Upon acquaintance with minerals, the inherent ability of many of them to take on the correct external outlines is involuntarily striking - to form crystals, that is, bodies bounded by a number of planes. In this regard, he constantly uses crystallographic terms and concepts. Therefore, brief information on crystallography should precede a systematic acquaintance with mineralogy.
PROPERTIES OF CRYSTAL SUBSTANCE
All homogeneous bodies according to the nature of the distribution of physical properties in them can be divided into two large groups: amorphous and crystalline bodies.
In amorphous bodies, all physical properties are statistically the same in all possible directions.
Such bodies are called isotropic (equivalent).
Amorphous bodies include liquids, gases, and from solid bodies - glasses, vitreous alloys, as well as hardened colloids (gels).
In crystalline bodies, many physical properties are associated with a certain direction: they are the same in parallel directions and are not the same, generally speaking, in non-parallel directions.
This nature of the properties is called anisotropy, and those with similar properties are anisotropic (unequal).
The majority of solids and, in particular, the vast majority of minerals belong to crystalline bodies.
Among the physical properties of any solid body is the force of adhesion between the individual particles that make up the body. This physical property in a crystalline medium changes with a change in direction. For example, in rock salt crystals (Fig. 1), occurring in the form of more or less regular cubes, this cohesion will be the least perpendicular tothe faces of the cube. Therefore, a piece of rock salt, upon impact, will split with the greatest ease in a certain direction - parallel to the face of the cube, and a piece of an amorphous substance, such as glass, of the same shape will split equally easily but in any direction.
The property of a mineral to split along a certain, previously known direction, with the formation of a split surface in the form of a smooth, shiny plane, is called cleavage (see below "Physical Properties of Minerals"). It is inherent in varying degrees in many minerals.
When isolated from a supersaturated solution, the same force of interparticle attraction causes deposition from the solution in certain directions; perpendicular to each of these directions, a plane is formed, which, as new portions settle on it, will move away from the center of the growing crystal parallel to itself. Fig 1. Perfect cleavage the density of such planes with rock salt gives the crystal inherenthim the correct polyhedral shape.
If the influx of matter to a growing crystal occurs unevenly from different sides, which is usually observed under natural conditions, in particular, if a crystal in its growth is constrained by the presence of neighboring crystals, the deposition of matter will also occur unevenly, and the crystal will receive a flattened or elongated shape, or will occupy only the free space that is located between previously formed crystals. It must be said that most often this happens, and regular, uniformly formed crystals for many minerals are rare.
With all this, however, the directions of the planes of each crystal remain unchanged, and therefore, the dihedral angles between the corresponding (equivalent) planes on different crystals of the same substance and the same structure should be constant values (Fig. 2).
This is the first fundamental law of crystallography, known as the law of constancy of dihedral angles, was first noticed by Kepler and expressed in general form by the Danish scientist N. Steno in 1669. In 1749, M.V. first connected the law of constancy of angles with the internal structure of a crystal using saltpeter as an example.
Finally, 30 years later, the French crystallographer J. Romet-Delille, after twenty years of work on measuring angles in crystals, confirmed the generality of this law and formulated it for the first time.
Rice. 2. Quartz crystals
This pattern, deduced by Steno-Lomonosov-Rome-Delisle, formed the basis of everything scientific research crystals of that time and served as a starting point for the further development of the science of crystals. If we imagine the faces of the crystal moved parallel to themselves so thatthe significant faces have moved to the same distance from the center, the resulting polyhedra will take on the ideal shape that would be achieved by a growing crystal in the case of ideal, i.e., not complicated by external influences, conditions.
SYMMETRY ELEMENTS
Symmetry. With seeming simplicity and routine, the concept of symmetry is rather complicated. In the simplest definition, symmetry is the correctness (pattern) in the arrangement of the same parts of the figure. This correctness is expressed: 1) in the regular repetition of parts during the rotation of the figure, and the latter, when turning, seems to be combined with itself; 2) in the mirror equality of the parts of the figure, when some parts of it are presented as a mirror image of others.
All these regularities will become much clearer after getting acquainted with the elements of symmetry.
Considering well-formed crystals or crystallographic models, it is easy to establish those regularities that are observed in the distribution in crystals of identical planes and equal angles. These regularities are reduced to the presence in crystals of the following symmetry elements (individually or in certain combinations): 1) symmetry planes, 2) symmetry axes, and 3) a center of symmetry.
Rice. 3. Plane of symmetry
1. An imaginary plane that divides a figure into two equal parts related to each other, like an object to its image in a mirror (or like a right hand to a left hand), is calledplane of symmetry and is denoted by the letter R(Fig. 3 - plane) AB).
2. The direction, when turning around which always at the same angle, all parts of the crystal repeat symmetrically P times, is called a simple or rotary axis of symmetry (Fig. 4 and 5). Number P, showing how many times the repetition of parts is observed with a complete (360 °) revolution of the crystal around the axis, is called the order or significance of the axis of symmetry.
On the basis of theoretical considerations, it is easy to prove that P - always an integer and that in crystals only symmetry axes of 2, 3, 4 and 6 orders can exist.
Rice. 4. Axis of symmetry of the 3rd order
The axis of symmetry is denoted by the letter L or g, and the order of the axis of symmetry - the indicator set at the top right. So L 3 denotes the axis of symmetry of the 3rd order; L 6- axis of symmetry of the 6th order, etc. If there are several axes or planes of symmetry in the crystal, then their number is indicated by a coefficient that is placed in front of the corresponding letter. So, 4L 3 3L 2 6P means that the crystal has four axes of symmetry of the 3rd order, three axes of symmetry of the 2nd order and 6 planes of symmetry.
In addition to simple axes of symmetry, complex axes are also possible. In the case of the so-called mirror-rotary axis, the alignment of the polyhedron with all its parts with the initial position occurs not as a result of only one rotation through some angle a, but also as a result of simultaneous reflection in an imaginaryperpendicular plane. The axis of complex symmetry is also denoted by the letter L but only the axis index is placed at the bottom, for example, L4. The study shows that crystalline polyhedra can have complex axes of 2, 4 and 6 names or orders, i.e. L 2 , L4 And L 6 .
Rice. 5. Polyhedron with an axis of symmetry of the 2nd order
The same kind of symmetry can be achieved using the inversion axis. In this case, the symmetrical operation consists of a combination of rotation around the axis by an angle of 90 or 60° and repetition through the center of symmetry.
The process of this symmetric operation can be illustrated by the following example: let there be a tetrahedron (tetrahedron) with edges AB And CD mutually perpendicular (Fig. 6). When the tetrahedron is rotated by 180° around the axis L i4 , the whole figure is aligned with the original position, i.e. the axis L i4 , is an axis of symmetry of the second order (L2). In fact, the figure is more symmetrical, since the rotation, about the same axis by 90 °
and subsequent movement of the point A according to the center of symmetry will translate it into a point D. In the same way, dot IN compatible with dot WITH. The whole figure will be aligned with its original position. Such a combination operation can be carried out each time when the figure is rotated around the axis L i4 by 90 °, but with the obligatory repetition through the center of symmetry. Selected axis direction L i4 and will be the direction of the 4th order inversion axis ( L i4 = G i4 ).
Rice. 6. Polyhedron with a quadruple inversion axis of symmetry (Li4)
The use of inversion axes is in some cases more convenient and visual than the use of mirror-rotary axes. They can also be referred to as G i3 ; G i4 ; G i6; or how L i3 ;L i4 ; L i6
point inside the crystal equal distance from which in opposite directions there are equal, parallel and generally inverse faces, is called the center of symmetry or the center of inverse equality and is denoted by the letter With(Fig. 7). It is very easy to prove that c =L i2
i.e., that the center of inverse equality appears in crystals, whichwhich have an axis of complex symmetry of the 2nd order. It should also be noted that the axes of complex symmetry are at the same time the axes of simple symmetry of half the name, i.e.designations are possible L 2 i4 ;L 3 i6 . However, the opposite conclusioncannot be done, since not every axis of simple symmetry will necessarily be an axis of complex symmetry twice as large denominations.
The Russian scientist A. V. Gadolin in 1869 proved that only 32 combinations (combinations) of the above symmetry elements, called crystallographic classes or types of symmetry, can exist in crystals. All of them are stated in natural or artificial crystals.
CRYSTALLOGRAPHIC AXES. PARAMETERS AND INDICES
When describing a crystal, in addition to indicating the symmetry elements, it is necessary to determine the position in space of its individual faces. To do this, they use the usual methods of analytical geometry, at the same time taking into account the features of natural crystalline polyhedra.
Rice. 7. Crystal having a center of symmetry
Crystallographic axes are drawn inside the crystal, intersecting in the center and in most cases coinciding with the symmetry elements (axes, crystal planes or perpendicular to them). With a rational choice of crystallographic axes, crystal faces that have the same shape and physical properties receive the same numerical value, and the axes themselves will run parallel to the observed or possible crystal edges. In most cases, they are limited to three axes I, II and III, less often it is necessary to carry out four axes - I, II, III and IV.
In the case of three axes, one axis is directed towards the observer and is indicated by the sign I (Fig. 8), the other axis is directed from left to right and is indicated by the sign II, and finally the third axis is directed vertically and is indicated by the sign III.
In some manuals, the I axis is called X,II axis - Y, and III axis - Z. In the presence of four axes, the I axis corresponds to the A axis, the II axis to the Y axis, the III axis to the axis U and IV axis -axes Z.
The ends of the axes directed towards the observer, to the right and up, are positive, and those directed from the observer to the left and down are negative.
Rice. 8. Crystal faces on coordinate axes
Let the plane R(Fig. 8) cuts off segments on the crystallographic axes a, b And With. Since crystal polyhedra are determined only by the facet angles and slope of each plane, and not by the dimensions of the planes, it is possible, by mixing any plane parallel to itself, to increase and decrease the size of the polyhedron (which happens during crystal growth). Therefore, to indicate the position of the plane R no need to know the absolute values of the segments a, b And With, but their attitude a: b: c. Any other plane of the same crystal will be denoted in the general case a' : b': c' or a": b": c".
Let's assume that a’-ta; b' = nb; c' = pc; a" = t'a; b" = n'b; c" = p's, i.e., the lengths of the segments along the crystallographic axes for these planes are expressed in numbers that are multiples of the lengths of the segments along the crystallographic axes of the plane R, called the original or singular. Quantities t, p, p, t', p', p' are called numerical parameters of the corresponding plane.
In crystalline polyhedra, the numerical parameters are simple and rational numbers.
This property of crystals was discovered in 1784 by the French scientist Ayui and is called the Law of Rationality of Parameters.
Rice. 9. Elementary parallelepiped and a single face
Usually the parameters are 1, 2, 3, 4; the larger the number that expresses the parameters, the rarer the corresponding faces.
If we choose the crystallographic axes so that they run elementary parallel to the edges of the crystal, then the segments of the boundarythese axes, which are cut off by the initial face of the crystal (face R), determine the basic cell of a given crystalline substance.
It should be borne in mind that for crystals with low symmetry it is often necessary to adopt an oblique system of crystallographic axes. In this case, it is necessary to indicate the angles between the crystallographic axes, denoting them as a (alpha), p (beta), and y (gamma). In this case, i is called the angle between the III and II axes, R-angle between III and I(the so-called monoclinic angle), am - the angle between the I and II axes (Fig. 9).
On fig. 8 reference plane R cuts segments on the corresponding axes a,b And With or their multiples.
Any other plane must cut along the I axis a segment that is a multiple of A, along the II axis - a multiple of b and along the III axis - a multiple With.
So plane R will cut off segments a, 2b and 2s, and the plane R" - segments 2a, 4b and 3c, etc. The coefficients of a, 6, and c, which are parameters, can only be rational values.
Quantities a, b and c or their ratios are characteristic constants for a given crystal and are called axial units.
Designations of planes along segments on crystallographic axes in general view dominated science until the last quarter of the 19th century, but then gave way to others.
At present, the Miller method is used to indicate the position of the crystal faces, as it is the most convenient for crystallographic calculations, although at first glance it seems somewhat complicated and artificial.
As stated above, the original or "unit" plane will determine the axial units, and, knowing the parameters t:n:p any other plane, it is possible to determine the position of this latter. For crystallographic calculations, it is more advantageous to characterize the position of any face not by the direct ratio of the segments made by it on the crystallographic axes of the crystal to the segments of the "single" face, but by the inverse ratio, i.e., by dividing the length of the segment made by the single face by the segment made by the face being determined.
Obviously, the ratios obtained will also be expressed by integers, denoted in the general case by the letters h, k And l. Thus, the position of any face can be expressed uniquely in terms of three quantities h, k And l, the ratio between which is inverse to the ratio of the lengths of the segments made by the face on three crystallographic axes, and along each axis, in the general case, those segments (single segments) that a single face makes on the corresponding axes should be taken. If we take for the crystallographic axes the directions that coincide with the axes of symmetry or normals to the planes of symmetry or, if there are no such symmetry elements, with the edges of the crystal, then the characteristics of the faces can be expressed in simple and integer numbers, while all faces of the same shape will be expressed in a similar way.
Quantities h, To And l are called the indexes of the face, and their combinations are called the symbol of the face. The face symbol is usually denoted by successive indices without any punctuation marks and enclosing them in parentheses (hbl). At the same time, the index h refers to the I axis, index k ko II and index l to III. It is obvious that the values of the index are inverse to the value of the segment made by the face on the axis. If the face is parallel to the crystallographic axis, then the corresponding index is zero. If all three indices can be reduced by the same amount,
then such a reduction must be done, remembering that the indices are always prime and integers.
The face symbol, if it is expressed in numbers, for example (210) reads: two, one, zero. If the face makes a segment in the negative direction of the axis, then a minus sign is placed over the corresponding index, for example (010). This symbol is read like this: zero, minus one, zero.
Rice. 10 Feldspar crystals
Law of constancy of angles
Under natural conditions, crystals do not always develop in favorable conditions and have such ideal shapes as shown in the figures.
Very often, crystals have incompletely developed forms, with underdeveloped limiting elements (faces, edges, corners). Often, in crystals of the same mineral, the size and shape of the faces can vary significantly (Fig. 9-11). Often in soils and rocks there are not whole crystals, but only their fragments. However, measurements have shown the angles between the corresponding faces (and edges) of crystals of different shapes of the same mineral always remain constant.
This is one of the basic laws of crystallography - the law of constancy of angles.
What explains such constancy of angles. This phenomenon is due to the fact that all crystals of the same have the same structure, that is, they are identical in their internal structure. The law is valid for the same physicochemical conditions in which the measured crystals are located, i.e., at the same temperatures, pressure, etc. A sharp change in the angles in crystals can occur during a polymorphic transformation (see Chapter III).
Rice. 11. Three quartz crystals with different development of the corresponding faces
The law of constancy of angles was first mentioned by a number of scientists: I. Kepler, E. Bartholin, X. Huygens, A. Leeuwenhoek. This law was expressed in general form in 1669 by the Danish scientist N. Stenop. In 1749, for the first time, he connected the law of constancy of angles with the internal structure of saltpeter. And finally, in 1772, the French mineralogist Rome de Lisle formulated this law for all crystals.
On fig. 10 shows two feldspar crystals of various shapes. The angles between the corresponding faces a and b of two crystals are equal to each other (they are denoted by the letter of the Greek alphabet a). On fig. 11, the angle between the faces t and r of quartz crystals of different external shape is 38° 13'. From what has been said, it is clear how important the measurement of the dihedral angles of crystals is for the accurate diagnosis of a mineral.
Rice. 12. 13. Measurement of the facet angle of a crystal using an applied goniometer.Schematic diagram of a reflective goniometer
Measurement of facet angles of crystals. Goniometers
To measure the dihedral angles of crystals, special instruments are used, called goniometers (Greek "gonos" - angle). The simplest goniometer used for approximate measurements is the so-called applied goniometer, or the Carangio goniometer (Fig. 12). For more accurate measurements, a reflective goniometer is used (Fig. 13).
Measurement of angles using a reflective goniometer is carried out as follows: a beam of light, reflected from the edge of the crystal, is captured by the eye of the observer; turning the crystal, fix the reflection of the light beam from the second face on the scale of the goniometer circle, count the angle between the two reflections, and, consequently, between the two faces of the crystal.
The measurement of the dihedral angle will be correct if the faces of the crystal, from which the light beam is reflected, are parallel to the axis of rotation of the goniometer. To ensure that this condition is always met, the measurement is made on a two-circle or theodolite goniometer, which has two circles of rotation: the crystal can rotate simultaneously around two axes - horizontal and vertical.
Rice. 14. Theodolite goniometer E. S. Fedorov
The theodolite goniometer was invented in late XIX V. the Russian crystallographer Fedorov and, independently of him, the German scientist W. Goldschmidt. The general view of the two-armed goniometer is shown in Fig. 14.
Crystal chemical analysis of E. S. Fedorova
The method of goniometric determination of the crystalline and, to a certain extent, its internal structure from the external forms of crystals allowed Fedorov to introduce crystal chemical analysis into the practice of diagnosing minerals.
The discovery of the law of constancy of angles made it possible, by measuring the facet angles of crystals and comparing the measurement data with the available tabular values, to establish that the crystal under study belongs to a particular substance. Fedorov did a great job of systematizing the huge literature on the measurement of crystals. Using it as well own measurements crystals, Fedorov wrote the monograph "The Kingdom of Crystals" (1920).
Rice. 15. Scheme of the ratio of angles in a crystal during its measurement
The students and followers of Fedorov - the Soviet crystallographer A. K. Boldyrev, the English scientist T. Barker (1881-1931) greatly simplified the methods for determining crystals. At present, “crystal chemical analysis is reduced to measuring the required angles on a goniometer and to determining the substance from reference tables.
In the goniometric measurement of crystals, the internal angle between the faces is directly determined (Fig. 15, ∠β). However, summary tables with measured angles of various substances always show the angle formed by the normals to the corresponding faces (Fig. 15, ∠α). Therefore, after the measurement, simple calculations should be made using the formula α= 180°-β (α=α1, as angles with mutually perpendicular sides) and determine the name of the mineral from the reference book.
Symmetry in crystals
We learn about the existence of symmetry in nature from early childhood. Butterfly and dragonfly wings, petals and leaves of various flowers and plants, snowflakes and convince us that there is symmetry in nature.
Symmetrical bodies are called bodies consisting of identical, symmetrical parts that can be combined. So, if a butterfly folds its wings, they will be completely combined with it. The plane that divides the butterfly into two parts will be the plane of symmetry. If we put a mirror in place of this plane, we will see in it a symmetrical reflection of the other wing of the butterfly. So the plane of symmetry has the property of mirroring - on both sides of this plane we see symmetrical, mirror-equal halves of the body.
As a result of studying the crystalline forms of minerals, it was found that there is symmetry in inanimate nature, in the world of crystals. In contrast to symmetry in living nature, it is called crystalline symmetry.
Crystal symmetry is the correct repeatability of limiting elements (edges, faces, corners) and other properties of crystals in certain directions.
The symmetry of crystals is most clearly revealed in their geometric form. Regular repetition geometric shapes can be seen if: 1) cut the crystal with a plane; 2) rotate it around a certain axis; 3) compare the location of the crystal limiting elements with respect to a point lying inside it.
Plane of symmetry of crystals
Let's cut a rock salt crystal into two halves (Fig. 16). The drawn plane divided the crystal into symmetrical parts. This plane is called the plane of symmetry.
Rice. 17 Planes of symmetry in a cube
The plane of symmetry of a crystalline polyhedron is a plane, on both sides of which there are identical limiting elements and are repeated same properties crystal.
The plane of symmetry has the property of specularity: each of the parts of the crystal, cut by the plane of symmetry, is combined with the other, that is, it is, as it were, its mirror image. In different crystals, a different number of symmetry planes can be drawn. For example, in a cube there are nine planes of symmetry (Fig. 17), in a hexagonal or hexagonal prism - seven planes of symmetry - three planes will pass through opposite edges (Fig. 18, plane a), three planes through the midpoints of opposite faces (parallel to the longitudinal axis of the polyhedron - in Fig. 18, plane b) and one plane is perpendicular to it (Fig. 18 plane
The plane of symmetry is denoted capital letter Latin alphabet P, and the coefficient in front of it shows the number of symmetry planes in the polyhedron. Thus, for a cube, you can write 9P, i.e., nine planes of symmetry, and for a hexagonal prism - 7 P.
Rice. 18. Planes of symmetry in a hexagonal prism (left) and the arrangement of axes of symmetry (in plan,on right)
Axis of symmetry
In crystalline polyhedra, one can find axes, when rotated around which the crystal will be aligned with its original position when rotated through a certain angle. Such axes are called axes of symmetry.
The axis of symmetry of a crystalline polyhedron is a line, during rotation around which the same limiting elements and other properties of the crystal are correctly repeated.
Axes of symmetry are denoted by the capital Latin letter L. When the crystal rotates around the axis of symmetry, the constraint elements and other properties of the crystal will repeat a certain number of times.
If, when the crystal is rotated by 360 °, the polyhedron is combined with its original position twice, we are dealing with a second-order symmetry axis, with four- and six-fold alignment, respectively, with the axes of the fourth and sixth orders. Axes of symmetry are designated: L 2 - axis of symmetry of the second order; L 3 - axis of symmetry of the third order; L 4 - axis of symmetry of the fourth order; L 6 - axis of symmetry of the sixth order.
The order of the axis is the number of combinations of the crystal with the original position when rotated through 360°.
Due to the homogeneity of the crystal structure and due to the regularities in the distribution of particles inside crystals, crystallography proves the possibility of the existence of only the above
axes of symmetry. The axis of symmetry of the first order is not taken into account, since it coincides with any direction of each figure. A crystalline polyhedron can have several axes of symmetry of different orders. The coefficient in front of the symmetry axis symbol indicates the number of symmetry axes of one order or another. Thus, in a cube there are three axes of symmetry of the fourth order 3L4 (through the midpoints of opposite faces); four axes of the third order - 4L3 (drawn through opposite vertices of trihedral angles) and six axes of the second order 6L2 (through the midpoints of opposite edges) (Fig. 19).
In a hexagonal prism, one axis of the sixth order and 6 axes of the second order can be drawn (Fig. 18 and 20). In crystals, along with the usual axes of symmetry described earlier, so-called inversion axes are distinguished.
The inversion axis of a crystal is a line, during rotation around which at some certain angle and subsequent reflection at the central point of the polyhedron (as in the center of symmetry), the same limiting elements are combined .
Rice. 20 Axes of symmetry of the sixth and second orders (L 6 6L 2) and planes of symmetry (7Р) in a hexagonal prism
The inversion axis is indicated by the symbol On crystal models, where it is usually necessary to define inversion axes, there is no center of symmetry. The possibility of existence of inversion axes of the following orders is proved: the first L i1 , the second L i2 , the third
L i3 , fourth L i4 , sixth L i6 . In practice, one has to deal only with inversion axes of the fourth and sixth orders (Fig. 21).
Sometimes inverted axes are indicated by a number to the right below the axis symbol. So, the inversion axis of the second order is denoted by the symbol of the third - L 3 , fourth L 4 sixth L 6 .
The inversion axis is, as it were, a combination of a simple axis of symmetry and the center of inversion (symmetry). The diagram below (Fig. 21) shows two inversion axes Li and L i4 . Let's analyze both cases of finding these axes in the models. In a trigonal prism (Fig. 21, I), the straight line LL is the third-order axis L 3. At the same time, it is simultaneously a sixth-order inversion axis. So, when turning by 60 ° around the axis of any parts of the polyhedron and then reflecting them at the central point, the figure is combined with itself. In other words, the rotation of the edge AB of this prism by 60° around LL brings it to position A 1 B 1 , the reflection of the edge A 1 B 1 through the center aligns it with DF.
In a tetragonal tetrahedron (Fig. 21, II), all faces consist of four completely identical isosceles triangles. Axis LL - axis of the second order L 2 When rotated around it by 180 °, the polyhedron is combined with its original position, and the face ABC goes to the place ABD. At the same time, the L2 axis is also a fourth-order inversion axis. If you rotate the face ABC by 90° around the LL axis, then it will take the position A 1 B 1 C 1 . When reflecting A 1 B 1 C 1 at the central point of the figure, the face will coincide with the position of BCD (point A1 coincides with C, B 1 with D and C 1 with B). Having done the same operation with all parts of the tetrahedron, we note that it is combined with itself. When the tetrahedron is rotated by 360°, we get four such combinations. Therefore, LL is a fourth-order inversion axis.
Center of symmetry
In crystalline polyhedra, in addition to planes and axes of symmetry, there can also be a center of symmetry (inversion).
The center of symmetry (inversion) of a crystalline polyhedron is a point lying inside the crystal, in diametrically opposite directions from which the same limiting elements and other properties of the polyhedron are located.
The center of symmetry is denoted by the letter C of the Latin alphabet. If there is a center of symmetry in the crystal, each face corresponds to another face, equal and parallel (reversely parallel) to the first. There cannot be more than one center of symmetry in a crystal. In crystals, any line passing through the center of symmetry is bisected.
The center of symmetry is easy to find in a cube, an octahedron in a hexagonal prism, since it is located in these polyhedra at the intersection point of the axes and planes of symmetry.
Disassembled elements found in crystalline polyhedra, planes, axes, center of symmetry - are called symmetry elements.
|
Table 1 32 type of symmetry of crystals Types of symmetry |
|||||||
| primitive | central | planned | axial | planaxial | inversion-primitive | inversion-plan | |
| Triclinic | |||||||
| Monoclinic |
R |
L 2PC |
|||||
| Rhombic |
L 2 2P |
3L2 |
3L 2 3PC |
||||
| Trigonal |
L 3 C |
L 3 3P |
L 3 3L 2 |
L 3 3L 23PC |
|||
| tetragonal |
L4PC |
L44P |
L 4 4P 2 |
L 4 4L 2 5PC |
19** L i4 = L 2 |
L i4 (=L2)2L 2 x2P |
|
| Hexagonal |
L 6 |
||||||
GEOMETRIC CRYSTALLOGRAPHY Crystallography is the science of crystals, their external form, internal structure, physical properties, the processes of their formation in the earth's crust, space and the laws of the development of the Earth as a whole. Any material object has different symmetry levels of structural organization. A mineral, as a natural object, is not an exception, but on the contrary, it is one of the main material objects. earth's crust, which has all the properties of a crystalline substance, on the example of which all the basic laws of symmetry of crystals of polyhedra were studied and derived. Crystals are called solids with an ordered internal structure, having a three-dimensional periodic spatial atomic structure and having, as a result, under certain conditions of formation, the shape of polyhedra.
CRYSTALLOGRAPHY A discipline of a fundamental nature, obligatory for students of all natural specialties (physicists, chemists, geologists). 1. 2. 3. Main literature Egorov-Tismenko EM Crystallography and crystal chemistry. M. : Publishing House of Moscow State University, 2006. 460 p. M. P. Shaskolskaya. Crystallography. M. : graduate School, 1976. 391 p. G. M. Popov, I. I. Shafranovsky. Crystallography. Moscow: Higher school, 1972. 346 p.
Crystallography as a science Crystallography is the science of crystals and the crystalline state of matter in general. The word "crystal" is of Greek origin and means "ice". rhinestone» . Crystallography studies the properties of crystals, their structure, growth and dissolution, application, artificial production, etc. Crystals are called solids in which material particles are arranged regularly in the form of spatial lattice nodes
Connection of crystallography with other sciences Crystallography Geometry Painting Architecture Physics Mineralogy Petrography Metallography Mechanics Electroacoustics Radio engineering Chemistry Geochemistry Biology
Significance of crystallography Theoretical significance - knowledge of the most general patterns of the structure of matter, in particular the earth's crust Practical significance - industrial growing of crystals (single-crystal industry)
The concept of the structure of crystals Under the structure of crystals is understood the regular arrangement of material particles (atoms, molecules, ions) inside a crystal chemical substance. To describe the arrangement of particles in space, they began to be identified with points. From this approach, the idea of a spatial or crystal lattice of mineral crystals gradually formed. Lomonosov, Hayuy, Bravais, Fedorov laid the foundations of the geometric theory of the structure of crystals. A spatial lattice is an infinite three-dimensional periodic formation, the elements of which are nodes, rows, flat grids, elementary cells. The main feature of crystal chemical structures is the regular repetition in the space of nodes, rows and flat grids.
The nodes of the spatial lattice are called points at which the material particles of a crystalline substance are located - atoms, ions, molecules, radicals. Spatial lattice rows - a set of nodes lying along a straight line and periodically repeating at regular intervals Flat grid of a spatial lattice - a set of nodes located in the same plane and located at the vertices of equal parallelograms oriented parallel and complex along integer sides. The elementary cell of the spatial lattice is the smallest parallelipiped in terms of volume formed by a system of 3 mutually intersecting flat grids.
14 types of Bravais lattices In 1855, O. Bravais deduced 14 spatial lattices, differing in the forms of elementary cells and symmetry. They represent a regular repetition of the nodes of the spatial lattice. These 14 lattices are grouped according to syngonies. Any spatial lattice can be presented in the form of parallelepipeds of repetition, which, moving in space in the direction of its edges and by their size, form an infinite spatial lattice. Parallelepipeds of repeatability (elementary cells of Bravais lattices) are chosen according to the following conditions: 1. the syngony of the selected parallelepiped 2. the number of equal edges and angles between the edges of the parallelepiped should be maximum 3. if there are right angles between the edges of the parallelepiped, their number should be the largest 4. if the first 3 conditions are met, the volume of the parallelepiped should be the smallest. When choosing a unit cell, the already known rules for installing crystals are used; Cell edges are the shortest distance along the coordinate axes between the corners of the lattice. To characterize the external shape of the elementary cell, the values of the edges of the cell a, b, c and the angles between these
Cubic - the shape of the elementary cell corresponds to a cube. Hexagonal - a hexagonal prism with a pinacoid. Trigonal - rhombohedron. Tetragonal - a tetragonal prism with a pinacoid. Rhombic - brick. Monoclinal - a parallelepiped with one oblique angle and 2 other straight lines. Triclinic - an oblique parallelepiped with unequal edges. In accordance with the additional lattice nodes located in different parts of the cells, all lattices are divided into: Primitive (P); Base-centered (C); Body-centered (U); face centered (F);
GEOMETRIC CRYSTALLOGRAPHY Elements of limitation of polyhedra A polyhedron is a three-dimensional geological body separated from the surrounding space by elements of limitation. Restriction elements are called geometric images that separate the polyhedron from the surrounding space. Polyhedron constraint elements include faces, edges, vertices, dihedral and polyhedral angles. Faces are flat surfaces that limit the polyhedron from the external environment. Edges are straight lines along which faces intersect. The vertices are the points where the edges intersect. Dihedral angles are the angles between two adjacent faces. Otherwise, these are corners at the edges. Polyhedral angles are angles between several faces converging at one vertex. Otherwise, these are vertex angles.
Among the polyhedral angles, right and wrong are distinguished. If, when connecting the ends of the edges emanating from the vertex of a polyhedral angle, a regular geometric figure is obtained ( right triangle, rectangle, rhombus, square, regular hexagon and their derivatives), then a regular polyhedral angle is formed. If during the same operation an irregular geometric figure (irregular polygon) is obtained, then such a polyhedral angle is called irregular. The following regular polyhedral angles are distinguished. 1. Trigonal - when the ends of the edges emanating from its vertex are connected, a regular triangle (trigon) is formed: 2. Rhombic of the 1st kind - the connection of the ends of the edges emanating from its vertex gives a figure in the form of a rhombus; 3. Rhombic of the 2nd kind - a figure obtained by connecting the ends of the edges emanating from its vertex - a rectangle: 4. Tetragonal - when connecting the ends of the edges emanating from its vertex, a square (tetragon) is formed:
5. Hexagonal - the connection of the ends of the edges emanating from its vertex gives a regular hexagon (hexagon): These five regular polyhedral angles are called basic. In addition, the following three derivatives of regular polyhedral angles are formed from trigonal, tetragonal and hexagonal angles by doubling them. 1. Ditrigonal - formed by doubling the faces that make up a trigonal angle (ditrigon): 2. Ditetragonal - formed by doubling the number of faces of a tetragonal angle (ditrigon): 3. Dihexagonal - formed by doubling the number of faces that bound a hexagonal angle (dihexagon):
In all derivatives of regular polyhedral angles, the dihedral angles are equal through one, and all sides of the figure formed by connecting the ends of the edges emanating from the vertex are equal. Thus, there are only 8 regular polyhedral angles. All other polyhedral angles are irregular. There may be an infinite number of them. There is a mathematical dependence between the elements of the restriction of polyhedra, characterized by the Euler formula. Descartes: G (faces) + V (vertices) = P (edges) + 2. For example, in a cube there are 6 faces, 8 vertices and 12 edges. Hence: 6+8=12+2. 2. Elements of symmetry of polyhedrons Elements of symmetry are auxiliary geometric images (point, line, plane and their combinations), with the help of which you can mentally combine equal faces of a crystal (polyhedron) in space. In this case, the symmetry of a crystal is understood as a regular repetition in space of its equal faces, as well as vertices and edges. There are three main symmetry elements of crystals - the center of symmetry, the plane of symmetry and the axis of symmetry.
The center of symmetry is an imaginary point inside the crystal, equidistant from its limiting elements (i.e., opposite vertices, midpoints of edges and faces). The center of symmetry is the point of intersection of the diagonals of a regular figure (cube, parallelepiped). The center of symmetry is denoted by the letter C, and by international system Herman-Mogen - I. The center of symmetry in a crystal can be only one. However, there are crystals in which there is no center of symmetry at all. When deciding whether there is a center of symmetry in your crystal, you must be guided by the following rule: “If there is a center of symmetry in a crystal, each of its faces corresponds to an equal and opposite face”. In practical exercises with laboratory models, the presence or absence of a center of symmetry in a crystal is established as follows. We put the crystal with one of its faces on the plane of the table. We check if there is an equal and parallel face on top. We repeat the same operation for each face of the crystal. If each face of the crystal corresponds from above to a face equal and parallel to it, then the center of symmetry is present in the crystal. If for at least one face of the crystal there is no face equal and parallel to it from above, then there is no center of symmetry in the crystal
The plane of symmetry (denoted by the letter P, according to international symbols - m) is an imaginary plane passing through the geometric center of the crystal and dividing it into two mirror equal halves. Crystals with a plane of symmetry have two properties. First, its two halves, separated by a plane of symmetry, are equal in volume; secondly, they are equal, like reflections in a mirror. To check the mirror equality of the halves of the crystal, it is necessary to draw an imaginary perpendicular to the plane from each of its vertices and continue it at the same distance from the plane. If each vertex corresponds to a vertex mirrored to it on the opposite side of the crystal, then the plane of symmetry is present in the crystal. When determining symmetry planes on laboratory models, the crystal is placed in a fixed position and then mentally cut into equal halves. The mirror equality of the resulting halves is checked. We consider how many times we can mentally cut the crystal into two mirror equal parts. Remember that the crystal must be motionless! The number of symmetry planes in crystals varies from 0 to 9. For example, in a rectangular parallelepiped we find three symmetry planes, that is, 3 R.
The axis of symmetry is an imaginary line passing through the geometric center of the crystal, when turning around which the crystal repeats its appearance several times in space, that is, it self-aligns. This means that after a rotation through a certain angle, some faces of the crystal are replaced by other faces equal to them. The main characteristic of the axis of symmetry is the smallest angle of rotation at which the crystal "repeats" in space for the first time. This angle is called the elementary angle of rotation of the axis and is denoted by α. For example: The elementary angle of rotation of any axis must be an integer number of times 360°, i.e. (an integer), where n is the order of the axis. Thus, the order of an axis is an integer showing how many times the elementary angle of rotation of a given axis is contained in 360 °. Otherwise, the order of the axis is the number of "repetitions" of the crystal in space when it is completely rotated around this axis. The axes of symmetry are denoted by the letter L. The order of the axis is indicated by a small number at the bottom right: for example, L 2. The following axes of symmetry and the corresponding elementary rotation angles are possible in crystals.
n α Designation Domestic L 1 International 1 1 360° 2 180° L 2 2 3 120° L 3 3 4 90° L 4 4 6 60° L 6 6
The axes of symmetry and the first order in any crystal are an infinite number. Therefore, in practice they are not defined. Axes of symmetry of the 5th and any order higher than the 6th in crystals do not exist at all. This feature of crystals is practiced as the law of crystal symmetry. The symmetry law of crystals is explained by the specificity of their internal structure, namely, the presence of a spatial lattice, which does not allow the possibility of axes of the 5th, 7th, 8th and so on orders. A crystal can have several axes of the same order. For example, in a cuboid there are three axes of the second order, that is, 3 L 2. In a cube there are 3 axes of the fourth order, 4 axes of the third order and 6 axes of the second order. Axes of symmetry of the highest order in a crystal are called principal. To find the axes of symmetry on the models during laboratory classes, they act in the following order. The crystal is taken with the fingertips of one hand at its opposite points (vertices, midpoints of edges or faces). An imaginary axis is placed in front of it vertically. We remember any characteristic appearance of the crystal. Then we rotate the crystal with the other hand around an imaginary axis until its original appearance "repeats" in space. We consider how many times the crystal "repeats" in space with a complete rotation around a given axis. This will be her order. Similarly, we check all other theoretically possible directions of passage of the symmetry axis in the crystal.
The combination of all symmetry elements of a crystal, written in conventional notation, is called its symmetry formula. In the symmetry formula, first the axes of symmetry are listed, then the planes of symmetry, and the last one shows the presence of a center of symmetry. There are no dots or commas between symbols. For example, the formula for the symmetry of a rectangular parallelepiped: 3 L 33 PC; cube - 3 L 44 L 36 L 29 PC.
3. Types of symmetry of crystals The types of symmetry are the possible combinations of symmetry elements in crystals. Each type of symmetry corresponds to a certain symmetry formula. In total, the presence of 32 types of symmetry has been theoretically proven for crystals. Thus, there are 32 crystal symmetry formulas in total. All types of symmetry are combined into 7 steps of symmetry, taking into account the presence of characteristic symmetry elements. Primitive - combines the types of symmetry, represented only by single axes of symmetry of different orders, for example: L 3, L 4, L 6. Central - in addition to single axes of symmetry, there is a center of symmetry; in addition, in the presence of even axes of symmetry, another plane of symmetry appears, for example: L 3 C, L 4 PC, L 6 PC. Planar (plan - plane, Greek) - there is a single axis and planes of symmetry: L 22 P, L 44 P. Axial (axis - axis, Greek) - only axes of symmetry are present: 3 L 2, L 33 L 2, L 66 L 2. Planaxial - there are axes, planes and a center of symmetry: 3 L 23 PC, L 44 L 25 PC. Inversion-primitive - the presence of a single inversion axis of symmetry: Li 4, Li 6. Inversion-planar - the presence, in addition to the inversion axis, of simple axes and planes of symmetry: Li 44 L 22 P, Li 63 L 23 P. Each symmetry level combines a different number of symmetry types: from 2 to 7.
A syngony is a group of types of symmetry that have the same name 4. Syngony of the main axis of symmetry and the same general level symmetry. Syn - similar, gonia - angle, literally: syngony - similarity (Greek). The transition from one syngony to another is accompanied by an increase in the degree of crystal symmetry. In total, 7 syngonies are distinguished. In the order of successive increase in the degree of symmetry of the crystals, they are arranged as follows. The triclinic syngony (wedge - angle, slope, in Greek) was named taking into account the peculiarity of crystals that the angles between all faces are always oblique. Apart from C, there are no other symmetry elements. Monoclinic (monos - one, in Greek) - in one direction between the faces of the crystals, the angle is always oblique. L 2, P and C can be present in crystals. None of the symmetry elements is repeated at least twice. Rhombic - got its name from the characteristic cross-section of crystals (remember the rhombic angles of the 1st and 2nd kind). Trigonal - named after the characteristic cross section (triangle) and polyhedral angles (trigonal, ditrigonal). One L 3 is obligatory present. Tetragonal - a square-shaped cross-section and polyhedral angles are characteristic - tetragonal and ditetragonal. L 4 or Li 4 is necessarily present. Hexagonal - a section in the form of a regular hexagon, polyhedral angles - hexagonal and dihexagonal. the presence of one L 6 or Li 6 is mandatory. Cubic - a typical cubic form of crystals. The combination of symmetry elements 4 L 3 is characteristic.
Syngonias are combined into 3 categories: lower, middle and higher. The triclinic, monoclinic and rhombic syngonies are combined into the lowest category. The middle category includes trigonal, tetragonal and hexagonal systems. One main axis of symmetry is characteristic. One cubic syngony belongs to the highest category. Unlike the previous categories, it is characterized by several main axes of symmetry.
5. The concept of a simple form, combination and habit In practical exercises with laboratory models, a set of equal crystal faces is considered as a simple form. If all the faces of a crystal are the same, then it is a simple shape as a whole. On the contrary, if all the faces of a crystal are not equal in shape and geometric outlines, then each of its faces is a separate simple form. Thus, a crystal will have as many simple forms as it has. geometric types faces, taking into account also their sizes. For example, in a cuboid there are 3 types of faces. Face types in a cuboid Therefore, it consists of 3 simple shapes. Each of them, in turn, consists of 2 equal parallel faces. The names of simple forms are given depending on the number of faces and their relative position. There are a total of 47 simple shapes, each of which
To determine simple forms in practical exercises, it is necessary to mentally continue the faces equal to each other until they intersect. The resulting imaginary figure will be the desired simple form. Among simple forms, two types are distinguished: open and closed. The edges of an open simple form do not close the space on all sides. On the contrary, the faces of a closed simple form, when they are mutually continued in space from all sides, will close some part of it. Combinations of simple forms that form crystals are called complex forms, or combinations. There will be as many simple shapes in a combination as there are face types in it. One open simple form can never form a crystal, it can only occur in combination with other simple forms. The combinations in nature are endless. The habit of a crystal is understood as the simple form prevailing in terms of facet area. The name of the habitus coincides with the name of the simple form, but is given as a definition (for example, the simple form is a cube, the habitus is cubic). If none of the facets with simple area predominates (or it is difficult to assess this), the habitus is called mixed or combined.
6. The procedure for analyzing crystal models When studying crystal models in practical classes, the following data are characterized: 1) crystal symmetry formula; 2) syngony; 3) type of symmetry; 4) simple forms; 5) habitus.
Crystallography Crystallography is one of the fundamental sciences of the Earth, it studies the process of formation, external form, internal structure and physical properties crystals. Recently, this science has gone far beyond its limits and is studying the patterns of the development of the Earth, its shape and processes occurring in the depths of the geospheres.
Crystals sparkle with symmetry. ES Fedorov The classical definition of a crystal (from the Greek "crystallos" - ice), a homogeneous solid body capable of self-cutting under certain conditions. Let's take a closer look at this definition...
Spatial lattice A spatial lattice is a geometric image that reflects a three-dimensional periodicity in the distribution of atoms in a crystal structure.
The term symmetry Crystallography, like any completely independent science, has its own method - the SYMMETRY METHOD. Symmetry from Greek. "symmetry" proportionality), as they say, was introduced by Pythagoras, designating with him the SPATIAL REGULARITY IN THE ARRANGEMENT OF THE SAME FIGURES OR THEIR PARTS. Symmetry - regularity, repetition of figures or their parts, in space !! In a figurative sense, symmetry is a synonym for harmony, beauty and perfection!
Symmetry and humanity The concept of symmetry has been treated with trepidation since ancient times. HF China - the circle is the most perfect figure, the dwelling of the gods is also a circle. In Christianity, the connection with the concept of the Trinity (God the Father, God the Son, God the Holy Spirit). In ancient Egypt - "The All-Seeing Eye"
Symmetry in geology Lithology - ripples in the sand Paleontology - due to the orientation of one plane of symmetry from another, it is possible to distinguish brachiopods from bivalves. . Planes of symmetry in underwater ridges (at the bottom of the World Ocean). Explanation of the concept of spreading
Symmetry in living matter The most important thing! Most biological objects have mirror symmetry. Sometimes there is a fifth-order symmetry axis L 5, not in crystals!!! According to the assumption of N. V. Belov, that they could not “petrify” because there are no fifth-order axes in the crystalline substance.
Concepts that are urgently needed when describing crystal models in the Bravais educational symbolism Elements of symmetry - geometric images (planes, straight lines, lines or points) with the help of which symmetrical transformations (symmetry operations) are set or carried out Plane of symmetry Axis of symmetry Center of symmetry
Axis of symmetry Rotary axes of symmetry are straight lines, when rotated through a certain angle, the figure (or crystal) is combined with itself. The smallest angle of rotation around such an axis is called the elementary angle of rotation. The value of this angle determines the order of the axis of symmetry (360 divided by the value of this angle). It is designated in educational symbolism as Ln, where n is the order of the axis of symmetry: L 2 L 3 L 4 L 6
Important I draw your attention to the fact that in crystallographic polyhedra the order of the axes is limited by the numbers 1, 2, 3, 4, 6. That is, in crystals, symmetry axes of the 5th and higher than the 6th orders are impossible. Anyone who can come up with a convincing proof of this fact will receive a Swiss chocolate bar right in class!
To the proof of this fact 1. "Spatially-lattice" proof 2. According to Nikolai Vasilyevich Belov
Mirror plane of symmetry Mirror plane of symmetry specifies the operation of reflection, in which the right side of the figure (figure), reflected in the plane as in a "two-sided mirror" is combined with its left side (figure). It is denoted by the letter P.
Center of symmetry (point of symmetry) It is like a "mirror point" in which the right figure not only passes into the left, but also, as it were, turns over. The inversion point in this case plays the role of a camera lens, and the figures connected by it are related as an object and its image on a film film. Referred to as C
Crystallographic systems (syngony) Symmetry classes with a single coordinate frame are combined into a family called a syngony or system (from Greek syn. “similar” and “gony” - angle. All thirty-two classes of symmetry of crystals are divided into three categories, each of which includes one or more syngonies. These are triclinic, monoclinic, rhombic, hexagonal, (a special case of trigonal), tetragonal and cubic syngony. Let's take a look at them by category.
Hexagonal syngony. Average category a=b≠c, α=β=90˚, γ=120 ˚ "hexa" - six Presence of L 6 main feature
And now let's practice describing the crystals in the Brava symbolism. YOU NEED TO FIND AND WRITE ITS FULL FORMULA IN THE EDUCATIONAL BRAVOY SYMBOL AND NAME THE SYNGONY TO WHICH IT RELATES. We look at the highest ORDER of the axis in the formula. In addition to the cubic system 4 L 3 - a sign of CUBIC SYNGONY L 6 - a sign of HEXAGONAL SUBSYGONY L 4 - A SIGN OF TETRAGONAL SING. L 3 - A FEATURE OF A TRIGONAMIC SYNGONY L 2, 3 L 2 - A FEATURE OF A RHOMBIC SYNGONY L 2 - A FEATURE OF A MONOCLINE SYNGONY Or L BIS. ORDER, or just C - A SIGN OF THE TRICLINE SYNGONY
In the next lesson, we will once again practice describing crystal models. We will learn how to determine the basic simple forms using a cheat sheet and talk about questions that may arise before you passing the crystallography room at the Olympiad, plus we will talk about the dependence of the shape of crystals (using quartz and calcite as an example) on the conditions for their formation Think about the next lesson. What shape will a crystal grown in space have!!!
electronic engineering
Lecture 2
Ph.D., Assoc. Maronchuk I.I.
Fundamentals of crystallography
INTRODUCTIONMost modern structural materials, including
and composite - these are crystalline substances. Crystal
is a collection of regularly arranged atoms,
forming a regular structure that arose spontaneously from
the disordered environment around him.
The reason for the symmetrical arrangement of atoms is
the tendency of the crystal to a minimum of free energy.
Crystallization (the emergence of order from chaos, that is, from a solution,
pair) occurs with the same inevitability as, for example, the process
falling bodies. In turn, the minimum free energy is reached
with the smallest fraction of surface atoms in the structure, therefore
external manifestation of the correct internal atomic structure
crystalline bodies is the faceting of crystals.
In 1669, the Danish scientist N. Stenon discovered the law of constancy of angles:
the angles between the corresponding crystal faces are constant and
characteristic of this substance. Every solid body is made up of
interacting particles. These particles, depending on
nature of matter, there may be individual atoms, groups of atoms,
molecules, ions, etc. Accordingly, the relationship between them is:
atomic (covalent), molecular (Van der Wals bond), ionic
(polar) and metallic. In modern crystallography, there are four
directions, which are to a certain extent connected with one
others:
- geometric crystallography, which studies various
forms of crystals and laws of their symmetry;
- structural crystallography and crystal chemistry,
who study the spatial arrangement of atoms in
crystals and its dependence on the chemical composition and
conditions for the formation of crystals;
- crystal physics, which studies the influence of internal
structure of crystals on their physical properties;
- physical and chemical crystallography, which studies
questions of the formation of artificial crystals. ANALYSIS OF SPATIAL LATTICES
The concept of a spatial lattice and elementary
cell
When studying the question of the crystal structure of bodies
First of all, you need to have a clear understanding of
terms: "spatial lattice" and "elementary
cell". These terms are used not only in
crystallography, but also in a number of related sciences for
descriptions of how they are arranged in space
material particles in crystalline bodies.
As is known, in crystalline bodies, in contrast to
amorphous, material particles (atoms, molecules,
ions) are arranged in a certain order, on
a certain distance from each other. A spatial grid is a diagram that shows
arrangement of material particles in space.
The spatial lattice (Fig.) actually consists of
sets
identical
parallelepipeds,
which
completely, without gaps, fill the space.
Material particles are usually located at nodes
lattice - the intersection points of its edges.
Spatial lattice The elementary cell is
least
parallelepiped, with
with which you can
build the whole
spatial lattice
through continuous
parallel transfers
(broadcasts) in three
directions of space.
Type of elementary cell
shown in fig.
Three vectors a, b, c, which are the edges of the elementary cell,
are called translation vectors. Their absolute value (a,
b, c) are the lattice periods, or axial units. Injected into
consideration and angles between translation vectors - α (between
vectors b, c), β (between a, c), and γ (between a, b). So
Thus, an elementary cell is defined by six quantities: three
period values (a, b, c) and three values of the angles between them
(α, β, γ). Unit cell selection rules
When studying the concepts of an elementary cell, one should
note that the magnitude and direction
translations in the spatial lattice can be chosen in different ways, so the shape and size of the unit cell
will be different.
On fig. the two-dimensional case is considered. Shown flat
lattice grid and different ways to choose flat
elementary cell.
Selection methods
elementary cell In the middle of the XIX century. French crystallographer O. Brave
proposed the following conditions for choosing an elementary
cells:
1) the symmetry of the elementary cell must correspond to
symmetries of the spatial lattice;
2) the number of equal edges and equal angles between the edges
should be maximum;
3) in the presence of right angles between the ribs, their number
should be maximum;
4) subject to these three conditions, the volume
elementary cell should be minimal.
Based on these rules, Bravais proved that there is
only 14 types of elementary cells, which received
the name of translational ones, since they are built by
translation - transfer. These grids are different from each other.
other by the magnitude and direction of the broadcasts, and from here
the difference in the shape of the elementary cell and in the number
nodes with material particles. Primitive and complex elementary cells
According to the number of nodes with material particles, elementary
cells are divided into primitive and complex. IN
primitive Bravais cells, material particles are
only at the vertices, in complex - at the vertices and additionally
inside or on the surface of the cell.
Complex cells include body-centered I,
face centered F and base centered C. In fig.
elementary Bravais cells are shown.
Bravais elementary cells: a - primitive, b -
base-centered, c – body-centered, d –
face-centered The body-centered cell has an additional node in
the center of the cell that belongs only to this cell, so
there are two nodes here (1/8x8+1 = 2).
In a face-centered cell, nodes with material particles
are, in addition to the vertices of the cell, also in the centers of all six faces.
Such nodes belong simultaneously to two cells: the given one and the
another adjacent to it. For the share of this cell, each of these
nodes belongs to 1/2 part. Therefore, in a face-centered
the cell will have four nodes (1/8x8+1/2x6 = 4).
Similarly, there are 2 nodes in a base-centered cell
(1/8х8+1/2х2 = 2) with material particles. Basic information
about elementary Bravais cells are given below in Table. 1.1.
The primitive Bravais cell contains translations a,b,c only
along the coordinate axes. In a body-centered cell
one more translation along the spatial diagonal is added -
to the node located in the center of the cell. in the face-centered
except for axial translations a,b,c there are additional
translation along the diagonals of the faces, and in the base-centered -
along the diagonal of a face perpendicular to the Z axis. Table 1.1
Basic information about primitive and complex Bravais cells
Basis
Grating type Brave
Number Major
translation nodes
Primitive R
1
a,b,c
Body centered 2
aya I
a,b,c,(a+b+c)/2
[]
face centered
F
a,b,c,(a+b)/2,(a+c)/2,
(b+c)/2
[]
a,b,c,(a+b)/2
[]
4
Base-centered С 2
The basis is understood as the set of coordinates
the minimum number of nodes, expressed in axial
units, by broadcasting which you can get the entire
spatial grid. The basis is written in double
square brackets. Basis coordinates for various
types of Bravais cells are given in Table 1.1. Bravais elementary cells
Depending on the shape, all Bravais cells are distributed between
seven crystal systems (sygonies). Word
“Syngonia” means similarity (from the Greek σύν - “according to,
together, side by side", and γωνία - "corner"). Each syngony corresponds
certain elements of symmetry. In table. ratios
between lattice periods a, b, c and axial angles α, β, γ for
each syngony
Syngonia
Triclinic
Monoclinic
Rhombic
tetragonal
Hexagonal
Relations between
lattice periods and angles
a ≠ c ≠ c, α ≠ β ≠ γ ≠ 90º
a ≠ b ≠ c, α = γ = 90º ≠ β
a ≠ b ≠ c, α = β = γ = 90º
a \u003d b ≠ c, α \u003d β \u003d γ \u003d 90º
a = b ≠ c, α = β =90º, γ =120º
Rhombohedral
cubic
a \u003d b \u003d c,
a = b = c,
α = β =γ ≠ 90º
α = β = γ = 90º On fig. all
fourteen types
elementary Bravais Cells,
distributed in syngonies.
Hexagonal Bravais cell
represents
base-centered
hexagonal prism. However
she is often portrayed
otherwise - in the form of a tetrahedral
prisms with a rhombus at the base,
which represents one of
three prisms that make up
hexagonal (in Fig. she
represented by solid
lines). Such an image
easier and more convenient, although associated with
violation of the principle
symmetry matching
(first selection principle
elementary cell according to Brava). For the rhombohedral syngony
elementary cell,
satisfying the conditions
Brave, is primitive
rhombohedron R for which a=b=c and
α=β=γ≠ 90º. Along with R-cell
to describe rhombohedral
structures are used and
hexagonal cell,
since the rhombohedral
cell can always be reduced to
hexagonal (fig.) and
imagine it as three
primitive hexagonal
cells. In this regard, in
rhombohedral literature
syngony sometimes not separately
Three primitive
consider, present, her
hexagonal Cells,
as a variety
equivalent to rhombohedral
hexagonal. It is accepted syngony with the same ratios between
axial units to combine in one category. That's why
triclinic, monoclinic and rhombic systems
combined into the lowest category (a≠b≠c), tetragonal,
hexagonal (and its derivative rhombohedral) - in
medium (a=b≠c), the highest category (a=b=c) is
cubic system.
The concept of coordination number
In complex cells, material particles are stacked more than
denser than in primitive ones, more fully fill the volume
cells are more connected to each other. To characterize
This introduces the concept of coordination number.
The coordination number of a given atom is the number
nearest neighboring atoms. If it's about
coordination number of the ion, then the number
ions closest to it of the opposite sign. The more
coordination number, those with a higher number of atoms or
ions is bound given, the more space is occupied by particles, the
more compact lattice. Spatial lattices of metals
The most common among metals are spatial
lattices are relatively simple. They mostly match
with translation gratings Bravais: cubic
body-centered and face-centered. At the nodes of these
lattices are metal atoms. In the lattice
body-centered cube (bcc - lattice) each atom
surrounded by eight nearest neighbors, and the coordination
number of CC \u003d 8. Metals have a bcc lattice: -Fe, Li, Na, K, V,
Cr, Ta, W, Mo, Nb, etc.
In the lattice of a face-centered cube (fcc - lattices) KN = 12:
any atom located at the top of the cell has
twelve nearest neighbors, which are atoms,
located in the centers of the edges. FCC lattice have metals:
Al, Ni, Cu, Pd, Ag, Ir, Pt, Pb, etc.
Along with these two, among metals (Be, Mg, Sc, -Ti, -Co,
Zn, Y, Zr, Re, Os, Tl, Cd, etc.) there is also a hexagonal
compact. This lattice is not a translational lattice
Brava, because it cannot be described by simple broadcasts. On fig. the unit cell of the hexagonal
compact lattice. Unit cell hexagonal
compact lattice is a hexagonal
prism, but most often it is depicted in the form
a tetrahedral prism whose base is a rhombus
(a=b) with angle γ = 120°. Atoms (Fig.b) are located at the vertices
and in the center of one of the two trihedral prisms forming
elementary cell. A cell has two atoms: 1/8x8 + 1
=2, its basis is [].
The ratio of the unit cell height c to the distance a, i.e.
c/a is equal to 1.633; the periods c and a for different substances
different.
Hexagonal
compact lattice:
a - hexagonal
prism, b -
tetrahedral
prism. CRYSTALLOGRAPHIC INDICES
Crystallographic indices of the plane
In crystallography it is often necessary to describe the mutual
arrangement of individual crystal planes, its
directions for which it is convenient to use
crystallographic indices. Crystallographic
indices give an idea of the location of the plane
or directions relative to a coordinate system. At
it doesn't matter if it's rectangular or oblique
coordinate system, same or different scale
segments along the coordinate axes. Imagine a series
parallel planes passing through the same
nodes of the spatial lattice. These planes
located at the same distance from each other and
form a family of parallel planes. They
equally oriented in space and therefore
have the same indexes. We choose some plane from this family and
we introduce into consideration the segments that the plane
clips along the coordinate axes (coordinate axes x,
y, z are usually combined with the edges of the elementary
cells, the scale on each axis is equal to
corresponding axial unit - period a, or b,
or c). The values of the segments are expressed in axial
units.
Crystallographic indices of the plane (indices
Miller) are the three smallest integers,
which are inversely proportional to the number of axial
units cut off by the plane on the coordinate
axes.
Plane indices are denoted by letters h, k, l,
are written in a row and concluded in round
brackets-(hkl). The indices (hkl) characterize all the planes of the family
parallel planes. This symbol means that
a family of parallel planes cuts the axial
unit along the x-axis into h parts, along the y-axis into k
parts and along the z axis into l parts.
In this case, the plane closest to the origin of coordinates,
cuts the segments 1/h on the coordinate axes (along the x axis),
1/k (along the y-axis), 1/l (along the z-axis).
Order of finding crystallographic indices
planes.
1. We find the segments cut off by the plane on
coordinate axes, measuring them in axial units.
2. We take the reciprocal values of these quantities.
3. We give the ratio of the obtained numbers to the ratio
the three smallest integers.
4. The resulting three numbers are enclosed in parentheses. Example. Find the indices of the plane that cuts off at
coordinate axes the following segments: 1/2; 1/4; 1/4.
Since the lengths of the segments are expressed in axial units,
we have 1/h=1/2; 1/k=1/4; 1/l=1/4.
Find the reciprocals and take their ratio
h:k:l = 2:4:4.
Reducing by two, we present the ratio of the obtained quantities
to the ratio of the three smallest integers: h: k: l = 1: 2:
2. Plane indices are written in parentheses
in a row, without commas - (122). They are read separately
"one, two, two".
If the plane intersects the crystallographic axis at
negative direction, above the corresponding
the minus sign is placed above the index. If the plane
is parallel to any coordinate axis, then in the symbol
plane index corresponding to this axis is zero.
For example, the symbol (hko) means that the plane
intersects the z-axis at infinity and the plane index
along this axis will be 1/∞ = 0. Planes clipping on each axis by an equal number
axial units are denoted as (111). in a cubic
their syngonies are called the planes of the octahedron, since the system
these planes, equidistant from the origin,
forms an octahedron - octahedron fig.
Octahedron Planes that cut along two axes an equal number of axial
units and parallel to a third axis (such as the z-axis)
denoted by (110). In the cubic syngony, similar
the planes are called the planes of the rhombic dodecahedron,
So
How
system
planes
type
(110)
forms
dodecahedron (dodeca - twelve), each face
which is a rhombus fig.
Rhombic
dodecahedron Planes that intersect one axis and are parallel to two
others (for example, the y and z axes), denote - (100) and
are called in the cubic syngony the planes of the cube, that is
a system of similar planes forms a cube.
When solving various problems related to the construction in
unit cell of planes, coordinate system
it is advisable to choose so that the desired plane
located in a given elementary cell. For example,
when constructing the (211) plane in a cubic cell, the beginning
coordinates can be conveniently transferred from node O to node O'.
Cube plane (211) Sometimes plane indices are written in curly braces
(hkl). This entry means the symbol of the set of identical
planes. Such planes pass through the same nodes
in a spatial lattice, symmetrically located in
space
And
characterized
the same
interplanar spacing.
The planes of the octahedron in the cubic syngony belong to
one set (111), they represent the faces of the octahedron and
have the following indices: (111) →(111), (111), (111), (111),
(111), (111), (111), (111).
The symbols of all constellation planes are found by
permutations and changes in the signs of individual
indexes.
For planes of a rhombic dodecahedron, the notation
set: (110) → (110), (110), (110),
(110), (101), (101), (101), (101), (011), (011), (011), (011).CRYSTALLOGRAPHIC INDICES OF THE NODE
The crystallographic indices of a node are its
coordinates taken in fractions of axial units and written in
double square brackets. In this case, the coordinate
corresponding to the x-axis, is generally denoted by the letter
u, for the y-axis - v, for the z-axis - w. The knot symbol looks like
[]. Symbols of some nodes in the elementary cell
shown in fig.
Some nodes in
elementary cell
(Sometimes a node is denoted
How []) Crystallographic direction indices
In a crystal where all parallel directions
identical to each other, the direction passing through
the origin of coordinates, characterizes the entire given family
parallel directions.
Position
V
space
directions,
passing through the origin, is determined
coordinates of any node lying on this
direction.
Coordinates
any
knot,
owned
direction, expressed in fractions of axial units and
reduced to the ratio of the three smallest integers
numbers,
And
There is
crystallographic
indices
directions. They are denoted by integers u, v, w
and are written together in square brackets. Order of Finding Direction Indexes
1. From the family of parallel directions, select
one that passes through the origin, or
move this direction parallel to itself
yourself to the origin, or move the origin
coordinates to a node lying in the given direction.
2. Find the coordinates of any node belonging to
given direction, expressing them in axial units.
3. Take the ratio of the coordinates of the node and bring it to
ratio of the smallest integers.
4. Conclude the resulting three numbers in square
brackets.
The most important directions in the cubic lattice and their
indices are presented in fig. Some directions in a cubic lattice THE CONCEPT OF CRYSTAL AND POLAR
COMPLEX
The method of crystallographic projections is based on
one of the characteristic features of crystals - the law
angle constancy: angles between certain faces and
the edges of the crystal are always constant.
So, when the crystal grows, the sizes of the faces change, their
shape, but the angles remain the same. Therefore, in
crystal, you can move all edges and faces in parallel
to ourselves at one point in space; corner
the ratio is preserved.
Such
totality
planes
And
directions,
parallel to planes and directions in the crystal and
passing through one point is called
crystal complex, and the point itself is called
center
complex.
At
building
crystallographic projections crystal always replace
crystalline complex. More often, not a crystalline complex is considered, but
polar (reverse).
Polar complex, obtained from crystalline
(direct) by replacing the planes with normals to them, and
directions - planes perpendicular to them.
A
b
Cube (a), its crystalline (b) and
polar complex (c)
V SYMMETRY OF CRYSTALLINE POLYHEDRONS
(CONTINUUM SYMMETRY)
THE CONCEPT OF SYMMETRY
Crystals exist in nature in the form of crystalline
polyhedra. Crystals of different substances are different
from each other in their forms. Rock salt is cubes;
rock crystal - hexagonal prisms pointed at
ends; diamond - most often regular octahedrons
(octahedra); garnet crystals - dodecahedrons (Fig.).
Such crystals are symmetrical. characteristic
feature
crystals
is
anisotropy of their properties: in different directions they
different, but identical in parallel directions, and
are also the same in symmetrical directions.
Crystals do not always have the shape of regular
polyhedra.
Under real growth conditions, at
difficulty in free growth symmetrical faces can
develop unevenly and correct external shape
may fail, but the correct internal
the structure is completely preserved, and also
the symmetry of the physical properties is preserved.
The Greek word "symmetry" means proportionality.
A symmetrical figure consists of equal, identical
parts. Symmetry is understood as a property of bodies or
geometric shapes to combine individual parts with each other
another under some symmetric transformations.
Geometric images, with the help of which are set and
symmetrical transformations are carried out, called
symmetry elements. Considering the symmetry of the outer faceting of the crystal,
crystalline
Wednesday
present
yourself
How
continuous, continuous, the so-called continuum (in
translated from Latin into Russian - means continuous,
solid). All points in such an environment are exactly the same.
The symmetry elements of the continuum describe the external
the shape of a crystalline polyhedron, so they are still
are called macroscopic symmetry elements.
Actually
same
crystalline
Wednesday
is
discrete. Crystals are made up of individual particles
(atoms, ions, molecules) that are located in
space
V
form
endlessly
extending
spatial grids. Symmetry in arrangement
of these particles is more complex and richer than the symmetry of the outer
forms of crystalline polyhedra. Therefore, along with
continuum
considered
And
discontinuum
-
discrete, real structure of material particles with
with its symmetry elements, called
microscopic symmetry elements. Elements of symmetry
IN
crystalline
polyhedra
meet
simple
elements
symmetry
(center
symmetry,
plane of symmetry, rotary axis) and complex element
symmetry (inversion axis).
Center of symmetry (or center of inversion) - singular point
inside the figure, when reflected in which any point
figure has an equivalent to itself, that is, both points
(for example, a pair of vertices) are located on the same straight line,
passing through the center of symmetry, and equidistant from
him. In the presence of a center of symmetry, each face
spatial
figures
It has
parallel
And
oppositely directed face, each edge
corresponds equidistant, equal, parallel, but
opposite edge. Therefore the center
symmetry is like a mirror point. A plane of symmetry is a plane that
divides the figure into two parts, located each
relative to a friend as an object and its mirror reflection,
that is, into two mirror equal parts
symmetry planes - Р (old) and m (international).
Graphically, the plane of symmetry is indicated by a solid
line. A figure can have one or more
planes of symmetry, and they all intersect each other
friend. A cube has nine planes of symmetry. The pivot axis is so straight, when turning around
which, at some definite angle, the figure
combines with itself. Angle of rotation
determines the order of the rotary axis n, which
shows how many times the figure will be combined with itself
with a full turn around this axis (360 °):
In isolated geometric shapes possible
symmetry axes of any order, but in crystalline
polyhedra, the axis order is limited, it can have
only the following values: n= 1, 2, 3, 4, 6. In
crystalline
polyhedra
impossible
axes
symmetries of the fifth and higher orders of the sixth. It follows
from the principle of continuity of the crystalline medium.
Symmetry axis designations: old - Ln (L1, L2, L3, L4, L6)
And
international
Arabic
numbers,
corresponding to the order of the rotary axis (1, 2, 3, 4, 6). Graphically
rotary
polygons:
axes
portrayed The concept of a symmetry class
Each crystalline polyhedron has a set
symmetry elements. Combining with each other, the elements
symmetries of a crystal necessarily intersect, and at the same time
the appearance of new symmetry elements is possible.
In crystallography, the following theorems are proved
addition of symmetry elements:
1. The line of intersection of two planes of symmetry is the axis
symmetry, for which the angle of rotation is twice the angle
between planes.
2. Through the point of intersection of two axes of symmetry passes
third axis of symmetry.
3. In
point
intersections
plane
symmetry
With
axis of symmetry of even order perpendicular to it
a center of symmetry appears.
4. The number of axes of the second order, perpendicular to the main
axes of symmetry of higher order (third, fourth,
sixth) is equal to the order of the main axis. 5. The number of planes of symmetry intersecting along
major axis of higher order, equal to the order of this axis.
The number of combinations of symmetry elements with each other
in crystals is strictly limited. All possible
combinations of symmetry elements in crystals are derived
strictly mathematical, taking into account the theorems
addition of symmetry elements.
A complete set of symmetry elements inherent in
given crystal is called its symmetry class.
Rigorous mathematical derivation shows that all
possible
For
crystalline
polyhedra
combinations
elements
symmetry
exhausted
thirty-two classes of symmetry. Relationship between the spatial lattice and elements
symmetry
The presence of certain symmetry elements determines
geometry
spatial
grids,
imposing
certain
conditions
on
mutual
location
coordinate axes and equality of axial units.
There are general rules for choosing coordinate axes,
taking into account the set of crystal symmetry elements.
1. Coordinate axes are combined with special or single
directions,
non-recurring
V
crystal
rotary or inversion axes, for which
the order of the axis is greater than one, and the normals to the plane
symmetry.
2. If there is only one special direction in the crystal, with it
combine one of the coordinate axes, usually the Z axis. Two
other axes are located in a plane perpendicular to
special direction parallel to the edges of the crystal.
3. In the absence of special directions, the coordinate axes
are chosen parallel to three not lying in the same plane
edges of the crystal. Based on these rules, you can get all seven
crystal systems, or syngonies. They differ
from each other by the ratio of scale units a, b, c and
axial angles. Three possibilities: a b c, a=b c, a=b=c
allow
distribute
All
crystallographic
coordinate systems (syngony) in three categories of lower, middle and higher.
Each category is characterized by the presence of certain
symmetry elements. So, for crystals of the lowest category
there are no higher order axes, that is, axes 3, 4 and 6, but there may be
axes of the second order, planes and the center of symmetry.
Crystals of the middle category have an axis of higher
order, and there may also be axes of the second order, planes
symmetry, center of symmetry.
The most symmetrical crystals belong to the highest
categories. They have several higher order axes
(third and fourth), there may be axes of the second order,
plane and center of symmetry. However, there are no axles
sixth order. The concept of the symmetry of the discontinuum and spatial
group
Availability
32
classes
symmetry
crystalline
polyhedra shows that the whole variety of external
crystal forms obey the laws of symmetry.
Symmetry of the internal structure of crystals, arrangement
particles (atoms, ions, molecules) inside crystals should
be more difficult because the external shape of the crystals
limited, and the crystal lattice extends
infinite in all directions of space.
The laws of arrangement of particles in crystals were
established by the great Russian crystallographer E.S.
Fedorov in 1891. They found 230 ways
arrangement of particles in a spatial lattice - 230
space symmetry groups. Elements of symmetry of spatial lattices
In addition to the symmetry elements described above (center
symmetry,
plane
symmetry,
rotary
And
inversion axes), in a discrete medium, other
elements
symmetry,
related
With
infinity
spatial lattice and periodic repetition
in the arrangement of the particles.
Consider new types of symmetry inherent only in
discountinuum. There are three of them: translation, sliding plane
reflections and helical axis.
Translation is the transfer of all particles along parallel
directions in the same direction to the same
size.
Translation is a simple element of symmetry,
inherent in each spatial lattice. Combination of translation with a plane of symmetry
leads to the appearance of a plane of grazing reflection,
the combination of translation with a rotary axis creates
screw axle.
Glide reflection plane, or plane
slip is such a plane, when reflected in
which, as in a mirror, followed by translation along
direction lying in a given plane, by the amount
equal to half the identity period for a given
directions, all points of the body are combined. Under period
identity, as before, we will understand the distance
between points along some direction (for example,
periods a, b, c in a unit cell are periods
identity along the coordinate axes X, Y, Z). The helical axis is a straight line, the rotation around which is
some
corner,
corresponding
order
axes,
With
subsequent translation along the axis by a multiple of
identity period t, combines the points of the body.
The designation of the helical axis in general form is nS, where n
characterizes the order of the rotary axis (n=1, 2, 3, 4, 6), and
St/n is the amount of translation along the axis. At the same time, S
translation is t/2, for the helical axis of the third
order of the smallest transfer t/3.
The designation of the helical axis of the second order will be 21.
The combination of particles will occur after rotation around the axis
180° followed by translation along the direction,
parallel to the axis, by t/2.
The designation of the helical axis of the third order will be 31.
However, axes with a translation that is a multiple of the smallest are possible.
Therefore, a helical axis 32 with translation 2t/3 is possible. Axes 31 and 32 mean rotation around the axis by 120° along
clockwise followed by a shift. These screw
axes are called right. If a turn is made
counterclockwise, then the center axes of symmetry
are called left. In this case, the action of the axis 31 of the right
identical to the action of the axis 32 left and 32 right - 31
left.
Helical axes of symmetry can also be considered
fourth and sixth orders: axes 41 and 43 axes 61 and 65, 62
and 64. can be right and left. Action of axes 21, 42 and
63 does not depend on the choice of the direction of rotation around the axis.
That's why
They
are
neutral.
Conditional
designations of helical axes of symmetry: Symmetry space group notation
The space group symbol contains the complete
information about the symmetry of the crystal structure. On
the first place in the space group symbol is put
letter characterizing the type of Bravais lattice: P primitive,
WITH
base-centered,
I
body-centered, F - face-centered. IN
rhombohedral syngony put the letter R in the first place.
Followed by one, two or three numbers or letters,
indicating
elements
symmetry
V
major
directions, similar to how it is done with
drawing up the notation of the symmetry class.
If in the structure in any of the main directions
both planes of symmetry and
axes of symmetry, preference is given to planes
symmetry, and into the space group symbol
planes of symmetry are written. If there are multiple axes, preference is given to
simple axes - rotary and inversion, since their
symmetry is higher than symmetry
screw axles.
Having a space group symbol, one can easily
determine the type of the Bravais lattice, the syngony of the cell, the elements
symmetry in the principal directions. Yes, spatial
group P42/mnm (Fedorov groups of ditetragonal dipyramidal
kind
symmetry,
135
group)
characterizes the primitive Bravais cell in the tetragonal
syngony (fourth-order helical axis 42 determines
tetragonal syngony).
The main directions are as follows:
symmetry elements. With direction - Z axis
coincides with the helical axis 42, which is perpendicular
symmetry m. In the and directions (X and Y axes)
the plane of grazing reflection of type n is located, in
direction passes the plane of symmetry m. Defects in the structure of crystalline bodies
Body defects are divided into dynamic
(temporary) and static (permanent).
1. Dynamic defects arise when
mechanical, thermal, electromagnetic
impact on the crystal.
These include phonons - time distortions
lattice regularity caused by thermal
the movement of atoms.
2. Static defects
Distinguish between point and extended imperfections
body structures. Point Defects: Unoccupied Lattice Sites
(vacancies); displacement of an atom from a node to an interstice;
introduction of a foreign atom or ion into the lattice.
Extended defects: dislocations (edge and
screw), pores, cracks, grain boundaries,
microinclusions of another phase. Some defects are shown
on the image. Basic properties
materials The main properties are: mechanical, thermal,
electrical, magnetic and technological, as well as their
corrosion resistance.
The mechanical properties of materials characterize the possibility of their
use in products exposed to
mechanical loads. The main indicators of such properties
serve as parameters of strength and hardness. They depend not only on
the nature of the materials, but also on the shape, size and condition
surface of samples, as well as test modes, first of all,
on the loading rate, temperature, exposure to media and other
factors.
Strength is the property of materials to resist fracture, and
also an irreversible change in the shape of the sample under the action of
external loads.
Tensile strength - stress corresponding to the maximum
(at the moment of destruction of the sample) to the value of the load. Attitude
the greatest force acting on the sample to the original area
its cross section is called breaking stress and
denote σv. Deformation is a change in the relative arrangement of particles in
material. Its simplest types are tension, compression, bending,
twist, shift. Deformation - a change in the shape and size of the sample in
the result of deformation.
Deformation parameters – relative elongation ε = (l– l0)/l0 (where
l0 and l are the original and after deformation lengths of the sample), the shear angle is
change in the right angle between rays emanating from one point in
sample, when it is deformed. The deformation is called elastic if
it disappears after removal of the load, or plastic, if it is not
disappears (irreversible). The plastic properties of materials at
small deformations are often neglected.
The elastic limit is the stress at which the residual deformations (i.e.
e. deformations detected during unloading of the sample) reach
the value set by the specifications. Usually the admission
residual deformation is 10–3 ÷10–2%. Elastic limit σy
limits the area of elastic deformations of the material.
The concept of the modulus as a characteristic of the elasticity of materials arose
when considering ideally elastic bodies, the deformation of which is linearly
depends on voltage. With simple stretching (compression)
σ = Eε
where E is Young's modulus, or modulus of longitudinal elasticity, which
characterizes the resistance of materials to elastic deformation (tensile, compression); ε is the relative strain. When shearing in the material in the direction of the shear and along the normal to it
only tangential stresses
where G is the shear modulus characterizing the elasticity of the material at
changing the shape of the sample, the volume of which remains constant; γ is the angle
shift.
With all-round compression in the material in all directions,
normal voltage
where K is the modulus of bulk elasticity, which characterizes
material resistance to sample volume change, not
accompanied by a change in its shape; ∆ - relative
bulk compression.
A constant value characterizing the elasticity of materials at
uniaxial tension, is the Poisson's ratio:
where ε' is the relative transverse compression; ε - relative
longitudinal elongation of the sample. Hardness is a mechanical characteristic of materials,
complex reflecting their strength, ductility, as well as
properties of the surface layer of the samples. She expresses herself
material resistance to local plastic
deformation that occurs when more than
solid body - indenter. Pressing the indenter into the sample with
subsequent measurement of the dimensions of the print is the main
technological method in assessing the hardness of materials. IN
depending on the features of the load application, design
indenters and determination of hardness numbers distinguish methods
Brinell, Rockwell, Vickers, Shore. When measuring
microhardness according to GOST 9450–76 on the sample surface
imprints of insignificant depth remain, therefore such
the method is used when the samples are made in the form of foil,
films, coatings of small thickness. Method of determination
plastic hardness is indentation into the sample
spherical tip by sequential application
various loads. Corrosion is a physical and chemical process of changing properties, damage
structure and destruction of materials due to the transition of their components into
chemical compounds with environmental components. Under
Corrosion damage refers to any structural defect
material resulting from corrosion. If mechanical
effects accelerate the corrosion of materials, and corrosion facilitates them
mechanical destruction, there is a corrosion-mechanical
material damage. Losses of materials due to corrosion and costs for
protection of machinery and equipment from it is continuously increasing
due to the intensification of human production activity and
pollution of the environment by production waste.
The resistance of materials to corrosion is most often characterized with
using the parameter of corrosion resistance - the value, the reciprocal
technical corrosion rate of the material in a given corrosion system.
The conditionality of this characteristic lies in the fact that it does not apply to
material, but to the corrosion system. Corrosion resistance of the material
cannot be changed without changing other parameters of the corrosion system.
Corrosion protection is a modification of the corrosion
system, leading to a decrease in the rate of corrosion of the material. Temperature characteristics.
Heat resistance - the property of materials to retain or slightly
change mechanical parameters at high temperatures. Property
metals resist the corrosive effects of gases at high
temperatures is called heat resistance. As a feature
heat resistance of fusible materials use temperature
softening.
Heat resistance - the property of materials to resist for a long time
deformation and fracture at high temperatures. This
the most important characteristic of materials used in
temperatures T > 0.3 Tm. Such conditions occur in engines
internal combustion, steam power plants, gas turbines,
metallurgical furnaces, etc.
At low temperatures (in technology - from 0 to -269 ° C) increases
static and cyclic strength of materials, their
ductility and toughness, increased susceptibility to brittle fracture.
Cold brittleness - an increase in the fragility of materials with a decrease in
temperature. The tendency of a material to brittle fracture is determined by
according to the results of impact tests of samples with a notch when lowering
temperature. Thermal expansion of materials is recorded by dimensional change
and the shape of the samples when the temperature changes. For gases, it is due
an increase in the kinetic energy of particles when heated, for liquids
and solid materials is associated with the asymmetry of thermal
vibrations of atoms, due to which the interatomic distances with increasing
temperatures increase.
Quantitatively, the thermal expansion of materials is characterized by
temperature coefficient of volume expansion:
and solid materials - and the temperature coefficient of the linear
extensions (TKLR):
- changes in the linear size, volume of samples and
temperature (respectively).
Index ξ serves to designate the conditions of thermal expansion (usually -
at constant pressure).
Experimentally, αV and αl are determined by dilatometry, which studies
dependence of changes in the size of bodies under the influence of external factors.
Special measuring instruments - dilatometers - differ
the device of sensors and sensitivity of systems of registration of the sizes
samples. Heat capacity - the ratio of the amount of heat received by the body during
an infinitesimal change in its state in any process, to
caused by the last temperature increment:
According to the signs of a thermodynamic process in which
heat capacity of the material, distinguish heat capacity at constant volume
and at constant pressure. During heating at constant
pressure (isobaric process) part of the heat is spent on expansion
sample, and part - to increase the internal energy of the material. Heat,
reported to the same sample at constant volume (isochoric process),
is spent only on increasing the internal energy of the material.
Specific heat capacity, J/(kg K)], is the ratio of heat capacity to mass
body. Distinguish between specific heat at constant pressure (cp) and
at constant volume (cv). The ratio of heat capacity to quantity
substances are called molar heat capacity (cm), J / (mol⋅K). For all
substances ср > сv, for rarefied (close to ideal) gases сmp – сmv =
R (where R = 8.314 J/(mol⋅K) is the universal gas constant). Thermal conductivity is the transfer of energy from hotter parts of the body to
less heated as a result of thermal motion and interaction
microparticles. This value characterizes the spontaneous
temperature equalization of solids.
For isotropic materials, the Fourier law is valid, according to which
the heat flux density vector q is proportional and opposite
in the direction of the temperature gradient T:
where λ is the thermal conductivity [W/(m K)] depending on
state of aggregation, atomic and molecular structure, structure,
temperature and other material parameters.
The thermal diffusivity (m2/s) is a measure
thermal insulation properties of the material:
where ρ is the density; cp is the specific heat capacity of the material at
constant pressure. Technological properties of materials characterize compliance
materials to technological influences during processing into products. Knowledge
these properties allows you to reasonably and rationally design and
carry out technological processes of manufacturing products. Main
technological characteristics of materials are machinability
cutting and pressure, casting parameters, weldability, tendency to
deformation and warping during heat treatment, etc.
Machinability is characterized by the following indicators:
the quality of material processing - the roughness of the machined surface
and dimensional accuracy of the sample, tool life, resistance
cutting - cutting speed and force, type of chip formation. Values
indicators are determined when turning samples and compared with
parameters of the material taken as a standard.
Machinability by pressure is determined in the process of technological
testing materials for plastic deformation. Assessment Methods
pressure machinability depends on the type of materials and their technology
processing. For example, technological tests of metals for bending
carried out by bending the samples to a predetermined angle. The sample is considered to have withstood
tests, if it does not appear fracture, delamination, tears, cracks.
Sheets and tapes are tested for extrusion using a special
press. A spherical hole is formed in the sample, stopping the drawing at the moment
achieving material flow. The result is determined by the maximum
well depth in undamaged specimens. Machinability by pressure of powder materials characterizes them
fluidity, compactability and formability. Method of determination
fluidity is based on registration of the expiration time of the powder sample in
the process of its spontaneous spilling through a calibrated
funnel hole. This parameter controls the fill rate.
powder materials molds for pressure treatment.
The compaction of the powder is characterized by the dependence of the sample volume
powder from pressure - pressing diagram. Formability - property
powder material to keep the shape obtained in the process
pressing.
Casting characteristics of materials - a set of technological
indicators characterizing the formation of castings by pouring
molten materials into a mold. Fluidity -
the property of the molten material to fill the mold depends on
on melt viscosity, melt and mold temperatures, degree
melt wetting of the walls of the mold, etc. It is evaluated by length
filling with melt a straight or spiral channel in
special mold. Shrinkage foundry - volume reduction
melt during the transition from liquid to solid state. Practically
shrinkage is defined as the ratio of the corresponding linear dimensions
molds and castings in the form of a dimensionless shrinkage coefficient,
individual for each material. Weldability - the property of a material to form
welded joint, the performance of which
corresponds to the quality of the base material,
welded. Weldability is judged by
test results of welded specimens and
characteristics of the base material in the zone of welded
seam. The rules for determining the following
indicators of weldability of metals: mechanical
properties of welded joints, permissible modes
arc welding and surfacing, the quality of welded
joints and welds, long-term strength
welded joints.