**6. Classic atomic thermal spheres and ellipsoids in hcp phase and the twin boundary**

Inside the perfect hcp phase, an atom is positioned in highly symmetric potential of neighbor atoms (see Eq. (9)) and quantum analogue Eq. (21). For any direction, the average thermal energy of an atom is *kBT=*2 where *kB* and *T* are the Boltzmann constant and temperature.

In isotropic harmonic approximation Eq. (9), the average thermal energy of an atom corresponds to the average potential isosurface (sphere of radius *R*):

$$\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2 = \mathbf{R}^2; \quad \mathbf{R}^2 = \frac{k\_B T}{k\_{\rm is}}; \tag{28}$$

Let us introduce the **isosurface deformation parameter** *q* as a geometric factor which describes the deformation of the atomic sphere Eqs. (22) and (28) into the

where *ε* is the eccentricity of the ellipse. Earlier in the paper [14], we introduced the quantum deformation parameter *qq*. Here we generalize the parameter *qq* to the cases of either quantum or thermal motion of an atom and introduce the isosurface

Now we present the self-consistent scheme of description for the twin boundary. **(0) Zero approximation**. An atom is a hard classic sphere Eq. (32) or quantum

**(1) The first approximation**. An atom is considered as a quantum anisotropic

approximation. In the general case, the ellipsoid parameters and the isosurface

respectively. The long ellipsoids axis is oriented along the shift direction 0*x*:

The further variations of **parameters** Eq. (31) can be obtained in the hard ellipsoid model. The hard ellipsoids have the isosurfaces with the same probability density *ρ*<sup>0</sup> as the hard spheres in the hcp phase, and the isosurface deformation parameter can be obtained. For a vacancy, the nearest neighbors form similar

**(2) The second approximation**. An atom is considered as an anisotropic threeaxis oscillator (the isosurface is three-axis ellipsoid). The first approximation gives the rigidity coefficients of the potential. Different ellipses are formed in

> 0; *ε*<sup>2</sup>

Now all three axes of the atomic ellipsoid are different. The softest potential and

2 *a*2 2

the longest axis *a*<sup>2</sup> are still oriented along the shift direction. The hard ellipsoid model Eq. (35) is used to obtain a new local atomic potential and a new ellipsoid

same way. The second and further steps are more cumbersome and

**8. Atom as anisotropic harmonic oscillator in the boundary, one axis**

In continual description inside the boundary, we have found a change of the atomic potential Eq. (10) with the corresponding rigidity constants. Therefore,

uniaxis oscillator. The potential Eq. (10) has been obtained in zero

deformation parameter are described by Eqs. (26), (27), and (33),

*<sup>b</sup>*<sup>1</sup> <sup>¼</sup> *<sup>c</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>a</sup>*1; *<sup>ρ</sup>* <sup>¼</sup> *<sup>ρ</sup>*0; *<sup>ε</sup>*<sup>2</sup>

the planes *ab* and *ac*, and their eccentricities equal

*<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *qb*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>b</sup>*<sup>2</sup>

*R*<sup>0</sup> ¼ *a* ¼ *b* ¼ *c*; *ρ* ¼ *ρ*0; *q* ¼ 0*:* (34)

*<sup>c</sup>*<sup>1</sup> <sup>¼</sup> *<sup>q</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> � *<sup>c</sup>*<sup>2</sup>

1 *a*2 1

*<sup>c</sup>*<sup>2</sup> <sup>¼</sup> *qc*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>c</sup>*<sup>2</sup>

th steps qualitatively replicate the previous steps in the

2 *a*2 2 >0*:* (36)

>0*:* (35)

; 0≤*ε*<sup>2</sup> ≤1*:* (33)

*<sup>a</sup>*<sup>2</sup> � *<sup>ε</sup>*<sup>2</sup>

*<sup>q</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>c</sup>*<sup>2</sup>

*Twin Boundary in hcp Crystals: Quantum and Thermal Behavior*

one-axis ellipsoid Eqs. (26) and (30):

*DOI: http://dx.doi.org/10.5772/intechopen.86909*

deformation parameter *q*.

ellipsoids [25].

shape.

**37**

*<sup>b</sup>*<sup>2</sup> 6¼ *<sup>c</sup>*<sup>2</sup> <sup>&</sup>lt; *<sup>a</sup>*2; *<sup>ε</sup>*<sup>2</sup>

(i) The third and further *i*

constants *λ<sup>i</sup>* in Eq. (25) take the following forms:

complicated.

isotropic oscillator:

The general anisotropic potential has form Eq. (10). In anisotropic harmonic case, the potential can be written with corresponding rigidity coefficients as (compare with Eq. (24))

$$\begin{aligned} U\_{anis}(\mathbf{r}) &= \frac{1}{2} \left( k\_X^2 x^2 + k\_y^2 y^2 + k\_x^2 x^2 \right); \\ k\_X &= k\_{xel}; \quad k\_\mathcal{y} = k\_{yel}; \quad k\_x = k\_{xel}. \end{aligned} \tag{29}$$

Then inside of the twin boundary, an atom is in the uniaxial potential of neighboring atoms Eq. (13): *kxel* ¼ *kb* ≤ *kyel* ¼ *kzel* ¼ *kis*.

The motion equation splits also into three independent equivalent equations. The equation of the potential isosurface is ellipsoid with semiaxes *a*≥*b*≥ *c* (compare with Eq. (26)):

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = \mathbf{1};$$

$$a^2 = \frac{k\_B T}{k\_X}; \quad b^2 = \frac{k\_B T}{k\_\gamma}; \quad c^2 = \frac{k\_B T}{k\_x}.\tag{30}$$

Thus, the relation Eq. (30) describes the atomic potential isosurfaces in the anisotropic case, i.e., inside TB. In the limit case *ki* ¼ *kis*, it corresponds to the isotropic case, i.e., hcp phase Eq. (28). The thermal potential isosurfaces (ellipsoids) have to be in order less than the quantum atomic spheres and ellipsoids normalized at *R*0. We emphasize that in this section the average thermal motion of atoms was considered.
