The experimental values of the elastic moduli of some hcp materials in the notation of Voigt Cik following .
The 180° twin boundary (TB) (stacking fault) is investigated in the hexagonal close-packed (hcp) light materials. It is shown that atomic symmetry inside the twin boundary is lower than in hcp phase due to interatomic interaction between neighbors. In the case of quantum or thermal behavior, for the isosurfaces, an initial spherical form (in hcp phase) transforms into ellipsoid (in the boundary). We introduce the isosurface deformation parameter. The self-consistent description is developed to estimate the parameters of the thermodynamic potential, and the models of hard spheres and ellipsoids are used. It is shown that the quantum or thermal behavior of the boundary atoms causes the following effects: (i) the increase of degree of overlap of the atomic wave functions or trajectories within the twin boundaries, (ii) the increase of diffusion inside the twin boundaries, and (iii) the decrease of energy and broadening of the quantum boundary in comparison with the classical case.
- solid helium
- twin boundary
- stacking fault
- hard ellipsoids
- quantum diffusionPACS numbers: 61.72.Mm
Helium crystals have unique quantum properties and are useful for the investigation of dynamic and kinetic behavior of atomic crystals . In experiments, the structure phase transition between body-centered cubic (bcc) and hcp phase was found for both metals and solid 4He [2, 3]. The coherent phase boundary (PB) and twin boundary (TB) or stacking fault (SF) was investigated theoretically in the frame of one order parameter (OP) model [4, 5]. The two-OP theory of PB [6, 7] was developed on the basis of the Burgers mechanism. In work  we proposed the three-OP theory that combines Sanati [6, 7] and Kaschenko  treatments; so we take into account the changes of volume and pressure under the phase transition. The three-OP and one-OP descriptions of PB and TB are uniquely related. In different models of coherent bcc-hcp boundary, the local oscillations spectra of OP in 4He were investigated in .
In the experiment , a glass formation under deformation of solid helium was investigated. Usually the deformation of crystals generates the different defects , including stacking faults. In the nuclear magnetic resonance (NMR) experiments , the great role of the interface in increasing the quantum diffusion was found. In work  for single hcp crystal 4He, the stacking fault energy was measured.
The present work is devoted to the development of the self-consistent description of quantum behavior of 4He atoms in twin boundary proposed in work . We apply this treatment to quantum and thermal description of twin boundary in some metals.
2. Model of the twin boundary
In the hcp phase of crystal 4He, 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).
where the integration is over the volume , square brackets contain the volume energy density, is a coordinate in the direction of heterogeneity, is a dispersion parameter responsible for the boundary width, and phenomenological parameters are positive. In hexagonal lattice, 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
where 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 of the potential barrier per unit volume:
For further analysis it is convenient to write the free energy Eq. (1) in terms of and :
where the boundary center is chosen at and 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:
It is expressed through parameters of the microscopic double-well potential or macroscopic parameters . The parameters of the thermodynamic potential Eq. (1) can be transformed into the microscopic ones:
Here h1, k21, and k41 are the barrier height Eq. (3), parameters k2 and k4Eq. (1) normalized per unit cell. These equalities are obtained by multiplying and or to the unit cell volume . The characteristic width Eq. (6)was obtained by molecular dynamic method in .
3. Atomic potential in continual description
In hcp lattice, one can find the symmetry axes (along ) of third and sixth orders. In the close-packed layers , hcp demonstrates isotropic properties of macroscopic tensors [11, 17]. The isotropic macroscopic tensors exist at appropriate relations of unit cell sizes [1, 11]. Inside the perfect hcp phase, an atom is in high symmetric (isotropic) potential:
where 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 . Nevertheless in helium crystals, the harmonic approximation is successfully applied [19, 20].
The isotropic rigidity can be divided into two contributions: , where is rigidity in the plane and 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
where the isotropic potential Eq. (9) splits into two terms. The first term is an anisotropic and nonlinear part of the potential in the shift direction . The second term is the rest of the isotropic part which is perpendicular to the shift direction. Further, the potential is divided too into , the isotropic part, and , the anisotropic one from the neighbor atomic planes. The last turn depends on the layer shift and the small deviation x. Therefore, only term changes inside TB which is shown in Figure 2. The analysis (see ) of the term allows to write the anisotropic atomic potential Eq. (10) in the following simple form:
where is rigidity coefficient inside TB, is a varied bottom level, and is the linear part. In the limit points , Eq. (11) transforms into isotropic hcp phase Eq. (9) with . Inside TB , the rigidity takes value . Thus, the rigidity coefficients in phase and in the middle of TB are represented by the rigidity coefficients inside the plane with two adjacent planes:
Inside the boundary the potential is considerably softer in direction because of (see Figure 2). The difference in these rigidity coefficients is too high . 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 describe the longitudinal deformation along the axes and 0z, respectively. In solid 4He, the ratio of the elastic modulus gives anisotropy of the rigidity coefficients in the basal plane and axis 0z. Uniaxial compression-tension in the basal plane of 0xy corresponds to the elastic modulus of C11 and atomic rigidity coefficient . The shuffle of the basal planes in an arbitrary direction corresponds to elastic modulus and atomic rigidity coefficients . Therefore, we have the following inequality:
4. The atomic potential and hard sphere model in hcp phase
where R0 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 centers of the shifting atomic plane B can move over the following four spherical surfaces:
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 ):
Signs – and + in 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
For the hard sphere model, the microscopic parameters are
where is the potential barrier between B to C position (see Figure 1). Coefficient 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 comparison, Eq. (11) allows us to find the rigidity coefficients in the phase and in the middle of the boundary .
5. Quantum atomic spheres and ellipsoids in hcp phase and in the twin boundary
where and are mass, radius vector of 4He atom and parameter of the quantum oscillator. The potential Eq. (12) gives .
The Schrodinger equation splits into three equivalent independent equations with the constant where are wave numbers. The ground state solution  has total zero-point energy . In isotropic harmonic approximation, a distribution of probability density of helium atom has spherical symmetry. Hence, the equation of probability isosurface (sphere of radius R) is
The probability density at a distance of R0 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
Here we have introduced the dimensionless parameter that is important for further consideration. This parameter is proportional to the atomic radius and depends on the isotropic rigidity of the atomic lattice . In respect to a huge change in the volume of solid helium , the parameter can vary widely.
An anisotropic harmonic potential can be written as 
The parameters are related to the rigidity coefficients:
In the hcp phase, an anisotropic harmonic approximation is more adequate. Then the rigidity coefficients satisfy inequality . If we use isotropic harmonic approximation in the hcp phase, then inside of the twin boundary, an atom 4He is in a uniaxial potential of neighboring atoms of Eq. (13): .
The equation splits also into three independent equations with known solutions . 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 :
Parameter describes the probability density. If the probability density equals at the atomic radius R0 in the hcp phase Eq. (23), then takes the following value:
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 where and 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 ):
Then inside of the twin boundary, an atom is in the uniaxial potential of neighboring atoms Eq. (13): .
The motion equation splits also into three independent equivalent equations. The equation of the potential isosurface is ellipsoid with semiaxes (compare with Eq. (26)):
Thus, the relation Eq. (30) describes the atomic potential isosurfaces in the anisotropic case, i.e., inside TB. In the limit case , 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 . We emphasize that in this section the average thermal motion of atoms was considered.
7. The self-consistent description of the twin boundary
The classic description of TB uses two coefficients of the thermodynamic potential Eq. (1):
The quantum and thermal description of TB is self-consistent, i.e., the parameters Eq. (31) are varied as a function of some parameter that, in its turn, is a function of these parameters:
where is the eccentricity of the ellipse. Earlier in the paper , we introduced the quantum deformation parameter . Here we generalize the parameter to the cases of either quantum or thermal motion of an atom and introduce the isosurface deformation parameter .
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:
(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 0x:
The further variations of parametersEq. (31) can be obtained in the hard ellipsoid model. The hard ellipsoids have the isosurfaces with the same probability density 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 .
(2) The second approximation. An atom is considered as an anisotropic three-axis 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
Now all three axes of the atomic ellipsoid are different. The softest potential and the longest axis a2 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 ith 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 in Eq. (25) take the following forms:
For fixed and reduced stiffness coefficient along axis 0x, the semiaxes of the ellipsoid change as follows: . Then the isosurface deformation parameter q1Eq. (35) takes the following dependence on the order parameter and coordinate
In He and Mg (see Table 2), the transverse components of the elastic module C44 are much smaller than the longitudinal ones C11. 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 are considerably greater. Moreover in Li, the parameter 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 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 :
Increasing overlapping volumes with high probability can be evaluated by segments of the crossing ellipsoids. Amplitude depends on two parameters and 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 . In the interphase boundaries in solid helium, NMR experiment  shows the quantum diffusion increasing. The interphase and twin boundaries are similar . 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 ; the found thermal ellipsoids’ deformation qualitatively explains these facts.
Now we can point out conditions when exchange integral Eq. (41) increases. We need minimal Eq. (42) in exponent Eq. (41). The parameter or can be defined by Eqs. (23) and (37) and analyzed in dependence on different factors. In  using atomic mass ma and evaluation of atomic radius R0, the parameter value was estimated:
where elastic module is related to the rigidity coefficient . In solid 4He, the atomic radius 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 gives rise in the exchange integral too. The van der Waals interaction in 4He is 3–4 orders of magnitude less than in metal (see Table 1).
Another way to estimate is to compare it with de Boer parameter , 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 ) :
The de Boer parameter for 4He  gives evaluation . Pressure growing leads to more difficult tunneling of atoms and different evaluations in Eqs. (42) and (43). Using the data in Table 1, for solid 4He we obtain , , and (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.
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:
Accounting these relations and condition (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:
where is defined in Eq. (19). These results are valid in the range . At we have and semiaxis relation . At the hard ellipsoid model needs transition in another state (see ). Therefore, inside TB, the change of the atomic wave function leads to the following change of 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
Thus, the coefficients of the potential (1) for the shift in the direction reduce and . 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):
For the classical and quantum cases, the free energy density relation Eq. (4) was analyzed analytically in . 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 grows by Eq. (6). The dependence causes further widening of region with and a minimal barrier height.
To estimate the energy of the twin boundary (stacking fault) from Eq. (7), we must know the following parameters: or .
The characteristic width (half width) of TB Eq. (6)nm was obtained by molecular dynamic method in . We estimate the dispersion parameter by comparing the differential equations for the transverse sound and shuffling waves:
where is the effective density of the oscillating shuffled subsystem, is the density of helium-4, is the shuffling order parameter, is macroscopic displacement, and is the transverse sound velocity in the shuffle direction (axis). The velocities of transverse sound and shuffling wave have close values. So the dispersion parameter is where value of module C44 is given in Table 1.
According to relation Eq. (5), it is possible to estimate the parameter of the thermodynamic potential . As follows from Eq. (2) to evaluate the parameter k4 of the potential, it is necessary to know the maximum displacement of the atom Eq. (17). Here atomic radius is related to atomic volume: . Then . 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):
The surface energy density calculated here for the classical model can be compared with the value found in the optical experiments at 0.2 K .
Above, we have predicted the local reduction of the barrier height h and a local increase in the width of the boundary in the quantum description of the twin boundary (stacking fault). In general for the defect, the surface energy density value 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 3He 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 4He and 3He, 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
The quantum self-consistent treatment to twin boundary (stacking faults), proposed in  for solid 4He, 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 (see Table 2). Meanwhile at 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 .
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 ). The estimation Eq. (50) of the defect energy is in good agreement with experiment . 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 4He were investigated in . For small values of the perturbations, dynamical differential equations (reduced to 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
where is an amplitude. For the local vibration ground state (51) and for the isosurface deformation parameter qEq. (39), both shapes coincide qualitatively. In the limit , 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.
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).