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Publicly Available Published by De Gruyter February 1, 2019

High-Pressure Third-Order Elastic Constants of MgO Single Crystal: First-Principles Investigation

  • Jianbing Gu , Chenju Wang EMAIL logo , Bin Sun , Weiwei Zhang and Dandan Liu

Abstract

High-pressure third-order elastic constants of materials have rarely been investigated experimentally and theoretically to date, so the predictive ability of the method of the volume-conserving, homogeneous deformations based on the first-principles total-energy calculations is tested for the first time in this work. Using this approach, the high-pressure third-order elastic constants C1113C112+2C123, C111/2+3C112+C123, C144C155, and C456 of the MgO single crystal are obtained successfully. Meanwhile, the reliability of this method is also verified by comparing the calculated structural properties and high-pressure second-order elastic constants of the MgO single crystal with the available experimental results and other theoretical predications. Results not only indicate the accuracy of our calculations but also reveal the feasibility of the present theoretical method. It is hoped that the present theoretical method and predictions on the high-pressure third-order elastic constants of the MgO single crystal would serve as a valuable guidance or reference for further related investigations.

1 Introduction

In the theory of nonlinear elasticity, the third-order elastic constants (TOECs) are important quantities for characterising the nonlinear elastic properties of materials, and numerous studies [1], [2], [3], [4], [5], [6] have been carried out to investigate them. Most of these studies, to the best of our knowledge, are restricted to the TOECs of materials at zero pressure, and little research on the high-pressure TOECs has been performed till now. Nevertheless, knowledge of the high-pressure TOECs of materials can assist us in predicting and understanding the interatomic forces, mechanical stability [7], [8], dynamic fracture [9], earthquakes, and the internal structures of the Earth. Therefore, investigations of the high-pressure TOECs are very necessary and interesting.

Experimentally, the subtle TOECs are quite sensitive to the experimental conditions and usually exhibit distinct divergence between the values measured by different teams (see [10], [11]). Up to now, experimental determination of the zero-pressure TOECs still presents significant difficulties [2], [3], to say nothing of the measurement of the high-pressure TOECs. However, the accuracy and reliability of theoretical calculations have substantially increased recently, and the computed results have been accepted by the research community almost at the same level of trust as the measured data. Naturally, theoretical calculations have become an important approach to predict the high-pressure TOECs of materials. At present, several theoretical methods [12], [13], [14], [15], [16] have been developed to calculate the zero-pressure TOECs of materials, such as the approach of homogeneous deformation based on empirical or first-principles total-energy calculations [12], the empirical interatomic force constant models [13], molecular dynamics simulation using the fluctuation formula [14], and other quantum methods [15], [16]. Among these approaches, the method of homogeneous deformation based on the first-principles total-energy calculations seems to be the most promising one and it has been successfully employed in the literature [1], [4], [5], [6]. Nevertheless, this approach has not been used to calculate the high-pressure TOECs of materials yet.

In this work, we take the MgO single crystal as an example and try to obtain its high-pressure TOECs by using the method of the homogeneous deformation based on first-principles total-energy calculations. The reasons for choosing the MgO single crystal are as follows: First, MgO, as one of the major earth-forming minerals, has great geophysical importance. Prediction of its high-pressure TOECs is very important for interpreting the seismically observed velocity variations in the Earth’s mantle and for constructing reliable mineralogical modes of the Earth’s interior [17]. Second, MgO, as one of the most stable oxides, can retain its simple NaCl-type structure under very high pressure (>227 GPa) [18], thereby making it an ideal candidate for the calculations of its high-pressure TOECs. Third, the MgO single crystal belongs to the cubic structure, which has the least number of independent TOECs among all single crystals; accordingly, the number of the homogeneous deformations required for obtaining the complete set of high-pressure TOECs is the least. Last but not least, many experimental and other theoretical results on its structural and elastic properties are available and they can be employed to verify the accuracy of the present calculations. The rest of this paper is organised as follows: In Section 2, the nonlinear elastic theory, special deformations required for the calculations of the high-pressure TOECs, and computational details are described briefly. Subsequently, the calculated results and discussion are given in Section 3. Finally, conclusions are drawn in Section 4.

2 Nonlinear Elastic Theory and Computational Details

2.1 Nonlinear Elasticity Theory

In the theory of nonlinear elasticity, we suppose that the un-deformed state X corresponding to a certain pressure P moves to the deformed state x when a crystal undergoes a finite deformation. Here, we denote the un-deformed coordinates of a point in configuration X by (X1, X2, X3) and represent those in configuration x by (x1, x2, x3). Then, the deformation gradient matrix can be defined as

(1)αij=xi/Xj,

where the subscripts i and j vary from 1 to 3. The Lagrangian strain ηij can be expressed as

(2)ηij=12(kαkiαkjδij),

in which δij is the Kronecker delta (unity when i = j, and zero otherwise). The volume ratio of the two configurations X and x can be given as

(3)V(x)/V(X)=J=det[αij],

so the condition describing the conservation of volume is

(4)det[αij]=1.

In addition, the corresponding change of the Helmholtz free energy F and the internal energy U per unit mass in the configuration x are [19], [20]

(5a)F(X,ηij,T)=F(X,0,T)+V(X)ijTij(X)ηij+12!V(X)ijklCijklT(X)ηijηkl+13!V(X)ijklCijklmnT(X)ηijηklηmn+

and

(5b)U(X,ηij,S)=U(X,0,S)+V(X)ijTij(X)ηij+12V(X)ijklCijklS(X)ηijηkl+13!V(X)ijklCijklmnS(X)ηijηklηmn+.

Then the corresponding isothermal and adiabatic stress and the second-order and third-order elastic constants at X state can be defined as

(6)TijT(X)=1V(X)Fηij|XηmnT,TijS(X)=1V(X)Uηij|XηmnS,
(7)CijklT(X)=1V(X)2Fηijηkl|XηuvT,CijklS(X)=1V(X)2Uηijηkl|XηuvS,

and

(8)CijklmnT(X)=1V(X)3Fηijηklηmn|XηuvT,CijklmnS(X)=1V(X)3Uηijηklηmn|XηuvS,

where V(X) is the volume of the system at X state, and the subscript ηuv denotes that the elements of the strain tensor other than those involved in the partial derivative are held constant. T and S, respectively, represent the isothermal and adiabatic processes, and Tij(X) is the component of the stress tensor applied on the X state. Specifically, when the X state is under hydrostatic pressure, Tij(X) can be expressed as TijS(X)=TijT(X)=Pδij [21]. Here we do not distinguish between the two types of the elastic constants because we perform the first-principles calculations at 0 K; i.e. F=UTS=U and CS=CT.

In the Voigt convention (η11η1, η22η2, η33η3, η23η4, η13η5, and η12η6), the cubic crystal has three independent second-order elastic constants (SOECs), namely C11, C12, and C44, and six independent TOECs, namely C111, C112, C123, C144, C166, and C456. Therefore, (5a) can be expressed as

(9)F(X,ηij)=F(X,0)+V(X)i=13Ti(X)ηi+V(X)φ2+V(X)φ3,

where

(10)φ2=12C11(η12+η22+η32)+C12(η1η2+η2η3+η1η3)+12C44(η42+η52+η62),

and

(11)φ3=16C111(η13+η23+η33)+12C112(η12η2+η12η3+η22η1+η22η3+η32η1+η32η2)+C123η1η2η3+12C144(η42η1+η52η2+η62η3)+12C166(η42η2+η42η3+η52η1+η52η3+η62η1+η62η2)+C456η4η5η6.

If we apply a series of proper, homogeneous deformations to the X state and calculate the total energies of the X and x states using the first-principles method, we can easily obtain the corresponding elastic constants by using (9)–(11). In the following, the specific deformations and computational details of the total energies are presented.

2.2 Deformation Types

For a cubic crystal, the calculation of the high-pressure TOECs is similar to that of the high-pressure SOECs. Specifically, when the SOECs of a cubic crystal under a certain pressure P are calculated, we first use the two volume-conserving homogeneous deformations displayed as matrixes (30) and (32) in [21] to obtain the two shear elastic constants (C11C12)/2 and C44, and then use the pure volume deformation to calculate the bulk elastic constant (C11+2C12+P)/3. Finally, the independent C11 and C12 are obtained, respectively, by connecting (C11C12)/2 with (C11+2C12+P)/3. Similarly, the following three volume-conserving homogeneous deformations [22] and one pure volume deformation are used to calculate the high-pressure TOECs of the MgO single crystal.

To determine the elastic constant C111 − 3C112 + 2C123, we adopt the volume-conserving homogeneous deformation

(12)αij=(1+η)(1/3)[1+η00010001],

which increases the length of all vectors parallel to [100] and uniformly reduces that of all vectors perpendicular to [100]. According to (2), we calculate the corresponding Lagrangian strain ηij. Then we substitute it into (5a) to obtain

(13)FA(η)=F(0)+13(C11C122P)Vη2+127[(C111 3C112+2C123)3(C11C12)+14P]Vη3+O(η4),

where η is the strain magnitude, F(η) and F(0), respectively, represent the total energies of the strained and unstrained unit cells, V is the volume of unstrained unit cell, and P denotes the pressure applied on the reference state (state X).

With the purpose of calculating the elastic constant C456, the deformation (14) is used:

(14)αij=(13η2+2η3)(1/3)[1ηηη1ηηη1].

It represents that all vectors parallel to [111] are increased while all vectors perpendicular to [111] are uniformly reduced. And the corresponding expression for the energy is

(15)FB(η)=F(0)+6(C44P)Vη2+(8C456+6C44+2P)Vη3+O(η4).

With the view to obtaining the elastic constant C144C155, we employ the deformation

(16)αij=(13η2η3)(1/3)[10η01+ηηηη1η]

and its related energy expansion

(17)FC(η)=F(0)+(C11C12+4C446P)Vη2+[2(C144C155)12(C11C12+4C44)+P]Vη3+O(η4).

Finally, we apply the uniform and isotropic (hydrostatic) deformation (18) to calculate C111/2+3C112+C123:

(18)αij=(1+η0001+η0001+η).

Accordingly, the total energy of the strained state can be expressed as

(19)FD(η)=F(0)+32(C11+2C12+P)Vη2+(C111/2+3C112+C123)Vη3+O(η4).

To the best of our knowledge, only the four homogeneous deformations (12), (14), (16), and (18) could be employed to calculate the high-pressure TOECs of the MgO single crystal so far. Consequently, we can only obtain C1113C112+2C123, C111/2+3C112+C123, C144C155, and C456 at present instead of the complete set of high-pressure TOECs. This may be the main reason why the high-pressure TOECs of crystals have not been investigated previously.

2.3 Computational Details

2.3.1 Density Functional Theory (DFT) Calculations

It is well known that the NaCl-type (B1) structure of the MgO single crystal will transform into the CsCl-type (B2) structure under high pressure. And knowledge of the structural stability of the B1 structure is a precondition for the calculations of the high-pressure TOECs, so the structural optimisations and calculations of the total energies for the two structures B1 and B2 are performed using the Vienna ab initio Simulation Package (VASP) [23]. In particular, the generalised gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) parameterisation [24] is used to treat the exchange-correlation effects. Simultaneously, the projector augmented-wave (PAW) potential [25] is employed to deal with the ion-electron interactions. In addition, careful convergence tests (tolerance for the total-energy difference in the self-consistent-field calculation is 1.0 × 10−9 eV and tolerance for the maximum force in the geometry optimisation is 1.0 × 10−5 eV/Å) are performed to determine the cut-off energy and the number of k-points. And the reasonable cut-off energy (700 eV for B1 and 650 eV for B2 structures) and rational k-points (17 × 17 × 17 for B1 and 15 × 15 × 15 for B2 structures) are ascertained. Afterwards, the total energies of various unit cell volumes in both B1 and B2 structures are calculated and a series of energy-volume data are obtained. Then, we obtain their equation of states (EOS) at 0 K, equilibrium volumes V0, bulk modulus B0, and their pressure derivatives B0 by fitting the energy-volume data with the fourth-order Brich-Murnaghan equation E(V)=n=14anV2n/3 [26]. Besides, the P-V data are also obtained from P=(E/V)T=0, which is the basis for the calculations of the TOECs at different pressures.

2.3.2 Determination of the Third-Order Elastic Constants

To calculate the TOECs of the single-crystal MgO in B1 structure at a certain pressure, we first ascertain the equilibrium state (un-deformed state) under this pressure by relaxing the cell parameters and then obtain the deformed state by using the equation r=iαijrj [27], where ri and rj, respectively, represent the deformed and un-deformed lattice vectors. As for the strain magnitude η, we set it from |ηmax| to +|ηmax| in steps of 0.001, where ηmax is the maximum value of the strain parameter. For each value of η, the total energies of the deformed and un-deformed states are calculated. Repeating this procedure, the relations of f(η) and η in (13), (15), (17), and (19) are obtained. Then by fitting these relationships with proper polynomials, the coefficients corresponding to the elastic constants can be obtained directly.

Since TOECs are very sensitive to the cut-off energy, k-points, the maximum strain magnitude |ηmax|, and the order of the polynomial used to fit the relationships (13), (15), (17), and (19), careful convergence tests are performed to obtain the high-pressure TOECs accurately. In the following, we take C1113C112+2C123, C144C155, and C456 as examples to examine the dependence of the TOECs on the four parameters mentioned above. First of all, the dependence of C1113C112+2C123, C144C155, and C456 on the cut-off energy is investigated, as displayed in Figure 1. As is clear from the figure, all the elastic constants converge well after the cut-off energy reaches 550 eV. Nevertheless, to determine the subtle TOECs of MgO under high pressure exactly, Ecutoff = 700 eV is adopted in our calculations. And we believe Ecutoff = 700 eV is enough to obtain accurate data of the high-pressure TOECs. Subsequently, convergence test of C1113C112+2C123, C144C155, and C456 as a function of the k-points is performed, as illustrated in Figure 2. As shown in Figure 2, all the elastic constants converge well when the k-point mesh size reaches 15 × 15 × 15 and the relative difference between two successive values of an examined constant is less than 1.0 %; even so, the 17 × 17 × 17 k-point is selected in our computations to guarantee the accuracy of the high-pressure TOECs. Then, we test the dependence of the TOECs on the maximum strain magnitude |ηmax|, which is displayed in Figure 3. From Figure 3, we can clearly find that all the TOECs converge well after the maximum strain parameters |ηmax| increase to 0.055. Therefore, |ηmax| = 0.055 is employed to calculate the high-pressure TOECs of the single-crystal MgO.

Figure 1: Dependence of the third-order elastic constants on the cut-off energy (17 × 17 × 17 k-point is applied for all points).
Figure 1:

Dependence of the third-order elastic constants on the cut-off energy (17 × 17 × 17 k-point is applied for all points).

Figure 2: Convergence test of the third-order elastic constants as a function of the k-points (cut-off energy of 700 eV is applied for all points).
Figure 2:

Convergence test of the third-order elastic constants as a function of the k-points (cut-off energy of 700 eV is applied for all points).

Figure 3: Dependence of the third-order elastic constants on the maximum strain magnitude |ηmax|$\left|{{\eta_{\max}}}\right|$.
Figure 3:

Dependence of the third-order elastic constants on the maximum strain magnitude |ηmax|.

Based on the parameters ascertained above, the relations of η and fα(η) (α = A, B, and C) are obtained. What remains is to fit these relationships via suitable polynomials. To determine the desirable order of the polynomials, the differences between the residuals, obtained, respectively, from the third-order polynomial fits and fourth-order polynomial fits, are presented in Figure 4. Figure 4 clearly shows that the residuals obtained from the fourth-order polynomial fits are smaller than those obtained from the third-order polynomial fits. Thus, the fourth-order polynomial is adopted to fit the relations of fα(η) and η. For example, we give the calculated fα(η)η (α = A, B, and C) data in Figure 5 and fit them using the fourth-order polynomial. Finally, the coefficients corresponding to the SOECs and TOECs are obtained directly.

Figure 4: Relative residuals as a function of the Lagrangian strain η. The dashed lines denote the residuals obtained from the fourth-order polynomial fits, the symbols represent the residuals obtained from third-order polynomial fits. (a), (b), and (c), respectively, represent the relative residuals obtained when the polynomial was fitted to the relations (13), (15), and (17).
Figure 4:

Relative residuals as a function of the Lagrangian strain η. The dashed lines denote the residuals obtained from the fourth-order polynomial fits, the symbols represent the residuals obtained from third-order polynomial fits. (a), (b), and (c), respectively, represent the relative residuals obtained when the polynomial was fitted to the relations (13), (15), and (17).

Figure 5: Strain energy density as a function of the Lagrangian strain η, where the discrete points and the solid lines represent, respectively, the results obtained from first-principles calculations and the nonlinear fits. fA, fB, and fC denote the energy expressions (13), (15), and (17), respectively.
Figure 5:

Strain energy density as a function of the Lagrangian strain η, where the discrete points and the solid lines represent, respectively, the results obtained from first-principles calculations and the nonlinear fits. fA, fB, and fC denote the energy expressions (13), (15), and (17), respectively.

3 Results and Discussion

3.1 Structural Properties

To verify the accuracy of our DFT calculations, we list the computed structural parameters (equilibrium lattice parameter a, bulk modulus B0, and its pressure derivations B0 and B0′′) for both the B1 and B2 structures of the MgO in Table 1, and compare them with the available experimental results [31], [29], [35], [39], [37] and other theoretical data [32], [34], [38], [28], [30], [33], [42], [43], [36]. Table 1 clearly shows that our structural parameters are in good agreement with the corresponding experimental as well as previous theoretical values. Experimentally, for the B1 structure of MgO, the calculated lattice constant 4.256 Å is slightly larger than the experimental data of 4.19 Å [29] and 4.213 Å [31], whereas the computed bulk modulus B0 (155.5 GPa) is somewhat smaller than the measured values (160–168.8 GPa) [29], [31], [35], [37]. Similarly, when the present value 4.14 of the B0 is compared with the corresponding experimental ones of 4.15 [31] and 4.5 [39], we find that it is also underestimated by about 0.2–8.0 %. Such phenomena can be attributed to the following two aspects. On one hand, the theoretical data is calculated at 0 K while the experimental results are usually obtained at 300 K. On the other hand, GGA calculations typically overestimate the equilibrium lattice parameters and underestimate the bulk modulus of materials at zero pressure. Theoretically, all the computed structural properties of the B1 structure are well consistent with the previous theoretical values. As regards the structural parameters of the B2 structure, though there is no corresponding experimental results for comparison at present, all the data obtained in this work show excellent agreement with the previous theoretical predications [30], [42], [43].

Table 1:

Equilibrium lattice parameters, bulk modulus B0 and its pressure derivations B0 and B0′′, as well as the transition pressure Ptr.

Present workTheoretical referencesExperimental references
NaCl-type structure
a (Å)4.2564.250[28]4.19[29]
4.240[30]4.213[31]
4.273[32]
4.259[33]
B0(GPa)155.5158[34]168.8[35]
153.9[28]160[31]
158.7[36]163[37]
153[32]164.6[29]
160[33]
B04.144.18[38]4.15[31]
4.26[34]4.5[39]
3.95[36]
4.00[30]
B0′′ (GPa−1)−0.028−0.025[30], [38]
−0.026[34]
−0.024[36]
Ptr (GPa)507490[30]>227[18]
510[40]
509[28]
515[41]
512[32]
CsCl-type structure
a (Å)3.3523.32[30]
B0 (GPa)137.493140.30[42]
134.33[43]
B04.1294.10[42]
4.24[43]
B0′′ (GPa−1)−0.03
  1. Previous experimental results [18], [29], [31], [35], [37], [39] and other theoretical calculations [28], [30], [32], [33], [34], [36], [38], [42], [43], [40], [41] are also included for comparison.

In addition, the pressure dependence of our volume ratio (V/V0, where V0 is the equilibrium unit cell volume of the B1 structure) is also compared with that obtained experimentally [18], [44], [45], [46], [47], [48], [49] and in previous calculations [38], [28], [50], [51], [52], [53], [54], which is illustrated in Figure 6a and b, respectively. From Figure 6a, we see that our results are slightly smaller than those measured by Mao and Bell [44] in the pressure range 50–95 GPa. This may be induced by the existence of the non-hydrostatic stresses in their experiment. However, it is delightful that our theoretical results are consistent with the other experimental values [18], [45], [46], [47], [48], [49]. Besides, comparison between our data and previous theoretical results [28], [38], [50], [51], [52], [53], [54] shown in Figure 6b also shows satisfactory agreement. These good consistencies indicate that the choice of the PAW pseudo-potential and the GGA-PBE approximation is reasonable for the current study and that the present calculations are reliable.

Figure 6: Volume ratio versus pressure. (a) Comparison between our calculations and available experimental values [18], [44], [45], [46], [47], [48], [49]. (b) Comparison between our results and other theoretical data [50], [51], [52], [53], [54].
Figure 6:

Volume ratio versus pressure. (a) Comparison between our calculations and available experimental values [18], [44], [45], [46], [47], [48], [49]. (b) Comparison between our results and other theoretical data [50], [51], [52], [53], [54].

To ascertain the highest pressure of TOECs of the B1 structure of the MgO single crystal, we calculate the enthalpies H = E + PV of the B1 and B2 structures at 0 K and display them in Figure 7. For the sake of clarity, we also give their enthalpy differences as a function of pressure in the insert of Figure 7. It is clear from Figure 7 that the phase transition (B1⟶B2) of the MgO single crystal occurs at about 507 GPa. For comparison, we list it together with the available experimental results [18] and other theoretical predications [28], [30], [32], [40], [41] in Table 1. Results show that our predicted value 507 GPa is very close to the previous data of 509 GPa [28], 490 GPa [30], 512 GPa [32], 510 GPa [40], and 515 GPa [41]. The only small differences (about 0.4–3.5 %) between them may originate from the different computational procedures and forms of exchange correlation functional used by different groups. That is to say, the transition pressure predicted in this study is reliable. Therefore, the high-pressure elastic constants of the MgO single crystal in the B1 structure are investigated only up to 500 GPa in the following.

Figure 7: Pressure dependence of the enthalpies of the NaCl and CsCl structures. In the inset are shown the enthalpy differences ΔH of the CsCl structure with respect to the NaCl structure as a function of pressure.
Figure 7:

Pressure dependence of the enthalpies of the NaCl and CsCl structures. In the inset are shown the enthalpy differences ΔH of the CsCl structure with respect to the NaCl structure as a function of pressure.

3.2 Elastic Constants

Since the experimental and previous theoretical values of the high-pressure SOECs of MgO in the B1 structure are available, and they can be used to test the accuracy of our high-pressure elastic constants, the high-pressure SOECs are considered first before the high-pressure TOECs are discussed. In Figure 8, pressure dependence of B44 (B44=C44P) [21] and BT (BT=(C11+2C12+P)/3) is displayed and compared with other theoretical data [28] and the existing experimental results [45], [55]. With regard to B44, the agreement between the present values and previous theoretical data is satisfactory, whereas the difference between our calculated results and the available experimental ones increases gradually with pressure. Such a phenomenon was also observed by Karki et al. [34] and Oganov and Dorogokupets [28]. We deem that it is independent of our calculation method and that it may be originating from the experimental techniques themselves, especially the non-hydrostatic technique used in the measurement of the high-pressure elastic constants. For BT, the calculated values show excellent agreement with the existing experimental results and other theoretical predications. These consistencies suggest that our calculation is performed at high accuracy and the elastic constants obtained in the present study are authentic.

Figure 8: Calculated B44 (upper) and BT (lower) of the MgO single crystal as a function of pressure. The existing experimental results [45], [55] and previous theoretical predication [28] are also displayed.
Figure 8:

Calculated B44 (upper) and BT (lower) of the MgO single crystal as a function of pressure. The existing experimental results [45], [55] and previous theoretical predication [28] are also displayed.

Having established the accuracy of the elastic constants calculations, we now turn to the TOECs of the MgO single crystal. Among the six independent TOECs, namely C111, C112, C123, C144, C155, and C456, only C456 is calculated directly through the four homogeneous deformations (12), (14), (16), and (18). So the zero-pressure value 106.49 GPa of C456 is first compared with the existing experimental data of 147 GPa [39] and other theoretical values of 307.9 GPa [56] and 356 GPa [57]. Comparisons show that our C456 agrees much better with the experimental data than the previous theoretical results. And the difference between the present result and the experimental data may be caused by the following two factors. First, it is rather difficult to measure TOECs experimentally, which leads to considerable uncertainties in the reported experimental results inevitably. Second, the theoretical result is calculated at 0 K while the experimental values are often determined in conditions that are far from the ideal case of 0 K. Particularly, the temperature effects on the TOECs of the MgO are distinguishable and should not be ignored (see [29]). Therefore, the deviation between them is reasonable and understandable.

With regard to the high-pressure C111 − 3C112 + 2C123, C111/2 + 3C112 + C123, C144C155, and C456, there is no related experiment or calculation till now, so our predications first obtained by using the method of the volume-conserving homogeneous deformation based on first-principles total-energy calculations are expected to serve as a valuable guidance or reference for further related investigations. As shown in Figure 9, the absolute values of C1113C112+2C123, C111/2+3C112+C123, C144C155, and C456 increase gradually with the increase in pressure, which is similar to the behaviour of high-pressure SOECs. In addition, we also find that the former two TOECs depend much more strongly on pressure than the latter two TOECs. Specifically, C1113C112+2C123 and C111/2+3C112+C123 increase by about 48 % and 23 %, respectively, per pressure difference of 20 GPa, whereas the increases of C144C155 and C456 are only about 5.5 % and 3.2 %, respectively, per pressure difference of 20 GPa. Besides, the rising behaviour of the four high-pressure TOECs displayed in Figure 9 also implies that (i) the anharmonic properties of the MgO single crystal under high pressure become more obvious than those at zero pressure; (ii) the electron interaction in the MgO single crystal at high pressures becomes stronger than at zero pressure; and (iii) the convergence of expressions (13), (15), (17), and (19) becomes more and more difficult with increasing pressure. Of course, there may be other explanations for this rising phenomenon of the high-pressure TOECs, and hence further related experimental and theoretical investigations are much needed.

Figure 9: Predicted third-order elastic constants of the MgO single crystal in the NaCl structure as a function of pressure.
Figure 9:

Predicted third-order elastic constants of the MgO single crystal in the NaCl structure as a function of pressure.

Finally, it is worth noting that though the six independent high-pressure TOECs C111, C112, C123, C144, C155, and C456 of the MgO single crystal could not be calculated separately in the present work, the first attempt to calculate the high-pressure TOECs of crystals by using the method of the volume-conserving homogeneous deformation based on first-principles total-energy calculations has been successful and encouraging. We hope that the present work would make a useful contribution to the development of nonlinear elasticity at high pressures.

4 Conclusions

The predictive ability of the method of the volume-conserving homogeneous deformation based on the first-principles total-energy calculations for computing the high-pressure TOECs of single crystals was tested successfully for the first time in the present work. Using this approach, the pressure dependence of C1113C112+2C123, C111/2+3C112+C123, C144C155, and C456 was obtained in the pressure range 0–500 GPa of the MgO single crystal with the B1 structure. Results showed that the absolute values of the four TOECs increase gradually with increasing pressure. In addition, the structural properties and high-pressure SOECs of the MgO obtained in this work were also compared with the available experimental results and other theoretical data to verify the reliability of the method. The good agreement between them not only indicates that our results are accurate and reliable but also reveals that the theoretical approach used in this study is practicable and can be expanded into other complex systems.

Award Identifier / Grant number: 11747062

Award Identifier / Grant number: 11747110

Award Identifier / Grant number: 11504035

Funding statement: This work was supported by the National Natural Science Foundation of China (Funder Id: 10.13039/501100001809, Grant Nos. 11747062, 11747110 and 11504035), the Science and Technology Tackling Project of the Education Department of Henan Province (Grant No. 172102210072), the Key Scientific Research Project of Higher Education of Henan Province (No. 17A140014), and the Science and Technology Research Project of the Chongqing Education Committee (No. KJ1703062).

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Received: 2018-11-07
Accepted: 2019-01-04
Published Online: 2019-02-01
Published in Print: 2019-05-27

©2019 Walter de Gruyter GmbH, Berlin/Boston

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