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The purpose of the research is to develop a dynamical theory of phase transitions in crystalline structures, when except for temperature, the pressure is acting. So, the phase diagram temperature-pressure (dimensions) must be constructed. In general case, it is a complicated question, which can be solved for simple models of crystal, as three atomic models, introduced in the work of Frenkel [ 1]. In this model, three identical atoms are placed on the straight line and interact with the forces, which can be described by the expression, given in the article of Lennard-Jones [ 2]. Such simple model may have success, when the crystalline structure is simple, which consists of one type of atoms, for example: carbon. The model was generalized to cubic cell model with a moving atom in the inner part of the cell. The rigorous calculation of phase diagram for transition graphite-diamond shows some similarity with results of numerous experimental investigations (which are not discussed here). So, the way of phase diagram calculation may attract attention.

As was mentioned in the work of Max Born, the linear or harmonic theory of crystalline lattice is insufficient to describe some properties of solid corps: structural transitions, flow, mechanical strength [

In

U ( x , y , z ) = U 0 + c 2 ( x 2 + y 2 + z 2 ) + b 4 ( x 4 + y 4 + z 4 ) (1)

with the cell edge length l. The form of potential energy is reproduced in neighboring cells, so in two points on the x-axis, symmetrical placed relative to the facet, the energy has the same value, so according to Roll theorem the derivative of the energy on the facet must be zero

( ∂ U ∂ x ) x = l 2 = l 2 ( c + b l 2 4 ) = 0 (2)

So

b = − 4 c l 2 (3)

Now we mark the values of potential energies in the centre of the cell U C and in the centre of facet U F . They have values

U C = U 0 (4)

U F = U 0 + l 2 8 ( c + b l 2 8 ) (5)

So it will be gained

c = 16 l 2 ( U F − U C ) (6)

b = − 64 l 4 ( U F − U C ) (7)

The dependence of the coefficients c, b from the cell dimensions is determined from the dependence of the Lennard-Jons potential from the distance between atoms. Taking into account only the nearest neighbors in the cell, it can be written

U C = 8 [ − A ( l 3 2 ) m + B ( l 3 2 ) n ] (8)

U F = 4 [ − A ( l 2 2 ) m + B ( l 2 2 ) n ] (9)

where, as it will be accepted, n = 2 m. From theses expressions it can be found

1 4 ( U F − U C ) = − A 2 m 2 l m [ 1 − 2 ( 2 3 ) m 2 ] + B 2 n 2 l n [ 1 − 2 ( 2 3 ) n 2 ] (10)

Because expressions in quadratic brackets are positive and n is greater, than m, it follows a change of sign from positive values to negative with increasing length of the cell side. Consequently, it exist some value of length l * , where elastic and anharmonic coefficients turns to zero, and the movement of the atom occurs unhampered (this question will be considered later). It can be put approximately

c ( l ) = ( d c d l ) * ( l − l * ) + ⋯ (11)

where the derivative here is negative. Further we consider the movement only along x-axis, because movements along all axes are identical. Taking into account (1) and (3) we write the expression of atom energy in the form

U = U 0 + c 2 x 2 − c l 2 x 4 (12)

If the thermal movement is absent, the coordinate is not changed with the time and determined through the external influence. The condition of equilibrium

∂ U ∂ x = c x − 4 c l 2 x 3 = 0 (13)

leads to the values of coordinate

x = 0 and x 2 = l 2 4 (14)

and these values correspond to the position in the centre of cell (cubic structure) and centre of facet (layered structure). The conditions of stability

∂ 2 U ∂ x 2 = c − 12 c l 2 x 2 > 0 (15)

for the position in the centre of the cell leads to inequality c > 0, and for the position in the centre of facet to the inequality c < 0. We consider now the question in general case, when the change in structure occurs through the action of thermal motion and mechanical action. Then in expression (1) the coordinate as a function of time t is represented in the form

x = s + u θ ( t ) (16)

where vibrations are described by means of simple function of time.

After the elevation of the coordinate in proper power and taking the mean time value the free energy is transformed to the form

F = U 0 + c 2 ( s 2 + u 2 ) + b 4 ( s 4 + 6 s 2 u 2 + θ 4 ¯ u 4 ) − k T ln u , θ 4 ¯ = 3 (17)

The general equations of state are

∂ F ∂ s = c s + b s 3 + 3 b s u 2 = 0 (18)

∂ F ∂ u = c u + 3 b s 2 u + 3 b u 3 − k T u = 0 (19)

The general conditions of stability

α ) ∂ 2 F ∂ s 2 = c + 3 b s 2 + 3 b u 2 > 0 (20)

β ) ∂ 2 F ∂ u 2 = c + 3 b s 2 + 9 b u 2 + k T u 2 > 0 (21)

γ ) ∂ 2 F ∂ s 2 ∂ 2 F ∂ u 2 − ( ∂ 2 F ∂ s ∂ u ) 2 = ( c + 3 b s 2 + 3 b u 2 ) ( c + 3 b s 2 + 9 b u 2 + k T u 2 ) − ( 6 b s u ) 2 > 0 (22)

From equation (18), equation (19) follows two expressions, describing the states conditionally called symmetrical ( s = 0 ) and nonsymmetrical ( s ≠ 0 ). Taking the amplitude as independent variable, one can write the expressions of displacement and temperature

s = 0 , k T = c u 2 + 3 b u 4 (23)

b s 2 = − c − 3 b u 2 , k T = − 2 c u 2 − 6 b u 4 (24)

In

The coordinates of maximum in

u M 2 = − c 6 b = l 2 24 ; k T M = − c 2 12 b = с l 2 48 (25)

The conditions of stability (20) and (21) are formulated as

u 2 < − c 3 b ; u 2 < − c 6 b (26)

From these inequalities essential is the second that marks the part of the curve to the left side of maximum as stable one and the inequality (22) is a consequence of precedents. The coordinates of maximum in

u M 2 = − c 6 b = l 2 24 ; k T M = c 2 6 b = − с l 2 24 ; s M 2 = − c 2 b = l 2 8 (27)

Excluding the displacement and reducing we receive the conditions of stability for non symmetrical structure in the form

− c − 3 b u 2 > 0 (28)

− 2 c − 3 b u 2 > 0 (29)

( − c − 3 b u 2 ) ( − c − 6 b u 2 ) > 0 (30)

So the inequality shown in formula (31)

u 2 < − c 6 b (31)

denotes the stable part of the curve. It is possible to express the temperature of maximum T M through cell dimension l, which is affected to the action of pressure. This temperature is the highest temperature, which can exist in considered structure without transition in the other state (it is possible a transition in the liquid state). In

According to diagram it can be concluded that when c is zero the liquid state exists at zero temperature. But such result is a consequence of simplifications in choice of model, where some potential barriers are not taken into account, but can be caused by defects of the structure. Their presence may be taken into account phenomenological through introducing additional terms in the energy and instead of (12) now will be written more general expression

U = U * + c + ε 2 x 2 + b + β 4 x 4 , ε > 0 , β < 0 (32)

without any relation between new constants ε , β . The following calculations will be the same, and for temperature of transition may be taken the expression (25), where for symmetrical state c > 0

k T M = − ( c + ε ) 2 12 ( b + β ) = l 2 48 ( c + ε ) 2 c − β l 2 4 (33)

The dependence of k T M from c is reproduced by means of mathematical function y(x) of the form

y = ( x + p ) 2 x + q , p > 0 , q > 0 (34)

graph of which is shown in

and may be equal to the corresponding value for non symmetric state through the choice of new constants. For non symmetrical state (c < 0) according to (27), we receive (new constants may have now opposite sign)

k T M = ( c + ε ) 2 6 ( b + β ) = l 2 24 ( − c − ε ) 2 − c + β l 2 4 (35)

Very schematic diagram with marked regions of diamond, graphite and liquid carbon existence is represented in

their mutual interaction is week, so they glide relative one another like molecules. But graphite in relation to liquid carbon behaves like a crystalline corps, which temperature of melting falls with increase of pressure. Graphite is a soft material and may be changed under the action of pressure. It exist an exploration, where graphite was exposed to γ radiation before utilization for diamond synthesis [

The result of calculations on the basis of dynamical theory of phase transitions is the phase diagram temperature-pressure (or cell dimensions) for elaborated here cubic model of Frenkel. Comparison with diagrams, constructed on the basis of numerous experimental investigations of phase transition graphite-diamond shows some resemblance between calculated and experimental diagrams. So, the model of Frenkel can serve not only as illustration possibility of phase transitions, but can be used in researches. It can be supposed that a combination of such models in linear chain can give possibility to come near to such phenomenon as rupture and some other anharmonic effects.

The author declares no conflicts of interest regarding the publication of this paper.

Kozlovskiy, V.Kh. (2018) Model of Cubic Cell for Description of Some Phase Transitions in Crystals. World Journal of Condensed Matter Physics, 8, 162-170. https://doi.org/10.4236/wjcmp.2018.84011