**2. Model of the twin boundary**

In the hcp phase of crystal <sup>4</sup> He, we consider the twin boundary under transition from the close packing layers ABAB … (see **Figure 1a,b**) to the close packaging ACAC …. The twin boundary (TB) corresponds to stacking faults (SF). The atomic plane A creates different positions (potential wells) B and C for neighbor layers (see **Figure 1a,b**).

The twin boundary was researched in works [4, 5] where the triple-well thermodynamic potential was used. Far from the bcc-hcp transition, the doublewell free energy can be applied:

$$F(\xi) = \int \left[ \frac{a}{2} \left( \frac{d\xi}{dx} \right)^2 + \frac{k\_4 \xi^4}{4} - \frac{k\_2 \xi^2}{2} \right] dv,\tag{1}$$

*ξmax* ¼ 0; *F ξmax* ð Þ¼ 0;

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

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

*<sup>h</sup>* <sup>¼</sup> <sup>1</sup> *V*

*F*ð Þ¼ *ξ*

twin boundary [9, 15] which has shape

by molecular dynamic method in [16].

**3. Atomic potential in continual description**

Z *α* 2

*dξ dz* � �<sup>2</sup>

*lT* ¼

*WT* ¼

potential Eq. (1) can be transformed into the microscopic ones:

*ξ*<sup>0</sup> and *h*:

further integration:

**31**

*<sup>ξ</sup>min* ¼ �*ξ*0; *<sup>F</sup> <sup>ξ</sup>min* ð Þ¼�*<sup>V</sup> <sup>k</sup>*4*ξ*<sup>4</sup>

0 <sup>4</sup> ; *<sup>ξ</sup>*<sup>0</sup> <sup>¼</sup>

> 0 <sup>4</sup> <sup>¼</sup> *<sup>k</sup>*2*ξ*<sup>2</sup> 0

*<sup>ξ</sup>*<sup>2</sup> � *<sup>ξ</sup>*<sup>2</sup> 0 � �<sup>2</sup> � *<sup>h</sup>*

*lT*

ffiffiffiffiffi 2*α k*4 r

where ∣*ξ*0∣ is the minimum position as displacement between the maximum and minimum positions (*B* and *C* in **Figure 1a, b**). The difference between the maximum and minimum energies gives the height *h* of the potential barrier per unit volume:

*<sup>F</sup> <sup>ξ</sup>max* ð Þ� *<sup>F</sup> <sup>ξ</sup>min* <sup>½</sup> ð Þ� ¼ *<sup>k</sup>*4*ξ*<sup>4</sup>

For further analysis it is convenient to write the free energy Eq. (1) in terms of

" #

þ *h ξ*4 0

The free energy Eq. (4) gives rise to such one-dimensional inhomogeneity as

*<sup>ξ</sup>* ¼ �*ξ*0tanh *<sup>z</sup>*

ffiffiffiffiffi 2*α k*2 r

¼ 1 *ξ*0

where the boundary center is chosen at *z* ¼ 0 and *lT* is the characteristic width of the boundary. The shear dependence on coordinate Eq. (5) can be substituted into relation Eq. (4). The surface energy density of the twin boundary is obtained by

> ffiffiffiffiffiffiffiffiffi 2*αk*<sup>3</sup> 2

¼ 4

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

*ξ*4 0 ;

3*k*<sup>4</sup>

It is expressed through parameters ð Þ *α; k*2*; k*<sup>4</sup> of the microscopic double-well potential or macroscopic parameters ð Þ *lT; h* . The parameters of the thermodynamic

*h*<sup>1</sup> ¼ *hv*1; *k*<sup>21</sup> ¼ *k*2*v*1; *k*<sup>41</sup> ¼ *k*4*v*1*:*

In hcp lattice, one can find the symmetry axes (along 0*z*) of third and sixth orders. In the close-packed layers ð Þ *x; y* , hcp demonstrates isotropic properties of

Here *h*1, *k*21, and *k*<sup>41</sup> are the barrier height Eq. (3), parameters *k*<sup>2</sup> and *k*<sup>4</sup> Eq. (1) normalized per unit cell. These equalities are obtained by multiplying *h* and *k*<sup>2</sup> or *k*<sup>4</sup> to the unit cell volume *v*1. The characteristic width Eq. (6) *lT* ≃1*:*5*nm* was obtained

q

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

ffiffiffiffiffi *k*2 *k*4

*:* (2)

<sup>4</sup> *:* (3)

*dv:* (4)

; (5)

*:* (6)

<sup>3</sup> *lTh:* (7)

(8)

s

where the integration is over the volume *v*, square brackets contain the volume energy density, *z* is a coordinate in the direction of heterogeneity, *α* is a dispersion parameter responsible for the boundary width, and phenomenological parameters *k*4*, k*<sup>2</sup> are positive. In hexagonal lattice, *ξ* is the order parameter which means the relative displacement of the atomic layers between positions B and C (see **Figure 1a,b**). For the homogeneous part of the free energy Eq. (1), the maximum and minima positions are

*(a, b) The close pack of the atomic layers (0001) ABAB… for hcp phase. Layer A is shown by solid lines, and layer B is shown by dotted lines. (a) The view perpendicular to the layers. (b) The view along the layers. Points B and C are atomic equilibrium positions in corresponding layer. (c, d) The change in the close packing of the atomic ellipsoids inside TB under quantum effects is accounted.*

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

The present work is devoted to the development of the self-consistent descrip-

We apply this treatment to quantum and thermal description of twin boundary in

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

from the close packing layers ABAB … (see **Figure 1a,b**) to the close packaging ACAC …. The twin boundary (TB) corresponds to stacking faults (SF). The atomic plane A creates different positions (potential wells) B and C for neighbor layers (see

The twin boundary was researched in works [4, 5] where the triple-well thermodynamic potential was used. Far from the bcc-hcp transition, the double-

> *dξ dz* � �<sup>2</sup>

þ *k*4*ξ*<sup>4</sup> <sup>4</sup> � *<sup>k</sup>*2*ξ*<sup>2</sup> 2

where the integration is over the volume *v*, square brackets contain the volume energy density, *z* is a coordinate in the direction of heterogeneity, *α* is a dispersion parameter responsible for the boundary width, and phenomenological parameters *k*4*, k*<sup>2</sup> are positive. In hexagonal lattice, *ξ* is the order parameter which means the relative displacement of the atomic layers between positions B and C (see **Figure 1a,b**). For the homogeneous part of the free energy Eq. (1), the maximum and minima

*(a, b) The close pack of the atomic layers (0001) ABAB… for hcp phase. Layer A is shown by solid lines, and layer B is shown by dotted lines. (a) The view perpendicular to the layers. (b) The view along the layers. Points B and C are atomic equilibrium positions in corresponding layer. (c, d) The change in the close packing of the*

*atomic ellipsoids inside TB under quantum effects is accounted.*

" #

He atoms in twin boundary proposed in work [14].

He, we consider the twin boundary under transition

*dv,* (1)

tion of quantum behavior of <sup>4</sup>

**2. Model of the twin boundary**

In the hcp phase of crystal <sup>4</sup>

well free energy can be applied:

*F*ð Þ¼ *ξ*

Z *α* 2

some metals.

**Figure 1a,b**).

positions are

**Figure 1.**

**30**

$$\begin{aligned} \xi\_{\text{max}} &= 0; & F(\xi\_{\text{max}}) &= 0; \\ \xi\_{\text{min}} &= \pm \xi\_0; & F(\xi\_{\text{min}}) &= -V \frac{k\_4 \xi\_0^4}{4}; & \xi\_0 &= \sqrt{\frac{k\_2}{k\_4}}. \end{aligned} \tag{2}$$

where ∣*ξ*0∣ is the minimum position as displacement between the maximum and minimum positions (*B* and *C* in **Figure 1a, b**). The difference between the maximum and minimum energies gives the height *h* of the potential barrier per unit volume:

$$h = \frac{1}{V} [F(\xi\_{\text{max}}) - F(\xi\_{\text{min}})] = \frac{k\_4 \xi\_0^4}{4} = \frac{k\_2 \xi\_0^2}{4}.\tag{3}$$

For further analysis it is convenient to write the free energy Eq. (1) in terms of *ξ*<sup>0</sup> and *h*:

$$F(\xi) = \int \left[ \frac{a}{2} \left( \frac{d\xi}{dz} \right)^2 + \frac{h}{\xi\_0^4} \left( \xi^2 - \xi\_0^2 \right)^2 - h \right] dv. \tag{4}$$

The free energy Eq. (4) gives rise to such one-dimensional inhomogeneity as twin boundary [9, 15] which has shape

$$
\xi = \pm \xi\_0 \tanh \frac{z}{l\_T};
\tag{5}
$$

$$l\_T = \sqrt{\frac{2a}{k\_2}} = \frac{1}{\xi\_0} \sqrt{\frac{2a}{k\_4}}.\tag{6}$$

where the boundary center is chosen at *z* ¼ 0 and *lT* is the characteristic width of the boundary. The shear dependence on coordinate Eq. (5) can be substituted into relation Eq. (4). The surface energy density of the twin boundary is obtained by further integration:

$$W\_T = \frac{\sqrt{2ak\_2^3}}{3k\_4} = \frac{4}{3}l\_T h. \tag{7}$$

It is expressed through parameters ð Þ *α; k*2*; k*<sup>4</sup> of the microscopic double-well potential or macroscopic parameters ð Þ *lT; h* . The parameters of the thermodynamic potential Eq. (1) can be transformed into the microscopic ones:

$$k\_{21} = \frac{4\mu\_1}{\xi\_0^2}; \quad k\_{41} = \frac{4\mu\_1}{\xi\_0^4};$$

$$h\_1 = hv\_1; \; k\_{21} = k\_2v\_1; \; k\_{41} = k\_4v\_1.$$

Here *h*1, *k*21, and *k*<sup>41</sup> are the barrier height Eq. (3), parameters *k*<sup>2</sup> and *k*<sup>4</sup> Eq. (1) normalized per unit cell. These equalities are obtained by multiplying *h* and *k*<sup>2</sup> or *k*<sup>4</sup> to the unit cell volume *v*1. The characteristic width Eq. (6) *lT* ≃1*:*5*nm* was obtained by molecular dynamic method in [16].

### **3. Atomic potential in continual description**

In hcp lattice, one can find the symmetry axes (along 0*z*) of third and sixth orders. In the close-packed layers ð Þ *x; y* , hcp demonstrates isotropic properties of macroscopic tensors [11, 17]. The isotropic macroscopic tensors exist at appropriate relations *<sup>c</sup>=<sup>a</sup>* <sup>¼</sup> ffiffiffi 8 <sup>p</sup> *<sup>=</sup>*3 of unit cell sizes [1, 11]. Inside the perfect hcp phase, an atom is in high symmetric (isotropic) potential:

$$U\_{\dot{s}i}(\mathbf{r}) = \frac{1}{2}k\_{\dot{s}i} \left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2\right). \tag{9}$$

where *kb*ð Þ*ξ* is rigidity coefficient inside TB, *U*0ð Þ*ξ* is a varied bottom level, and *c*ð Þ*ξ x* is the linear part. In the limit points *ξ* ¼ �*ξ*0, Eq. (11) transforms into isotropic hcp phase Eq. (9) with *kpn ξ*<sup>0</sup> ð Þ¼þ2*k*21. Inside TB *ξ* ¼ 0, the rigidity takes value *kpn*ð Þ¼� 0 *k*21. Thus, the rigidity coefficients in phase ð Þ *kis* and in the middle of TB

Inside the boundary the potential is considerably softer in direction *Ox* because of *kb*ð Þ*ξ* < *kis* (see **Figure 2**). The difference in these rigidity coefficients is too high *kis* � *kb*ð Þ¼ 0 3*k*21. For further analysis, we need especially the quadric form in

The ratio of the rigidity coefficients in the relation Eq. (10) can be related to the ratio of the elastic modules which are shown in **Table 1**. The macroscopic tensor components *C*11*C*<sup>33</sup> describe the longitudinal deformation along the axes 0*x* and 0*z*,

anisotropy of the rigidity coefficients *kelz=kis* in the basal plane and axis 0*z*. Uniaxial compression-tension in the basal plane of 0*xy* corresponds to the elastic modulus of *C*<sup>11</sup> and atomic rigidity coefficient *kis*. The shuffle of the basal planes in an arbitrary direction corresponds to elastic modulus *C*<sup>44</sup> and atomic rigidity coefficients 2*k*21.

> <sup>¼</sup> <sup>2</sup>*k*<sup>21</sup> *kp* þ 2*k*<sup>21</sup>

He, the ratio of the elastic modulus *C*33*=C*<sup>11</sup> ¼ 1*:*37 gives

≲ *C*<sup>44</sup> *C*<sup>11</sup>

ffiffiffi 3 <sup>p</sup> *;* <sup>0</sup>*;* <sup>0</sup>

The geometry of the hcp lattice is shown in **Figure 1a**. In the hard sphere model

where *R*<sup>0</sup> is the atomic radius, *x* is a coordinate along the shift direction of the atomic plane B, *z* is a coordinate along the direction perpendicular to the atomic plane, and *y* is a coordinate along the atomic plane perpendicular to the shift direction. (0,0,0) is the touch point of the spheres in plane A. Then the sphere

**Element** *C***11, GPa** *C***33, GPa** *C***13, GPa** *C***44, GPa**

He [21]<sup>a</sup> 4:05�10�<sup>2</sup> 5:54�10�<sup>2</sup> 1:05�10�<sup>2</sup> 1:24�10�<sup>2</sup>

Li [22]b 14.2 — — 10.7

Be [23]b 292 349 6 163 24Mg [23]b 59.3 61.5 21.4 16.4

*The experimental values of the elastic moduli of some hcp materials in the notation of Voigt* Cik *following [11].*

*He are found at T* � 1K *and molar volume 20.97*�10�6m3*=*mo1 *[21]. <sup>b</sup>*

*kis* ¼ *kp* þ *kpn ξ*<sup>0</sup> ð Þ¼ *kp* þ 2*k*21; (12) *kb*ð Þ¼ 0 *kp* þ *kpn*ð Þ¼ 0 *kp* � *k*21*:* (13)

� � with two

*:* (14)

� �; (15)

ð Þ *kb*ð Þ 0 are represented by the rigidity coefficients inside the plane *kp*

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

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

adjacent ð Þ *k*<sup>21</sup> planes:

respectively. In solid <sup>4</sup>

Therefore, we have the following inequality:

2*k*<sup>21</sup> *kis*

**4. The atomic potential and hard sphere model in hcp phase**

for the hcp plane A (see **Figure 1a,b**), the coordinates of atomic centers are

ð Þ 0*;* �*R*0*;* 0 ; �*R*<sup>0</sup>

Eq. (11).

4

7

9

**Table 1.**

*The elastic moduli of hcp <sup>4</sup>*

*At room temperature.*

*a*

**33**

where *kis* is isotropic rigidity. The harmonic approximation Eq. (9) is satisfied better for heavier inert atoms or light metals; however, the helium crystal has pronouncedly anharmonic atomic potential [18]. Nevertheless in helium crystals, the harmonic approximation is successfully applied [19, 20].

The isotropic rigidity *kis* can be divided into two contributions: *kis* ¼ *kp* þ *kpn*, where *kp* is rigidity in the plane and *kpn* is rigidity from the interaction with the neighbor planes.

Inside the twin boundary, the neighbor layers are shifted from the symmetric positions, and it causes an anisotropic atomic potential. The previous spherical potential is broken. Then inside the twin boundary, the initial isotropic atomic potential transforms into

$$\begin{aligned} U\_{\text{an1}}(\mathbf{r}) &= U\_{\text{it}}(y, \mathbf{z}) + U\_{\text{an1}}(\mathbf{x}); \\ U\_{\text{it}}(y, \mathbf{z}) &= \frac{1}{2} k\_{\text{is}} \left( y^2 + \mathbf{z}^2 \right); \ U\_{\text{an1}}(\mathbf{x}) = U\_p(\mathbf{x}) + U\_{pn}(\mathbf{x}, \boldsymbol{\xi}); \\ U\_p(\mathbf{x}) &= \frac{1}{2} k\_{\text{p}} \mathbf{x}^2; \ U\_{\text{pn}}(\mathbf{x}, \boldsymbol{\xi}) = \frac{k\_{41} \left( \boldsymbol{\xi} - \mathbf{x} \right)^4}{4} - \frac{k\_{21} \left( \boldsymbol{\xi} - \mathbf{x} \right)^2}{2}. \end{aligned} \tag{10}$$

where the isotropic potential Eq. (9) splits into two terms. The first term *Uan*1ð Þ *x* is an anisotropic and nonlinear part of the potential in the shift direction *Ox*. The second term *Uis*ð Þ *y; z* is the rest of the isotropic part which is perpendicular to the shift direction. Further, the potential *Uan*1ð Þ *x* is divided too into *Up*ð Þ *x* , the isotropic part, and *Upn*ð Þ *x; ξ* , the anisotropic one from the neighbor atomic planes. The last turn depends on the layer shift *ξ* and the small deviation *x*. Therefore, only term *Upn*ð Þ *x; ξ* changes inside TB which is shown in **Figure 2**. The analysis (see [14]) of the term *Upn*ð Þ *x; ξ* allows to write the anisotropic atomic potential Eq. (10) in the following simple form:

$$U\_{m1}(\mathbf{r}, \boldsymbol{\xi}) \simeq U\_0(\boldsymbol{\xi}) + c(\boldsymbol{\xi})\mathbf{x} + \frac{1}{2}k\_b(\boldsymbol{\xi})\mathbf{x}^2 + \frac{1}{2}k\_{ii}(\boldsymbol{y}^2 + \boldsymbol{z}^2);$$

$$k\_b(\boldsymbol{\xi}) = k\_p + k\_{pn}(\boldsymbol{\xi}) = k\_{ii} + 3k\_{21} \left(\frac{\boldsymbol{\xi}^2}{\xi\_0^2} - 1\right);\ k\_{pn}(\boldsymbol{\xi}) = +k\_{21} \left(3\frac{\boldsymbol{\xi}^2}{\xi\_0^2} - 1\right). \tag{11}$$

$$\left\langle \sum\_{i=1}^{n} \left\langle \sum\_{j=1}^{n} \left\langle \sum\_{i=1}^{n} \frac{\partial}{\partial \xi\_i} \right\rangle \right\rangle \right\rangle$$

#### **Figure 2.**

*Smooth changed parts of the potential in dependence on the coordinates ξ and* x *according to Eqs. (10) and (11): Upn*ð Þ 0*; ξ is a lower double-well curve and Uan*1ð Þ *x; ξ is a set of parabolas.*

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

macroscopic tensors [11, 17]. The isotropic macroscopic tensors exist at appropriate

where *kis* is isotropic rigidity. The harmonic approximation Eq. (9) is satisfied better for heavier inert atoms or light metals; however, the helium crystal has pronouncedly anharmonic atomic potential [18]. Nevertheless in helium crystals,

The isotropic rigidity *kis* can be divided into two contributions: *kis* ¼ *kp* þ *kpn*, where *kp* is rigidity in the plane and *kpn* is rigidity from the interaction with the

Inside the twin boundary, the neighbor layers are shifted from the symmetric positions, and it causes an anisotropic atomic potential. The previous spherical potential is broken. Then inside the twin boundary, the initial isotropic atomic

*kis <sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> � �; *Uan*1ð Þ¼ *<sup>x</sup> Up*ð Þþ *<sup>x</sup> Upn*ð Þ *<sup>x</sup>; <sup>ξ</sup>* ;

where the isotropic potential Eq. (9) splits into two terms. The first term *Uan*1ð Þ *x* is an anisotropic and nonlinear part of the potential in the shift direction *Ox*. The second term *Uis*ð Þ *y; z* is the rest of the isotropic part which is perpendicular to the shift direction. Further, the potential *Uan*1ð Þ *x* is divided too into *Up*ð Þ *x* , the isotropic part, and *Upn*ð Þ *x; ξ* , the anisotropic one from the neighbor atomic planes. The last turn depends on the layer shift *ξ* and the small deviation *x*. Therefore, only term *Upn*ð Þ *x; ξ* changes inside TB which is shown in **Figure 2**. The analysis (see [14]) of the term *Upn*ð Þ *x; ξ* allows to write the anisotropic atomic potential Eq. (10) in the

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

2 <sup>2</sup> *:*

> *ξ*2 0 � 1 !

*:*

(11)

(10)

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

1 2

*kb*ð Þ*<sup>ξ</sup> <sup>x</sup>*<sup>2</sup> <sup>þ</sup>

*ξ*2 *ξ*2 0 � 1 !

*Smooth changed parts of the potential in dependence on the coordinates ξ and* x *according to Eqs. (10) and*

*(11): Upn*ð Þ 0*; ξ is a lower double-well curve and Uan*1ð Þ *x; ξ is a set of parabolas.*

1 2

*kis <sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> � �;

; *kpn*ð Þ¼þ *<sup>ξ</sup> <sup>k</sup>*<sup>21</sup> <sup>3</sup> *<sup>ξ</sup>*<sup>2</sup>

*Uis*ð Þ¼ **r**

the harmonic approximation is successfully applied [19, 20].

*Uan*1ð Þ¼ **r** *Uis*ð Þþ *y; z Uan*1ð Þ *x* ;

1 2

1 2 *kpx*<sup>2</sup> 1 2

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

<sup>p</sup> *<sup>=</sup>*3 of unit cell sizes [1, 11]. Inside the perfect hcp phase, an atom

*kis <sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> � �*:* (9)

relations *<sup>c</sup>=<sup>a</sup>* <sup>¼</sup> ffiffiffi

neighbor planes.

potential transforms into

following simple form:

**Figure 2.**

**32**

*Uan*1ð Þ **r***; ξ* ≃ *U*0ð Þþ *ξ c*ð Þ*ξ x* þ

*kb*ð Þ¼ *ξ kp* þ *kpn*ð Þ¼ *ξ kis* þ 3*k*<sup>21</sup>

*Uis*ð Þ¼ *y; z*

*Up*ð Þ¼ *x*

8

is in high symmetric (isotropic) potential:

where *kb*ð Þ*ξ* is rigidity coefficient inside TB, *U*0ð Þ*ξ* is a varied bottom level, and *c*ð Þ*ξ x* is the linear part. In the limit points *ξ* ¼ �*ξ*0, Eq. (11) transforms into isotropic hcp phase Eq. (9) with *kpn ξ*<sup>0</sup> ð Þ¼þ2*k*21. Inside TB *ξ* ¼ 0, the rigidity takes value *kpn*ð Þ¼� 0 *k*21. Thus, the rigidity coefficients in phase ð Þ *kis* and in the middle of TB ð Þ *kb*ð Þ 0 are represented by the rigidity coefficients inside the plane *kp* � � with two adjacent ð Þ *k*<sup>21</sup> planes:

$$k\_{\rm is} = k\_p + k\_{pn}(\xi\_0) = k\_p + 2k\_{21};\tag{12}$$

$$k\_b(\mathbf{0}) = k\_p + k\_{pn}(\mathbf{0}) = k\_p - k\_{21}.\tag{13}$$

Inside the boundary the potential is considerably softer in direction *Ox* because of *kb*ð Þ*ξ* < *kis* (see **Figure 2**). The difference in these rigidity coefficients is too high *kis* � *kb*ð Þ¼ 0 3*k*21. For further analysis, we need especially the quadric form in Eq. (11).

The ratio of the rigidity coefficients in the relation Eq. (10) can be related to the ratio of the elastic modules which are shown in **Table 1**. The macroscopic tensor components *C*11*C*<sup>33</sup> describe the longitudinal deformation along the axes 0*x* and 0*z*, respectively. In solid <sup>4</sup> He, the ratio of the elastic modulus *C*33*=C*<sup>11</sup> ¼ 1*:*37 gives anisotropy of the rigidity coefficients *kelz=kis* in the basal plane and axis 0*z*. Uniaxial compression-tension in the basal plane of 0*xy* corresponds to the elastic modulus of *C*<sup>11</sup> and atomic rigidity coefficient *kis*. The shuffle of the basal planes in an arbitrary direction corresponds to elastic modulus *C*<sup>44</sup> and atomic rigidity coefficients 2*k*21. Therefore, we have the following inequality:

$$\frac{2k\_{21}}{k\_{is}} = \frac{2k\_{21}}{k\_p + 2k\_{21}} \lesssim \frac{C\_{44}}{C\_{11}}.\tag{14}$$

### **4. The atomic potential and hard sphere model in hcp phase**

The geometry of the hcp lattice is shown in **Figure 1a**. In the hard sphere model for the hcp plane A (see **Figure 1a,b**), the coordinates of atomic centers are

$$(\mathbf{0}, \pm \mathbf{R}\_0, \mathbf{0}); \quad \left(\pm \mathbf{R}\_0 \sqrt{3}, \mathbf{0}, \mathbf{0}\right); \tag{15}$$

where *R*<sup>0</sup> is the atomic radius, *x* is a coordinate along the shift direction of the atomic plane B, *z* is a coordinate along the direction perpendicular to the atomic plane, and *y* is a coordinate along the atomic plane perpendicular to the shift direction. (0,0,0) is the touch point of the spheres in plane A. Then the sphere


*a The elastic moduli of hcp <sup>4</sup> He are found at T* � 1K *and molar volume 20.97*�10�6m3*=*mo1 *[21]. <sup>b</sup> At room temperature.*

#### **Table 1.**

*The experimental values of the elastic moduli of some hcp materials in the notation of Voigt* Cik *following [11].*

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

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

$$\begin{aligned} \mathbf{x}^2 + \left(\mathbf{y} \pm \mathbf{R}\_0\right)^2 + \mathbf{z}^2 &= \left(2\mathbf{R}\_0\right)^2; \\ \left(\mathbf{x} \pm \mathbf{R}\_0 \sqrt{\mathbf{3}}\right)^2 + \mathbf{y}^2 + \mathbf{z}^2 &= \left(2\mathbf{R}\_0\right)^2; \end{aligned} \tag{16}$$

helium atom has spherical symmetry. Hence, the equation of probability isosurface

; *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> *<sup>N</sup><sup>ρ</sup>is*

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

� � ; *<sup>N</sup><sup>ρ</sup>is*ð Þ� *<sup>R</sup>*<sup>0</sup> *<sup>κ</sup>*<sup>2</sup>

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>

to a huge change in the volume of solid helium [1], the parameter *κ*<sup>0</sup> can vary widely.

*Xx*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*yy*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *zz*<sup>2</sup> � � ;

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

*<sup>z</sup>* <sup>¼</sup> *<sup>m</sup>*

*<sup>λ</sup>* ; *Npis* <sup>¼</sup> ln

ffiffiffiffiffiffiffiffiffi *λ*3 *ρ*2*π*<sup>3</sup>

*:* (22)

<sup>0</sup>*λ:* (23)

1*=*4

<sup>ℏ</sup><sup>2</sup> *kzel:* (25)

*is* . In respect

(24)

s

<sup>0</sup> <sup>¼</sup> *<sup>R</sup>*<sup>2</sup>

ℏ *:*

(sphere of radius *R*) is

crystal) is

an atom <sup>4</sup>

value:

**35**

*kxel* ¼ *kb* ≤*kyel* ¼ *kzel* ¼ *kis*.

with semiaxes *a*≥*b*≥*c*:

*<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>¼</sup> *<sup>R</sup>*<sup>2</sup>

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

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

ffiffiffiffiffi *λ*3 *π*3

exp �*κ*<sup>2</sup> 0

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>*

An anisotropic harmonic potential can be written as [24]

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

<sup>ℏ</sup><sup>2</sup> *kxel*; *<sup>λ</sup>*<sup>2</sup>

; *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *<sup>N</sup><sup>ρ</sup> λ*0 ; *c*

*<sup>N</sup><sup>ρ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>κ</sup>*<sup>2</sup>

1 2 *m ω*<sup>2</sup>

<sup>ℏ</sup> ; *<sup>λ</sup><sup>y</sup>* <sup>¼</sup> *<sup>m</sup>ω*<sup>3</sup>

*<sup>y</sup>* <sup>¼</sup> *<sup>m</sup>*

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,

He is in a uniaxial potential of neighboring atoms of Eq. (13):

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

> <sup>2</sup> <sup>¼</sup> *<sup>N</sup><sup>ρ</sup> λz*

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

r

<sup>0</sup> þ ln

isosurfaces to find an atom in the anisotropic case. On appropriate limit *λ<sup>i</sup>* ¼ *λ*, these

Thus, the relations Eqs. (26) and (27) describe the probability density

; *Np* ¼ ln

ffiffiffiffiffiffiffiffiffiffiffiffiffi *λXλ*0*λ<sup>z</sup> λ*3

ffiffiffiffiffiffiffiffiffiffiffiffiffi *λXλ*0*λ<sup>z</sup> ρ*<sup>2</sup>*π*<sup>3</sup>

*:* (27)

*:* (26)

s

<sup>ℏ</sup><sup>2</sup> *kyel*; *<sup>λ</sup>*<sup>2</sup>

*Uanis*ð Þ¼ **r**

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

*λ*2 *<sup>X</sup>* <sup>¼</sup> *<sup>m</sup>*

*<sup>a</sup>*<sup>2</sup> <sup>¼</sup> *<sup>N</sup><sup>ρ</sup> λX*

relations describe the isotropic case.

s

*ρ*<sup>0</sup> ¼

The equilibrium points for the atom of the shifting neighbor atomic plane B can be found from the geometry of the system (Eq. (16) at y ¼ 0):

$$\mathbf{x}\_{\text{Re}} = \pm R \mathbf{o} \frac{\mathbf{1}}{\sqrt{3}}; \quad \mathbf{y}\_{\text{Re}} = \mathbf{0}; \quad \mathbf{z}\_{\text{Re}} = R \mathbf{o} \sqrt{\frac{8}{3}}.\tag{17}$$

Signs – and + in *xRe* describe positions B and C in plane B, respectively. From the first Eq. (16), the saddle point coordinates for an atom of plane B are

$$\mathbf{x}\_{\mathrm{R}} = \mathbf{0}; \qquad \mathbf{y}\_{\mathrm{R}} = \mathbf{0}; \qquad \mathbf{z}\_{\mathrm{R}\mathfrak{s}} = \mathbf{R}\_{0} \sqrt{\mathbf{3}}.\tag{18}$$

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

$$\mathfrak{z}\_{\rm OR} = |\varkappa\_{\rm Re}|; \quad h\_{1-R} = \frac{1}{2} \text{g}k\_{\rm is} (\mathbf{z}\_{\rm Rs} - \mathbf{z}\_{\rm Re})^2 \tag{19}$$

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 reason for the quasielastic energy behavior.

For the hard sphere model, the substitution of relations (19) into Eqs. (3) and (8) gives the parameters of the microscopic interatomic potential:

$$k\_{21-R} = \frac{4h\_{1-R}}{\xi\_{0R}^2}; \quad k\_{41-R} = \frac{4h\_{1-R}}{\xi\_{0R}^4}.\tag{20}$$

For comparison, Eq. (11) allows us to find the rigidity coefficients in the phase *kis* and in the middle of the boundary *kb*ð Þ 0 .
