbach's law for hemispherical indenters. From the results, using the Frank and Lawn theory of 1967, an estimate of the surface energy of TiC is obtained. In the case of sliding contact the load to fracture varies markedly with the coefficient of friction. A reasonable estimate of the sliding load to fracture is obtained by assuming that the fracture stress, as calculated using the analysis of Hamilton and Goodman in 1966, is the same as in the case of static loading for the same indenter. This is at some variance with the Lawn's 1967 predictions.

Surface processes in electrocrystallization

Abstract A short review of the theories of crystal growth of perfect and imperfect planeS is given. Experimental evidence of two-dimensional nucleation mechanism of growth of perfect crystal planes is provided for the first time and some crystallographic parameters of this process are determined. A quantitative verification of Frank's theory is given by investigating the growth phenomena on nearly perfect planes. The effect of screw dislocations on crystal growth is studied and some phenomena concerning the influence of surface active impurities on crystal growth and dissolution are reported. The theory of heterogeneous phase formation is briefly discussed and an experimental verification of the basic relation following this theory is given. The effect of the crystallographic structure upon nucleation is closely studied. A method permitting the determination of the activity distribution curve of nucleation sites on the basis of their critical supersaturation is also given.

Kinetics of Crystallization of Anthracene from the Vapor

Abstract Anthracene was purified by zone melting, chromatography and sublimation. It was again zone melted in a closed system within which it was transferred to a crystal-growth tube. A sublimation apparatus was constructed allowing direct, automatic control of temperature-difference δT between sublimand and crystallization site as well as photography of the growing crystal. In evacuated tubes and in the presence of propane at 1 torr, it was not possible to control nucleation at 120°C. In the presence of N2 at 1 torr, single nuclei were formed at δ T=9.5° ± 0.5° and these were grown to 1 cm diameter crystals of good optical quality. The basal plane of the crystals was the ab plane. Cinematographic observation of growth and evaporation at under-saturations and supersaturations up to 6% showed that growth rate R varies linearly with supersaturation σ. According to the Frank theory, R should be proportional to σ2 for small σ when growth takes place by a single dislocation. Evidently this very unlikely condition is not fulfilled.

Epitaxial theory of polytypism; observations on the growth of PbI 2 crystals

Equilibrium thermodynamic theories of polytypism are discussed and rejected in favor of a non-equilibrium screw-dislocation theory. It is suggested that the parent dislocation originates through epitaxial growth on a foreign nucleus, whereas Frank's theory assumes that the dislocation originates through buckling and slip of an otherwise perfect protoplate of the crystal. Observations on gel-grown PbI 2 crystals support the present view, and observed discrepancies, which are difficult to explain on Frank's theory, can now be accounted for in a simple manner.

Orientation-Dependent Dissolution of Iron Whiskers

The dissolution kinetics of iron filamentary single crystals in dilute acidic environments were shown to obey the topographical theorems proposed by Frank to describe etch-rate anisotropy. Successive dissolution profiles of iron whiskers were observed in situ. by using time-lapse motion picture photomicroscopy. Reluctance diagrams constructed from these dissolution shapes agreed very favorably with experimentally measured dissolution rates and in this way demonstrated the validity of Frank's theory to metallic substrates. A three-dimensional reluctance surface for iron whiskers having 〈111〉 growth axes inferred from the experimental data was consistent with known relative magnitudes of anisotropic dissolution rates; in fact, the reluctance plots constructed according to the kinematic theory were seen to be two-dimensional sections of this surface.

Radiation resistance of an oscillating dipole in a moving medium

The radiation resistance of an oscillating electric dipole immersed in a moving medium is calculated by Brillouin's EMF method. Two cases are studied in detail: in case 1) the moving medium is simple, and in case 2) the moving medium is an ionized gas (plasma). It is found that in case 1), the motion of the medium tends to increase the radiation resistance, whereas in case 2), the motion of the medium tends to decrease the radiation resistance. A comparison is made between Frank's well-known theory of the radiation from an oscillating dipole moving through a material medium and the results obtained in this paper.

Light Scattering from Diffusion-Controlled Phase Separations in Glass

Light-scattering measurements were made on a number of phase-separated glasses in which the number and size of the particles of the new phase were varied. Plots of scattering intensities versus scattering angles showed two unique features: (1) higher scattering intensities were found in the back angles and (2) in some cases sharp maxima in intensities were found. In the first case the experimental data were in excellent quantitative agreement with Goldstein's theory for scattering from systems in which the growth of the new phase is diffusion controlled. In the second case the data were explained using Rayleigh-Gans theory of light scattering for spheres and Frank's theory for diffusion-controlled growth. The results confirmed the existence as well as the extent of the composition gradient around the growing particles as predicted by diffusion-controlled growth theory.

Vapour growth patterns of CdTe crystals

Abstract CdTe single crystals were grown from the vapour phase by the sublimation method in order to study crystal habits and growth patterns. For this purpose a new technique for revealing the growth patterns was developed. This technique is based on an oxidation of the surface and results in colourful growth patterns. Observations of the growth spirals provide further evidence to support Frank's theory of screw-dislocation-induced crystal growth.

Growth Rates of Zinc Crystals from the Vapor Phase

The rate of growth of zinc crystals from the vapor phase has been studied as a function of the vapor supersaturation σ at a temperature of 390°C. The σ values ranged between 0.009 and 0.09. The rate R was found to vary approximately linearly with σ, permitting estimates to be made of the two parameters σ1 and βC0, appearing in the Burton, Cabrera, and Frank theory of crystal growth. The results indicate that surface nucleation was not operative in either the nucleation or the growth of these crystals.

On the Spiral Growth of Graphite Crystals

Layer growth patters, concentric hexagonal and circular loops, have been observed on the artificial graphite prepared by heat-treatment of carbon black seast 116. Some electron microscope and diffraction data are given. And it is concluded that these patterns are taken as the spiral growth steps, in accordance with the prediction of Frank's theory, expected when some screw dislocation rows emerge on the crystal faces.From the view point of dislocation theory a brief discussion is tried on certain features of graphite. The numerical values of radii of the critical nuclei are calculated 80-270Aring; as well as the supersaturation of vapour pressure 2.5-8%. The step heights in spirals have been estimated to be 150-250A, which may suggest that scores of initial dislocation with unit strength are crowded in the core. of each crystal.A new concepts, termed as “turbinal layer structure”, is proposed to explain the characteristic structure of those crystals which produce circular growth patterns in micrograph and sporty Debye-Scherrer rings, accompanied by ellipses in diffraction diagram.

The crystallography of the β-α transformation in titanium

The orientation relationships, habit plane and surface contour associated with the martensitic transformation in iodide titanium were investigated. The transformation markings studied were those on the natural facets of the specimens, which had originally been single crystals of β-phase. The orientation relationship was close to but not quite identical with the Burgers relationship: (0001) β | | {110} β; {H20} α || {lll} β; the deviations were 1 2−1° . The habit plane was close to {8, 9, 12}, to an accuracy of 1°. The orientations associated with individual habit plane variants were analysed and fitted into a consistent scheme which was close to that expected if the transformation mechanism proposed by Burgers operated. Each habit plane was 10° from the variant of {112} predicted as shear plane by Burgers. Data on the surface contour showed that the direction of atom motion was close to the variant of 〈111〉 required by Burgers, though the magnitude of the surface tilt was smaller than predicted. The precise contour of individual plates, as revealed by multiple beam interferometry, showed that accommodation slip had occurred which would reduce the surface tilt. The results could not be satisfactorily interpreted on the basis of Frank's theory of the martensitic transformation, but they provide some support for Geisler's views.

Čerenkov Radiation from a Co^60 Source in Water

The spectral distribution of the Čerenkov radiation induced in deionized water by gamma rays from Co60 has been obtained. A 3400-curie Co60 source provided adequate light emission in the water surrounding it to allow a direct measurement with a grating spectrometer. In the observed spectral range, 3200A to 6000A, the spectral distribution is in agreement with the prediction of the Frank and Tamm theory and is offered as a quantitative check on this theory.

LI. Growth spirals originating from screw dislocations on gold crystals

Summary The present paper gives a more detailed account of the observation of growth spirals on (111) faces of gold crystals, obtained by precipitation. The observations prove that Frank's theory also holds for metal crystals; moreover they constitute the first visual proof for the presence in metal crystals of dislocations in general, and screw dislocations in particular. They further show that the Burgers vector of dislocations in metals is not necessarily limited to one unit of a few simple lattice displacements, as is generally assumed. The shape of the growth spirals is briefly discussed.

Growth Spiral Patterns on Carborundum Crystals

OBSERVATIONS on growth spirals on carborundum crystals have been reported earlier1, affording experimental evidence of Frank's theory of crystal growth depending upon dislocations. Some new and unpredicted features of these growth spirals are reported here.

CI. Observations on carborundum of growth spirals originating from screw dislocations

Abstract Numerous growth spirals with shapes ranging from circular to regular hexagonal have been observed on carborundum crystals of types I. and II., by coating the crystal faces with a thin film of silver of reflectivity nearly 90 per cent and then working in reflection, using phase contrast illumination. The shapes of these spirals, in relation to the crystal structure, are shown to be in accordance with the predictions of Frank's theory. On a type II. crystal, hexagonal interlaced spirals have been observed. The annihilation of growth fronts where they meet, and other properties of growth fronts are illustrated and hence the observed patterns for two or more screw dislocations are explained. The observed density of dislocations ranges from small values up to a maximum of 104 per sq. cm. For a typical circular spiral the calculated values of the radius of the critical nucleus is 2μ and supersaturation is 0·2 per cent. By the application of multiple-beam interference, using both Fizeau fringes and fringes of equal chromatic order, the step heights in spirals have been measured accurately for type II. crystal and found to be 15 Å. This is equal to the height of the unit cell, proving that these are growth spirals originating from screw dislocations, in exact accordance with theoretical prediction.