A New Condition of Formation and Stablity of All Crystalline Systems in a Good Agreement with Experimental Data

In this paper we derived a new condition of formation and stability of all crystalline systems and we checked its validity and it is found to be in a good agreement with experimental data. This condition is derived directly from the quantum conditions on the free electron Fermi gas inside the crystal. The new condition relates both the volume of Fermi sphere VF and volume of Brillouin zone VB by the valence electron concentration VEC as ; VF VB = n VEC 2 for all crystalline systems (where n is the number of atoms per lattice point).

The new condition for crystalline phase formation and stability derived by T. El Ashram [1] for a cubic, hexagonal and tetragonal systems is found to be in a good agreement with experimental data.This condition is derived directly from the quantum conditions on the free electron Fermi gas inside the crystal and it is not an assumption.In this condition the ratio of volume of Fermi sphere VF to volume of Brillouin zone VB is related to valence electron concentration VEC as; In the present work we will extend this condition to the other crystalline systems such as orthorhombic, rhombohedral, monoclinic and triclinic.Also we will start by considering Fermi wavevector or the radius of Fermi sphere kF, which is given by   =    .This is a general condition for all crystalline systems, it does not depend on the type of the crystal system.However it depends on n, the number of atoms per lattice point or primitive cell.This condition does not contradict the condition derived previously for cubic, hexagonal and tetragonal systems but can be reduced to it if we take n into account.Therefore the aim of the present work is to check the validity of this condition for some of pure elements with different crystal structure and valencies such as O, Li, Na, Cu, Ag, Al, Pb, Cd, Zn, In, Sn, Bi, Ga and P. The experimental XRD data were obtained from reference [2] and the data for kF were taken from reference [3].

Hexagonal System (Hex. P)
The primitive cell of Hexagonal system is described by the primitive vectors [4]; a1=(a/2, -a /2, 0), a2=(a/2, a /2, 0), and a3=(0, 0, c).Also the volume of the primitive cell VP is given by; VP= a1.a2xa3, substituting we get VP=a 2 c /2.Let us take the Cadmium (Cd) element as an example for this structure.From [2] a =2.9793Å and c =5.6181 Å this gives for VP = 43.1865Å 3 and for VB=5.7436Å -3 .For Cd kF is calculated to be 1.4 Å -1 from which VF =11.4940Å -3 therefore VF/VB =2.001Now let us calculate nVEC/2; for Cd, n =2 and VEC=2 therefore nVEC/2= 2.0 which is in a good agreement with the calculated value for VF/VB.By the same method of calculations we get the same result for Zinc (Zn) element.(see Table 1) D e c e m b e r 2 8 , 2 0 1 5

Volume Centered Tetragonal System (Tet. I)
The primitive cell of the volume centered Tetragonal system is described by the primitive vectors [4]; a1=(-a/2, a/2, c/2), a2=(a/2, -a/2, c/2), and a3=(a/2, a/2, -c/2).Also the volume of the primitive cell VP is given by; VP= a1.a2xa3,by substituting we get for VP =a 2 c/2.Let us take the Tin (white Sn) element as an example for this structure.From [2] a =5.831Å and c =3.182 Å this gives for VP = 54.094Å 3 and for VB=4.5854Å - .For Sn kF is calculated to be 1.62 Å -1 from which VF =17.808Å -3 therefore VF/VB =3.883Now let us calculate nVEC/2; for Sn, n =2 and VEC=4 therefore nVEC/2= 4.0 which is in a good agreement with the calculated value for VF/VB.By the same method of calculations we get the same result for Indium (In) element.(see Table 1)

Triclinic System (Tri. P)
The primitive cell of Triclinic system is described by the primitive vectors [4]  -3 .For P kF is calculated to be 1.775 Å -1 from which VF =23.425Å -3 therefore VF/VB = 60.003Now let us calculate nVEC/2; for P, n =24 and VEC =5 therefore nVEC/2= 60.0 which is in a good agreement with the calculated value for VF/VB.(see Table 1)

CONCLUSION
It is found that the theoretically derived condition is in a very good agreement with the experimental data which confirm the validity of this condition for crystalline phase formation and stability.Therefore we can conclude that the alloy adapts its crystal structure in such a way, the ratio of volume of Fermi sphere VF to volume of Brillouin zone VB is related to valence electron concentration VEC as; , where N is the number of electrons in the volume V.In terms of the volume of primitive cell VP and valence electron concentration VEC we can rewrite kF as   =        , where n is the number of atoms per lattice point.Since the primitive cell contains only one lattice point, this lattice point may contain 1, or 2,.....or n atoms and every atom has VEC of electrons.Now let us calculate the volume of Fermi sphere VF from the equation of kF as the following;