A crystal is different from an amorphous solid. Crystalline and amorphous bodies: structure and properties. Coordination number c.n.

Solids are crystalline and amorphous bodies. Crystal is what ice was called in ancient times. And then they began to call quartz a crystal and considered these minerals to be petrified ice. Crystals are natural and are used in the jewelry industry, optics, radio engineering and electronics, as supports for elements in ultra-precision instruments, as an ultra-hard abrasive material.

Crystalline bodies are characterized by hardness and have a strictly regular position in space of molecules, ions or atoms, resulting in the formation of a three-dimensional periodic crystal lattice (structure). Outwardly, this is expressed by a certain symmetry of the shape of a solid body and its certain physical properties. In their external form, crystalline bodies reflect the symmetry inherent in the internal “packing” of particles. This determines the equality of the angles between the faces of all crystals consisting of the same substance.

In them, the distances from center to center between neighboring atoms will also be equal (if they are located on the same straight line, then this distance will be the same along the entire length of the line). But for atoms lying on a straight line with a different direction, the distance between the centers of the atoms will be different. This circumstance explains the anisotropy. Anisotropy is the main difference between crystalline bodies and amorphous ones.

More than 90% of solids can be classified as crystals. In nature they exist in the form of single crystals and polycrystals. Monocrystals are single crystals, the faces of which are represented by regular polygons; They are characterized by the presence of a continuous crystal lattice and anisotropy of physical properties.

Polycrystals are bodies consisting of many small crystals, “grown together” somewhat chaotically. Polycrystals are metals, sugar, stones, sand. In such bodies (for example, a fragment of a metal), anisotropy usually does not appear due to the random arrangement of elements, although anisotropy is characteristic of an individual crystal of this body.

Other properties of crystalline bodies: strictly defined temperature (presence of critical points), strength, elasticity, electrical conductivity, magnetic conductivity, thermal conductivity.

Amorphous - having no shape. This is how this word is literally translated from Greek. Amorphous bodies are created by nature. For example, amber, wax. Humans are involved in the creation of artificial amorphous bodies - glass and resins (artificial), paraffin, plastics (polymers), rosin, naphthalene, var. do not have due to the chaotic arrangement of molecules (atoms, ions) in the structure of the body. Therefore, for any amorphous body they are isotropic - the same in all directions. For amorphous bodies there is no critical melting point; they gradually soften when heated and turn into viscous liquids. Amorphous bodies are assigned an intermediate (transitional) position between liquids and crystalline bodies: at low temperatures they harden and become elastic, in addition, they can split into shapeless pieces upon impact. At high temperatures, these same elements exhibit plasticity, becoming viscous liquids.

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Introduction

Chapter 1. Crystalline and amorphous bodies

1.1 Ideal crystals

1.2 Single crystals and crystalline aggregates

1.3 Polycrystals

Chapter 2. Elements of symmetry of crystals

Chapter 3. Types of defects in solids

3.1 Point defects

3.2 Linear defects

3.3 Surface defects

3.4 Volumetric defects

Chapter 4. Obtaining crystals

Chapter 5. Properties of Crystals

Conclusion

Bibliography

Introduction

Crystals are one of the most beautiful and mysterious creations of nature. Currently, the science of crystallography is studying the diversity of crystals. She reveals signs of unity in this diversity, studies the properties and structure of both single crystals and crystalline aggregates. Crystallography is a science that comprehensively studies crystalline matter. This work is also devoted to crystals and their properties.

Currently, crystals are widely used in science and technology, as they have special properties. Such areas of use of crystals as semiconductors, superconductors, quantum electronics and many others require a deep understanding of the dependence of the physical properties of crystals on their chemical composition and structure.

Currently, methods for artificially growing crystals are known. A crystal can be grown in an ordinary glass; this requires only a certain solution and the care with which it is necessary to care for the growing crystal.

There are a great variety of crystals in nature, and there are also many different forms of crystals. In reality, it is almost impossible to provide a definition that would apply to all crystals. Here, the results of X-ray analysis of crystals can be used to help. X-rays make it possible to feel the atoms inside a crystalline body, and determine their spatial location. As a result, it was found that absolutely all crystals are built from elementary particles located in strict order inside the crystalline body.

In all crystalline structures without exception, many identical atoms can be distinguished from atoms, located like nodes of a spatial lattice. To imagine such a lattice, let’s mentally fill the space with many equal parallelepipeds, parallel oriented and touching along entire faces. The simplest example of such a building is a masonry of identical bricks. If we select the corresponding points inside the bricks, for example, their centers or vertices, then we will obtain a model of a spatial lattice. All crystalline bodies without exception are characterized by a lattice structure.

Crystals are called " all solids in which the constituent particles (atoms, ions, molecules) are arranged strictly regularly like nodes of spatial lattices". This definition is as close as possible to the truth; it is suitable for any homogeneous crystalline bodies: boules (a form of crystal that has no faces, edges, or protruding vertices), grains, and flat-faced figures.

Chapter 1.Crystalline and amorphous bodies

Based on their physical properties and molecular structure, solids are divided into two classes - amorphous and crystalline solids.

A characteristic feature of amorphous bodies is their isotropy, i.e. independence of all physical properties (mechanical, optical, etc.) from direction. Molecules and atoms in isotropic solids are arranged randomly, forming only small local groups containing several particles (short-range order). In their structure, amorphous bodies are very close to liquids.

Examples of amorphous bodies include glass, various hardened resins (amber), plastics, etc. If an amorphous body is heated, it gradually softens, and the transition to a liquid state takes a significant temperature range.

In crystalline bodies, particles are arranged in a strict order, forming spatial periodically repeating structures throughout the entire volume of the body. To visually represent such structures, spatial crystal lattices, at the nodes of which the centers of atoms or molecules of a given substance are located.

In each spatial lattice, one can distinguish a structural element of minimal size, which is called unit cell.

Rice. 1. Types of crystal lattices: 1 - simple cubic lattice; 2 - face-centered cubic lattice; 3 - body-centered cubic lattice; 4 - hexagonal lattice

In a simple cubic lattice, the particles are located at the vertices of the cube. In a face-centered lattice, particles are located not only at the vertices of the cube, but also at the centers of each of its faces. In a body-centered cubic lattice, an additional particle is located at the center of each cubic unit cell.

It should be remembered that the particles in crystals are tightly packed, so that the distance between their centers is approximately equal to the size of the particles. In the image of crystal lattices, only the position of the centers of the particles is indicated.

1. 1 Perfect Crystals

The correct geometric shape of crystals attracted the attention of researchers even in the early stages of the development of crystallography and gave rise to the creation of certain hypotheses about their internal structure.

If we consider an ideal crystal, we will not find any violations in it; all identical particles are located in identical parallel rows. If we apply three elementary translations that do not lie in the same plane to an arbitrary point and repeat it endlessly in space, we get a spatial lattice, i.e. three-dimensional system of equivalent nodes. Thus, in an ideal crystal, the arrangement of material particles is characterized by strict three-dimensional periodicity. And in order to get a clear idea of ​​the patterns associated with the geometrically correct internal structure of crystals, in laboratory classes in crystallography they usually use models of ideally formed crystals in the form of convex polyhedra with flat faces and straight edges. In fact, the faces of real crystals are not perfectly flat, since as they grow they become covered with tubercles, roughness, grooves, growth pits, vicinals (faces that deviate entirely or partially from their ideal position), spirals of growth or dissolution, etc. .

Perfect Crystal- this is a physical model, which is an infinite single crystal that does not contain impurities or structural defects. The difference between real crystals and ideal ones is due to the finiteness of their sizes and the presence of defects. The presence of some defects (for example, impurities, intercrystalline boundaries) in real crystals can be almost completely avoided using special methods of growth, annealing or purification. However, at a temperature T>0K, crystals always have a finite concentration of (thermally activated) vacancies and interstitial atoms, the number of which in equilibrium decreases exponentially with decreasing temperature.

Crystalline substances can exist in the form of single crystals or polycrystalline samples.

A single crystal is a solid in which a regular structure covers the entire volume of the substance. Single crystals are found in nature (quartz, diamond, emerald) or are produced artificially (ruby).

Polycrystalline samples consist of a large number of small, randomly oriented, crystals of different sizes, which can be interconnected by certain interaction forces.

1. 2 Monocrystalloys and crystalline aggregates

Monocrystal- a separate homogeneous crystal that has a continuous crystal lattice and sometimes has anisotropy of physical properties. The external shape of a single crystal is determined by its atomic crystal lattice and the conditions (mainly speed and uniformity) of crystallization. A slowly grown single crystal almost always acquires a well-defined natural cut; under nonequilibrium conditions (average growth rate) of crystallization, the cut appears weakly. At an even higher rate of crystallization, instead of a single crystal, homogeneous polycrystals and polycrystalline aggregates are formed, consisting of many differently oriented small single crystals. Examples of faceted natural single crystals include single crystals of quartz, rock salt, Iceland spar, diamond, and topaz. Single crystals of semiconductor and dielectric materials grown under special conditions are of great industrial importance. In particular, single crystals of silicon and artificial alloys of elements of group III (third) with elements of group V (fifth) of the periodic table (for example, GaAs gallium arsenide) are the basis of modern solid-state electronics. Single crystals of metals and their alloys do not have special properties and are practically not used. Single crystals of ultrapure substances have the same properties regardless of the method of their preparation. Crystallization occurs near the melting point (condensation) from gaseous (for example, frost and snowflakes), liquid (most often) and solid amorphous states with the release of heat. Crystallization from gas or liquid has a powerful purifying mechanism: the chemical composition of slowly grown single crystals is almost ideal. Almost all contaminants remain (accumulate) in liquid or gas. This happens because as the crystal lattice grows, a spontaneous selection of the required atoms (molecules for molecular crystals) occurs not only according to their chemical properties (valency), but also according to size.

Modern technology no longer lacks the limited set of properties of natural crystals (especially for creating semiconductor lasers), and scientists have come up with a method for creating crystal-like substances with intermediate properties by growing alternating ultra-thin layers of crystals with similar crystal lattice parameters.

Unlike other states of aggregation, the crystalline state is diverse. Molecules of the same composition can be packaged in crystals in different ways. The physical and chemical properties of the substance depend on the packaging method. Thus, substances with the same chemical composition often actually have different physical properties. Such diversity is not typical for a liquid state, but impossible for a gaseous state.

If we take, for example, ordinary table salt, it is easy to see individual crystals even without a microscope.

If we want to emphasize that we are dealing with a single, separate crystal, then we call it single crystal, to emphasize that we are talking about an accumulation of many crystals, the term is used crystalline aggregate. If individual crystals in a crystalline aggregate are almost not faceted, this may be explained by the fact that crystallization began simultaneously in many points of the substance and its speed was quite high. Growing crystals are an obstacle to each other and prevent the correct cutting of each of them.

In this work we will mainly talk about single crystals, and since they are components of crystalline aggregates, their properties will be similar to the properties of the aggregates.

1. 3 Polycrystals

Polycrystal- an aggregate of small crystals of any substance, sometimes called crystallites or crystal grains because of their irregular shape. Many materials of natural and artificial origin (minerals, metals, alloys, ceramics, etc.) are polycrystalline.

Properties and getting. The properties of polycrystals are determined by the properties of the crystalline grains that make it up, their average size, which ranges from 1-2 microns to several millimeters (in some cases up to several meters), the crystallographic orientation of the grains and the structure of the grain boundaries. If the grains are randomly oriented and their sizes are small compared to the size of the polycrystal, then the anisotropy of physical properties characteristic of single crystals does not appear in the polycrystal. If a polycrystal has a predominant crystallographic orientation of grains, then the polycrystal is textured and, in this case, has anisotropy of properties. The presence of grain boundaries significantly affects the physical, especially mechanical, properties of polycrystals, since scattering of conduction electrons, phonons, braking of dislocations, etc. occurs at the boundaries.

Polycrystals are formed during crystallization, polymorphic transformations and as a result of sintering of crystalline powders. A polycrystal is less stable than a single crystal; therefore, during prolonged annealing of a polycrystal, recrystallization occurs (predominant growth of individual grains at the expense of others), leading to the formation of large crystalline blocks.

Chapter 2. Crystal symmetry elements

The concepts of symmetry and asymmetry have appeared in science since ancient times as an aesthetic criterion rather than strictly scientific definitions. Before the idea of ​​symmetry appeared, mathematics, physics, and natural science in general resembled separate islands of ideas, theories, and laws that were hopelessly isolated from each other and even contradictory. Symmetry characterizes and marks the era of synthesis, when disparate fragments of scientific knowledge merge into a single, holistic picture of the world. One of the main trends in this process is the mathematization of scientific knowledge.

Symmetry is usually considered not only as a fundamental picture of scientific knowledge, establishing internal connections between systems, theories, laws and concepts, but also attribute it to attributes as fundamental as space and time, movement. In this sense, symmetry determines the structure of the material world and all its components. Symmetry has a multifaceted and multi-level character. For example, in the system of physical knowledge, symmetry is considered at the level of phenomena, laws that describe these phenomena, and the principles underlying these laws, and in mathematics - when describing geometric objects. Symmetry can be classified as:

· structural;

· geometric;

· dynamic, describing, respectively, crystallographic,

mathematical and physical aspects of this concept.

The simplest symmetries can be represented geometrically in our ordinary three-dimensional space and are therefore visual. Such symmetries are associated with geometric operations that bring the body in question into coincidence with itself. They say that symmetry is manifested in the immutability (invariance) of a body or system in relation to a certain operation. For example, a sphere (without any marks on its surface) is invariant under any rotation. This shows its symmetry. A sphere with a mark, for example, in the form of a point, coincides with itself only when rotated, after which the mark on it returns to its original position. Our three-dimensional space is isotropic. This means that, like a sphere without marks, it coincides with itself at any rotation. Space is inextricably linked with matter. Therefore, our Universe is also isotropic. The space is also homogeneous. This means that it (and our Universe) has symmetry with respect to the shift operation. Time has the same symmetry.

In addition to simple (geometric) symmetries, very complex, so-called dynamic symmetries are widely encountered in physics, that is, symmetries associated not with space and time, but with a certain type of interaction. They are not visual, and even the simplest of them, for example, the so-called gauge symmetries, it is difficult to explain without using a rather complex physical theory. Some conservation laws also correspond to gauge symmetries in physics. For example, the gauge symmetry of electromagnetic potentials leads to the law of conservation of electric charge.

In the course of social practice, humanity has accumulated many facts indicating both strict orderliness, balance between parts of the whole, and violations of this orderliness. In this regard, the following five categories of symmetry can be distinguished:

· symmetry;

· asymmetry;

· dissymmetry;

· antisymmetry;

· supersymmetry.

Asymmetry . Asymmetry is asymmetry, i.e. a state where there is no symmetry. But Kant also said that negation is never a simple exception or absence of corresponding positive content. For example, movement is a negation of its previous state, a change in an object. Movement denies rest, but rest is not the absence of movement, since there is very little information and this information is erroneous. There is no absence of rest, just like there is no movement, since these are two sides of the same essence. Rest is another aspect of movement.

There is also no complete absence of symmetry. A figure that does not have an element of symmetry is called asymmetrical. But, strictly speaking, this is not so. In the case of asymmetrical figures, the disorder of symmetry is simply brought to an end, but not to the complete absence of symmetry, since these figures are still characterized by an infinite number of first-order axes, which are also elements of symmetry.

Asymmetry is associated with the absence of all elements of symmetry in an object. Such an element is indivisible into parts. An example is the human hand. Asymmetry is a category opposite to symmetry, which reflects imbalances existing in the objective world associated with change, development, and restructuring of parts of the whole. Just as we talk about movement, meaning the unity of movement and rest, so symmetry and asymmetry are two polar opposites of the objective world. In real nature there is no pure symmetry and asymmetry. They are always in unity and continuous struggle.

At different levels of development of matter, there is either symmetry (relative order) or asymmetry (a tendency to disturb peace, movement, development), but these two tendencies are always united and their struggle is absolute. Real, even the most perfect crystals are far from the crystals of ideal shape and ideal symmetry considered in crystallography in their structure. They contain significant deviations from ideal symmetry. They also have elements of asymmetry: dislocations, vacancies, which affect their physical properties.

The definitions of symmetry and asymmetry indicate the universal, general nature of symmetry and asymmetry as properties of the material world. Analysis of the concept of symmetry in physics and mathematics (with rare exceptions) tends to absolutize symmetry and interpret asymmetry as the absence of symmetry and order. The antipode of symmetry appears as a purely negative concept, but worthy of attention. Significant interest in asymmetry arose in the middle of the 19th century in connection with L. Pasteur's experiments in the study and separation of stereoisomers.

Dissymmetry . Dissymmetry is internal, or upset, symmetry, i.e. the object lacks some elements of symmetry. For example, rivers flowing along the earth's meridians have one bank higher than the other (in the Northern Hemisphere, the right bank is higher than the left, and in the Southern Hemisphere, vice versa). According to Pasteur, a disymmetrical figure is one that cannot be combined with its mirror image by simple superposition. The amount of symmetry of a disymmetric object can be arbitrarily high. Dissymmetry in the broadest sense of its understanding could be defined as any form of approximation from an infinitely symmetrical object to an infinitely asymmetrical one.

Antisymmetry . Antisymmetry is called opposite symmetry, or symmetry of opposites. It is associated with a change in the sign of the figure: particles - antiparticles, convexity - concavity, black - white, tension - compression, forward - backward, etc. This concept can be explained by the example of two pairs of black and white gloves. If two pairs of black and white gloves are sewn from a piece of leather, the two sides of which are painted white and black, respectively, then they can be distinguished on the basis of rightism - leftism, by color - blackness and whiteness, in other words, on the basis of sign informatism and some other sign. The antisymmetry operation consists of ordinary symmetry operations, accompanied by a change in the second attribute of the figure.

Supersymmetry In the last decades of the 20th century, a model of supersymmetry began to develop, which was proposed by Russian theorists Gelfand and Lichtman. Simply put, their idea was that, just as there are ordinary dimensions of space and time, there must be extra dimensions that can be measured in the so-called Grassmann numbers. As S. Hawking said, even science fiction writers have not thought of anything as strange as the Grassmann dimensions. In our ordinary arithmetic, if the number 4 multiplied by 6 is the same as 6 multiplied by 4. But the strange thing about Grassmann numbers is that if X is multiplied by Y, then it is equal to minus Y multiplied by X. You feel How far is this from our classical ideas about nature and methods of describing it?

Symmetry can also be considered by forms of movement or so-called symmetry operations. The following symmetry operations can be distinguished:

· reflection in a plane of symmetry (reflection in a mirror);

rotation around the axis of symmetry ( rotational symmetry);

· reflection at the center of symmetry (inversion);

transfer ( broadcast) figures at a distance;

· screw turns;

· permutation symmetry.

Reflection in the plane of symmetry . Reflection is the most famous and most often found type of symmetry in nature. The mirror reproduces exactly what it “sees,” but the order considered is reversed: your double’s right hand will actually be his left hand, since the fingers are arranged in the reverse order. Everyone is probably familiar with the film “The Kingdom of Crooked Mirrors” from childhood, where the names of all the characters were read in reverse order. Mirror symmetry can be found everywhere: in the leaves and flowers of plants, architecture, ornaments. The human body, if we talk only about its appearance, has mirror symmetry, although not quite strict. Moreover, mirror symmetry is inherent in the bodies of almost all living creatures, and such a coincidence is by no means accidental. The importance of the concept of mirror symmetry can hardly be overestimated.

Anything that can be divided into two mirror-like halves has mirror symmetry. Each of the halves serves as a mirror image of the other, and the plane separating them is called the plane of mirror reflection, or simply the mirror plane. This plane can be called a symmetry element, and the corresponding operation can be called a symmetry operation . We come across three-dimensional symmetrical patterns every day: these are many modern residential buildings, and sometimes entire blocks, boxes and boxes piled up in warehouses; atoms of a substance in a crystalline state form a crystal lattice - an element of three-dimensional symmetry. In all these cases, the correct location allows for economical use of space and ensures stability.

A remarkable example of mirror symmetry in literature is the “changeling” phrase: “And the rose fell on Azor’s paw.” . In this line, the center of mirror symmetry is the letter “n”, relative to which all other letters (not taking into account the spaces between words) are located in mutually opposite order.

Rotational symmetry . The appearance of the pattern will not change if it is rotated at a certain angle around its axis. The symmetry that arises in this case is called rotational symmetry . An example is the children's game "pinwheel" with rotational symmetry. In many dances, figures are based on rotational movements, often performed only in one direction (i.e. without reflection), for example, round dances.

The leaves and flowers of many plants exhibit radial symmetry. This is a symmetry in which a leaf or flower, turning around the axis of symmetry, turns into itself. In cross sections of tissues forming the root or stem of a plant, radial symmetry is clearly visible. The inflorescences of many flowers also have radial symmetry.

Reflection at the center of symmetry . An example of an object of the highest symmetry, characterizing this symmetry operation, is a ball. Spherical forms are quite widespread in nature. They are common in the atmosphere (fog droplets, clouds), hydrosphere (various microorganisms), lithosphere and space. Spores and pollen of plants, drops of water released in a state of weightlessness on a spaceship have a spherical shape. At the metagalactic level, the largest spherical structures are spherical galaxies. The denser a galaxy cluster, the closer it is to a spherical shape. Star clusters are also spherical.

Translation, or transfer of a figure over a distance . Translation, or parallel transfer of a figure over a distance, is any unlimitedly repeating pattern. It can be one-dimensional, two-dimensional, three-dimensional. Translation in the same or opposite directions forms a one-dimensional pattern. Translation in two non-parallel directions forms a two-dimensional pattern. Parquet floors, wallpaper patterns, lace ribbons, paths paved with bricks or tiles, crystalline figures form patterns that have no natural boundaries. When studying the patterns used in book printing, the same elements of symmetry were discovered as in the design of tiled floors. Ornamental borders are associated with music. In music, elements of symmetrical construction include the operations of repetition (translation) and reversal (reflection). It is these elements of symmetry that are found in borders. Although most music is not strictly symmetrical, many pieces of music are based on symmetry operations. They are especially noticeable in children's songs, which, apparently, are so easy to remember. Operations of symmetry are found in the music of the Middle Ages and the Renaissance, in the music of the Baroque era (often in a very sophisticated form). During the time of I.S. Bach, when symmetry was an important principle of composition, a kind of musical puzzle game became widespread. One of them was to solve the mysterious "canons". Kanon is a form of polyphonic music based on carrying out a theme led by one voice in other voices. The composer would propose a theme, and the listeners would have to guess the operations of symmetry that he intended to use in repeating the theme.

Nature sets puzzles of the opposite type: we are offered a completed canon, and we must find the rules and motives underlying existing patterns and symmetry, and vice versa, look for patterns that arise when repeating a motive according to different rules. The first approach leads to the study of the structure of matter, art, music, and thinking. The second approach confronts us with the problem of design or plan, which has concerned artists, architects, musicians, and scientists since ancient times.

Helical turns . Translation can be combined with reflection or rotation, which creates new symmetry operations. A rotation by a certain number of degrees, accompanied by a translation over a distance along the axis of rotation, generates helical symmetry - the symmetry of a spiral staircase. An example of helical symmetry is the arrangement of leaves on the stem of many plants. The sunflower head has shoots arranged in geometric spirals, unwinding from the center outward. The youngest members of the spiral are in the center. In such systems, one can notice two families of spirals, unwinding in opposite directions and intersecting at angles close to straight lines. But no matter how interesting and attractive the manifestations of symmetry in the plant world are, there are still many secrets that control development processes. Following Goethe, who spoke about the tendency of nature towards a spiral, we can assume that this movement is carried out along a logarithmic spiral, each time starting from a central, fixed point and combining translational movement (stretching) with a rotation.

Commutation symmetry . Further expansion of the number of physical symmetries is associated with the development of quantum mechanics. One of the special types of symmetry in the microcosm is permutation symmetry. It is based on the fundamental indistinguishability of identical microparticles, which do not move along specific trajectories, and their positions are estimated according to probabilistic characteristics associated with the square of the modulus of the wave function. Commutation symmetry lies in the fact that when quantum particles are “rearranged,” the probabilistic characteristics do not change; the squared modulus of the wave function is a constant value.

Symmetry of similarity . Another type of symmetry is similarity symmetry, associated with the simultaneous increase or decrease of similar parts of the figure and the distances between them. An example of this kind of symmetry is the matryoshka doll. Such symmetry is very widespread in living nature. It is demonstrated by all growing organisms.

Symmetry issues play a decisive role in modern physics. The dynamic laws of nature are characterized by certain types of symmetry. In a general sense, symmetry of physical laws means their invariance with respect to certain transformations. It should also be noted that the types of symmetry considered have certain limits of applicability. For example, the symmetry of right and left exists only in the region of strong electromagnetic interactions, but is violated in weak ones. Isotopic invariance is valid only when electromagnetic forces are taken into account. To apply the concept of symmetry, you can introduce a certain structure that takes into account four factors:

· object or phenomenon that is being studied;

· transformation in relation to which symmetry is considered;

· Invariance of any properties of an object or phenomenon, expressing the symmetry in question. The connection between the symmetry of physical laws and conservation laws;

· limits of applicability of various types of symmetry.

The study of the symmetry properties of physical systems or laws requires the use of special mathematical analysis, primarily the concepts of group theory, which is currently most developed in solid state physics and crystallography.

Chapter 3. Types of defects in solids

All real solids, both single-crystalline and polycrystalline, contain so-called structural defects, types, concentrations, and behavior of which are very diverse and depend on the nature, conditions of obtaining materials and the nature of external influences. Most defects created by external influences are thermodynamically unstable, and the state of the system in this case is excited (nonequilibrium). Such external influences can be temperature, pressure, irradiation with particles and high-energy quanta, the introduction of impurities, phase hardening during polymorphic and other transformations, mechanical effects, etc. The transition to an equilibrium state can take place in different ways and, as a rule, is realized through a series metastable states.

Defects of the same types, interacting with defects of the same or another type, can annihilate or form new associations of defects. These processes are accompanied by a decrease in the energy of the system.

Based on the number of directions N in which the violation of the periodic arrangement of atoms in the crystal lattice, caused by a given defect, extends, defects are distinguished:

· Point (zero-dimensional, N=0);

· Linear (one-dimensional, N=1);

· Surface (two-dimensional, N=2);

· Volume (three-dimensional, N=3);

Now we will consider each defect in detail.

3.1 Point defects

To zero-dimensional (or point) crystal defects include all defects that are associated with the displacement or replacement of a small group of atoms, as well as with impurities. They arise during heating, doping, during crystal growth and as a result of radiation exposure. They can also be introduced as a result of implantation. The properties of such defects and the mechanisms of their formation have been best studied, including motion, interaction, annihilation, and evaporation.

· Vacancy - a free, unoccupied atom, node of the crystal lattice.

· Proper interstitial atom - an atom of the main element located in the interstitial position of the unit cell.

· Impurity atom substitution - replacement of an atom of one type with an atom of another type in a node of the crystal lattice. Substitution positions may contain atoms that differ relatively little in size and electronic properties from the base atoms.

· Interstitial impurity atom - the impurity atom is located in the interstices of the crystal lattice. In metals, interstitial impurities are usually hydrogen, carbon, nitrogen and oxygen. In semiconductors, these are impurities that create deep energy levels in the bandgap, such as copper and gold in silicon.

Complexes consisting of several point defects are also often observed in crystals, for example, a Frenkel defect (vacancy + own interstitial atom), bivacancy (vacancy + vacancy), A-center (vacancy + oxygen atom in silicon and germanium), etc.

Thermodynamics of point defects. Point defects increase the energy of the crystal, since a certain amount of energy was expended to form each defect. Elastic deformation causes a very small fraction of the vacancy formation energy, since ion displacements do not exceed 1% and the corresponding deformation energy is tenths of an eV. During the formation of an interstitial atom, displacements of neighboring ions can reach 20% of the interatomic distance, and the corresponding energy of elastic deformation of the lattice can reach several eV. The main share of the formation of a point defect is associated with a violation of the periodicity of the atomic structure and the bonding forces between atoms. A point defect in a metal interacts with the entire electron gas. Removing a positive ion from a site is equivalent to introducing a point negative charge; conduction electrons are repelled from this charge, which causes an increase in their energy. Theoretical calculations show that the energy of formation of a vacancy in the fcc lattice of copper is about 1 eV, and that of an interstitial atom is from 2.5 to 3.5 eV.

Despite the increase in crystal energy during the formation of its own point defects, they can be in thermodynamic equilibrium in the lattice, since their formation leads to an increase in entropy. At elevated temperatures, the increase in the entropy term TS of the free energy due to the formation of point defects compensates for the increase in the total crystal energy U, and the free energy turns out to be minimal.

Equilibrium concentration of vacancies:

Where E 0 - energy of formation of one vacancy, k- Boltzmann constant, T- absolute temperature. The same formula is valid for interstitial atoms. The formula shows that the concentration of vacancies should strongly depend on temperature. The formula for calculation is simple, but exact quantitative values ​​can be obtained only by knowing the energy value of defect formation. It is very difficult to calculate this value theoretically, so one has to be content with only approximate estimates.

Since the energy of defect formation is included in the exponent, this difference causes a huge difference in the concentration of vacancies and interstitial atoms. Thus, at 1000 °C in copper, the concentration of interstitial atoms is only 10?39, which is 35 orders of magnitude less than the concentration of vacancies at this temperature. In dense packings, which are characteristic of most metals, it is very difficult for interstitial atoms to form, and vacancies in such crystals are the main point defects (not counting impurity atoms).

Migration of point defects. Atoms undergoing vibrational motion continuously exchange energy. Due to the randomness of thermal motion, energy is unevenly distributed between different atoms. At some point, an atom may receive such an excess of energy from its neighbors that it will occupy a neighboring position in the lattice. This is how the migration (movement) of point defects occurs in the bulk of the crystals.

If one of the atoms surrounding a vacancy moves to a vacant site, then the vacancy will correspondingly move to its place. Consecutive elementary acts of displacement of a certain vacancy are carried out by different atoms. The figure shows that in a layer of close-packed balls (atoms), in order to move one of the balls to a vacant place, it must move balls 1 and 2 apart. Consequently, to move from a position in a node, where the energy of the atom is minimal, to an adjacent vacant node, where the energy is also is minimal, the atom must pass through a state with increased potential energy and overcome the energy barrier. For this, it is necessary for the atom to receive from its neighbors an excess of energy, which it loses while “squeezing” into a new position. The height of the energy barrier E m is called vacancy migration activation energy.

Sources and sinks of point defects. The main source and sink of point defects are linear and surface defects. In large perfect single crystals, the decomposition of a supersaturated solid solution of its own point defects is possible with the formation of the so-called. microdefects.

Complexes of point defects. The simplest complex of point defects is a bivacancy (divacancy): two vacancies located at adjacent lattice sites. Complexes consisting of two or more impurity atoms, as well as impurity atoms and their own point defects, play a major role in metals and semiconductors. In particular, such complexes can significantly affect the strength, electrical and optical properties of solids.

3.2 Linear defects

One-dimensional (linear) defects are crystal defects, the size of which in one direction is much larger than the lattice parameter, and in the other two - comparable to it. Linear defects include dislocations and disclinations. General definition: dislocation is the boundary of an area of ​​incomplete shear in a crystal. Dislocations are characterized by a shear vector (Burgers vector) and an angle μ between it and the dislocation line. When μ = 0, the dislocation is called a screw dislocation; at c=90° - edge; at other angles it is mixed and can then be decomposed into helical and edge components. Dislocations arise during crystal growth; during its plastic deformation and in many other cases. Their distribution and behavior under external influences determine the most important mechanical properties, in particular such as strength, ductility, etc. Disclination is the boundary of the region of incomplete rotation in the crystal. Characterized by a rotation vector.

3.3 Surface defects

The main representative defect of this class is the surface of the crystal. Other cases are grain boundaries of a material, including low-angle boundaries (representing associations of dislocations), twinning planes, phase interfaces, etc.

3.4 Volumetric defects

These include clusters of vacancies that form pores and channels; particles deposited on various defects (decorating), for example, gas bubbles, mother liquor bubbles; accumulations of impurities in the form of sectors (hourglasses) and growth zones. As a rule, these are pores or inclusions of impurity phases. They are a conglomerate of many defects. Origin: disruption of crystal growth regimes, decomposition of a supersaturated solid solution, contamination of samples. In some cases (for example, during precipitation hardening), volumetric defects are specially introduced into the material to modify its physical properties.

Chapter 4. Receivedno crystals

The development of science and technology has led to the fact that many precious stones or simply crystals rarely found in nature have become very necessary for the manufacture of parts of devices and machines, for scientific research. The demand for many crystals has increased so much that it was impossible to satisfy it by expanding the scale of production of old and searching for new natural deposits.

In addition, many branches of technology and especially scientific research increasingly require single crystals of very high chemical purity with a perfect crystal structure. Crystals found in nature do not meet these requirements, since they grow in conditions that are very far from ideal.

Thus, the task arose of developing a technology for the artificial production of single crystals of many elements and chemical compounds.

The development of a relatively simple method of making a “gem” leads to the fact that it ceases to be precious. This is explained by the fact that most precious stones are crystals of chemical elements and compounds widespread in nature. Thus, diamond is a carbon crystal, ruby ​​and sapphire are aluminum oxide crystals with various impurities.

Let's consider the main methods of growing single crystals. At first glance, it may seem that crystallization from a melt is very simple. It is enough to heat the substance above its melting point, obtain a melt, and then cool it. In principle, this is the correct way, but if special measures are not taken, then at best you will end up with a polycrystalline sample. And if the experiment is carried out, for example, with quartz, sulfur, selenium, sugar, which, depending on the rate of cooling of their melts, can solidify in a crystalline or amorphous state, then there is no guarantee that an amorphous body will not be obtained.

In order to grow one single crystal, slow cooling is not enough. It is necessary to first cool one small area of ​​the melt and obtain a “nucleation” of a crystal in it, and then, sequentially cooling the melt surrounding the “nucleation”, allow the crystal to grow throughout the entire volume of the melt. This process can be achieved by slowly lowering a crucible containing the melt through an opening in a vertical tube furnace. The crystal nucleates at the bottom of the crucible, since it first enters the region of lower temperatures, and then gradually grows throughout the entire volume of the melt. The bottom of the crucible is specially made narrow, pointed to a cone, so that only one crystalline nucleus can be located in it.

This method is often used to grow crystals of zinc, silver, aluminum, copper and other metals, as well as sodium chloride, potassium bromide, lithium fluoride and other salts used in the optical industry. In one day you can grow a rock salt crystal weighing about a kilogram.

The disadvantage of the described method is contamination of the crystals by the crucible material. crystal defect symmetry property

The crucibleless method of growing crystals from a melt, which is used to grow, for example, corundum (rubies, sapphires), does not have this drawback. The finest aluminum oxide powder from grains 2-100 microns in size is poured out in a thin stream from the hopper, passes through an oxygen-hydrogen flame, melts and falls in the form of drops onto a rod of refractory material. The temperature of the rod is maintained slightly below the melting point of aluminum oxide (2030°C). Drops of aluminum oxide cool on it and form a crust of sintered corundum mass. The clock mechanism slowly (10-20 mm/h) lowers the rod, and an uncut corundum crystal gradually grows on it, shaped like an inverted pear, the so-called boule.

As in nature, obtaining crystals from solution comes down to two methods. The first of these consists of slowly evaporating the solvent from a saturated solution, and the second of slowly decreasing the temperature of the solution. The second method is more often used. Water, alcohols, acids, molten salts and metals are used as solvents. A disadvantage of methods for growing crystals from solution is the possibility of contamination of the crystals with solvent particles.

The crystal grows from those areas of the supersaturated solution that immediately surround it. As a result, the solution near the crystal turns out to be less supersaturated than far from it. Since a supersaturated solution is heavier than a saturated one, there is always an upward flow of “used” solution above the surface of the growing crystal. Without such stirring of the solution, crystal growth would quickly cease. Therefore, the solution is often additionally stirred or the crystal is fixed on a rotating holder. This allows you to grow more advanced crystals.

The lower the growth rate, the better the crystals obtained. This rule applies to all growing methods. Sugar and table salt crystals can be easily obtained from an aqueous solution at home. But, unfortunately, not all crystals can be grown so easily. For example, the production of quartz crystals from solution occurs at a temperature of 400°C and a pressure of 1000 at.

Chapter 5. Properties of Crystals

Looking at various crystals, we see that they are all different in shape, but each of them represents a symmetrical body. Indeed, symmetry is one of the main properties of crystals. We call bodies symmetrical if they consist of equal, identical parts.

All crystals are symmetrical. This means that in each crystalline polyhedron one can find planes of symmetry, axes of symmetry, centers of symmetry and other symmetry elements so that identical parts of the polyhedron fit together. Let us introduce another concept related to symmetry - polarity.

Each crystalline polyhedron has a certain set of symmetry elements. The complete set of all symmetry elements inherent in a given crystal is called a symmetry class. Their number is limited. It has been mathematically proven that there are 32 types of symmetry in crystals.

Let us consider in more detail the types of symmetry in a crystal. First of all, crystals can have symmetry axes of only 1, 2, 3, 4 and 6 orders. Obviously, symmetry axes of the 5th, 7th and higher orders are not possible, because with such a structure, atomic rows and networks will not fill the space continuously; voids and gaps will appear between the equilibrium positions of the atoms. The atoms will not be in the most stable positions, and the crystal structure will collapse.

In a crystalline polyhedron you can find different combinations of symmetry elements - some have few, others have many. According to symmetry, primarily along the axes of symmetry, crystals are divided into three categories.

The highest category includes the most symmetrical crystals; they may have several symmetry axes of orders 2, 3 and 4, no axes of the 6th order, they may have planes and centers of symmetry. These shapes include cube, octahedron, tetrahedron, etc. They all have a common feature: they are approximately the same in all directions.

Crystals of the middle category can have axes of 3, 4 and 6 orders, but only one at a time. There can be several axes of order 2; planes of symmetry and centers of symmetry are possible. The shapes of these crystals: prisms, pyramids, etc. Common feature: a sharp difference along and across the main axis of symmetry.

Crystals in the highest category include: diamond, quartz, germanium, silicon, copper, aluminum, gold, silver, gray tin, tungsten, iron. To the middle category: graphite, ruby, quartz, zinc, magnesium, white tin, tourmaline, beryl. To the lowest: gypsum, mica, copper sulfate, Rochelle salt, etc. Of course, this list did not list all existing crystals, but only the most famous of them.

The categories are in turn divided into seven systems. Translated from Greek, "syngony" means "similar angle." Crystals with identical axes of symmetry, and therefore with similar angles of rotation in the structure, are combined into a crystal system.

The physical properties of crystals most often depend on their structure and chemical composition.

First, it is worth mentioning two basic properties of crystals. One of them is anisotropy. This term means a change in properties depending on the direction. At the same time, crystals are homogeneous bodies. The homogeneity of a crystalline substance consists in the fact that its two sections of the same shape and the same orientation have identical properties.

Let's talk first about electrical properties. In principle, the electrical properties of crystals can be considered using the example of metals, since metals, in one of their states, can be crystalline aggregates. Electrons, moving freely in the metal, cannot go out; this requires energy. If radiant energy is expended in this case, the effect of electron abstraction causes the so-called photoelectric effect. A similar effect is observed in single crystals. An electron torn from the molecular orbit, remaining inside the crystal, causes metallic conductivity in the latter (internal photoelectric effect). Under normal conditions (without irradiation), such connections are not conductors of electric current.

The behavior of light waves in crystals was studied by E. Bertolin, who was the first to notice that the waves behave non-standardly when passing through a crystal. One day Bertalin was sketching the dihedral angles of Iceland spar, then he put the crystal on the drawings, then the scientist saw for the first time that each line bifurcated. He was convinced several times that all spar crystals bifurcate light, only then did Bertalin write a treatise “Experiments with a birefringent Icelandic crystal, which led to the discovery of a wonderful and extraordinary refraction” (1669). The scientist sent the results of his experiments to individual scientists and academies in several countries. The works were accepted with complete distrust. The English Academy of Sciences allocated a group of scientists to test this law (Newton, Boyle, Hooke, etc.). This authoritative commission recognized the phenomenon as accidental and the law as non-existent. The results of Bertalin's experiments were forgotten.

Only 20 years later, Christiaan Huygens confirmed the correctness of Bertalin’s discovery and himself discovered birefringence in quartz. Many scientists who subsequently studied this property confirmed that not only Iceland spar, but also many other crystals bifurcate light.

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4. . 5. . 6. . 7. .

Everyone can easily divide bodies into solid and liquid. However, this division will only be based on external signs. In order to find out what properties solids have, we will heat them. Some bodies will begin to burn (wood, coal) - these are organic substances. Others will soften (resin) even at low temperatures - these are amorphous. A special group of solids consists of those for which the dependence of temperature on heating time is presented in Figure 12. These are crystalline solids. This behavior of crystalline bodies when heated is explained by their internal structure. Crystal bodies- these are bodies whose atoms and molecules are arranged in a certain order, and this order is preserved over a fairly large distance. The spatial periodic arrangement of atoms or ions in a crystal is called crystal lattice. The points of the crystal lattice at which atoms or ions are located are called lattice nodes.

Crystalline bodies are either single crystals or polycrystals. Monocrystal has a single crystal lattice throughout its entire volume.

Anisotropy single crystals lies in the dependence of their physical properties on direction. Polycrystal It is a combination of small, differently oriented single crystals (grains) and does not have anisotropy of properties. Most solids have a polycrystalline structure (minerals, alloys, ceramics).

The main properties of crystalline bodies are: certainty of melting point, elasticity, strength, dependence of properties on the order of arrangement of atoms, i.e., on the type of crystal lattice.

Amorphous are substances that have no order in the arrangement of atoms and molecules throughout the entire volume of this substance. Unlike crystalline substances, amorphous substances isotropic. This means that the properties are the same in all directions. The transition from an amorphous state to a liquid occurs gradually; there is no specific melting point. Amorphous bodies do not have elasticity, they are plastic. Various substances are in an amorphous state: glass, resins, plastics, etc.

Elasticity- the property of bodies to restore their shape and volume after the cessation of external forces or other reasons that caused the deformation of bodies. According to the nature of the displacement of particles of a solid body, the deformations that occur when its shape changes are divided into: tension - compression, shear, torsion and bending. For elastic deformations, Hooke's law is valid, according to which elastic deformations are directly proportional to the external influences that cause them. For tensile-compressive deformation, Hooke's law has the form: , where is mechanical stress, is relative elongation, is absolute elongation, is Young's modulus (elastic modulus). Elasticity is due to the interaction and thermal movement of the particles that make up the substance.

Depending on the physical properties and molecular structure, there are two main classes of solids - crystalline and amorphous.

Definition 1

Amorphous bodies have such a feature as isotropy. This concept means that they are relatively independent of optical, mechanical and other physical properties and the direction in which external forces act on them.

The main feature of aphmoric bodies is the chaotic arrangement of atoms and molecules, which gather only in small local groups, no more than a few particles in each.

This property brings amorphous bodies closer to liquids. Such solids include amber and other hard resins, various types of plastic and glass. Under the influence of high temperatures, amorphous bodies soften, but strong heat is required to convert them into liquid.

All crystalline bodies have a clear internal structure. Groups of particles in the same order are periodically repeated throughout the entire volume of such a body. To visualize such a structure, spatial crystal lattices are usually used. They consist of a certain number of nodes that form the centers of molecules or atoms of a particular substance. Typically, such a lattice is built from ions that are part of the desired molecules. Thus, in table salt, the internal structure consists of sodium and chlorine ions, combined in pairs into molecules. Such crystalline bodies are called ionic.

Figure 3. 6. 1 . Crystal lattice of table salt.

Definition 2

In the structure of each substance, one minimal component can be distinguished - unit cell.

The entire lattice of which the crystalline body is composed can be composed by translation (parallel transfer) of such a cell in certain directions.

The number of types of crystal lattices is not infinite. There are 230 species in total, most of which are artificially created or found in natural materials. Structural lattices can take the form of body-centered cubes (for example, for iron), face-centered cubes (for gold, copper), or a prism with six faces (magnesium, zinc).

In turn, crystalline bodies are divided into polycrystals and single crystals. Most substances belong to polycrystals, because they consist of so-called crystallites. These are small crystals fused together and randomly oriented. Monocrystalline substances are relatively rare, even among artificial materials.

Definition 3

Polycrystals have the property of isotropy, that is, the same properties in all directions.

The polycrystalline structure of the body is clearly visible under a microscope, and for some materials, such as cast iron, even with the naked eye.

Definition 4

Polymorphism– is the ability of a substance to exist in several phases, i.e. crystal modifications that differ from each other in physical properties.

The process of switching to another modification is called polymorphic transition.

An example of such a phenomenon could be the transformation of graphite into diamond, which in industrial conditions occurs at high pressure (up to 100,000 atmospheres) and high temperatures
(up to 2000 K).

To study the lattice structure of a single crystal or polycrystalline sample, X-ray diffraction is used.

Simple crystal lattices are shown in the figure below. It must be taken into account that the distance between the particles is so small that it is comparable to the size of the particles themselves. For clarity, the diagrams show only the positions of the centers.

Figure 3. 6. 2. Simple crystal lattices: 1 – simple cubic lattice; 2 – face-centered cubic lattice; 3 – body-centered cubic lattice; 4 – hexagonal lattice.

The simplest is the cubic lattice: such a structure consists of cubes with particles at the vertices. A face-centered lattice has particles not only at the vertices, but also on the faces. For example, the crystal lattice of table salt consists of two face-centered lattices nested inside each other. A body-centered lattice has additional particles at the center of each cube.

Metal gratings have one important feature. The ions of a substance are held in place by interaction with a gas of free electrons. The so-called electron gas is formed by one or more electrons given up by atoms. Such free electrons can move throughout the entire volume of the crystal.

Figure 3. 6. 3. Structure of a metal crystal.

If you notice an error in the text, please highlight it and press Ctrl+Enter

Like liquid, but also form. They are predominantly in a crystalline state.
Crystals- these are solid bodies, the atoms or molecules of which occupy certain, ordered positions in space. Therefore, the crystals have flat edges. For example, a grain of ordinary table salt has flat edges that form right angles with each other ( Fig.12.1).

This can be seen by examining the salt with a magnifying glass. And how geometrically correct the shape of a snowflake is! It also reflects the geometric correctness of the internal structure of a crystalline solid - ice ( Fig.12.2).

Anisotropy of crystals. However, the correct external shape is not the only or even the most important consequence of the ordered structure of the crystal. The main thing is dependence of the physical properties of the crystal on the direction chosen in the crystal.
First of all, the different mechanical strength of the crystals in different directions is striking. For example, a piece of mica easily exfoliates in one direction into thin plates ( Fig.12.3), but it is much more difficult to break it in the direction perpendicular to the plates.

A graphite crystal also easily exfoliates in one direction. When you write with a pencil, this delamination occurs continuously and thin layers of graphite remain on the paper. This happens because the graphite crystal lattice has a layered structure. The layers are formed by a series of parallel networks consisting of carbon atoms ( Fig.12.4). The atoms are located at the vertices of regular hexagons. The distance between the layers is relatively large - about 2 times the length of the side of the hexagon, so the bonds between the layers are less strong than the bonds within them.

Many crystals conduct heat and electricity differently in different directions. The optical properties of crystals also depend on the direction. Thus, a quartz crystal refracts light differently depending on the direction of the rays incident on it.
The dependence of physical properties on the direction inside the crystal is called anisotropy. All crystalline bodies are anisotropic.
Single crystals and polycrystals. Metals have a crystalline structure. It is metals that are mainly used today for the manufacture of tools, various machines and mechanisms.
If you take a relatively large piece of metal, then at first glance its crystalline structure does not appear in any way either in the appearance of this piece or in its physical properties. Metals in their normal state do not exhibit anisotropy.
The point here is that metal usually consists of a huge number of small crystals fused together. Under a microscope or even with a magnifying glass it is easy to see them, especially on a fresh metal fracture ( Fig.12.5). The properties of each crystal depend on the direction, but the crystals are randomly oriented relative to each other. As a result, in a volume significantly larger than the volume of individual crystals, all directions within metals are equal and the properties of metals are the same in all directions.

A solid consisting of a large number of small crystals is called polycrystalline. Single crystals are called single crystals.
By taking great precautions, it is possible to grow a large metal crystal - a single crystal.
Under normal conditions, a polycrystalline body is formed as a result of the fact that the growth of many crystals that has begun continues until they come into contact with each other, forming a single body.
Polycrystals include not only metals. A piece of sugar, for example, also has a polycrystalline structure.
Most crystalline solids are polycrystals, as they consist of many intergrown crystals. Single crystals - single crystals have a regular geometric shape, and their properties are different in different directions (anisotropy).

???
1. Are all crystalline bodies anisotropic?
2. Wood is anisotropic. Is it a crystalline body?
3. Give examples of monocrystalline and polycrystalline solids not mentioned in the text.

G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics 10th grade

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