DESCRIPTION OF THE PHYSICAL PROPERTIES OF SOLID STRUCTURES AND PROCESSES IN THEM ON THE BASIS OF ENERGY PARAMETERS OF MATERIALS

A. A. VOROB’EV

Crystalline and glass-like structures with an ionic type of bond, as well as materials containing both of these phases, are widely distributed in nature and find extensive application in technology and the national economy. An enormous number of ceramic and other artificial materials with ionic bonds have been developed, used in electrical and radio engineering, construction, and other branches of the economy and technology.

In principle, the fundamental laws for describing the structure and properties of materials are now known. Previously, the classification of the electrical, mechanical, thermal, optical, chemical, and other properties of materials with an ionic type of bond was carried out by separately describing the processes under consideration, occurring in structures under given conditions and external action. Isolated consideration of the properties of ionic structures led to the creation of a theory of a given process—for example, electrical breakdown and the characteristic of the electric strength of a dielectric—and did not make it possible to evaluate the properties of the material as a whole. The absence of a unified physical theory of the properties of materials, of the interrelation of properties with the chemical composition and structure of the substance, made it difficult to develop engineering calculations and to select materials for use under complex loading conditions, as well as to develop new materials with predetermined properties.

M. Born and Huang Kun, in the dynamic theory of the crystal lattice, advanced the proposition that the normal mechanical, thermal, optical, and electrical properties of a substance can be described by means of the free energy. However, the magnitude of the free energy of a system cannot be measured or calculated. Without knowing the value of this quantity, it is impossible to evaluate and compare with one another the properties of bodies of different chemical composition and after different treatments. A. A. Vorob’ev, using thermodynamic concepts, showed theoretically and experimentally that the properties of a substance are characterized by the energy of interaction between the weakly bound structural elements into which a body decomposes when it disintegrates. For ionic crystalline structures such a characteristic is the lattice energy; for bodies with covalent bonds, the atomization energy; for glass-like structures, the heat or temperature of vitrification. The smallest interaction energy in layered structures is characterized by the cleavage energy, and so on. For ionic crystalline structures, by means of the lattice energy one can describe properties such as ne-

both damaged and defective crystalline structures, since the equilibrium concentration of defects and their stability are also related to the lattice energy. The equilibrium concentration of defects increases as the lattice energy decreases. The stability of defects decreases as the lattice energy decreases. The greater the lattice energy, the higher the stability of the structure under external actions, and the higher its mechanical, electrical, optical, chemical, thermal, and radiation stability. These simple regularities, obtained in studies, made it possible to carry out a scientific classification of materials with an ionic type of bond and to predict directions for choosing the chemical composition in order to obtain materials of high stability under external action. Such a correlation is well observed for real dielectrics, since the degree of disruption of their structure and the equilibrium concentration of defects, which determine many properties, are governed by the energy of the crystal lattice. When the energy of the external action is equal to or greater than the lattice energy, the stability of the structure is disrupted; defects are formed, and their concentration begins to increase. Simultaneously with the formation of defects, their annealing takes place. Under certain conditions of external action, the rate of defect formation may exceed the rate of their annealing. The concentration of defects will begin to increase. It will exceed the equilibrium value for the given temperature, and the process of defect accumulation will begin. This condition may be selected as a criterion of stability of the body. The destruction of a body under external actions, for example in a radiation field or an electric field, is described not as a catastrophic process, but as a dynamic process developing in time. The rate of development of the process of defect accumulation and destruction of the body increases with increasing intensity of the external action. However, the rate of growth of the crack of destruction of the body cannot exceed a certain limiting value, whose magnitude is the greater, the greater the lattice energy. It is assumed that the process of destruction of undamaged structures passes through the following stages: formation and healing of structural disturbances, accumulation of disturbances and formation of microdamage in the body, accumulation and coagulation of microdamage and formation of the initial crack of destruction; its development and the completion of destruction of the body. With an increase in the intensity of the external action of the electric field, mechanical stress, the time required for each stage decreases, which entails a decrease in the total time of destruction of the dielectric. A general energetic description of the processes of accumulation of structural damage and destruction of the body made it possible to solve particular problems as well—destruction in a high-strength electric field, and also under the action of irradiation. A description of mechanical destruction by the accumulation of disturbances was given by S. N. Zhurkov.

The theory of the electrical strength of solid dielectrics with ionic bonds, developed by A. A. Vorob’ev and E. K. Zavadovskaya, gives the correct dependence of strength on lattice energy, loading rate, thickness, and other parameters.

Radiation disturbances of solids at small doses, when radiation defects of the structure are formed, constitute the first stage of destruction. At large doses, the second stage of structural damage and the third stage of the destruction process may be reached, up to the disintegration of the body into separate pieces.

The developed theory of the properties of ionic crystalline structures leads to practically important and interesting conclusions concerning the relationship between the properties of solids with predominantly ionic bonds, their chemical composition, and the technology of their manufacture.

It has been shown theoretically that with an increase in lattice energy, practically important mechanical, electrical, and thermal properties increase.

and so forth—the properties of crystalline bodies with an ionic type of bond. The correlation between lattice energy and the properties of bodies has been traced experimentally for a wide range of pressures, temperatures, and irradiation conditions.

Regularities have been formulated for changes in the properties of ionic structures under normal conditions, as well as at elevated temperatures and pressures in the Earth’s interior and under irradiation. Regularities have been found that determine the stability of bodies under mechanical or electrical loading, heating, dissolution, or irradiation. The relation of electrical, mechanical, thermal, and other properties to lattice energy has been traced in a large number of types of ionic compounds and agrees satisfactorily with the conclusions of the theory.

It has been shown that the stability of bodies against external actions increases with increasing lattice energy, while their workability decreases.

The weakening of a body under loading is determined by the energy absorbed by the body, and the residual changes in the properties of bodies by the value of the energy stored under external action. With increasing lattice energy, the value of the absorbed energy decreases and the value of the stored energy increases. Other conditions being equal, the values of the energy absorbed and stored by a body determine its strength and workability, as well as the technological properties of materials.

The irreversible change in the properties of solid dielectrics under the conditions of their service, called aging, is of considerable practical interest, since aging of insulation in many cases is accompanied by disruption of the operation of the corresponding structures.

A general theoretical concept has been developed for the aging of dielectrics under the action of electrical and mechanical loads, irradiation, temperature, and the influence of chemical agents. The thermodynamic approach to the description of aging of materials is based on the fluctuation concepts set forth above. During the time of action of a given external factor that lowers the potential barriers in the structure, defects are formed and, simultaneously, their temperature annealing—healing—takes place. Aging of a dielectric develops when defects accumulate and damage to the body is formed, or when new chemical compounds are formed.

In order to create physicochemical and physicoenergetic scientific foundations for describing the properties of materials and their aging and processing with the use of high-intensity factors acting on a body—mechanical, chemical, and physical factors, including radiation—it was necessary to develop a theory of the interaction with a solid body of continuous and pulsed high-intensity energy fluxes; to develop scientific foundations for the processes occurring in materials under the action on them of strong electric, static, and pulsed fields and other external factors. Scientific data were needed on the basis of which methods could be developed for calculating processes occurring in a solid body, for determining pathways of directed action in order to obtain a specified effect in the processing of materials and bodies, and for determining the rates of their aging. Such studies have been carried out in Tomsk since the 1930s.

The basis of the quantum structure of any system is its energy spectrum. The development of methods of physical and chemical action on materials must be constructed with allowance for the structure of its electronic spectrum and the distribution of electron density. Therefore, the chemical bond was studied on the basis of investigations of the electron density, electrical, magnetic, thermal, and luminescent properties of solid dielectrics and semiconductors, and the interrelation between the chemical composition of materials, their structure, and their properties was established.

The systematization of materials according to the results of studying the electronic and ionic properties of crystals considered together has made it possible to understand and describe radiation properties more fully and to formulate the scientific foundations of the doctrine of aging and processing of materials. Tomsk physicists have experimentally shown that the energetics of ionic, electronic, and hole processes and of ionic compounds is connected with, or correlated with, the chemical composition of the energy lattice and with the experimental conditions.

The study of changes in the properties and aging of dielectrics under the action of heat, radiation, an electric field, and chemical factors is of practical and theoretical interest. In our studies of various materials and their compositions under operating conditions, a thermodynamic assessment of the stability of compounds was applied; it was performed using thermochemical parameters, lattice energy, Debye temperature, heat of vitrification, heat of atomization, and other characteristics.

As an energy characteristic in the aging of a dielectric, one may take the change in the internal energy of a unit volume of a solid dielectric during its phase transition from the standard state to a new state—for example, during breakdown, when the substance in the volume of the channel passes into the state of a partially ionized gas plasma.

The action of high-energy particles is accompanied by the formation of radiation defects of the structure. The processes of defect formation may be reversible, when their annealing occurs due to temperature fluctuations, and irreversible. In the latter case, color centers or radiolysis, or phase transformations, are formed in the material.

Structural defects cause changes in certain practically important physical and chemical properties of solid dielectrics, including electrical conductivity and losses; they alter adsorption and other properties and may significantly affect their processability in general and the process of destruction in particular.

Fluxes of high-energy particles cause disturbances in the structure of a solid body and the formation of its defects. An increase in defect concentration is accompanied by their coagulation and the formation of damage. The accumulation of damage is accompanied by macroscopic destruction of the body.

Radiation aging of dielectrics proper, that is, an irreversible change in properties, is determined by the value of the concentration of defects, the radiation-chemical yield, and phase transformations in the materials. The characteristics of the listed processes are determined by the radiation resistance of the dielectric. By using radiation action on dielectric materials, one can conveniently change the concentration of defects and, consequently, the degree of damage to their structure.

At present, theories have been proposed that make it possible to calculate the accumulation of the concentration of the simplest radiation defects—\(F\)-color centers—in insulators. The aging of crystals of halide salts, oxides, fluorides, nitrides, oxynitrides, sulfates of alkali and alkaline-earth metals, glasses and polymers of various compositions, rubbers, ceramic and other dielectric materials has been studied.

The formation of defects in dielectrics is accompanied by the absorption of part of the flux of the supplied energy. After irradiation ceases, some of the defects recombine and the free energy of the body decreases. The change in the properties of a body under irradiation is determined by the value of the absorbed energy. The change in the properties of a dielectric after irradiation op-

is determined by the energy stored in defects. Since the value of the absorbed energy is much higher than the value of the stored energy, the change in the properties of dielectrics under irradiation will accordingly be much stronger than the change in the properties of bodies after irradiation. Therefore, measuring changes in the properties of materials after their irradiation cannot characterize the stability of bodies in a radiation field.

In studying the dynamics of the destruction of a body, as of any other process, theory must explain and make it possible to calculate the possibility of a body’s transition from one state to another and the rates of such transitions. For solving these questions, a thermodynamic approach is convenient.

It is possible to formulate an equation for describing the change in the energy of a system when a body is destroyed into pieces or structural particles of the material, and the rate of transition of systems from one state to another under different external conditions, for example under the action of strong electric fields.

The rate of transition can be calculated and the time required for the establishment of thermodynamic equilibrium can be determined. There has been no such macroscopic engineering description, for example, of the process of electrical destruction.

In the theory of breakdown, in fact, only the initial stage of discharge, the formation of free electrons and the appearance of a significant current, has been considered.

The phenomenon of breakdown of a solid in an electric field is the melting, mechanical, or electrochemical destruction of that solid. Destruction of the body occurs at the expense of the energy of the electric field and can be described as the formation of defects and the accumulation of disturbances in materials under external action. In the known theories of breakdown, the process of the initiation and development of a discharge in the body is developed. As a rule, it is assumed that breakdown ends with melting of the body. Brittle mechanical destruction during breakdown cannot be explained by melting.

There has been no theory of the destruction of dielectrics and semiconductors under pulsed electrical breakdown.

For the development of a theory of electropulse destruction of bodies, their brittle destruction in an electric field is of interest; this can occur as a result of the action of ponderomotive forces. The theory of this phenomenon was developed by A. A. Vorob’ev and E. K. Zavadovskaya for an ideal crystal lattice.

In our studies, the connection between mechanical strength in brittle fracture and pulsed energy strength has been physically substantiated. A theory has been developed for the dependence of strength on the prolonged action of a load.

Destruction of a body can occur through the accumulation of structural damage at the expense of field energy when:

a) in fields of high intensity, strong electron currents arise in dielectrics, which, together with the ions of the structure, form plasma in the solid body; in the regions where plasma of high concentration is formed, the body loses strength;

b) in regions of higher field intensity, the potential barrier separating ions is lowered, which promotes their leaving their places in the lattice sites; at high field intensity, the activation processes of ion transitions into interstitial positions are accelerated;

c) the attachment of electrons turns ions into atoms, which, due to thermal fluctuations, then move into interstitial positions. Under the action of impact ionization by electrons, structural damage occurs in an electric field. At the same time, as a result of the thermal motion of atoms, annealing of defects takes place. When the state of dyna-

chemical equilibrium between the rates of formation of defects and their annealing, an avalanche-like accumulation of defects begins, structural disturbances are formed, and the process enters the stage of damage to the material and destruction of the body. The field strength at which this process arises measures the electrical strength of a solid dielectric.

The following hypothesis of multistage destruction of compounds with an ionic type of bond is also possible. Under the action of accelerated electrons or radiation quanta from the discharge region, electrons in anions are excited and an exciton is formed, or an electron is detached and the atom is ionized. Such a particle loses its bonds in the lattice. By the action of thermal fluctuations, the excited anion or atom, having become an atom or a cation, is removed from its position at the lattice site.

The processes described above of excitation of electronic states in atoms propagate in the dielectric at the speed of light. Therefore any rate of discharge development less than the speed of light in this medium can be explained.

The optical excitation of an electron associated with exciton absorption in ionic compounds is based on the same physical processes, that is, the transition of an electron from an anion to a cation, as in the reaction of thermal decomposition—detonation of explosives with an ionic type of bond.

Detonation, like the destruction of a body during breakdown, can occur as a result of ionization and loss of chemical bonds in molecules.

Breakdown of detonating substances will be accompanied by the release of energy in an exothermic reaction and by an explosive effect.

Yu. N. Vershinin developed a method of energy analysis and applied it to the development of a quantitative method for determining the pulsed electrical strength of solid dielectrics with a complex molecular structure. His calculation method is based on considering phase transformations of a substance in the primary breakdown channel and describing the process of destruction of the substance by means of a specific energy characteristic numerically equal to the change in internal energy of a unit volume of the dielectric when it passes from a stationary state into the state of a partially ionized gas plasma.

The modification of the energy description of processes of electrical destruction of dielectrics, developed in the studies of Yu. N. Vershinin and proposed earlier by A. A. Vorob'ev, makes it possible, with accuracy surprising for practice, to calculate the pulsed electrical strength of solid dielectrics of complex chemical composition at microsecond exposure and a layer thickness of 100 microns and more.

The energy supplied to a dielectric is expended on processes of aging of the material and formation of structural defects, and at the beginning of destruction also on the formation of new surfaces and various secondary processes, such as elastic vibrations and the kinetic energy of particles of the body flying apart from the zone of its destruction, light, heat, sound, plastic deformation, and other processes on which energy is expended. To carry out effective destruction with a high efficiency it is essential to choose the operating regime, to increase the fraction of energy going to cleavage and formation of new surfaces, and to reduce energy losses in secondary processes. On the basis of the method of experimental analysis, one can study the influence of the regimes of action on a body and choose the regime in which the energy efficiency of the processes of destruction of a solid dielectric will be greatest.

More stable dielectrics under the action of mechanical, thermal, chemical, electrical, and radiation loads are structures with a greater bond energy between particles. In particular, the stability of crystalline ionic dielectrics increases with

with an increase in lattice energy; for covalent crystals—with an increase in the heat of atomization; for glasses and polymers—with an increase in the glass-transition or softening temperatures of the materials.

The proposed general thermodynamic description of the aging processes of dielectrics is made more concrete in the description of particular problems. Suitable methods have been found for describing the aging of homogeneous and inhomogeneous multicomponent dielectrics. V. S. Dmitrievskii, on the basis of ideas concerning the accumulation of disturbances and damage in an electric field, developed a theory of the longevity of insulation for flexible cables and made very important practical recommendations for increasing the service life of cables.

In presenting the problems under consideration—aging and processability—there is a general approach from the standpoint of the dynamic theory of the crystal lattice, according to which all normal properties of solid crystals are associated with changes in the free energy of the system, or, as shown in our works, with the energy of the crystal lattice.

An increase in the binding energy of particles in a body, characterized by an increase in the lattice energy, the glass-transition temperature, etc., is accompanied by an improvement in practically important properties of bodies: increased stability, poorer processability, and reduced aging of bodies under mechanical, electrical, chemical, thermal, or radiation action.

The indicated correlations between the energetics of the lattice and the stability of bodies with an ionic structure have been traced over a wide range of variation in the action and properties of bodies. The energetics of ionic, electronic, and other processes in an ionic compound is associated with the chemical composition of the compounds.

The doctrine of the interrelation of the properties of dielectrics and their dependence on the energetics of the structure, advanced and developed in our investigations, will make it possible to determine the scientific foundations for the development of new electrical-insulating materials on the basis of the properties of the elements of D. I. Mendeleev’s periodic system. High-strength dielectric materials are substances with a large binding energy between particles.

The regularities determining the interrelations of the properties of bodies and their correlations with the energetics of the structure provide the scientific basis for describing the processability, stability, and aging of dielectrics from the general standpoint of the theory of the solid state.

The aging of dielectrics, as an irreversible residual change in their properties occurring under the external action of various factors on a body, is also satisfactorily described when the processes are considered energetically.

In compounds whose particles have a lower binding energy during the action of external factors, greater changes in properties are observed than in compounds with a higher binding energy between particles. After the cessation of the action, in bodies with weaker bonds, a more rapid annealing of the changes that have occurred takes place through thermal fluctuations than in bodies with a higher binding energy. In bodies with a lower binding energy between particles, the residual changes are smaller than in bodies with a higher binding energy between particles. The residual changes in bodies with a higher binding energy are relatively smaller, but they persist longer than in bodies with a lower binding energy. Bodies with a higher binding energy are more difficult to process, but they better preserve the results of the action and have a greater memory.

These ideas, expressed and developed by Tomsk scientists, have received satisfactory confirmation in practice and are of great importance for the systematization and correct description of the properties of materials under intense external actions, for example, those operating in

field of radiation, or high temperatures and mechanical stresses, high electric-field strengths.

At present, the study of the ionic and electronic properties of solids is acquiring ever greater importance. The electronic and ionic properties of crystalline structures have received a scientific description from a unified standpoint. Knowing only one quantity—the energy of the crystal lattice—on the basis of the data of the theory under consideration one can say which chemical compound in the series considered will have the larger coefficient of reflection or absorption of electromagnetic waves, the absorption of electrons, positrons, and other particles, the value of ionic electrical conductivity, losses, defect concentration, thermal conductivity, solubility, melting temperature, etc.

Such possibilities for classifying the properties of materials by only one physical characteristic present great practical convenience and are widely used.

The interrelation of the properties of solid dielectrics and their dependence on the energy of the lattice constitute a scientific idea of very great significance.

In the course of research work many interesting physical regularities were discovered, subsequently described in the literature, and numerous practical recommendations were made, often applied in practice even without reference to the original scientific ideas put forward in Tomsk.

Among them we may mention, for example, the proposal on fluorination of polymers as a means of increasing their stability, published by us in 1945; ways of increasing the strength of solid dielectrics by selecting materials with a greater bond energy between particles, in particular, for increasing the strength of glasses by replacing potassium or sodium with lithium, or barium with magnesium, and much else. In the research results cited above, in the energetic description of physical processes in materials, they were considered from both the macroscopic and the microscopic sides.

Many years of research on the microscopic and macroscopic description of physical processes—in particular, the thermodynamic description of the same processes in different materials—made it possible to establish the fruitfulness of each of them and the areas of their most successful application.

It is interesting to note that the energy of the crystal lattice, chosen by us to describe the properties of ionic crystalline processes in their correlation between the properties of bodies and their microscopic structure, reflects both aspects. As a thermodynamic characteristic, the lattice energy is a macroscopic quantity. On the other hand, in the theory of the solid state the energy of the crystal lattice is determined through ionic radii and valence; therefore it is a microscopic characteristic of a substance.

In the last decade, the microscopic direction has been developing successfully in the description of physical processes, as well as of the properties of materials and the design of new substances on an atomic-molecular basis. Using the energy spectrum of electrons in atoms, one can describe the energetics of a molecule and the energy spectrum of electrons in undistorted crystals built from these atoms. The corresponding theories have made it possible to calculate the melting temperature and boiling temperature of simple bodies. Knowing the chemical composition of certain compounds, it is often possible, geometrically, to precompute the interatomic distance and the number of nearest neighbors on the basis of the experimental determination of the radii of atoms or ions and their packing in the structure. Physical theory made it possible to introduce the division of solids into metals, insulators, and semiconductors. The classification was made according to the type of ener-

of the optical spectrum of electrons in these substances. Microscopic theory has made it possible to describe the corresponding transitions from the densely packed structure of ionic crystals to more complex framework structures of the type SiO₄, TiO₆, where polyhedra, tetrahedra, octahedra, etc., are assembled at special angles at the nodes of the structure. By filling the voids in the frameworks with chains of various ions, still more complex ionic structures can be obtained. There may be several variants of the arrangement of different atoms for the sought structure with specified properties. Therefore one may speak of the optimal design of materials and schemes on an atomic–molecular basis, including the design of living systems and the control of the vital functions of the simplest organisms.

Previously, the activity of materials scientists and physicists was characterized by their specialization and separateness. At present, the development of new materials on the basis of a molecular–atomic understanding of processes in bodies and of their properties unites specialists from various technologies and sciences.

In the last decade, lively discussions have unfolded concerning the possible role, in determining the properties of bodies, of crystal structure, chemical composition, short-range order, interaction energy, and the correlation between the structure and properties of a substance. The importance of short-range forces in the crystal structure is evident from the example of the role of localized \(d\)-electrons in creating materials with specified properties. The outer electrons of the \(S\) and \(P\) shells form the principal energy bands in crystals when the atomic energy levels of electrons split as a result of interaction effects between terms. A partially filled \(d\)-shell determines the magnetic properties of a substance. The electronic conductivity of a material is formed by energetically allowed electronic transitions. The crystal structure is determined by the distribution of electron density in the system under consideration.

Of considerable interest is the glassy state, in which, as in a liquid, there is no initial long-range order in the arrangement of particles. The high viscosity of glass provides experimental possibilities for preserving differences among metastable states and phases, which opens broad opportunities for producing materials with various properties—for example, the dielectric strength of glass-ceramic insulation for cables and capacitors, the manufacture of light filters with interesting properties, and much else. The glassy state can also provide great possibilities in the study of the molecular structure of living systems.

The modern physicist has remarkable experimental means that make it possible to observe microdefects in a crystal structure; studies of ionic conductivity and optical absorption, and comparison of the density of a body determined macroscopically with the density determined by X-ray methods, provide information about vacancies, interstitial and displaced atoms in the structure. Significant possibilities for determining microdefects of the structure are opened by the use of the ion microscope and by the phenomena of channeling of protons, electrons, and positrons. These means and methods make it possible to observe the formation of vacancies, their clusters, dislocations, and certain other microdisturbances in the lattice in the course of their appearance under the action of external factors—for example, temperature, high mechanical stresses, and high electric fields. The energy analysis of the physical processes governing the properties of dielectric materials, and the results found with its aid, not only proved to be a necessary stage in the development of the theory and practice of the physics of dielectrics, but have retained their scientific significance up to the present time.

For the intensification of production processes, the improvement of old methods and the development of new methods for processing metals are acquiring ever greater importance.

The general approach developed in our investigations to the description of the properties of a solid from the standpoint of the dynamic theory of crystalline structures—according to which all normal properties of bodies are described by the free energy or, approximately, by the energy of the crystal lattice—will make it possible to formulate certain general propositions for describing the machinability of dielectrics. It should be noted that the extraction of materials in the mining industry is associated with their destruction; in the processing industry, mechanical machining is associated with the removal of allowances, excess material from the blank, that is, again with the destruction of bodies. The energy supplied to the body is expended on the formation of defects, their coagulation, the creation of microcracks, their growth, and the accumulation of free energy. At the same time, recombination of defects occurs in the material, accompanied by dissipation of part of the stored energy. The shorter the duration and the higher the intensity of the external action, the less energy will be dissipated into the surrounding medium. Therefore, the efficiency of the process of material destruction increases with a decrease in the time of action of the external destructive factor. This general theoretical proposition leads to the conclusion that methods and means of pulsed destruction of materials have high efficiency, ensuring the maximum rate of propagation of a brittle-fracture crack in the given material. It has been established that the rate of propagation of a fracture crack increases with an increase in the strength of the material. A thermodynamic consideration of the processes occurring in stressed materials also made it possible to explain the processes of their strengthening during annealing, radiation treatment, and spark treatment of the surface, which have wide practical application.

REFERENCES

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