**7. The self-consistent description of the twin boundary**

The **classic description** of TB uses two coefficients of the thermodynamic potential Eq. (1):

$$k\_{21} = \text{const}; \qquad k\_{41} = \text{const}; \quad \text{or} \quad h\_1 = \text{const}; \qquad \xi\_0 = \text{const}.\tag{31}$$

They can be corresponded to the hard sphere model (see Eqs. (15)–(20)).

**The quantum and thermal description** of TB is self-consistent, i.e., the parameters Eq. (31) are varied as a function of some parameter *q* that, in its turn, is a function of these parameters:

$$h\_1 = h\_1(q); \qquad \xi\_0 = \xi\_0(q); \qquad q = q(h\_1, \ \xi\_0). \tag{32}$$

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

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

atom corresponds to the average potential isosurface (sphere of radius *R*):

*<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>

1 2 *k*2 *Xx*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

*Uanis*ð Þ¼ **r**

neighboring atoms Eq. (13): *kxel* ¼ *kb* ≤ *kyel* ¼ *kzel* ¼ *kis*.

*<sup>a</sup>*<sup>2</sup> <sup>¼</sup> *kBT kX*

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

In isotropic harmonic approximation Eq. (9), the average thermal energy of an

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

*kX* ¼ *kxel*; *ky* ¼ *kyel*; *kz* ¼ *kzel:*

The motion equation splits also into three independent equivalent equations.

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

*ky*

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.

; *c*

<sup>2</sup> <sup>¼</sup> *kBT kz :*

Then inside of the twin boundary, an atom is in the uniaxial potential of

; *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *kBT*

The **classic description** of TB uses two coefficients of the thermodynamic

They can be corresponded to the hard sphere model (see Eqs. (15)–(20)). **The quantum and thermal description** of TB is self-consistent, i.e., the param-

eters Eq. (31) are varied as a function of some parameter *q* that, in its turn, is a

*k*<sup>21</sup> ¼ *const*; *k*<sup>41</sup> ¼ *const*; or *h*<sup>1</sup> ¼ *const*; *ξ*<sup>0</sup> ¼ *const:* (31)

*h*<sup>1</sup> ¼ *h*1ð Þ*q* ; *ξ*<sup>0</sup> ¼ *ξ*0ð Þ*q* ; *q* ¼ *q h*1*; ξ*<sup>0</sup> ð Þ*:* (32)

The equation of the potential isosurface is ellipsoid with semiaxes *a*≥*b*≥ *c*

*x*2 *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> *<sup>b</sup>*<sup>2</sup> <sup>þ</sup>

**7. The self-consistent description of the twin boundary**

; *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> *kBT*

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

*kis*

;

; (28)

(29)

(30)

**and the twin boundary**

constant and temperature.

pare with Eq. (24))

(compare with Eq. (26)):

potential Eq. (1):

**36**

function of these parameters:

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

$$q = \mathbf{1} - \frac{c^2}{a^2} \equiv e^2; \quad \mathbf{0} \le e^2 \le \mathbf{1}. \tag{33}$$

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 deformation parameter *q*.

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 isotropic oscillator:

$$R\_0 = a = b = c; \quad \rho = \rho\_0; \qquad q = \mathbf{0}.\tag{34}$$

**(1) The first approximation**. An atom is considered as a quantum anisotropic uniaxis oscillator. The potential Eq. (10) has been obtained in zero approximation. In the general case, the ellipsoid parameters and the isosurface deformation parameter are described by Eqs. (26), (27), and (33),

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

$$b\_1 = c\_1 \lhd a\_1; \qquad \rho = \rho\_0; \quad \varepsilon\_{c1}^2 = q\_1 = 1 - \frac{c\_1^2}{a\_1^2} > 0. \tag{35}$$

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 ellipsoids [25].

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

$$b\_2 \neq c\_2 \lhd a\_2; \qquad e\_{b2}^2 = q\_{b2} = \mathbf{1} - \frac{b\_2^2}{a\_2^2} > 0; \qquad e\_{c2}^2 = q\_{c2} = \mathbf{1} - \frac{c\_2^2}{a\_2^2} > 0. \tag{36}$$

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

(i) The third and further *i* th steps qualitatively replicate the previous steps in the same way. The second and further steps are more cumbersome and complicated.

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

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

$$
\lambda\_X = \frac{1}{\hbar} \sqrt{m \left[ k\_{\rm is} + 3k\_{21} \left( \frac{\xi^2}{\xi\_0^2} - 1 \right) \right]} \le \lambda; \quad \lambda\_\text{y} = \lambda\_x = \lambda = \frac{1}{\hbar} \sqrt{mk\_{\rm is}}.\tag{37}
$$

Using Eqs. (26) and (27), the atomic isosurface can be described by ellipsoid with semiaxes:

$$a\_1^2 = \frac{N\_{\rho\_0 1}}{\lambda\_X}; \quad b\_1^2 = c\_1^2 = \frac{N\_{\rho\_0 1}}{\lambda}; \quad N\_{\rho\_0 1} = \kappa\_0^2 + \ln\sqrt{\frac{\lambda\_X}{\lambda}}.\tag{38}$$

For fixed *λy, λ<sup>z</sup>* ¼ *λ* and reduced stiffness coefficient *λ<sup>X</sup>* along axis 0*x*, the semiaxes of the ellipsoid change as follows: *a*<sup>1</sup> >*R*0; *b*<sup>1</sup> ¼ C<sup>1</sup> < *R*0. Then the isosurface deformation parameter *q*<sup>1</sup> Eq. (35) takes the following dependence on the order parameter *ξ* and coordinate

$$q\_1 = 1 - \sqrt{1 - 3 \frac{k\_{21}}{k\_{\rm is}} \frac{1}{\cosh^2(z/l\_T)}}.\tag{39}$$

if considerably gentle upper parabolas (stronger interaction between the crystal planes in comparison with in-plane interaction) are taken into account which are

In the quantum case, we can evaluate the minimal increase of the exchange integral due to the increase of overlapping wave functions caused by the elliptic

cosh <sup>4</sup>ð Þ *z=lT*

Increasing overlapping volumes Δ*V* with high probability can be evaluated by segments of the crossing ellipsoids. Amplitude Δ*I*<sup>0</sup> depends on two parameters *κ*<sup>0</sup>

In the basal hcp plane, the exchange integral is varied depending on the quantum deformation parameter *q*; the wave function tails are the most sensitive, especially in the overlapping region. Evaluations Eq. (41) take into account only space changing but not the amplitude one. The amplitude changing can achieve several orders because of exponential dependence. The exchange integral *I* uniquely defines the diffusion coefficient [26]. In the interphase boundaries in solid helium, NMR experiment [12] shows the quantum diffusion increasing. The interphase and twin boundaries are similar [5]. So for the quantum diffusion case in TB, the predicted and the experimentally observed arising values are closely related. Experiments show thermal diffusion arising at boundaries [11]; the found thermal ellipsoids'

Now we can point out conditions when exchange integral Eq. (41) increases. We

r

where elastic module *E* ¼ *C*<sup>11</sup> is related to the rigidity coefficient *kis* ≃*πR*0*E=*2. In

So, a high value of exchange integral can be achieved. Compressibility is small in metals, first of all, in light ones (lithium, beryllium, magnesium). Minimal rigidity *kis* gives rise in the exchange integral too. The van der Waals interaction in <sup>4</sup>

fundamental characteristic of quantum crystal. The de Boer parameter gives the probability density to find an atom in the site of a neighboring atom (at distance

> Λ � �

sure growing leads to more difficult tunneling of atoms and different *κ*<sup>2</sup>

He, the atomic radius *R*<sup>0</sup> is the soft parameter, especially under low pressure.

<sup>¼</sup> exp �*λa*<sup>2</sup>

*l* � � ; *κ*<sup>2</sup>

defined by Eqs. (23) and (37) and analyzed in dependence on different factors. In [14] using atomic mass *ma* and evaluation of atomic radius *R*0, the parameter *κ*<sup>2</sup>

> *κ*2 <sup>0</sup> <sup>≃</sup> <sup>1</sup> <sup>ℏ</sup> *<sup>R</sup>*<sup>5</sup>*=*<sup>2</sup> 0

3–4 orders of magnitude less than in metal (see **Table 1**).

*<sup>ρ</sup>*ð Þ� *al* exp � <sup>1</sup>

in Eqs. (42) and (43). Using the data in **Table 1**, for solid <sup>4</sup>

The de Boer parameter <sup>Λ</sup> <sup>¼</sup> <sup>0</sup>*:*45 for <sup>4</sup>

<sup>0</sup> ≃ 3*:*77, and Λ≃0*:*07 (see **Table 2**).

Another way to estimate *κ*<sup>2</sup>

<sup>0</sup> Eq. (42) in exponent Eq. (41). The parameter *κ*<sup>0</sup> or *λ* can be

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 *πmaE*

<sup>0</sup> is to compare it with de Boer parameter Λ, the

He [26] gives evaluation *κ*<sup>2</sup>

<sup>0</sup> <sup>¼</sup> <sup>1</sup>

<sup>4</sup>Λ*:* (43)

He we obtain *R*<sup>0</sup> [14],

<sup>0</sup> ≃ 0*:*59. Pres-

<sup>0</sup> evaluations

1

4 ffiffiffiffiffi *κ*5 0 q

;

exp �*κ*<sup>2</sup> 0 � �; (41)

0

He is

*:* (42)

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

<sup>Δ</sup>*<sup>I</sup>* <sup>¼</sup> <sup>Δ</sup>*I*<sup>0</sup> <sup>¼</sup> <sup>1</sup>

16 ffiffiffi *π* p

<sup>Δ</sup>*I*<sup>0</sup> <sup>¼</sup> <sup>3</sup>

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

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

deformation qualitatively explains these facts.

shown in **Figure 2**.

deformations [14]:

and *k*21*=kis* only.

need minimal *κ*<sup>2</sup>

solid <sup>4</sup>

*al* ¼ 2*R*0) [26]:

*κ*2

**39**

value was estimated:

We obtain the same result for the thermal excitations; however, instead of relation Eq. (37), we use the rigidity constants Eq. (30):

$$k\_X = k\_{i\circ} + 3k\_{21} \left(\frac{\xi^2}{\xi\_0^2} - 1\right) \le k\_{i\circ}.\tag{40}$$

In **Table 2**, evaluations of different parameters are shown according to **Table 1** and relation Eqs. (14), (39), (42), and (43); the sources are shown in round brackets on top of columns.

In He and Mg (see **Table 2**), the transverse components of the elastic module *C*<sup>44</sup> are much smaller than the longitudinal ones *C*11. Accordingly in these materials, the isosurface deformation parameters in the middle point of TB *qmax* take relatively small value.

In Li and Be (see **Table 2**), the transverse and longitudinal components of the elastic moduli are closer. Hence, in these materials, the parameters *qmax* are considerably greater. Moreover in Li, the parameter *qmax* can reach 1 or even take complex (imaginary root) values. This indicates a possible instability of Li crystal lattice (see further consideration). This, seemingly unexpected, result is quite understandable


*a Evaluation of the de Boer parameter <sup>Λ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>45</sup> *for <sup>4</sup> He at* �*1 K [2].*

*b At room temperature.*

*c Evaluation of the de Boer parameter at* � *1K (present work).*

*For all materials the parameters κ*<sup>2</sup> <sup>0</sup> *and Λ are evaluated with the same R*0*.*

#### **Table 2.**

*Evaluation of the elastic moduli relations, rigidity relations, the isosurface deformation parameter in the middle point of TB qmax, and the de Boer parameter Λ of some hcp materials.*

*<sup>λ</sup><sup>X</sup>* <sup>¼</sup> <sup>1</sup> *ħ*

> *a*2 <sup>1</sup> <sup>¼</sup> *<sup>N</sup><sup>ρ</sup>*<sup>01</sup> *λX*

order parameter *ξ* and coordinate

brackets on top of columns.

small value.

**Element** *<sup>C</sup>***<sup>44</sup>**

*At room temperature.*

*For all materials the parameters κ*<sup>2</sup>

*a*

*b*

*c*

**38**

**Table 2.**

*<sup>C</sup>***<sup>11</sup> , (Table 1, (14)) <sup>3</sup>** *<sup>k</sup>***<sup>21</sup>**

*Evaluation of the de Boer parameter <sup>Λ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>45</sup> *for <sup>4</sup>*

*Evaluation of the de Boer parameter at* � *1K (present work).*

*point of TB qmax, and the de Boer parameter Λ of some hcp materials.*

with semiaxes:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

; *b*<sup>2</sup> <sup>1</sup> ¼ *c* 2 <sup>1</sup> <sup>¼</sup> *<sup>N</sup><sup>ρ</sup>*<sup>01</sup>

*q*<sup>1</sup> ¼ 1 �

relation Eq. (37), we use the rigidity constants Eq. (30):

*ξ*2 *ξ*2 0 � 1 vu " # ! ut <sup>≤</sup>*λ*; *<sup>λ</sup><sup>y</sup>* <sup>¼</sup> *<sup>λ</sup><sup>z</sup>* <sup>¼</sup> *<sup>λ</sup>* <sup>¼</sup> <sup>1</sup>

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

Using Eqs. (26) and (27), the atomic isosurface can be described by ellipsoid

For fixed *λy, λ<sup>z</sup>* ¼ *λ* and reduced stiffness coefficient *λ<sup>X</sup>* along axis 0*x*, the semiaxes of the ellipsoid change as follows: *a*<sup>1</sup> >*R*0; *b*<sup>1</sup> ¼ C<sup>1</sup> < *R*0. Then the

> 1 � 3 *k*<sup>21</sup> *kis*

We obtain the same result for the thermal excitations; however, instead of

*ξ*2 *ξ*2 0 � 1 !

In **Table 2**, evaluations of different parameters are shown according to **Table 1**

In He and Mg (see **Table 2**), the transverse components of the elastic module *C*<sup>44</sup> are much smaller than the longitudinal ones *C*11. Accordingly in these materials, the isosurface deformation parameters in the middle point of TB *qmax* take relatively

In Li and Be (see **Table 2**), the transverse and longitudinal components of the elastic moduli are closer. Hence, in these materials, the parameters *qmax* are considerably greater. Moreover in Li, the parameter *qmax* can reach 1 or even take complex (imaginary root) values. This indicates a possible instability of Li crystal lattice (see further consideration). This, seemingly unexpected, result is quite understandable

4He ≃0*:*306 ≤ 0*:*46 ≃ 0*:*27 ≃3*:*77 ≃0*:*0663<sup>a</sup> 7Li<sup>b</sup> <sup>≃</sup>0*:*<sup>75</sup> <sup>≤</sup>1*:*<sup>13</sup> ! <sup>1</sup> <sup>≃</sup>151*:*<sup>3</sup> <sup>≃</sup> <sup>0</sup>*:*0017<sup>c</sup> 9Be<sup>b</sup> ≃0*:*558 ≤0*:*84 ≃0*:*60 ≃ 127*:*4 ≃0*:*0020<sup>c</sup> 24Mg<sup>b</sup> ≃0*:*277 ≤0*:*41 ≃0*:*23 ≃353*:*4 ≃ 0*:*0007<sup>c</sup>

*He at* �*1 K [2].*

<sup>0</sup> *and Λ are evaluated with the same R*0*.*

*Evaluation of the elastic moduli relations, rigidity relations, the isosurface deformation parameter in the middle*

*<sup>k</sup>is* **, Eq. (14)** *<sup>q</sup>max***, Eq. (39)** *<sup>κ</sup>***<sup>2</sup>**

and relation Eqs. (14), (39), (42), and (43); the sources are shown in round

s

*kX* ¼ *kis* þ 3*k*<sup>21</sup>

isosurface deformation parameter *q*<sup>1</sup> Eq. (35) takes the following dependence on the

*<sup>λ</sup>* ; *<sup>N</sup><sup>ρ</sup>*<sup>01</sup> <sup>¼</sup> *<sup>κ</sup>*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cosh <sup>2</sup>ð Þ *z=lT* ℏ

<sup>0</sup> þ ln

ffiffiffiffiffiffiffiffiffi *mkis*

ffiffiffiffiffi *λX λ* r

*:* (39)

≤*kis:* (40)

**0, Eq. (42)** Λ**, Eq. (43)**

p *:* (37)

*:* (38)

*m kis* þ 3*k*<sup>21</sup>

if considerably gentle upper parabolas (stronger interaction between the crystal planes in comparison with in-plane interaction) are taken into account which are shown in **Figure 2**.

In the quantum case, we can evaluate the minimal increase of the exchange integral due to the increase of overlapping wave functions caused by the elliptic deformations [14]:

$$\begin{split} \Delta I &= \Delta I\_0 = \frac{1}{\cosh^4(\mathbf{z}/l\_T)};\\ \Delta I\_0 &= \frac{3}{16\sqrt{\pi}} \frac{1}{4\sqrt{\kappa\_0^5}} \left(\frac{k\_{21}}{k\_{is}}\right)^2 \exp\left(-\kappa\_0^2\right); \end{split} \tag{41}$$

Increasing overlapping volumes Δ*V* with high probability can be evaluated by segments of the crossing ellipsoids. Amplitude Δ*I*<sup>0</sup> depends on two parameters *κ*<sup>0</sup> and *k*21*=kis* only.

In the basal hcp plane, the exchange integral is varied depending on the quantum deformation parameter *q*; the wave function tails are the most sensitive, especially in the overlapping region. Evaluations Eq. (41) take into account only space changing but not the amplitude one. The amplitude changing can achieve several orders because of exponential dependence. The exchange integral *I* uniquely defines the diffusion coefficient [26]. In the interphase boundaries in solid helium, NMR experiment [12] shows the quantum diffusion increasing. The interphase and twin boundaries are similar [5]. So for the quantum diffusion case in TB, the predicted and the experimentally observed arising values are closely related. Experiments show thermal diffusion arising at boundaries [11]; the found thermal ellipsoids' deformation qualitatively explains these facts.

Now we can point out conditions when exchange integral Eq. (41) increases. We need minimal *κ*<sup>2</sup> <sup>0</sup> Eq. (42) in exponent Eq. (41). The parameter *κ*<sup>0</sup> or *λ* can be defined by Eqs. (23) and (37) and analyzed in dependence on different factors. In [14] using atomic mass *ma* and evaluation of atomic radius *R*0, the parameter *κ*<sup>2</sup> 0 value was estimated:

$$
\kappa\_0^2 \simeq \frac{1}{\hbar} R\_0^{5/2} \sqrt{\frac{1}{2} \pi m\_a E}. \tag{42}
$$

where elastic module *E* ¼ *C*<sup>11</sup> is related to the rigidity coefficient *kis* ≃*πR*0*E=*2. In solid <sup>4</sup> He, the atomic radius *R*<sup>0</sup> is the soft parameter, especially under low pressure. So, a high value of exchange integral can be achieved. Compressibility is small in metals, first of all, in light ones (lithium, beryllium, magnesium). Minimal rigidity *kis* gives rise in the exchange integral too. The van der Waals interaction in <sup>4</sup> He is 3–4 orders of magnitude less than in metal (see **Table 1**).

Another way to estimate *κ*<sup>2</sup> <sup>0</sup> is to compare it with de Boer parameter Λ, the fundamental characteristic of quantum crystal. The de Boer parameter gives the probability density to find an atom in the site of a neighboring atom (at distance *al* ¼ 2*R*0) [26]:

$$\rho(a\_l) \sim \exp\left(-\frac{1}{\Lambda}\right) = \exp\left(-\lambda a\_l^2\right); \quad \kappa\_0^2 = \frac{1}{4\Lambda}.\tag{43}$$

The de Boer parameter <sup>Λ</sup> <sup>¼</sup> <sup>0</sup>*:*45 for <sup>4</sup> He [26] gives evaluation *κ*<sup>2</sup> <sup>0</sup> ≃ 0*:*59. Pressure growing leads to more difficult tunneling of atoms and different *κ*<sup>2</sup> <sup>0</sup> evaluations in Eqs. (42) and (43). Using the data in **Table 1**, for solid <sup>4</sup> He we obtain *R*<sup>0</sup> [14], *κ*2 <sup>0</sup> ≃ 3*:*77, and Λ≃0*:*07 (see **Table 2**).

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 quantum diffusion along the boundary plane increases.
