**9. The self-consistent correspondence of the potential and the uniaxial hard ellipsoid model**

Inside the twin boundary, the arising anisotropic atomic potential transforms an atomic probability isosurface from sphere to ellipsoid. Let us introduce the hard ellipsoid model as analogue of the hard sphere model. Then coefficients' local values for the potential can be found inside TB. We suppose that the twin boundary does not change symmetry and positions of the atomic centers inside a shifting plane. So, the atomic plane A keeps the atomic centers' coordinates Eq. (15) under shifting (see **Figure 1c, d**). In the shifting neighbor atomic plane B, the atomic isosurface equation is defined by Eq. (22). Then for the shifting atomic plane B, the atomic (ellipsoids) center moves over the great ellipsoidal surfaces:

$$\begin{aligned} \left(\frac{\varkappa}{2a\_1}\right)^2 + \left(\frac{y \pm R\_0}{2c\_1}\right)^2 + \left(\frac{z}{2c\_1}\right)^2 &= \mathbf{1};\\ \left(\frac{\varkappa \pm R\_0\sqrt{3}}{2a\_1}\right)^2 + \left(\frac{y}{2c\_1}\right)^2 + \left(\frac{z}{2c\_1}\right)^2 &= \mathbf{1};\end{aligned} \tag{44}$$

the interaction potential: the equilibrium displacement and the potential barrier height decrease (see **Figure 3**). However, the potential barrier height decreases much faster. The resulting evolution of the potential Eq. (46) is shown in **Figure 4**.

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

Thus, the coefficients of the potential (1) for the shift in the direction 0*x* reduce *k*<sup>21</sup> >*k*21�1ð Þ*q* and *k*<sup>41</sup> > *k*41�1ð Þ*q* . It means softening of the potential in the direction of the plane shuffle. The correspondence between the hard ellipsoid model and the atomic microscopic potential Eqs. (4), (8), and (46) is shown in **Figure 3**. Elliptical

*Uan*2ð Þ¼ **r** *Uan*2ð Þþ *y; z Up*2ð Þþ *x Upn*2ð Þ *x; ξ* ;

4

*Comparison of the hard ellipsoids model and the atomic microscopic potential. The red double-well curve shows the potential as a function of ξ*<sup>0</sup>�<sup>1</sup> *and h*<sup>1</sup>�<sup>1</sup>*. Small solid red ellipsoids show atomic isosurfaces at κ*0*. Big dot black ellipsoids show the cross sections of the surfaces Eq. (44) at y* ¼ 0 *and quantum parameter values*

*The cross sections of the potential density according to Eqs. (46), (47), and (39). The quantum boundary has lower potential peak and shorter distance between shallower wells ξ*<sup>0</sup>�<sup>1</sup> ð Þ *as q grows (0, 0.2, 0.4, 0.6). The*

<sup>4</sup> � *<sup>k</sup>*<sup>21</sup>�<sup>1</sup>ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*

1 2 *kye*2*y*<sup>2</sup>

deformation of the probability isosurface leads to the transformation of the

*kxe*2*z*<sup>2</sup> <sup>þ</sup>

*ξ*4 01

; *Up*2ð Þ¼ *x*

1 2 *kp*2*x*<sup>2</sup> ;

2 <sup>2</sup> ;

*:* (47)

(48)

Then from Eqs. (3) and (46), the coefficients of the potential are

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

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

*<sup>k</sup>*21�<sup>1</sup> <sup>¼</sup> <sup>4</sup>*h*1�<sup>1</sup> *ξ*2 01

potential energy of the atom in Eq. (10):

*(a) q = 0, (b) q = 0.2, (c) q = 0.4, and (d) q = 0.6.*

*barrier in the middle of wall (TB) decreases.*

**Figure 3.**

**Figure 4.**

**41**

*Uan*2ð Þ¼ *y; z*

1 2

*Upn*2ð Þ¼ *<sup>x</sup>; <sup>ξ</sup> <sup>k</sup>*<sup>41</sup>�<sup>1</sup>ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*

where the equilibrium and saddle points for an atom are located. Only four ellipsoids with centers 0ð Þ *;* �*R*0*;* 0 and �*R*<sup>0</sup> ffiffiffi <sup>3</sup> <sup>p</sup> *;* <sup>0</sup>*;* <sup>0</sup> � � are described. Axis 0*<sup>x</sup>* is directed along the shift (see **Figure 1c, d**).

Relations Eq. (26), (27), and (35) define the ellipsoid's semiaxes as function of *R*0*, q*:

$$\begin{aligned} a\_1^2 &= \frac{N\_p}{\lambda\_X}; \ q\_1 = 1 - \frac{\lambda\_X}{\lambda};\\ b\_1^2 &= c\_1^2 = R\_0^2 \chi\_1(q\_1) \ ; \ \chi\_1(q\_1) = 1 + \frac{1}{\kappa\_0^2} \ln \sqrt{1 - q\_1}. \end{aligned} \tag{45}$$

Accounting these relations and condition *y* ¼ 0 (see **Figure 1**), we obtain solution for the equation system Eq. (44) and the equilibrium point coordinates for the atom of the plane B. So, in the hard ellipsoid model, we find the microscopic parameters Eqs. (2) and (3) of the atomic potential:

$$\begin{aligned} \left| \xi\_{0-1} = \left| \mathbf{x}\_{1-\epsilon} \right| &= \xi\_0 \frac{2 - 3q}{2(1 - q)}; \quad h\_{1-1} \sim \frac{1}{2} k\_{ii} \left( \mathbf{z}\_{1-\epsilon} - \mathbf{z}\_{1-\epsilon} \right)^2 = \\\ &= \frac{3h\_{1-R}}{\left( 3 - \sqrt{8} \right)^2} \left[ \sqrt{4\gamma\_1(q) - 1} - \sqrt{4\gamma\_1(q) - 1 - \frac{1}{12} \frac{\left( 2 - 3q \right)^2}{1 - q}} \right]^2 \end{aligned} \tag{46}$$

where *h*<sup>1</sup>�*<sup>R</sup>* is defined in Eq. (19). These results are valid in the range 0 ≤*q*≤2*=*3. At *<sup>q</sup>*<sup>1</sup> ! <sup>2</sup>*=*3 we have *<sup>ξ</sup>*<sup>0</sup>�<sup>1</sup>*, h*<sup>1</sup>�<sup>1</sup> ! 0 and semiaxis relation *<sup>a</sup>*1*=c*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffi <sup>3</sup> <sup>p</sup> . At *<sup>q</sup>* <sup>¼</sup> <sup>2</sup>*=*<sup>3</sup> the hard ellipsoid model needs transition in another state (see [14]). Therefore, inside TB, the change of the atomic wave function leads to the following change of

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

the interaction potential: the equilibrium displacement and the potential barrier height decrease (see **Figure 3**). However, the potential barrier height decreases much faster. The resulting evolution of the potential Eq. (46) is shown in **Figure 4**. Then from Eqs. (3) and (46), the coefficients of the potential are

$$k\_{21-1} = \frac{4h\_{1-1}}{\xi\_{01}^2}; \quad k\_{41-1} = \frac{4h\_{1-1}}{\xi\_{01}^4}.\tag{47}$$

Thus, the coefficients of the potential (1) for the shift in the direction 0*x* reduce *k*<sup>21</sup> >*k*21�1ð Þ*q* and *k*<sup>41</sup> > *k*41�1ð Þ*q* . It means softening of the potential in the direction of the plane shuffle. The correspondence between the hard ellipsoid model and the atomic microscopic potential Eqs. (4), (8), and (46) is shown in **Figure 3**. Elliptical deformation of the probability isosurface leads to the transformation of the potential energy of the atom in Eq. (10):

$$U\_{an2}(\mathbf{r}) = U\_{an2}(\mathcal{y}, \mathbf{z}) + U\_{p2}(\mathbf{x}) + U\_{pn2}(\mathbf{x}, \xi) \quad ;$$

$$U\_{an2}(\mathcal{y}, \mathbf{z}) = \frac{1}{2}k\_{\mathbf{x}c2}\mathbf{z}^2 + \frac{1}{2}k\_{\mathbf{y}c2}\mathbf{y}^2; \ U\_{p2}(\mathbf{x}) = \frac{1}{2}k\_{p2}\mathbf{x}^2; \tag{48}$$

$$U\_{pn2}(\mathbf{x}, \xi) = \frac{k\_{41-1}(\xi - \mathbf{x})^4}{4} - \frac{k\_{21-1}(\xi - \mathbf{x})^2}{2};$$

**Figure 3.**

We can make the following conclusion. The softening of the effective atomic potential is anisotropic inside the twin boundary which increases the exchange integral and tunneling probability in the selected shear direction. As a result the

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

**9. The self-consistent correspondence of the potential and the uniaxial**

Inside the twin boundary, the arising anisotropic atomic potential transforms an atomic probability isosurface from sphere to ellipsoid. Let us introduce the hard ellipsoid model as analogue of the hard sphere model. Then coefficients' local values for the potential can be found inside TB. We suppose that the twin boundary does not change symmetry and positions of the atomic centers inside a shifting plane. So, the atomic plane A keeps the atomic centers' coordinates Eq. (15) under shifting (see **Figure 1c, d**). In the shifting neighbor atomic plane B, the atomic isosurface equation is defined by Eq. (22). Then for the shifting atomic plane B, the atomic

quantum diffusion along the boundary plane increases.

(ellipsoids) center moves over the great ellipsoidal surfaces:

ffiffiffi 3 p

; *<sup>q</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> � *<sup>λ</sup><sup>X</sup>*

<sup>0</sup>*γ*<sup>1</sup> *q*<sup>1</sup>

<sup>þ</sup> *<sup>y</sup>* � *<sup>R</sup>*<sup>0</sup> 2*c*<sup>1</sup> � �<sup>2</sup>

> <sup>þ</sup> *<sup>y</sup>* 2*c*<sup>1</sup> � �<sup>2</sup>

where the equilibrium and saddle points for an atom are located. Only four

Relations Eq. (26), (27), and (35) define the ellipsoid's semiaxes as function of

Accounting these relations and condition *y* ¼ 0 (see **Figure 1**), we obtain solution for the equation system Eq. (44) and the equilibrium point coordinates for the atom of the plane B. So, in the hard ellipsoid model, we find the microscopic

2

where *h*<sup>1</sup>�*<sup>R</sup>* is defined in Eq. (19). These results are valid in the range 0 ≤*q*≤2*=*3.

*λ* ;

� � ; *<sup>γ</sup>*<sup>1</sup> *<sup>q</sup>*<sup>1</sup>

; *<sup>h</sup>*<sup>1</sup>�<sup>1</sup> � <sup>1</sup>

�

the hard ellipsoid model needs transition in another state (see [14]). Therefore, inside TB, the change of the atomic wave function leads to the following change of

þ

þ

ffiffiffi

� � <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

1 *κ*2 0

*kis*ð Þ *<sup>z</sup>*<sup>1</sup>�*<sup>s</sup>* � *<sup>z</sup>*<sup>1</sup>�*<sup>e</sup>* <sup>2</sup> <sup>¼</sup>

<sup>4</sup>*γ*1ð Þ� *<sup>q</sup>* <sup>1</sup> � <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12

ð Þ 2 � 3*q* 2

3 5

<sup>2</sup> (46)

<sup>3</sup> <sup>p</sup> . At *<sup>q</sup>* <sup>¼</sup> <sup>2</sup>*=*<sup>3</sup>

1 � *q*

*z* 2*c*<sup>1</sup> � �<sup>2</sup>

*z* 2*c*<sup>1</sup> � �<sup>2</sup> ¼ 1;

(44)

(45)

¼ 1;

<sup>3</sup> <sup>p</sup> *;* <sup>0</sup>*;* <sup>0</sup> � � are described. Axis 0*<sup>x</sup>* is

ln ffiffiffiffiffiffiffiffiffiffiffiffi 1 � *q*<sup>1</sup> p *:*

*x* 2*a*<sup>1</sup> � �<sup>2</sup>

*x* � *R*<sup>0</sup>

<sup>1</sup> <sup>¼</sup> *<sup>R</sup>*<sup>2</sup>

parameters Eqs. (2) and (3) of the atomic potential:

2 � 3*q* 2 1ð Þ � *q*

q

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4*γ*1ð Þ� *q* 1

2 s

At *<sup>q</sup>*<sup>1</sup> ! <sup>2</sup>*=*3 we have *<sup>ξ</sup>*<sup>0</sup>�<sup>1</sup>*, h*<sup>1</sup>�<sup>1</sup> ! 0 and semiaxis relation *<sup>a</sup>*1*=c*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffi

ellipsoids with centers 0ð Þ *;* �*R*0*;* 0 and �*R*<sup>0</sup>

directed along the shift (see **Figure 1c, d**).

*a*2 <sup>1</sup> <sup>¼</sup> *Np λX*

*b*2 <sup>1</sup> <sup>¼</sup> *<sup>c</sup>*<sup>2</sup>

*ξ*<sup>0</sup>�<sup>1</sup> ¼ ∣*x*<sup>1</sup>�*<sup>e</sup>*∣ ¼ *ξ*<sup>0</sup>

<sup>¼</sup> <sup>3</sup>*h*<sup>1</sup>�*<sup>R</sup>* <sup>3</sup> � ffiffiffi <sup>8</sup> � � <sup>p</sup> <sup>2</sup>

*R*0*, q*:

**40**

2*a*<sup>1</sup> � �<sup>2</sup>

**hard ellipsoid model**

*Comparison of the hard ellipsoids model and the atomic microscopic potential. The red double-well curve shows the potential as a function of ξ*<sup>0</sup>�<sup>1</sup> *and h*<sup>1</sup>�<sup>1</sup>*. Small solid red ellipsoids show atomic isosurfaces at κ*0*. Big dot black ellipsoids show the cross sections of the surfaces Eq. (44) at y* ¼ 0 *and quantum parameter values (a) q = 0, (b) q = 0.2, (c) q = 0.4, and (d) q = 0.6.*

#### **Figure 4.**

*The cross sections of the potential density according to Eqs. (46), (47), and (39). The quantum boundary has lower potential peak and shorter distance between shallower wells ξ*<sup>0</sup>�<sup>1</sup> ð Þ *as q grows (0, 0.2, 0.4, 0.6). The barrier in the middle of wall (TB) decreases.*

All terms are changed in the potential Eq. (48) in comparison with Eq. (10). Isotropy is broken in atomic planes A or B-C due to superposition of the ellipsoids in the shear direction.

For the classical and quantum cases, the free energy density relation Eq. (4) was analyzed analytically in [14]. It was shown that the classical and quantum boundaries have different properties. In particular, from **Figure 5**, it is qualitatively clear why the classical and quantum boundaries have different potential barrier and energy density. In TB both the width and the height of the barrier decrease to zero, according to Eqs. (46) and (47) (see **Figure 4**). In **Figure 3**, they are shown as higher smooth curves. Simultaneously the space width of the boundary *lT* <sup>¼</sup> <sup>1</sup>*=ξ*<sup>0</sup> ð Þ ffiffiffiffiffi <sup>2</sup>*<sup>α</sup>* <sup>p</sup> *<sup>=</sup>k*<sup>4</sup> grows by Eq. (6). The dependence *lT*ð Þ*<sup>q</sup>* causes further widening of region with *q* ! *qmax* and a minimal barrier height.

To estimate the energy of the twin boundary (stacking fault) from Eq. (7), we must know the following parameters: *α, k*2*, k*<sup>4</sup> or *lT, h*.

The characteristic width (half width) of TB Eq. (6) *lT* ≃1*:*5 nm was obtained by molecular dynamic method in [16]. We estimate the dispersion parameter *α* by comparing the differential equations for the transverse sound and shuffling waves:

$$
\rho^\* \frac{\partial^2 \xi}{\partial t^2} - a \frac{\partial^2 \xi}{\partial x^2} = 0; \quad \frac{\partial^2 \iota}{\partial t^2} - s^2 \frac{\partial^2 \iota}{\partial x^2} = 0; \tag{49}
$$

; (50)

The surface energy density calculated here for the classical model can be compared with the value *WSFex* <sup>¼</sup> ð Þ <sup>0</sup>*:*<sup>07</sup> � <sup>0</sup>*:*<sup>02</sup> mJ*=*m2 found in the optical experiments

Above, we have predicted the local reduction of the barrier height *h* and a local increase in the width *lT* of the boundary in the quantum description of the twin boundary (stacking fault). In general for the defect, the surface energy density value *WT* � *lTef hTef* in Eq. (50) can be close to the classical case. In different experiments and theoretical estimates, a wide variation of the values may be caused

We have discussed the change of zero vibrations of atoms in the twin boundary

The difference between quantum statistics of the isotopes should address deeper and more delicate quantum properties of the defects. We note briefly below only the most striking manifestation of different statistics and problems arising in this

The quantum self-consistent treatment to twin boundary (stacking faults), pro-

In the hcp phase, the potential of an atom, created by its neighbors, has spherical symmetry (initial approximation). In the hcp phase, an atom is an isotropic quantum oscillator. In the twin boundary, an atom is an anisotropic quantum oscillator. It is shown that in the twin boundary, the potential of the atom is softer in the

The quantum parameter *qq* and its generalization and the isosurface deformation parameter *q* are introduced. These parameters have simple and visual meaning: *q* equals to the square of the eccentricity of the cross section of the probability density ellipsoid (or the thermal ellipsoid). We have shown that parameter *q* is associated with de Boer parameter, the fundamental characteristic of quantum crystal, and anisotropy in the boundary. Evaluations for different materials show that the isosurface deformation parameter *q* can achieve values 0*:*2÷1 (see **Table 2**). Meanwhile at *q* ¼ 2*=*3 the structure instability takes place in the system of the atomic ellipsoids. From this point of view, the properties of TB in lithium are especially

The overlap of the atomic wave functions and the exchange integral value can be

described in terms of the quantum parameter *q*. Inside the twin boundary, the quantum diffusion increases which was observed in the phase boundary (see experiment [12]). The estimation Eq. (50) of the defect energy is in good agreement with experiment [13]. We have shown that the quantum deformation of atoms leads to the space broadening of the twin boundary and to its energy decreasing. In conclusion we note that local oscillations spectra of the order parameter in

small values of the perturbations, dynamical differential equations (reduced to

thermal description. The relation between discrete models of hard spheres and continuum interatomic potential is used as a sample for a similar relationship in the case of the hard ellipsoid models. As we move deeper into the defect, the transition

He, is developed here for metals and their quantum and

magnitude for all parameters of the twin boundary (stacking fault). The qualitative

He we can expect the same order of

He, apparently,

He were investigated in [9]. For

He and <sup>3</sup>

at 0.2 K [13].

regard.

**43**

by variations of temperatures and pressures.

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

(stacking fault) and the related effects. For <sup>3</sup>

cannot be obtained in the proposed model.

**10. Discussion and conclusion**

from one model to another is accomplished.

direction of shuffle of the atomic planes.

interesting because the parameter achieves high value *q* ! 1.

different models of coherent bcc-hcp boundary in <sup>4</sup>

posed in [14] for solid <sup>4</sup>

difference between the pure hcp crystals of isotopes <sup>4</sup>

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

where *<sup>ρ</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>ρ</sup>=*2 is the effective density of the oscillating shuffled subsystem, *ρ* is the density of helium-4, *ξ* is the shuffling order parameter, *u* is macroscopic displacement, and *<sup>s</sup>* <sup>¼</sup> ffiffiffiffiffiffiffi *<sup>C</sup>*<sup>44</sup> <sup>p</sup> *<sup>=</sup>p*<sup>≃</sup> 255m*=*s is the transverse sound velocity in the shuffle direction (*Oz* axis). The velocities of transverse sound and shuffling wave have close values. So the dispersion parameter is *<sup>α</sup>* <sup>≃</sup>*C*44*=*<sup>2</sup> <sup>¼</sup> <sup>6</sup>*:*<sup>2</sup> � <sup>10</sup><sup>6</sup>*J=m*<sup>3</sup> where value of module *C*<sup>44</sup> is given in **Table 1**.

According to relation Eq. (5), it is possible to estimate the parameter of the thermodynamic potential *k*<sup>2</sup> ¼ 2*α=l* 2 *<sup>T</sup>* <sup>≃</sup> <sup>8</sup>*:*<sup>27</sup> � <sup>10</sup>24J*=*m5. As follows from Eq. (2) to evaluate the parameter *k*<sup>4</sup> of the potential, it is necessary to know the maximum displacement of the atom Eq. (17) *<sup>ξ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>R</sup>*0*<sup>=</sup>* ffiffiffi <sup>3</sup> <sup>p</sup> <sup>≃</sup>1*:*<sup>17</sup> � <sup>10</sup>�10m. Here atomic radius is related to atomic volume: *Vm=NA* <sup>≃</sup>ð Þ <sup>4</sup>*=*<sup>3</sup> *<sup>π</sup>R*<sup>3</sup> 0. Then

*k*<sup>4</sup> ¼ 2*α= ξ*<sup>0</sup> ð Þ *lT* <sup>2</sup> <sup>¼</sup> <sup>6</sup>*:*<sup>04</sup> � <sup>10</sup>44J*=*m7. So, for the classical model of the twin boundary (stacking fault), it is possible to estimate bulk density of the barrier height *h* and the surface energy density *WT* according to Eqs. (3) and (7):

; *WT* <sup>¼</sup> <sup>4</sup>

#### **Figure 5.**

*<sup>h</sup>* <sup>¼</sup> *<sup>k</sup>*2*ξ*<sup>2</sup> 0

*The smooth double-well potential according to Eqs. (10) and (11). Instead of a set of parabolas in Figure 2, we see only ones at the bottom and the peaks of the potential and their quantum levels. The relationship between the barriers for the atomic displacement in the classical hc and quantum hq boundaries is hc* >*hq.*

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

The surface energy density calculated here for the classical model can be compared with the value *WSFex* <sup>¼</sup> ð Þ <sup>0</sup>*:*<sup>07</sup> � <sup>0</sup>*:*<sup>02</sup> mJ*=*m2 found in the optical experiments at 0.2 K [13].

Above, we have predicted the local reduction of the barrier height *h* and a local increase in the width *lT* of the boundary in the quantum description of the twin boundary (stacking fault). In general for the defect, the surface energy density value *WT* � *lTef hTef* in Eq. (50) can be close to the classical case. In different experiments and theoretical estimates, a wide variation of the values may be caused by variations of temperatures and pressures.

We have discussed the change of zero vibrations of atoms in the twin boundary (stacking fault) and the related effects. For <sup>3</sup> He we can expect the same order of magnitude for all parameters of the twin boundary (stacking fault). The qualitative difference between the pure hcp crystals of isotopes <sup>4</sup> He and <sup>3</sup> He, apparently, cannot be obtained in the proposed model.

The difference between quantum statistics of the isotopes should address deeper and more delicate quantum properties of the defects. We note briefly below only the most striking manifestation of different statistics and problems arising in this regard.

### **10. Discussion and conclusion**

All terms are changed in the potential Eq. (48) in comparison with Eq. (10). Isotropy is broken in atomic planes A or B-C due to superposition of the ellipsoids in

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

For the classical and quantum cases, the free energy density relation Eq. (4) was analyzed analytically in [14]. It was shown that the classical and quantum boundaries have different properties. In particular, from **Figure 5**, it is qualitatively clear why the classical and quantum boundaries have different potential barrier and energy density. In TB both the width and the height of the barrier decrease to zero, according to Eqs. (46) and (47) (see **Figure 4**). In **Figure 3**, they are shown as higher smooth curves. Simultaneously the space width of the boundary

<sup>2</sup>*<sup>α</sup>* <sup>p</sup> *<sup>=</sup>k*<sup>4</sup> grows by Eq. (6). The dependence *lT*ð Þ*<sup>q</sup>* causes further widening

To estimate the energy of the twin boundary (stacking fault) from Eq. (7), we

The characteristic width (half width) of TB Eq. (6) *lT* ≃1*:*5 nm was obtained by molecular dynamic method in [16]. We estimate the dispersion parameter *α* by comparing the differential equations for the transverse sound and shuffling waves:

where *<sup>ρ</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>ρ</sup>=*2 is the effective density of the oscillating shuffled subsystem, *ρ* is the density of helium-4, *ξ* is the shuffling order parameter, *u* is macroscopic

shuffle direction (*Oz* axis). The velocities of transverse sound and shuffling wave have close values. So the dispersion parameter is *<sup>α</sup>* <sup>≃</sup>*C*44*=*<sup>2</sup> <sup>¼</sup> <sup>6</sup>*:*<sup>2</sup> � <sup>10</sup><sup>6</sup>*J=m*<sup>3</sup> where

According to relation Eq. (5), it is possible to estimate the parameter of the

evaluate the parameter *k*<sup>4</sup> of the potential, it is necessary to know the maximum

(stacking fault), it is possible to estimate bulk density of the barrier height *h* and the

; *WT* <sup>¼</sup> <sup>4</sup>

*The smooth double-well potential according to Eqs. (10) and (11). Instead of a set of parabolas in Figure 2, we see only ones at the bottom and the peaks of the potential and their quantum levels. The relationship between the*

*barriers for the atomic displacement in the classical hc and quantum hq boundaries is hc* >*hq.*

2

*∂*<sup>2</sup><sup>U</sup> *<sup>∂</sup>t*<sup>2</sup> � *<sup>s</sup>*

<sup>2</sup> *∂*<sup>2</sup><sup>U</sup>

<sup>p</sup> *<sup>=</sup>p*<sup>≃</sup> 255m*=*s is the transverse sound velocity in the

0. Then

<sup>2</sup> <sup>¼</sup> <sup>6</sup>*:*<sup>04</sup> � <sup>10</sup>44J*=*m7. So, for the classical model of the twin boundary

*<sup>T</sup>* <sup>≃</sup> <sup>8</sup>*:*<sup>27</sup> � <sup>10</sup>24J*=*m5. As follows from Eq. (2) to

<sup>3</sup> *lTh*≃0*:*057*mJ=m*<sup>2</sup>

<sup>3</sup> <sup>p</sup> <sup>≃</sup>1*:*<sup>17</sup> � <sup>10</sup>�10m. Here atomic radius

; (50)

*<sup>∂</sup>z*<sup>2</sup> <sup>¼</sup> <sup>0</sup>; (49)

*ξ <sup>∂</sup>z*<sup>2</sup> <sup>¼</sup> <sup>0</sup>;

the shear direction.

*lT* <sup>¼</sup> <sup>1</sup>*=ξ*<sup>0</sup> ð Þ ffiffiffiffiffi

displacement, and *<sup>s</sup>* <sup>¼</sup> ffiffiffiffiffiffiffi

*<sup>h</sup>* <sup>¼</sup> *<sup>k</sup>*2*ξ*<sup>2</sup> 0

*k*<sup>4</sup> ¼ 2*α= ξ*<sup>0</sup> ð Þ *lT*

**Figure 5.**

**42**

value of module *C*<sup>44</sup> is given in **Table 1**.

displacement of the atom Eq. (17) *<sup>ξ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>R</sup>*0*<sup>=</sup>* ffiffiffi

is related to atomic volume: *Vm=NA* <sup>≃</sup>ð Þ <sup>4</sup>*=*<sup>3</sup> *<sup>π</sup>R*<sup>3</sup>

surface energy density *WT* according to Eqs. (3) and (7):

<sup>4</sup> <sup>≃</sup>2*:*<sup>83</sup> � <sup>10</sup><sup>4</sup>*J=m*<sup>3</sup>

thermodynamic potential *k*<sup>2</sup> ¼ 2*α=l*

of region with *q* ! *qmax* and a minimal barrier height.

must know the following parameters: *α, k*2*, k*<sup>4</sup> or *lT, h*.

*<sup>ρ</sup>* <sup>∗</sup> *<sup>∂</sup>*<sup>2</sup> *ξ <sup>∂</sup>t*<sup>2</sup> � *<sup>α</sup> <sup>∂</sup>*<sup>2</sup>

*C*<sup>44</sup>

The quantum self-consistent treatment to twin boundary (stacking faults), proposed in [14] for solid <sup>4</sup> He, is developed here for metals and their quantum and thermal description. The relation between discrete models of hard spheres and continuum interatomic potential is used as a sample for a similar relationship in the case of the hard ellipsoid models. As we move deeper into the defect, the transition from one model to another is accomplished.

In the hcp phase, the potential of an atom, created by its neighbors, has spherical symmetry (initial approximation). In the hcp phase, an atom is an isotropic quantum oscillator. In the twin boundary, an atom is an anisotropic quantum oscillator. It is shown that in the twin boundary, the potential of the atom is softer in the direction of shuffle of the atomic planes.

The quantum parameter *qq* and its generalization and the isosurface deformation parameter *q* are introduced. These parameters have simple and visual meaning: *q* equals to the square of the eccentricity of the cross section of the probability density ellipsoid (or the thermal ellipsoid). We have shown that parameter *q* is associated with de Boer parameter, the fundamental characteristic of quantum crystal, and anisotropy in the boundary. Evaluations for different materials show that the isosurface deformation parameter *q* can achieve values 0*:*2÷1 (see **Table 2**). Meanwhile at *q* ¼ 2*=*3 the structure instability takes place in the system of the atomic ellipsoids. From this point of view, the properties of TB in lithium are especially interesting because the parameter achieves high value *q* ! 1.

The overlap of the atomic wave functions and the exchange integral value can be described in terms of the quantum parameter *q*. Inside the twin boundary, the quantum diffusion increases which was observed in the phase boundary (see experiment [12]). The estimation Eq. (50) of the defect energy is in good agreement with experiment [13]. We have shown that the quantum deformation of atoms leads to the space broadening of the twin boundary and to its energy decreasing.

In conclusion we note that local oscillations spectra of the order parameter in different models of coherent bcc-hcp boundary in <sup>4</sup> He were investigated in [9]. For small values of the perturbations, dynamical differential equations (reduced to

#### **Figure 6.**

*Local modes of the order parameter at TB [9]. Dash dot line shows the local potential which has local energy levels 0, 1, and 2 (dash). Solid lines show corresponding local oscillations' shape dependence on normalized coordinate z* <sup>∗</sup> *.*

Schrodinger equations) were obtained and solved. The characteristic frequencies (energy levels) and shape were found and estimated (see **Figure 6**). For the ground state in TB, the local vibration shape can be written as

$$\eta\_0(\boldsymbol{z}^\*) = \frac{A\_0}{\cosh^2 \boldsymbol{z}^\*}; \qquad \boldsymbol{z}^\* = \frac{\boldsymbol{z}}{l\_T}. \tag{51}$$

**Author details**

Victor A. Lykah<sup>1</sup>

**45**

\* and Eugen S. Syrkin<sup>2</sup>

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

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

\*Address all correspondence to: lykahva@yahoo.com

Sciences of Ukraine, Kharkiv, Ukraine

provided the original work is properly cited.

University "Kharkiv Polytechnic Institute", Kharkiv, Ukraine

1 Educational-scientific Institute of Physical Engineering National Technical

2 B.I. Verkin Institute for Low Temperature Physics of National Academy of

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

where *A*<sup>0</sup> is an amplitude. For the local vibration ground state (51) and for the isosurface deformation parameter *q* Eq. (39), both shapes coincide qualitatively. In the limit *q* < < 1, both coincide completely. The local vibration of the order parameter describes a correlated motion of the atomic layers in twin boundary. Meanwhile, the quantum and thermal treatments give probabilistic descriptions of the atomic motion. The results (the found smooth arising of the atomic motion amplitude in TB) give evidence that different probabilistic (quantum and thermal) and dynamic methods lead to qualitatively identical features of the atomic basic state inside TB.

### **Acknowledgements**

This research is supported by the FFI National Academy of Sciences of Ukraine, grant 4/18-H, Ministry of Science and Education of Ukraine under the Projects M05486 (0118U002048).

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