About what condensed matter is and how theoretical physics deals with them 01/18/2002. Section VIII

A typical condensed medium is when there are a lot of particles, and each particle “lives” not its own separate life or even paired with a neighbor, but in “peace and harmony” with a whole set of nearest neighbors.

School examples of condensed matter: solid (such as crystal) and liquids. More exotic media: electronic and others quantum liquids , superfluid helium , liquid crystals, various dispersed systems(gels, pastes, emulsions, suspensions), neutron matter , quark-gluon plasma. And finally, crowd of people in a state of panic, a dense flow of cars on the roads, and that complex computer network that we call the Internet are all also examples of condensed matter.

Why is condensed matter physics such an interesting and active field of research? The fact is that due to the fact that the movement of each individual particle in a condensed medium is highly correlated with the movement of many neighbors; the equations describing the movement of particles are strongly “intertwined” with each other. You will not be able, for example, to first solve the equation of motion of the first particle, then the second, etc. It is necessary to solve all the equations of motion at once, for billions, quintillions, etc. individual particles. Such systems of equations are not easy to solve, but even difficult to imagine.

This situation is depressing, isn't it? But theoretical physicists are an inventive people, and little by little they learned to describe systems that are unimaginably complex at first glance. (In fact, in my opinion, the awareness of this impasse and attempts to get out of it are the moment of the birth of real theoretical physics; but I will write about this sometime later.)

The most famous example of how to solve trillions of equations at once is the story of phonons. Imagine that we have a crystal. Each atom in it feels several of its closest neighbors, and feels it very, very strongly. One atom cannot vibrate on its own; it will definitely pull its neighbors along with it. As a result, having “vibrated” an individual particle, we immediately involve its immediate neighbors in motion, so that after some time all matter, all particles will begin to move.

Let's look at it completely differently Of what consists of a crystal, How he lives. Vibrations of individual atoms are some not very convenient way to talk about the life of crystals. But if we talk about certain coordinated vibrations of all particles at once phonons when the whole movement crystal lattice understands a traveling sine wave, then everything becomes amazingly simple. Individual phonons, it turns out, live an independent life: they can “run” across a crystal for a long time, passing through each other. And this means that the equations describing each individual phonon are solved independently and therefore in.

Of course, this is all true for an ideal crystal, when the lattice is strictly periodic, when there are no defects, when the boundaries of the crystal do not affect its internal life, and finally, when the vibrations can be considered linear (which entails non-interaction of phonons). Real crystals are not like this, and therefore the properties described above are not strictly satisfied for it, but only approximately. But this is also quite sufficient to explain many phenomena occurring in a crystal.

Of course, one can argue that, in reality, we have vibrations of atoms, and not phonons. But, say, when describing the thermodynamic properties of a crystal, it is easiest to perceive it as a gas of phonons. And to be honest, I don’t know whether it is possible to construct the entire statistical physics of a crystal without ever turning to the concept of phonons.

In fact, the transition from individual atoms to phonons is nothing more than Fourier transform from coordinates to (quasi)momenta. It just turns out that in the impulse representation the crystal looks much simpler than in the coordinate representation.

The life of a crystal, of course, is not limited to vibrations of the crystal lattice alone. Therefore, the phonons described here are only the simplest of a whole family of quasiparticles that inhabit a solid body.

TYPES OF BONDS IN CRYSTALS

The existence of stable bonds between atoms in a solid implies that the total energy of the crystal is less than the total energy of the corresponding number of free atoms (distant from each other at large distances). The difference between these two energies is called chemical bond energy or simply bond energy.

The forces that bind atoms together are almost entirely electrical in nature, the role of magnetic interactions is insignificant (eV/atom), and gravitational interactions are almost zero. Even for the heaviest atoms it will be eV/atom.

However, it should be noted that taking into account only electrostatic interaction does not explain the stability of the crystal. Indeed, according to Earnshaw's theorem, a stable static configuration of electric charges is impossible. Therefore, it is necessary to take into account forces that are of a quantum mechanical nature.

Classification of condensed matter by types of bonds

Of the aggregate states of matter, two - solid and liquid - are called condensed.

All types of connections between atoms are caused by the attraction or repulsion of electrical charges. The type and strength of the bond are determined by the electronic structure of the interacting atoms. Regardless of the nature of the forces that arise when atoms approach each other, their nature remains the same: at large distances, attractive forces are predominant, at small distances, repulsive forces prevail. At a certain (equilibrium) distance, the resulting force becomes zero, and the interaction energy reaches a minimum value (Fig. 2.1).

A solid is a state of aggregation of a substance, which is characterized by stability of shape and the oscillatory nature of the thermal movement of atoms. Consequently, the latter have kinetic energy.

The problem of the interaction of even the simplest atoms is very complex, because we have to consider the behavior of many particles - nuclei and electrons. It is necessary to take into account the wave properties of microparticles, primarily electrons, and solve the corresponding Schrödinger equation using approximate methods.

Interatomic bonding is accompanied by a significant rearrangement of the valence electrons of the atoms, and the nature of the rearrangement is determined by the nature of the atoms themselves and the state of the electrons that take part in the formation of the chemical bond. The main contribution to the energy of formation of a solid body from atoms is made by valence electrons; the contribution of electrons of the inner shells is insignificant.

As a result of the interaction of valence electrons, common electron pairs are formed. Covalent a bond occurs when an electron pair is not completely displaced towards one of the atoms, but is localized in an orbit common to both electrons.

When a pair of electrons is almost completely shifted to one of the atoms, we have the example ionic communications. That is, ionic bonding can be considered an extreme case of covalent bonding. In this case, the interaction energy in crystals with such a bond can be calculated based on the Coulomb interaction of positive and negative ions that were formed in the crystal as a result of the redistribution of electrons between atoms.

Metal connection can also be considered as an extreme case of covalent bonding, when the valence electrons become itinerant, that is, simultaneously belong to many atoms .

In atoms with filled valence shells, the distribution of electric charge is spherical, so they do not have a constant electric moment. But due to the movement of electrons, an atom can turn into an instantaneous electric dipole, which leads to the emergence of so-called van der Waals forces. For example, in a hydrogen atom the average electrical torque is zero, while the instantaneous torque can reach 2.5 D (debye). When atoms approach each other, the interaction of instantaneous atomic dipoles occurs.

The main characteristics of a chemical bond are energy, length, polarity, multiplicity, direction, and saturation. For ionic bonding, the effective charge of the ions must be taken into account.

Based on the nature of the bonding forces, solids can be divided into the following classes: atomic, ionic, metallic, molecular crystals and crystals with hydrogen bonds.

Atomic crystals

Atomic(by polarity type - homeopolar) crystals are formed due to covalent bonds. It is predetermined by electrostatic and exchange interactions. Understanding the nature of the covalent bond can only be achieved using quantum mechanical concepts that take into account the wave properties of the electron. In a covalent bond, neighboring atoms form common electron shells by exchanging electrons. As follows from quantum mechanical calculations, when common electron shells are formed, the potential energy of the system decreases due to the so-called exchange effects. The decrease in energy is equivalent to the emergence of attractive forces.

Let us consider the mechanism of the occurrence of exchange interaction using the example of the formation of a hydrogen molecule, in which two electrons move in the field of two nuclei (Fig. 2.2).

The potential energy of interaction between two atoms consists of two parts: the energy of interaction of nuclei and the energy of electrons, which depends on the distance between the two nuclei R:

. (2.1)

To find the eigenfunctions and eigenvalues ​​of the energy of such a system, it is necessary to solve the stationary Schrödinger equation:

. (2.2)

The Hamiltonian of the hydrogen molecule can be given as follows:

Where corresponds to the movement of the first electron (1) around the nucleus ( A)

, (2.4)

corresponds to the movement of the second electron (2) around the nucleus ( b)

, (2.5)

A represents the energy of electrostatic interaction of electrons with “foreign” nuclei and among themselves

. (2.6)

It is impossible to carry out an exact solution of the Schrödinger equation with Hamiltonian (2.3). Let's use the perturbation method. Let's look at long distances first. Let the first electron be near the nucleus, and the second - near the nucleus. Then the value in (2.3) can be neglected and we obtain the equation

As an initial approximation for the wave function, we use the wave functions of non-interacting hydrogen atoms:

Where And are found from solving the equations

, (2.9)

. (2.10)

The energy value corresponding to solution (2.8) will be .

If there were no degeneracy, then solution (2.8) would be the zero approximation. In fact, in this case we have the so-called exchange degeneracy. Obviously, in addition to solution (2.8), such a solution is also possible when in the first atom ( A) there is a second electron (2), and in the second atom ( b) – the first electron (1). The Hamiltonian will have the same form as (2.3), only the electrons will change places (1-2). The solution will look like

Thus, for large ones, equation (2.2) has two solutions (2.8) and (2.11), which belong to the energy . When taking into account the interaction between atoms, the zero approximation to will be a linear combination of and:

where and are the coefficients that need to be determined, and is a small addition to the zero approximation.

Let's represent the energy in the form

, (2.13)

Where – additives that determine the change in electron energy as atoms approach each other.

Substituting (2.12) and (2.13) into (2.2) and neglecting small quantities , , , we get

Let us use (2.3) and the last expression, but taking into account the rearrangement of electrons. Then (2.14) takes the form

(2.15)

Let us substitute into (2.15) and from (2.8) and (2.11) and neglect the small terms , . We get

(2.16)

This is an inhomogeneous equation for determining corrections to the wave function and to the energy eigenvalue.

An inhomogeneous equation has a solution in the case when its right-hand side is orthogonal to the solution of a homogeneous equation (such an equation arises if the right-hand side in (2.16) is equal to zero). That is, the condition must be satisfied

Where , .

In a similar way we obtain the second equation (orthogonality to the solution)

Let us introduce the following abbreviated notations

The functions and are not orthogonal to each other, so we introduce the following integral

. (2.21)

Using these notations, equations (2.17) and (2.18) can be written as follows

From these equations, we first obtain the equation for:

It has two roots

, (2.25)

. (2.26)

Substituting these values ​​into (2.22), we find for

(2.27)

and for

. (2.28)

Therefore, the solutions will be written in the following form:

(2.29)

(antisymmetric solution) and

(2.30)

(symmetrical solution).

Let us consider the physical meaning of the integrals and . Using (2.19), (2.6) and (2.11), we obtain

. (2.31)

Let's use the normalization conditions And , we denote the average density of the electronic charge that is created by electron (1) in the atom ( A), through , electron (2) in the atom ( b) through . In this case we get for:

The first integral is the average potential energy of the electron (2) of the atom ( b) in the core field ( A), the second integral is the same value for the electron (1) of the atom ( A) in the core field ( b) and the third integral is the average potential energy of electrons that are in different atoms. So there is average energy of electrostatic interaction of atoms , except for the nuclear interaction energy, which is calculated separately (2.1).

Integral (2.20) is called exchange integral. Designating the exchange density

(2.33)

let's write it in the form

The last term represents the exchange energy, which has no analogues in classical mechanics. It is due to the fact that each of the electrons can be partially located near the atom ( a), partially – about ( b).

The first two terms on the right side of (2.34) represent corrections to the exchange energy due to the non-orthogonality of the wave functions, in fact,

At wave functions and due to the exponential decrease with increasing distance from the nuclei ( a) And ( b) overlap slightly, therefore, . When , kernels ( a) And ( b) match up. Then and are wave functions of the same hydrogen atom. Due to normalization and equals 1. Therefore,

. (2.36)

The integral also changes within these limits.

Using (2.1), (2.12) (2.29) and (2.30) and performing some transformations, we obtain

, (2.37)

. (2.38)

Members represent the average Coulomb energy of two hydrogen atoms that are located at a distance from each other - the exchange energy. The last term c includes corrections for non-orthogonality of the wave functions, which were used as a zero approximation.

Using formulas (2.32) and (2.34), both the Coulomb and exchange energy can be calculated if we use for and the wave function of the normal state of hydrogen:

, (2.39)

where is the distance of the electron from the nucleus, and is the radius of the first Bohr orbit.

The and integrals contain wave functions that belong to different atoms and each of these functions decreases exponentially with distance. Therefore, both integrals differ from zero only because the wave functions, and, consequently, the electron shells of the atoms overlap. As a result, both integrals decrease with increasing distance between the atoms, as . Figure 2.3 shows the mutual energy of atoms And as a function of the distance between them. The value is taken as 0 when counting energy.

Fig.2.3. Energy of symmetric and antisymmetric states

As can be seen from the figure, for an antisymmetric state the energy corresponds to the mutual repulsion of two hydrogen atoms, and therefore a molecule cannot be formed. On the contrary, for a symmetric state the energy has a minimum, in this case the hydrogen atoms are at a distance and form a molecule. The wave function depends only on the coordinates. The complete wave function must also depend on the electron spins and . Since we neglected the interaction of spins with orbital motion and the interaction of spins with each other, the total wave function should be the product of the coordinate function and the spin function . Electrons obey the Pauli principle, so the wave function must be asymmetric with respect to the rearrangement of electrons. We have a coordinate function that is either symmetric or antisymmetric.

The complete wave function will be antisymmetric for symmetric coordinate and antisymmetric spin, as well as for antisymmetric coordinate and symmetric spin.

Therefore, two hydrogen atoms that have electrons with opposite spins (singlet state) are attracted to each other. Hydrogen atoms, which have electrons with parallel spins (triplet state), repel each other.

If an atom of a substance has several unpaired electrons, then a corresponding number of exchange bonds can occur. For example, in crystals with a diamond lattice (Fig. 1.9, A) each atom is connected to four nearest neighbors.

A covalent bond is formed when electron shells overlap; therefore, it is observed at small distances between atoms. Moreover, the density of the “electron cloud” increases in the directions that connect the atoms, that is, the electrons are, as it were, drawn into the space between the nuclei and their field ensures their attraction. This implies the directionality and saturation of covalent bonds: they act only in certain directions and between a certain number of neighbors.

Covalent bonds predominate in atomic crystals and are close in order of magnitude to ionic bonds. Such crystals have low compressibility and high hardness. Electrically, they are dielectrics or semiconductors.

Substances with covalent bonds include:

– most organic compounds;

– halogens in solid and liquid states;

– hydrogen, nitrogen, oxygen (bonds in the molecule);

– elements of group VI, group V and IV (crystals of diamond, silicon, germanium, );

– chemical compounds that obey the rule ( ), if the elements included in their composition are not located at different ends of the row of the periodic system (for example, ).

Solids with covalent bonds can crystallize in several structural modifications. This property, called polymorphism, was discussed in Chapter 1.

Ionic crystals

Such substances are formed through a chemical bond, which is based on electrostatic interaction between ions. Ionic bond (by type of polarity - heteropolar) is mainly limited to binary systems like NaCl(Fig. 1.10, A), that is, it is established between the atoms of elements that have the greatest affinity for electrons, on the one hand, and the atoms of elements that have the lowest ionization potential, on the other. When an ionic crystal is formed, the nearest neighbors of a given ion are ions of opposite sign. With the most favorable ratio of sizes of positive and negative ions, they touch each other, and an extremely high packing density is achieved. A small change in the interionic distance towards its decrease from the equilibrium one causes the emergence of repulsive forces between the electron shells.

The degree of ionization of the atoms that form an ionic crystal is often such that the electron shells of the ions correspond to the electron shells characteristic of noble gas atoms. A rough estimate of the binding energy can be made by assuming that most of it is due to Coulomb (that is, electrostatic) interaction. For example, in a crystal NaCl the distance between the nearest positive and negative ions is approximately 0.28 nm, which gives the value of the potential energy associated with the mutual attraction of a pair of ions of about 5.1 eV. Experimentally determined energy value for NaCl is 7.9 eV per molecule. Thus, both quantities are of the same order, and this makes it possible to use this approach for more accurate calculations.

Ionic bonds are non-directional and unsaturated. The latter is reflected in the fact that each ion tends to bring the largest number of ions of the opposite sign closer to itself, that is, to form a structure with a high coordination number. Ionic bonding is common among inorganic compounds: metals with halides, sulfides, metal oxides, etc. The binding energy in such crystals is several electron volts per atom, therefore such crystals have greater strength and high melting temperatures.

Let's calculate the ionic bond energy. To do this, let us recall the components of the potential energy of an ionic crystal:

Coulomb attraction of ions of different signs;

Coulomb repulsion of ions of the same sign;

quantum mechanical interaction when electronic shells overlap;

van der Waals attraction between ions.

The main contribution to the binding energy of ionic crystals is made by the electrostatic energy of attraction and repulsion; the role of the last two contributions is insignificant. Therefore, if we denote the interaction energy between ions i And j through , then the total energy of the ion, taking into account all its interactions, will be

. (2.40)

Let us present it as the sum of the repulsion and attraction potentials:

, (2.41)

where the “plus” sign is taken in the case of identical, and the “minus” sign in the case of unlike charges. The total lattice energy of an ionic crystal, which consists of N molecules (2 N ions), will be

. (2.42)

When calculating the total energy, each interacting pair of ions should be counted only once. For convenience, we introduce the following parameter , where is the distance between two neighboring (opposite) ions in the crystal. Thus

, (2.43)

Where Madelung constant α and constant D are defined as follows:

, (2.44)

. (2.45)

Sums (2.44) and (2.45) must take into account the contribution of the entire lattice. The plus sign corresponds to the attraction of unlike ions, the minus sign to the repulsion of like ions.

We define the constant as follows. In the equilibrium state, the total energy is minimal. Hence, , and therefore we have

, (2.46)

where is the equilibrium distance between neighboring ions.

From (2.46) we obtain

, (2.47)

and the expression for the total energy of the crystal in an equilibrium state takes the form

. (2.48)

Magnitude represents the so-called Madelung energy. Since the indicator , then the total energy can be almost completely identified with the Coulomb energy. A small value indicates that the repulsive forces are short-range and change sharply with distance.

As an example, let's calculate the Madelung constant for a one-dimensional crystal - an endless chain of ions of the opposite sign, which alternate (Fig. 2.4).

By choosing any ion, for example, with the “–” sign as the initial one, we will have two ions with the “+” sign at a distance r 0 from it, two ions of the “–” sign at a distance of 2 r 0 and so on.

Therefore, we have

,

.

Using series expansion
, we obtain in the case of a one-dimensional crystal the Madelung constant

. (2.49)

Thus, the expression for the energy per molecule takes the following form

. (2.50)

In the case of a three-dimensional crystal, the series converges conditionally, that is, the result depends on the method of summation. The convergence of the series can be improved by selecting groups of ions in the lattice so that the group is electrically neutral, and, if necessary, dividing the ion between different groups and introducing fractional charges (Evjen’s method ( Evjen H.M.,1932)).

We will consider the charges on the faces of the cubic crystal lattice (Fig. 2.5) as follows: the charges on the faces belong to two neighboring cells (in each cell the charge is 1/2), the charges on the edges belong to four cells (1/4 in each cell), the charges at the vertices belong to eight cells (1/8 in each cell). Contribution to the α t of the first cube can be written as a sum:

If we take the next largest cube, which includes the one we considered, we get , which coincides well with the exact value for a lattice like . For a structure like received , for a structure of type – .

Let's estimate the binding energy for the crystal , assuming that the lattice parameter and elastic modulus IN known. The elastic modulus can be determined as follows:

, (2.51)

where is the volume of the crystal. Bulk modulus of elasticity IN is a measure of compression during all-round compression. For a face-centered cubic (fcc) structure of the type the volume occupied by the molecules is equal to

. (2.52)

Then we can write

From (2.53) it is easy to obtain the second derivative

. (2.54)

In the equilibrium state, the first derivative vanishes, therefore, from (2.52–2.54) we determine

. (2.55)

Let us use (2.43) and obtain

. (2.56)

From (2.47), (2.56) and (2.55) we find the bulk modulus of elasticity IN:

. (2.57)

Expression (2.57) allows us to calculate the exponent in the repulsive potential using the experimental values ​​of and . For crystal
, , . Then from (2.57) we have

. (2.58)

Note that for most ionic crystals the exponent n in the potential of repulsive forces varies within 6–10.

Consequently, a large magnitude of the degree determines the short-range nature of the repulsive forces. Using (2.48), we calculate the binding energy (energy per molecule)

eV/molecule. (2.59)

This agrees well with the experimental value of -7.948 eV/molecule. It should be remembered that in the calculations we took into account only Coulomb forces.

Crystals with covalent and ionic bond types can be considered as limiting cases; between them there is a series of crystals that have intermediate types of connection. Such a partially ionic () and partially covalent () bond can be described using the wave function

, (2.60)

in this case, the degree of ionicity can be determined as follows:

. (2.61)

Table 2.1 shows some examples for crystals of binary compounds.

Table 2.1. Degree of ionicity in crystals

Crystal	Degree of ionicity	Crystal	Degree of ionicity	Crystal	Degree of ionicity
SiC ZnO ZnS ZnSe ZnTe CdO CdS CdSe CdTe	0,18 0,62 0,62 0,63 0,61 0,79 0,69 0,70 0,67	InP InAs InSb GaAs GaSb CuCl CuBr AgCl AgBr	0,44 0,35 0,32 0,32 0,26 0,75 0,74 0,86 0,85	AgI MgO MgS MgSe LiF NaCl RbF	0,77 0,84 0,79 0,77 0,92 0,94 0,96

Metal crystals

Metals are characterized by high electrical conductivity, which is determined by the collectivization of valence electrons. From the point of view of electronic theory, a metal consists of positive ions immersed in a medium formed by itinerant electrons. The latter can move freely in the volume of the crystal, since they are not associated with specific atoms. Moreover, the kinetic energy of itinerant electrons decreases compared to the kinetic energy of valence electrons in a free atom.

Bonding in metal crystals results from the interaction of positive ions with collectivized electrons. The free electrons that are between the ions seem to pull them together, balancing the repulsive forces between ions of the same sign. As the distance between the ions decreases, the density of the electron gas increases, and, consequently, the attractive forces increase. However, at the same time, the repulsive forces begin to increase. When a certain distance between the ions is reached, the forces are balanced and the lattice becomes stable.

Thus, the energy of a metal crystal can be represented in the form of the following terms:

– electrostatic energy of free electrons in the field of positive ions (crystal lattice);

– kinetic energy of electrons;

– mutual electrostatic potential energy of positive ions;

– mutual electrostatic potential energy of electrons.

It can be shown that only the first two terms are significant. As an example, consider sodium metal, which has a bcc lattice. Let us select the volume per atom in the lattice by drawing planes perpendicular to the lines connecting this atom with its neighbors and dividing the indicated segments in half. We obtain the so-called Wigner-Seitz cell, which for a given lattice has the shape of a cuboctahedron (see Chapter 1).

Although electrons move throughout the crystal, near each atom, that is, in the Wigner-Seitz cell, the electron density is on average constant. This means that if a metal has one electron per atom, then on average there is one electron near each atom. Cuboctahedra turn out to be electrically neutral and weakly interact with each other electrostatically. The main part of the interaction is concentrated inside cubectahedrons, that is, it corresponds to the energy of free electrons in the field of positive ions.

The probability of finding an electron at a distance between and from a given ion is determined by the following expression

,

Where – probability density (squared modulus of the radial part of the wave function). Then the electron energy in the field of a given ion is equal to

,

that is, the value averaged over all possible positions of the electron. Since the integration region is equal to the entire volume of the metal, the result of integration will determine the energy of all free electrons in the field of a given ion if represents the average charge density in the lattice.

From the above it follows that the energy term corresponding to the mutual potential energy of electrons and ions will have the form

, (2.62)

where is the volume of metal, A– some constant ( A<0).

Let's determine the kinetic energy of electrons. This issue will be discussed in Chapter 4, and now we will use the results obtained there. The average kinetic energy of electrons is determined in terms of the Fermi energy and is

,

Where ; – electron concentration. The latter is determined by the number of atoms and volume of the metal. Finally, the energy can be represented in the form

. (2.63)

The total energy of a metal crystal, according to the previous one, is determined by two terms

If we plot the dependence as a function of the distance between atoms, that is, a value proportional to , we get a curve with a minimum at the point (Fig. 2.6). The value at this minimum determines the binding energy, and the second derivative at this point determines the compressibility modulus. The role of repulsive forces in the case of metal crystals is played by the kinetic energy of electrons, which increases with decreasing interatomic distances.

Calculation of the binding energy (heat of evaporation) of metallic sodium according to the above scheme gives a value of about 1 eV/atom, which agrees well with the experimental data - 1.13 eV/atom.

Due to the fact that purely metallic bonding is non-directional, metals crystallize into relatively densely packed structures with large coordination numbers: face-centered cubic (fcc), hexagonal close-packed (hcp), body-centered cubic. For fcc and hcp crystals, the packing density and coordination number are the same: 0.74 and 12, respectively. Consequently, the closeness of the parameters indicates the closeness of the binding energy values ​​in such crystals. Indeed, a number of metals can, under a relatively weak external influence, change the structure from fcc to hcp and vice versa.

In some metals, not only metallic bonds, caused by itinerant electrons, operate, but also covalent bonds, which are characterized by the localization of atomic orbitals in space. In crystals of transition metals, the covalent bond predominates, the emergence of which is associated with the presence of unfilled internal shells, and the metallic bond is of subordinate importance. Therefore, the binding energy in such crystals is significantly higher compared to alkali metals. For example, for nickel it is four times higher than for sodium.

Such metals may also have lower symmetry lattices than alkali and noble metals.

It should be noted that many substances, which under normal conditions are dielectrics or semiconductors, experience phase transitions with increasing pressure and acquire metallic properties. The forced approach of atoms increases the overlap of electron shells, which contributes to the sharing of electrons. For example, a semiconductor becomes a metal at a pressure of ~4 GPa, – at 16 GPa, – at 2 GPa. There are hypotheses that at a pressure of ~2000 GPa, molecular hydrogen can transform into the metallic state, and the phase may turn out to be stable after the pressure is removed and may turn out to be superconducting.

Molecular crystals

In such crystals, van der Waals coupling forces operate, which are of an electrical nature and are the most universal. Molecular forces consist of different types of interactions: orientation(between polar molecules), induction(at high polarizability of molecules) and dispersive.

Dispersion interaction is characteristic of all molecules and is practically unique in the case of non-polar molecules. This connection was first explained on the basis of a quantum mechanical solution to the problem of the interaction of two oscillators (F. London, 1930). The presence in the oscillator of minimal, non-zero energy, which decreases as the oscillators approach each other, leads to the appearance of dispersion interaction forces, which are classified as short-range.

A non-polar molecule, due to the movement of the electrons entering it, can acquire an instantaneous dipole moment - the molecule becomes polarized. Under the influence of this polarization, an induced moment occurs in the neighboring molecule, and an interaction is established between them.

In addition to dispersive forces, two more types of forces can act in molecular crystals: orientational in the case of polar molecules and inductive in the presence of molecules with a high ability to be polarized. Typically, all three types of interaction are observed in crystals, although the contribution of each may be different. The binding energy of molecular crystals is low and amounts to less than 0.1 eV/atom. Therefore, the corresponding substances have a low melting point and a low boiling point. The crystal structure of such substances is often characterized by close packing. Noble gases, when converted to a solid state, form crystals of a densely packed cubic structure.

Each molecule is a kind of quantum oscillator, therefore quantitative characteristics of dispersion interaction can be obtained by solving the quantum mechanical problem of the interaction of two linear harmonic oscillators with dipole moments and located at a distance. Potential energy of such a system

, (2.65)

where is the coefficient of elasticity of the dipole, and is the potential energy of interaction between two dipoles.

Let us define (in absolute system units)

. (2.66)

Expanding in a series and preserving the third terms of the expansion (provided ), we get

. (2.67)

Let's introduce normal coordinates

(2.68)

and transform :

. (2.69)

Solution of the stationary Schrödinger equation for a system of two oscillators

(2.70)

carried out using the method of separation of variables. The solvability conditions for each of the equations determine the discrete energy spectrum of the system

Where ; ; .

Let us define the “zero” energy ( ) of two interacting oscillators, arranging the radicals in a series up to third terms:

. (2.72)

Considering that the “zero” energy of two non-interacting oscillators , we obtain the energy of dispersion interaction

(GHS), (2.73)

(SI). (2.74)

From the last expression we obtain the strength of the dispersion interaction

. (2.75)

Hence, the existence of dispersion forces is due to the presence of “zero” energy of atoms and molecules, which decreases as they approach each other. Dispersion forces, as can be seen from (2.75), are short-range.

If molecules have permanent dipole moments or induced dipoles arise in them due to the high polarizability of molecules, then an additional dipole interaction appears. Under the influence of electrical forces, molecules tend to orient themselves relative to each other in such a way that the interaction energy of the dipoles decreases. This orientation is disrupted by chaotic thermal motion.

At sufficiently high temperatures, when the interaction energy of two dipoles , the energy of orientation interaction is equal to

, (2.76)

where is the dipole moment.

At low temperatures , when full orientation of the dipoles is achieved, the energy of the dipole interaction is equal to

. (2.77)

In molecules with high polarizability, induced dipole moments arise under the influence of an electric field. . The interaction energy of induced dipoles does not depend on temperature and is

. (2.78)

In the general case, the interaction energy of molecules can consist of various parts corresponding to orientational, inductive and dispersion interactions. The contribution of each of them is different depending on the type of molecules (Table 2.2).

The most universal are dispersion forces, which act not only between atoms with filled shells, but also between any atoms, ions and molecules.

Table 2.2. Characteristics of intermolecular interaction (%)

In the presence of strong bonds, dispersion interaction plays the role of a small additive. In other cases, dispersion interaction constitutes a significant proportion of the total intermolecular interaction, and in some cases, for example, for crystals of inert elements, it is the only type of attractive forces.

Similar information.

Physics of Condensed Matter

Physics of condensed matter- a large branch of physics that studies the behavior of complex systems (that is, systems with a large number of degrees of freedom) with strong coupling. The fundamental feature of the evolution of such systems is that it (the evolution of the entire system) cannot be “divided” into the evolution of individual particles. You have to “understand” the entire system as a whole. As a result, collective oscillations often have to be considered instead of the movement of individual particles. In a quantum description, these collective degrees of freedom become quasiparticles.

Condensed matter physics is a rich field of physics, both in terms of mathematical models and in terms of applications to reality. Condensed matter with a wide variety of properties are found everywhere: ordinary liquids, crystals and amorphous bodies, materials with a complex internal structure (which include soft condensed matter), quantum liquids (electronic liquid in metals, neutron liquid in neutrino stars, superfluids, atomic nuclei), spin chains, magnetic moments, complex networks, etc. Often their properties are so complex and multifaceted that it is necessary to first consider their simplified mathematical models. As a result, the search and study of exactly solvable mathematical models of condensed matter has become one of the most active areas in condensed matter physics.

Main areas of research:

• soft condensed matter
• highly correlated systems
  • spin chains
  • high temperature superconductivity
• physics of disordered systems

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• General physics of condensed matter, Evgeniy Zalmanovich Meilikhov. The textbook is part of a course in general physics for its special field (condensed matter physics). The manual assumes knowledge within the physics and mathematics programs...
• Solid State Physics for Engineers Textbook, Gurtov V., Osaulenko R.. The textbook is a systematic and accessible presentation of the course in solid state physics, containing the basic elements of condensed matter physics and its applications for…

Condensed matter is a concept that combines solids and liquids as opposed to gas. Atomic particles (atoms, molecules, ions) in a condensed body are interconnected. Wed. The energy of thermal motion of particles is not enough to spontaneously break the bond, so the condensed body retains its volume. A measure of the connection of atomic particles is the heat of evaporation (in a liquid) and the heat of sublimation (in a solid).

Unlike the gaseous state, a substance in a condensed state has order in the arrangement of particles (ions, atoms, molecules). Crystalline solids have a high degree of order - long-range order in the arrangement of particles. Particles of liquids and amorphous solids are located more chaotically and are characterized by short-range order. The properties of substances in the condensed state are determined by their structure and the interaction of particles.

1. Amorphous compounds

Amorphous compounds, in addition to highly elastic ones, can be found in two other physical compounds. states: glassy state and viscous-fluid state. high-molecular compounds that transform from a highly elastic state into a glassy state at temperatures below room temperature are classified as elastomers; at higher temperatures, they are classified as plastics. Crystalline high molecular weight compounds are usually plastics.

1. Crystals and their types

Crystals- from Greekκρύσταλλος, originally - ice, further - rhinestone, crystal) - solids in which atoms are arranged regularly, forming a three-dimensional periodic spatial arrangement - a crystal lattice.

Crystals are solid substances that have a natural external shape of regular symmetrical polyhedra, based on their internal structure, that is, on one of several specific regular arrangements of the particles that make up the substance (atoms, molecules, ions).

Types of crystals

It is necessary to separate the ideal and real crystal.

Perfect Crystal

It is, in fact, a mathematical object that has complete, inherent symmetry, idealized smooth smooth edges, etc.

Real crystal

It always contains various defects in the internal structure of the lattice, distortions and irregularities on the faces and has a reduced symmetry of the polyhedron due to the specific growth conditions, heterogeneity of the feeding medium, damage and deformations. A real crystal does not necessarily have crystallographic faces and a regular shape, but it retains its main property - the regular position of atoms in the crystal lattice.

The main distinguishing feature of crystals is their inherent property of anisotropy, that is, the dependence of their properties on direction, whereas in isotropic (liquids, amorphous solids) or pseudo-isotropic (polycrystals) properties properties do not depend on directions.

1. Properties of crystals depending on the type of chemical bonds

Types of chemical bonds in crystals. Depending on the nature of the particles and the nature of the interaction forces, four types of chemical bonds in crystals are distinguished: covalent, ionic, metallic and molecular.

Types of chemical bonding are a convenient simplification. More precisely, the behavior of an electron in a crystal is described by the laws of quantum mechanics. When talking about the type of connection in a crystal, you need to keep in mind the following:

a bond between two atoms never completely corresponds to one of the described types. An ionic bond always contains an element of covalent bond, etc.

in complex substances, the bonds between different atoms can be of different types. For example, in a protein crystal, the bond in the protein molecule is covalent, and between molecules (or different parts of the same molecule) is hydrogen.

Condensed matter physics is one of the richest areas in modern physics in terms of mathematical models and formulas.

Figure 1. Condensed matter. Author24 - online exchange of student work

Note 1

Condensed matter with a wide variety of characteristics are found absolutely everywhere: crystals, ordinary liquids and amorphous bodies, materials with an internal complex structure (which may include soft condensed elements), quantum liquids, spin constant chains, magnetic moments, complex spaces, and so on.

Often the properties of these substances are so complex and multifaceted that scientists have to initially consider simplified mathematical options. As a result, the study of exactly solvable condensed matter equations has become an active area in science.

The movement of each elementary particle in a condensed medium is in close relationship with the movement of its neighbors; Consequently, the formulas describing this process are strongly “intertwined” with each other.

Among the classical sections of condensed matter physics, the following can be distinguished:

• solid mechanics;
• theory of plasticity and cracks;
• hydrodynamics;
• plasma physics;
• electrodynamics of continuous media.

The common starting point in the above sections is the concept of continuum. The transition from a specific set of individual particles (ions or atoms) to a stable state consists of a complex averaging of the properties of the concept.

Main areas of study

Figure 2. Physical forms of condensed matter. Author24 - online exchange of student work

Basically, the different physical forms fall into three categories: gaseous, liquid and solid. In these three states of matter, the subject matter of condensed study determines progress at every stage of the discipline along with all areas of human life. Traditional ideal metals, ceramics and composite elements actively participate in all structures that involve the emission of light and electricity.

Heat and other characteristics of physical bodies are based on research in condensed matter physics, which directly provides the basis for many branches of high science and nanotechnology itself. Today, the implementation of the principles of this scientific direction is on the rise with the development of microelectronics, laser technology and optical communication technologies.

Main areas of condensed matter physics:

• theory of disordered systems;
• nanotechnology;
• continuum mechanics;
• electrodynamics of continuous media;
• structure of a solid;
• movement of liquids;
• condensed soft matter;
• quantum Hall effect;
• superconductivity of heat.

In condensed matter physics, all elements are divided into atoms for the purpose of studying various structures in detail. This area of ​​physics has only begun to gain popularity in recent decades. It is necessary to note the significance of the phenomenon that comes from the study of a crystalline solid during its transformation into a liquid state. In these two long-term experiments, the researchers were able to build some confidence, and gradually introduce some workable methods to facilitate further scientific research.

Quantum theory of condensed matter

The quantum hypothesis allowed inventors not only to explain atomic nuances and spectra, but also to solve many complex mysteries in the behavior of solid physical bodies, especially ideal crystals. It would seem that a crystal containing millions of atoms is millions of times more difficult to study than an individual elementary particle. However, the task is not so difficult if you look at it from a completely different point of view.

Definition 1

The structure of any crystal is very ordered - it is an ordinary crystal lattice.

Inside it, along each straight line, the same atoms (or molecules and ions) are located at equal intervals. The crystal is equipped with the unique property of periodicity in any direction considered.

That is why, when studying crystals, it is orderliness that helps primarily, and not the properties of individual elements. As in the molecular spectra hypothesis, the methods of theoretical groups and their general representations are used here. If a molecule in a crystal is moved, a force will instantly arise that will ultimately push it away from neighboring particles and return it to its original position.

Thanks to this, the crystal is stable under any conditions: its ions and atoms can experience only minor fluctuations relative to the position of stability and equilibrium. Another thing is the electrons of the atoms themselves. A certain part of them, which is located at lower energy levels, always remains in its atom. But elements from the upper levels move quite freely from one atom to another and belong to the entire crystal.

Note 2

The movement of such electrons is characterized not so much by the characteristics of individual particles, but by the characteristics of the crystal lattice.

Therefore, a crystal can be considered as a combination of two physical subsystems. The first of them is the crystal lattice itself in the form of a periodic structure of molecules that are devoid of valence elements, and therefore positively charged in any position. The second is the commonality of electrons in the periodic electric field of a positively charged lattice.

Any external influence on the crystal (electrical, mechanical, magnetic, thermal) results in the fact that in one of the concepts waves propagate chaotically - like from a thrown stone into water. The property of periodicity eliminates the need for researchers to study such vibrations of individual ions in a crystal. It is enough to study the wave as a whole: according to the quantum hypothesis, any such process corresponds to a particle - a wave quantum; in the theory of solid physical bodies, it is called a quasiparticle. There are many types of quasiparticles. One of the most common is quanta or photons of elastic vibrations of the crystal lattice, which are responsible for the propagation of heat and sound in the crystal.

Remark 3

Thus, it can be stated that quantum theory is a unique scientific tool that allows you to quickly conduct quantitative and qualitative studies of physical matter at any level - from atoms to continuous media.

Prospects for the development of condensed matter physics

The physics of condensed matter is currently in its brightest period of flowering. And, since fundamental research in this field of science and the practical use of technology are often closely interconnected, the results of experiments represent a series of new universal technologies, materials and devices, which play an irreplaceable key role in the modern world of high technology.

In recent years, experiments in the field of condensed matter physics, methods and technologies of study are increasingly penetrating neighboring disciplines associated with the development of chemical, biophysical and geophysical sciences.

Today, the physics of condensed matter is actively developing and being introduced into all areas of human life. However, since this direction is the source of quantum theory and the movements of crystalline solids, today it is still the main object of study of the structures of continuous spaces. After all, scientists are faced with the same nature, in which many laws and phenomena are universal. It is through in-depth study that it is possible to understand and realize such patterns.