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
Let us present it as the sum of the repulsion and attraction potentials:
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
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
Where Madelung constant α and constant D are defined as follows:
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. Therefore, , and therefore we have
where is the equilibrium distance between neighboring ions.
From (2.46) we obtain
and the expression for the total energy of the crystal in an equilibrium state takes the form
The value represents the so-called Madelung energy. Since the exponent is , 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 the series expansion, we obtain the Madelung constant in the case of a one-dimensional crystal
Thus, the expression for the energy per molecule takes the following form
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 have considered, we obtain , which coincides well with the exact value for a lattice of type . For a type structure, , and for a type structure, .
Let us 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:
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) type structure, the volume occupied by the molecules is equal to
Then we can write
From (2.53) it is easy to obtain the second derivative
In the equilibrium state, the first derivative vanishes, therefore, from (2.52–2.54) we determine
Let us use (2.43) and obtain
From (2.47), (2.56) and (2.55) we find the bulk modulus of elasticity IN:
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
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
in this case, the degree of ionicity can be determined as follows:
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 |
In complex crystals consisting of elements of different valencies, the formation of an ionic type of bond is possible. Such crystals are called ionic.
When atoms come closer and valence energy bands overlap between elements, electrons are redistributed. An electropositive element loses valence electrons, turning into a positive ion, and an electronegative element gains it, thereby completing its valence band to a stable configuration, like that of inert gases. Thus, ions are located at the nodes of the ionic crystal.
A representative of this group is an oxide crystal whose lattice consists of negatively charged oxygen ions and positively charged iron ions.
The redistribution of valence electrons during an ionic bond occurs between the atoms of one molecule (one iron atom and one oxygen atom).
For covalent crystals, the coordination number K, the crystalline number, and the possible lattice type are determined by the valence of the element. For ionic crystals, the coordination number is determined by the ratio of the radii of the metallic and nonmetallic ions, since each ion tends to attract as many ions of the opposite sign as possible. The ions in the lattice are arranged like balls of different diameters.
The radius of the nonmetallic ion is greater than the radius of the metallic ion, and therefore metallic ions fill the pores in the crystal lattice formed by the nonmetallic ions. In ionic crystals the coordination number
determines the number of ions of the opposite sign that surround a given ion.
The values given below for the ratio of the radius of a metal to the radius of a non-metal and the corresponding coordination numbers follow from the geometry of the packing of spheres of different diameters.
For the coordination number will be equal to 6, since the indicated ratio is 0.54. In Fig. Figure 1.14 shows the crystal lattice. Oxygen ions form an fcc lattice, iron ions occupy pores in it. Each iron ion is surrounded by six oxygen ions, and, conversely, each oxygen ion is surrounded by six iron ions. In connection with this, in ionic crystals it is impossible to isolate a pair of ions that could be considered a molecule. Upon evaporation, such a crystal disintegrates into molecules.
When heated, the ratio of ionic radii can change, since the ionic radius of a nonmetal increases more rapidly than the radius of a metal ion. This leads to a change in the type of crystal structure, i.e., to polymorphism. For example, when an oxide is heated, the spinel crystal lattice changes to a rhombohedral lattice (see section 14.2),
Rice. 1.14. Crystal lattice a - diagram; b - spatial image
The binding energy of an ionic crystal is close in magnitude to the binding energy of covalent crystals and exceeds the binding energy of metallic and, especially, molecular crystals. In this regard, ionic crystals have a high melting and evaporation temperature, a high elastic modulus and low coefficients of compressibility and linear expansion.
The filling of energy bands due to the redistribution of electrons makes ionic crystals semiconductors or dielectrics.
An ideal ionic crystal consists of positively and negatively charged spherical ions. This idea is most consistent with, if not all, then at least some alkali halide compounds, i.e. salts formed by one of the alkali metals (lithium, sodium, potassium, rubidium, cesium) and one of the halogens (fluorine, chlorine, bromine, iodine). There is evidence that the crystals of these salts are indeed formed by positive metal ions and negatively charged halogen ions. The most direct of them is X-ray diffraction data, on the basis of which the electronic charge distribution is calculated (cm. rice. 9 for the case of NaCl).(22.74 Kb)The fact that such solids consist of ions rather than atoms can be explained as follows. First of all, all alkali metal atoms have one outer valence electron, while the outer shell of halogen atoms contains seven valence electrons. When a valence electron passes from an alkali metal atom to a halogen atom, two ions are formed, each of which has a stable electronic configuration characteristic of noble gas atoms. Even more important is the energy gain due to the Coulomb attraction between positive and negative ions. Let's take sodium chloride (NaCl) as an example. To remove an outer (valence) electron from a Na atom, you need to expend 5.14 eV (ionization energy). When this electron is added to the Cl atom, there is an energy gain of 3.61 eV (electron affinity energy). Thus, the energy required for the transfer of a valence electron from Na to Cl is (
5,14 - 3.61) eV = 1.53 eV. The Coulomb energy of attraction between two emerging Na ions+ and Cl - with a distance between them (in the crystal) equal to 2.18, is 5.1 eV. This value more than compensates for the total electron transition energy and leads to a decrease in the total energy of the ion system compared to a similar system of free atoms. This is the main reason that alkali halide compounds consist of ions rather than atoms.Calculating the energy of ionic crystals is actually more complicated than it might seem from the above discussions. But at least for alkali halide crystals there is good agreement between the theoretical and experimental values of the binding energy. The ionic bonds are quite strong, as indicated, for example, by the high melting point of 1074 K for NaCl.
Due to the high degree of stability of the electronic structure, ionic crystals fall into the category of dielectrics. Because positive and negative ions interact with electromagnetic waves, ionic crystals exhibit strong optical absorption in the infrared region of the spectrum. (The frequency of the oscillating external electric field in this region of the spectrum is close to the natural frequency of transverse lattice waves, in which positive and negative ions of the crystal move in opposite directions.) In the visible region of the spectrum, the oscillation frequencies are too high for massive ions to have time to respond to the influence of such waves Therefore, light waves pass through the crystal without interaction, i.e. such crystals are transparent. At even higher frequencies in the ultraviolet region of the spectrum field quanta can have sufficient energy to excite valence electrons, ensuring the transition of valence electrons of negative ions to unoccupied states of positive ions. This results in strong optical absorption.
Covalent crystals. The most famous covalent crystals are diamond, silicon and germanium. Each atom in such crystals is surrounded by four neighboring atoms located at the vertices of a regular tetrahedron. Free atoms of each of these elements have four valence electrons, and this is enough to form four paired electronic bonds (between this atom and its four nearest neighbors). Thus, two electrons are collectivized by two atoms forming a bond and are located in space along the line connecting the atoms. This is almost the same bond as between the two hydrogen atoms in the hydrogen molecule H 2 . In diamond, these bonds are very strong, and because they have a strictly defined direction relative to each other, diamond is an extremely hard material. The strength of the covalent bond between an electron and a crystal is characterized by the so-called energy gap the minimum energy that must be transferred to an electron so that it can move freely in the crystal and create an electric current. For diamond, silicon and germanium, the width of this gap is 5.4, 1.17 and 0.744 eV, respectively. Therefore, diamond is a good dielectric; the energy of thermal vibrations in it at room temperature is too low to release valence electrons. In silicon and especially in germanium, due to the relatively small width of the energy gap, thermal excitation of a certain number of valence electrons at room temperature is possible. Thus, they conduct current, but since their conductivity is much less than that of metals, silicon and germanium are classified as semiconductors.Ionic crystals are compounds with a predominant ionic nature of the chemical bond, which is based on the electrostatic interaction between charged ions. Typical representatives of ionic crystals are alkali metal halides, for example, with a structure such as NaCl and CaCl.
When crystals like rock salt (NaCl) are formed, halogen atoms (F, Cl, Br, I), which have a high electron affinity, capture valence electrons of alkali metals (Li, Na, K, Rb, I), which have low ionization potentials, while positive and negative ions are formed, the electron shells of which are similar to the spherically symmetric filled s 2 p 6 shells of the nearest inert gases (for example, the N + shell is similar to the Ne shell, and the Cl shell is similar to the Ar shell). As a result of the Coulomb attraction of anions and cations, the six outer p-orbitals overlap and a lattice of the NaCl type is formed, the symmetry of which and the coordination number of 6 correspond to the six valence bonds of each atom with its neighbors (Fig. 3.4). It is significant that when the p-orbitals overlap, there is a decrease in the nominal charges (+1 for Na and -1 for Cl) on the ions to small real values due to a shift in the electron density in six bonds from the anion to the cation, so that the real charge of the atoms in the compound It turns out, for example, that for Na it is equal to +0.92e, and for Cl- the negative charge also becomes less than -1e.
A decrease in the nominal charges of atoms to real values in compounds indicates that even when the most electronegative electropositive elements interact, compounds are formed in which the bond is not purely ionic.
Rice. 3.4. Ionic mechanism of formation of interatomic bonds in structures likeNaCl. Arrows indicate the directions of electron density shift
According to the described mechanism, not only alkali metal halides are formed, but also nitrides and carbides of transition metals, most of which have a NaCl type structure.
Due to the fact that the ionic bond is non-directional and unsaturated, ionic crystals are characterized by large coordination numbers. The main structural features of ionic crystals are well described on the basis of the principle of dense packing of spheres of certain radii. Thus, in the NaCl structure, large Cl anions form a cubic close packing, in which all octahedral voids are occupied by smaller Na cations. These are the structures of KCl, RbCl and many other compounds.
Ionic crystals include most dielectrics with high electrical resistivity values. The electrical conductivity of ionic crystals at room temperature is more than twenty orders of magnitude less than the electrical conductivity of metals. Electrical conductivity in ionic crystals is carried out mainly by ions. Most ionic crystals are transparent in the visible region of the electromagnetic spectrum.
In ionic crystals, attraction is mainly due to the Coulomb interaction between charged ions. - In addition to the attraction between oppositely charged ions, there is also repulsion, caused, on the one hand, by the repulsion of like charges, on the other, by the action of the Pauli exclusion principle, since each ion has stable electronic configurations of inert gases with filled shells. From the point of view of the above, in a simple model of an ionic crystal, it can be assumed that the ions are hard, impenetrable charged spheres, although in reality, under the influence of the electric fields of neighboring ions, the spherically symmetrical shape of the ions is somewhat disrupted as a result of polarization.
Under conditions where both attractive and repulsive forces exist simultaneously, the stability of ionic crystals is explained by the fact that the distance between unlike charges is less than between like ones. Therefore, the forces of attraction prevail over the forces of repulsion.
Again, as in the case of molecular crystals, when calculating the cohesion energy of ionic crystals, one can proceed from the usual classical concepts, assuming that the ions are located at the nodes of the crystal lattice (equilibrium positions), their kinetic energy is negligible and the forces acting between the ions are central .
What is ionic polarization
Ionic polarization consists of the displacement of ions in an external electric field and the deformation of electronic shells. Let's consider a crystal of type $M^+X^-$. The crystal lattice of such a crystal can be considered as two cubic lattices, one of which is built from $M^+$ ions, the other from $X^-$, and they are inserted into one another. Let us direct the external uniform electric field ($\overrightarrow(E)$) along the Z axis. The ion lattices will shift in opposite directions by segments $\pm z$. If we accept that $m_(\pm )(\omega )^2_0$ is a quasi-elastic force that returns an ion with mass $m_(\pm )$ to the equilibrium position, then the force ( $F_(upr)$), which is equal to:
In this case, the electric force ($F_e$), which acts on ions of the same lattice, is equal to:
Equilibrium conditions
In this case, the equilibrium conditions will take the form:
For positive ions:
For negative ions:
In this case, the total relative displacement of the ions is equal to:
Ionic polarization is equal to:
where $V_0$ is the volume of one molecule.
If we take, for example, the structure of $NaCl$, in which each ion is surrounded by six ions of the opposite sign, which are located at a distance a from it, we obtain:
and, therefore, using (5) and (6), we obtain that:
Ionic polarization is established in a very short time, approximately $(10)^(-13)sec.$ It does not lead to energy dissipation and does not cause dielectric losses. When the external field is removed, the electronic shells return to their previous state.
Ionic lattice polarization is described by formula (9). In most cases, such polarization is anisotropic.
where $\left\langle \overrightarrow(p)\right\rangle $ is the average value of the dipole moments of ions that are equal in magnitude but oppositely directed, $\overrightarrow(p_i)$ are the dipole moments of individual ions. In isotropic dielectrics, the average dipole moments coincide in direction with the strength of the external electric field.
Local field strength for crystals
The local field strength ($\overrightarrow(E")\ or\ sometimes\ \overrightarrow(E_(lok))\ $) for cubic crystals can be expressed by the formulas:
where $\overrightarrow(E)$ is the average macroscopic field in the dielectric. Or:
If equation (10) is applicable for calculating the local field for cubic crystals, then the Clausius-Mossotti formula can be applied to such crystals:
where $\beta$ is the polarizability of the molecule, $n$ is the concentration of molecules.
The relationship between the polarizability ($\beta $) of a molecule and the dielectric susceptibility ($\varkappa$) for cubic crystals can be given by the expression:
Example 1
Assignment: The dielectric constant of the crystal is $\varepsilon =2.8$. How many times is the local strength ($\overrightarrow(E")$) of the cubic system field greater than the average macroscopic field strength in the dielectric ($E$)?
As a basis, we will take the formula for calculating the local field strength, namely:
\[\overrightarrow(E")=\frac(\varepsilon +2)(3)\overrightarrow(E)\left(1.1\right).\]
Therefore, for the desired tension ratio we can write that:
\[\frac(E")(E)=\frac(\frac(\varepsilon +2)(3)E)(E)=\frac(\varepsilon +2)(3)\left(1.2\right) .\]
Let's carry out the calculations:
\[\frac(E")(E)=\frac(2.8+2)(3)=1.6.\]
Answer: 1.6 times.
Example 2
Assignment: Determine the polarizability of carbon atoms in diamond ($\beta $), if the dielectric constant of diamond is $\varepsilon =5.6$, and its density is $(\rho )_m=3.5\cdot (10)^3\ frac(kg)(m^3.)$
As a basis for solving the problem, we take the Clausius-Mossotti equation:
\[\frac(\varepsilon -1)(\varepsilon +2)=\frac(n\beta )(3)\left(2.1\right).\]
where the particle concentration $n$ can be expressed as:
where $(\rho )_m$ is the mass density of the substance, $\mu =14\cdot (10)^(-3)\frac(kg)(mol)$ is the molar mass of carbon, $N_A=6.02\cdot (10)^(23)mol^(-1)$ is Avogadro's constant.
Then expression (2.1) will take the form:
\[\frac(\varepsilon -1)(\varepsilon +2)=\frac(\beta )(3)\frac((\rho )_mN_A)(\mu )\ \left(2.3\right).\]
From expression (2.3) we express the polarizability $\beta $, we obtain:
\[\ \beta =\frac(3\mu (\varepsilon -1))((\rho )_mN_A(\varepsilon +2))\left(2.4\right).\]
Let's substitute the available numerical values and carry out the calculations:
\[\beta =\frac(3\cdot 14\cdot (10)^(-3)(5.6-1))(3.5\cdot (10)^3\cdot 6.02\cdot (10 )^(23)(5.6+2))=\frac(193.2\cdot (10)^(-3))(160.132\cdot (10)^(26))=1.2\cdot ( 10)^(-29)m^3\]
Answer: $\beta =1.2\cdot (10)^(-29)m^3$.
