**5. Quantum atomic spheres and ellipsoids in hcp phase and in the twin boundary**

Inside the perfect hcp phase, a <sup>4</sup> He atom is in highly symmetric potential of neighbor atoms. In isotropic harmonic approximation [19, 20], the atomic potential can be presented as [24]

$$U\_{\dot{\nu}}(\mathbf{r}) = \frac{1}{2}m\rho^2 \mathbf{r}^2; \ \lambda = \frac{m\rho}{\hbar}. \tag{21}$$

where *m,* **r** and *λ* are mass, radius vector of <sup>4</sup> He atom and parameter of the quantum oscillator. The potential Eq. (12) gives *<sup>m</sup>ω*<sup>2</sup> <sup>¼</sup> *kis*; *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *kism=*ℏ<sup>2</sup> .

The Schrodinger equation splits into three equivalent independent equations with the constant *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> *<sup>X</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *<sup>y</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *<sup>z</sup>* <sup>¼</sup> <sup>2</sup>*mW=*ℏ<sup>2</sup> where *ki* are wave numbers. The ground state solution [24] has total zero-point energy *<sup>W</sup>*0*is* <sup>¼</sup> <sup>3</sup> <sup>2</sup> ℏ*ω*. In isotropic harmonic approximation, a distribution of probability density *ρ* ¼ j j *ψ*ð Þ *x; y; z* <sup>2</sup> of

*Twin Boundary in hcp Crystals: Quantum and Thermal Behavior DOI: http://dx.doi.org/10.5772/intechopen.86909*

centers of the shifting atomic plane B can move over the following four spherical

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>¼</sup> ð Þ <sup>2</sup>*R*<sup>0</sup>

The equilibrium points for the atom of the shifting neighbor atomic plane B can

<sup>3</sup> <sup>p</sup> ; *yRe* <sup>¼</sup> <sup>0</sup>; *zRe* <sup>¼</sup> *<sup>R</sup>*<sup>0</sup>

Signs – and + in *xRe* describe positions B and C in plane B, respectively. From the

2

*gkis*ð Þ *zRs* � *zRe*

*xRs* ¼ 0; *yRs* ¼ 0; *zRs* ¼ *R*<sup>0</sup>

where *h*<sup>1</sup>�*<sup>R</sup>* is the potential barrier between B to C position (see **Figure 1**). Coefficient *g* � 1 evaluates the quasielastic energy. In the middle of TB, the neighbor number is 4, which is less than 6 once inside the phase. This is a microscopic

For the hard sphere model, the substitution of relations (19) into Eqs. (3) and

For comparison, Eq. (11) allows us to find the rigidity coefficients in the phase

neighbor atoms. In isotropic harmonic approximation [19, 20], the atomic potential

The Schrodinger equation splits into three equivalent independent equations

; *<sup>k</sup>*<sup>41</sup>�*<sup>R</sup>* <sup>¼</sup> <sup>4</sup>*h*<sup>1</sup>�*<sup>R</sup>*

; *<sup>λ</sup>* <sup>¼</sup> *<sup>m</sup><sup>ω</sup>*

*ξ*4 0*R*

He atom is in highly symmetric potential of

*<sup>z</sup>* <sup>¼</sup> <sup>2</sup>*mW=*ℏ<sup>2</sup> where *ki* are wave numbers. The

<sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>¼</sup> ð Þ <sup>2</sup>*R*<sup>0</sup>

2 ;

> 2 ;

> > ffiffiffi 8 3 r

ffiffiffi 3 *:* (17)

<sup>p</sup> *:* (18)

<sup>2</sup> (19)

*:* (20)

<sup>ℏ</sup> *:* (21)

.

<sup>2</sup> of

<sup>2</sup> ℏ*ω*. In isotropic

He atom and parameter of the

(16)

*<sup>x</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>y</sup>* � *<sup>R</sup>*<sup>0</sup>

ffiffiffi <sup>3</sup> � � <sup>p</sup> <sup>2</sup>

*x* � *R*<sup>0</sup>

be found from the geometry of the system (Eq. (16) at y ¼ 0):

1 ffiffiffi

first Eq. (16), the saddle point coordinates for an atom of plane B are

For the hard sphere model, the microscopic parameters *ξ*0*, h*<sup>1</sup> are

*<sup>ξ</sup>*0*<sup>R</sup>* <sup>¼</sup> <sup>∣</sup>*xRe*∣; *<sup>h</sup>*<sup>1</sup>�*<sup>R</sup>* <sup>¼</sup> <sup>1</sup>

(8) gives the parameters of the microscopic interatomic potential:

*<sup>k</sup>*<sup>21</sup>�*<sup>R</sup>* <sup>¼</sup> <sup>4</sup>*h*<sup>1</sup>�*<sup>R</sup> ξ*2 0*R*

**5. Quantum atomic spheres and ellipsoids in hcp phase**

*Uis*ð Þ¼ **r**

*<sup>y</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

ground state solution [24] has total zero-point energy *<sup>W</sup>*0*is* <sup>¼</sup> <sup>3</sup>

where *m,* **r** and *λ* are mass, radius vector of <sup>4</sup>

*<sup>X</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

1 2 *mω*<sup>2</sup> **r**2

quantum oscillator. The potential Eq. (12) gives *<sup>m</sup>ω*<sup>2</sup> <sup>¼</sup> *kis*; *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *kism=*ℏ<sup>2</sup>

harmonic approximation, a distribution of probability density *ρ* ¼ j j *ψ*ð Þ *x; y; z*

*xRe* ¼ �*R*<sup>0</sup>

reason for the quasielastic energy behavior.

*kis* and in the middle of the boundary *kb*ð Þ 0 .

**and in the twin boundary**

can be presented as [24]

with the constant *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

**34**

Inside the perfect hcp phase, a <sup>4</sup>

surfaces:

helium atom has spherical symmetry. Hence, the equation of probability isosurface (sphere of radius *R*) is

$$\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2 = \mathbf{R}^2; \quad \mathbf{R}^2 = \frac{\mathbf{N}\_{\rho is}}{\lambda}; \quad \mathbf{N}\_{pis} = \ln\sqrt{\frac{\lambda^3}{\rho^2 \pi^3}}.\tag{22}$$

The probability density at a distance of *R*<sup>0</sup> that equals to the radius of the atom in the hcp phase (half the distance between the centers of neighboring atoms in the crystal) is

$$\rho\_0 = \sqrt{\frac{\lambda^3}{\pi^3}} \exp\left(-\kappa\_0^2\right) \; ; \; N\_{\rho \dot{m}}(R\_0) \equiv \kappa\_0^2 = R\_0^2 \lambda. \tag{23}$$

Here we have introduced the dimensionless parameter *κ*<sup>0</sup> that is important for further consideration. This parameter is proportional to the atomic radius *κ*<sup>0</sup> � *R*<sup>0</sup> and depends on the isotropic rigidity of the atomic lattice *<sup>κ</sup>*<sup>0</sup> � *<sup>λ</sup>*<sup>1</sup>*=*<sup>2</sup> � *<sup>k</sup>* 1*=*4 *is* . In respect to a huge change in the volume of solid helium [1], the parameter *κ*<sup>0</sup> can vary widely.

An anisotropic harmonic potential can be written as [24]

$$\begin{aligned} U\_{\text{anti}}(\mathbf{r}) &= \frac{1}{2}m \left( \alpha\_X^2 \mathbf{x}^2 + \alpha\_y^2 \mathbf{y}^2 + \alpha\_x^2 \mathbf{z}^2 \right); \\ \lambda\_X &= \frac{m\alpha\_X}{\hbar}; \quad \lambda\_\mathbf{y} = \frac{m\alpha\_\mathbf{0}}{\hbar}; \quad \lambda\_x = \frac{m\alpha\_x}{\hbar}. \end{aligned} \tag{24}$$

The parameters *λ<sup>i</sup>* are related to the rigidity coefficients:

$$
\lambda\_X^2 = \frac{m}{\hbar^2} k\_{\text{xel}}; \quad \lambda\_y^2 = \frac{m}{\hbar^2} k\_{\text{yl}}; \quad \lambda\_x^2 = \frac{m}{\hbar^2} k\_{\text{xel}}.\tag{25}
$$

In the hcp phase, an anisotropic harmonic approximation is more adequate. Then the rigidity coefficients satisfy inequality *kxel* ¼ *kyel* ¼ *kis* < *kzel*. If we use isotropic harmonic approximation in the hcp phase, then inside of the twin boundary, an atom <sup>4</sup> He is in a uniaxial potential of neighboring atoms of Eq. (13): *kxel* ¼ *kb* ≤*kyel* ¼ *kzel* ¼ *kis*.

The equation splits also into three independent equations with known solutions [24]. Inside TB for the ground state, the distribution of the probability density of the helium atom loses its spherical symmetry. The probability isosurface is ellipsoid with semiaxes *a*≥*b*≥*c*:

$$a^2 = \frac{N\_\rho}{\lambda\_X}; \quad b^2 = \frac{N\_\rho}{\lambda\_0}; \quad c^2 = \frac{N\_\rho}{\lambda\_x}; \quad N\_p = \ln\sqrt{\frac{\lambda\_X \lambda\_0 \lambda\_x}{\rho^2 \pi^3}}.\tag{26}$$

Parameter *N<sup>ρ</sup>* describes the probability density. If the probability density equals *ρ*<sup>0</sup> at the atomic radius *R*<sup>0</sup> in the hcp phase Eq. (23), then *N*ð Þ*ρ* takes the following value:

$$N\_{\rho\_0} = \kappa\_0^2 + \ln \sqrt{\frac{\lambda\_X \lambda\_0 \lambda\_x}{\lambda^3}}. \tag{27}$$

Thus, the relations Eqs. (26) and (27) describe the probability density isosurfaces to find an atom in the anisotropic case. On appropriate limit *λ<sup>i</sup>* ¼ *λ*, these relations describe the isotropic case.
