*3.3.4. CrystalsAmorhous and dispersive bodies*

The radii of atoms can be associated with periods of their location in the crystal space. In Table 2, for example, the cubic lattice provides relationships between the lattice period *а* and radius of atoms in the dense packing, number Z of atoms in the elemental cell of the crystal.


**Table 2.** Correlation between atoms of the dense packing with parameters of cubic lattices.

**substance**

114 Solar Cells - New Approaches and Reviews

\*Density of substance at boiling point.

**Table 1.** Values of density and density of amount of substance for some simple matters

Kirkwood-Yvon (BBGKY) which describes the melting of a solid body.

*3.3.4. CrystalsAmorhous and dispersive bodies*

According to Fig. 2b and Table 1, the points showing the density of amount of substance of liquids and radii of the atoms are arranged between straight lines for solids and gases. The position of these points allows giving a qualitative characteristic of the structure bodies. So that, liquid bromine and liquid mercury with their structure are similar to solids. In opposite, liquid helium and liquid nitrogen are maximally removed from the line of correlation. Absence of own correlation among them confirms the fact that liquids are the transitional state in condensation of gas to solid. Therefore, one should not expect the existence of a universal equationof state for liquids. These states should be described by special solutions of the equation of state of the real gas and solid body. An example is the equation of van der Waals describing condensation of gas to liquid, and a chain of equations by Bogoliubov-Born-Green-

The radii of atoms can be associated with periods of their location in the crystal space. In Table 2, for example, the cubic lattice provides relationships between the lattice period *а* and radius

of atoms in the dense packing, number Z of atoms in the elemental cell of the crystal.

**Atom radius Rat Density of the amount of substance σ<sup>0</sup>**

Н2, gas 2.22 / 1.1 9.654 / 9.559 2,0158 0,08987 Н2, liq. 0.3707 10.43 2,0158 708\* Не, gas 1.82 9.740 4,0026 0,1785 Не, liq. 0.53 10.276 4,0026 126\* N2, gas 3.22 / 1.5 9.492 / 9.824 28,0134 1,2506 N2, liq. 0.547 10.262 28,0134 808 Li 1.520 9.823 6,94 534 Cs 2.655 9.576 132,91 1900 Mg 1.599 9.796 24,31 1740 Ba 2.174 9.663 137,34 3760 Al 1.432 9.844 26,98 2699 W 1.371 9.863 183,85 19320 Hg, liq. 1.503 9.823 200,59 13600 Br2, liq. 1.1415 9.943 159,81 3102 B 0.795 10.10 10,81 2340 Be 1.113 9.953 9,01 1850 C 0.771 10.11 12,01 1880

**or mole mass М0, 103**

**кg/ m<sup>3</sup> 10-10 m -lgRat**

 **кg/mol**

**Density of substance ρ,**

There are other relationships for crystals of another syngonies. So that, there is a possibility to move from the dependence of density ρ to the density of the amount of substance σ as a function of the volume of elemental cell of the crystal Vcell and number Z of formula units of substance in the cell.

The upper part of Fig. 3 presents points linking based (picknometry) density p and volume Vcell of elemental cells of the crystal determined from x-ray structural analysis data. Each point corresponds to the crystal of certain chemical composition. Chaotic arrangement of points gives place of their correlation, if the density of the crystal will be divided by the mass of the mole of formula units. It is shown in the middle part of Fig. 3. The ratio ρ/M0 is the density of a standard amount of substance σ<sup>0</sup> or density of standard amount of crystal substance. It was calculated by the formula (1)

$$\sigma\_0 = \rho / \,\mathbf{M}\_0.$$

If chemical identity of the crystal was registered by the mass of mole of formula units M0, the structural identity of the crystal can be taken into account by the number Z of formula units of the substance in the cell of the crystal. Lower part of Fig. 3 shows a linear correlation between the volume of the unit cell and the density of amount of crystalline substance σ=ρ / (Z M0). It corresponds to the formula

$$\mathbf{V}\_{\rm cell} = \mathbf{M}\_0 \mathbf{Z} \;/\; \text{(\(\mathbb{Q}\,\text{N}\_A\))}$$

which is known in X-ray structural analysis for calculating Z. The join solution is

$$
\sigma = 1 / \left( \mathcal{V}\_{\text{cell}} \mathcal{N}\_{\text{A}} \right) / \ .
$$

showing that the tangent of the slope of the straight line is numerically equal to the reverse value of the Avogadro number NA.

Note some features of gradual transition from the density of the crystal p to its relation to molecular weight M0 and density of amount of crystalline substance σ=*ρ* / (Z M0).

First, in Fig. 3 the properties correlation (or absence of it) completes within the error of measurement of density of substance about 400 chemical compounds of different classes.

**Figure 3.** Evolution of correlation between the volume of unit cell of the crystal in series of properties "density – mole mass – number of formula units in the cell of the crystal".

Electrical properties present here conductors, semiconductors and dielectrics. Chemical composition except for simple substances presents here oxides, halides, chalcogenides, oxidocompounds, nitrates, nitrites, sulphates, carbonates, iodates, bromates, chlorates, wolframites, carbides, nitrides, hydrides, hydroxides and other chemical compounds. Optical properties present here transparent and colored crystals, as well as metals. On structural properties here are the crystals of all systems (syngonies). By type of chemical bond here are presented crystals with metallic, ionic and covalent bond, and molecular crystals.

Second, the density of amount of substance σ=*ρ* / (Z M0) and the volume of the elementary cell was calculated from the experimental data. We used tables of molecular weights of chemical compounds that are defined with sufficient precision, tables of densities found by picknometry method, the cell sizes of the crystals were determined by X-ray structural analysis of crystals.

Third, the tangent of the slope of a line is numerically equal to the inverse value of Avogadro's number NA. This result obtained in 1997 [43, 44, 51, 52] corresponds to the Resolution XXIV of the General conference on weights and measures (General Conference on Weights and Measures, 2011), which proposed in the future to determine mol by fixation of the numerical value of the Avogadro constant.

These features allow stating that following expressions for calculation

$$
\sigma\_0 = \rho / \,\mathbf{M}\_0$$

$$
\mathbf{v}\_{\rm m0} = (\mathbf{M}\_0)^{-1} \mathbf{v}$$

and the standard density of amount of substance of crystalline σ0 as well as its specific amount νm,0 are corresponding each other (Table 3).


\* The lowest density of natural diamonds and carbons (graphite)

**Table 3.** Density of standard amount of substance for some chemical elements and compounds.

Opposite, the calculation expressions

Electrical properties present here conductors, semiconductors and dielectrics. Chemical composition except for simple substances presents here oxides, halides, chalcogenides, oxidocompounds, nitrates, nitrites, sulphates, carbonates, iodates, bromates, chlorates, wolframites, carbides, nitrides, hydrides, hydroxides and other chemical compounds. Optical properties present here transparent and colored crystals, as well as metals. On structural properties here are the crystals of all systems (syngonies). By type of chemical bond here are

**Figure 3.** Evolution of correlation between the volume of unit cell of the crystal in series of properties "density – mole

Second, the density of amount of substance σ=*ρ* / (Z M0) and the volume of the elementary cell was calculated from the experimental data. We used tables of molecular weights of chemical compounds that are defined with sufficient precision, tables of densities found by picknometry method, the cell sizes of the crystals were determined by X-ray structural analysis of crystals.

Third, the tangent of the slope of a line is numerically equal to the inverse value of Avogadro's number NA. This result obtained in 1997 [43, 44, 51, 52] corresponds to the Resolution XXIV of the General conference on weights and measures (General Conference on Weights and

presented crystals with metallic, ionic and covalent bond, and molecular crystals.

mass – number of formula units in the cell of the crystal".

116 Solar Cells - New Approaches and Reviews

$$
\boldsymbol{\sigma} = \boldsymbol{\varrho} / \,\mathrm{(Z \, M\_0)},
$$

$$
\mathbf{v}\_{\mathrm{m}} = \mathbf{v} / \,\mathrm{m} = \mathrm{(Z \, M\_0)^{-1}}
$$

correspond to the density σ of real amount of crystal substances and its specific amount νm. Table 4 shows comparison of the calculation results σ and ν<sup>m</sup> of diamonds, graphite, copper, and other compounds. These crystals are found in nature, and they are grown artificially.

One can see from Tables 3, 4 that specific amount of crystal substances of four carbon structures varies, although one kg of each allotropic form of carbon contains an equal number of atoms. However, a unit mass of a diamond and lonsdeylite, two-layer α-graphite (2H) and three-layer β-graphite (3R) has a different number of cells. Therefore, the change in the amount of crystalline substances illustrates kinetic difference in polymorphic transformations of carbon (Table 5).


**Table 4.** Density of amount of crystalline substance of some chemical elements and compounds


**Table 5.** Balance of amounts of standard and crystalline substances in polymorphic transformations and sublimation of carbon.

Crystal-chemical formulation of phase transitions of graphite is given in Table 5 in short form. It does not reflect the difference in structures of α-graphite and lonsdeylite if writing ithem as the form C4. The full form takes into account the number of regular systems of points (the equivalent of locations) in the crystal structures of carbon. So, in 2H cell of double-layer αgraphite four atoms are located at the points of two regular systems. In 3R cell of β-graphite six carbon atoms are arranged on two points of regular systems. Full form of polymorphic transformations in graphite is given in Table 6. Atoms of diamond and lonsdeylite are located on the same system of equivalent points. Therefore, the forms of their entries are given in Tables 5, 6.

β-graphite (3R) has a different number of cells. Therefore, the change in the amount of crystalline substances illustrates kinetic difference in polymorphic transformations of carbon

**kg/m3 <sup>Z</sup>**

Сu 8950 4 3.935 35220 NaCl 2170 4 4.278 9280 CsCl 3980 1 5.940 26640 Diamond С 3470\* 8 10.41 36110 Lonsdeylite С 3510 4 20.82 73050 Bilayer α-graphite (С-2Н) 2090\* 4 20.82 43500 Trilayer β-graphite (С-3R) 2090\* 6 13.88 29000

Wurtzite ZnS 4087 2 5.132 20970 Sphalerite ZnS 4090 4 2,566 10490 CdJ2 5670 1 2.730 15480

> **Crystal-chemical form of writing phase transition (short)**

**Table 4.** Density of amount of crystalline substance of some chemical elements and compounds

**Change Δν0 of standard amount of substance, mole**

Diamond → α-graphite 2Н 0 С<sup>8</sup> → 2 С<sup>4</sup> 2 Diamond → β-graphite 3R 0 3 С<sup>8</sup> → 4 С<sup>6</sup> 1 Diamond → lonsdeylite 0 С<sup>8</sup> → 2 С<sup>4</sup> 2 β-graphite → α-graphite 0 2С6 <sup>→</sup> 3С<sup>4</sup> 1 Diamond → Сgas 0 С<sup>8</sup> → 8 Сgas 8 α-graphite → Сgas 0 С4 <sup>→</sup> 4 Сgas 4 β-graphite → Сgas 0 С6 <sup>→</sup> 6 Сgas 6 Lonsdeylite → Сgas 0 С4 <sup>→</sup> 4 Сgas 4

**Table 5.** Balance of amounts of standard and crystalline substances in polymorphic transformations and sublimation of

**density of amount of crystalline substance**

**Volume density σ, mole/m3**

**Change Δν of amount of crystalline substance, mole**

**Specific density νm0, mole/kg**

(Table 5).

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**Natural or artificial crystal Density of mass ρ,**

\* The lowest density of natural diamonds and graphite

**Traditional form of writing phase transition**

carbon.


**Table 6.** Change of specific amounts of standard and crystalline substance under polymorphic transformations and sublimation of carbon

Suppose that 0.012 kg of α-graphite 3R turn on the β-graphite 2H. The initial amount of crystalline substances is 1/6 mole, as Z=6, and in 3R cell of β-graphite six carbon atoms are arranged on points of two regular systems. The amount of crystalline substance obtained from the phase transition 3R → 2H, will be ¼ mole, because Z=4, and in 2H cell of α-graphite four atoms are located at the points of two regular systems. So, 0.012 kg of graphite from a form 3R turns into 2H with decrease of crystalline substance 1/4-1/6=1/12 mole. The increase in the amount of crystalline substance for other phase transitions is shown in Table 6.

Suppose now that 0.012 kg of diamond turned into Lonsdeylite. The initial amount of crystal‐ line substances is 1/8 mole, as Z=8, and eight carbon atoms are arranged by points of one regular system. The amount of crystalline substance obtained for the phase transition diamond → lonsdeylite will be 1/8 mole, because four atoms in the cell of lonsdeylite are located on the points of one regular system. Table 5 shows the changes in the amount of crystalline substances in all possible polymorphic transformations of diamond and graphite.
