High-Pressure Elastic Constant of Some Materials of Earth’s Mantle

1 Introduction

The behaviour of earth materials at high pressure is central to our understanding of structure, dynamics, and origin of the earth. Over the range of conditions that exist within the earth’s mantle, the physical properties of condensed matter depend more strongly on pressure than on other factors such as temperature [1]. Elastic constants and their pressure derivatives of the mantle materials are very important to predict the effect of pressure on crystal lattice of minerals and to construct mineralogical models of the earth’s interiors consistent with seismological information.

Seismological observations of the earth’s interior in principle contain a wealth of information concerning its composition, thermal state, and dynamics. This information has proved difficult to extract, primarily because the elastic constants of mantle minerals under the relevant condition are essentially unknown [2]. Measurements of the elastic constants of mantle materials exist at ambient pressure and up to mantle temperatures for some materials. Experimental advances have led to the first measurements of elastic constants of mantle minerals at relevant pressures [3–5]. A variety of techniques including light-scattering and ultrasonic methods hold tremendous promise for revealing the elasticity of mantle minerals at mantle pressure and temperature conditions. These methods have not yet been applied to the major minerals of the lower mantles.

It is worth mentioning that, in 2005, Stixrude and Lithgow-Bertelloni [2] presented a theory for the computation of phase equilibria and physical properties of multicomponent assemblages relevant to the mantle of the earth. The theory is based on the concept of fundamental thermodynamic relations appropriately generalised to anisotropic strain and in encompassing elasticity in addition to the usual isotropic thermodynamic properties. Under the framework of this theory, they extrapolate the elasticity of materials to high pressure and temperature [2].

Major advances in theory and computation now make it possible to calculate the full elastic constant tensor of mantle minerals from the first principles methods based on the density functional theory [6–8]. The density functional theory [9] has become a powerful tool for examining the behaviour of earth materials at high pressure. Central to the theory is the proof that ground-state properties are a unique functional of the charge density; it is not necessary to solve for the complete 10²³-dimensional total wave function.

Whereas the first principles methods seek to reduce approximations to a bare minimum, ab initio methods construct an approximate model of some aspects of the relevant physics, such as the charge density or the interactions between orbitals. The ab initio model has been applied to the calculation of elastic constants of earth materials [10]. Moreover, these models often yield insight that is sometimes difficult to extract from more complex and elaborate first principles calculations.

Semi-empirical methods differ from those discussed thus far in that they often do not view the solids as being composed of nuclei and electrons, but of larger entities. The primary advantage of these models is that they are simple and rapid [11–13]. They are useful to the extent that they faithfully interpolate or extrapolate existing experimental or theoretical results or provide additional insight not otherwise available. Thus, it is legitimate and may be useful to propose a simple and straightforward method for the determination of the pressure dependence of the elastic constants of mantle materials, which is the purpose of the present work.

2 Method of Analysis

The parameter δ_T, known as the Anderson–Grüneisen parameter [14], is defined as follows:

where α and B_T are the volume thermal expansion coefficient and the bulk modulus, respectively. Knowledge of δ_T is important in many high-pressure studies, in particular those involving α, B_T, γ (Grüneisen parameter), S (entropy), and P_TH, the thermal pressure. Combining (1), a definition, with the well-known identity

It can be shown that the following identity exists:

This equation is an exact thermodynamic relation. However, if we start from the most widely used approximation αB_T=constant [15], we will get the following equation:

This shows that volume dependence of B_T is controlled by δ_T. Suppose that δ_T is independent of volume; then by integration of (4), we have along an isotherm:

However, it was shown by Anderson et al. based on investigations made by earlier worker that δ_T is dependent on V. Anderson et al. investigated the following relationship for the variation of δ_T along isotherms [16, 17]:

where k is a dimensionless thermoelastic parameter whose value is about 1.4 [16, 17]. It is interesting to mention here that for k=1, this equation reduces to the basic assumption considered in his formulation by Tallon [18]. In this paper, the value of k is taken 1.4.

If using (6) in (4), we obtain the following expression:

This equation is a formula for the volume dependence of bulk modulus [11]. The compression ratio V(T, P)/ V(T, 0) in this equation can be calculated from the high-pressure equation of state (EOS) for condensed matter. In this paper, we have chosen the well-known usual Tait EOS [19, 20] because it is simple and inverted, which can be expressed as follows:

where B′T(T, 0) is the first pressure derivative of the bulk modulus at zero pressure. Substituting (8) into (7), we get the expression for the pressure dependence of bulk modulus. This is expressed as follows:

Grover et al. [21] used a different definition of δ and called it the parameter a, as given in the following:

where V₀ was taken as a constant. When generalised, (10) reads as follows:

where M represents any of the elastic moduli such as C₁₁, C₁₂, C₄₄, C_s, or B_T. Grover et al. [21] expressed (11) as follows:

The application of this equation for the determination of elastic constants has been advocated by Tallon [18] and repeated by Kumar et al. [12, 13] by changing the crystal.

Following the method of generalisation as used by Grover et al. [21] to get (12), (9) may be used to estimate the pressure dependence of elastic constants. At T=T₀=300 K, (9) may be rewritten as follows:

Here δ_M(T₀, 0) are evaluated using the relation δ_M(T₀, 0)=(∂M/∂P) [13]. This equation is a simple and straightforward relation for the determination of the pressure dependence of the elastic constants of solids. Using the ambient (zero pressure) value, we can immediately use (13) to get elastic constants at higher pressure.

3 Results and Discussion

To study the pressure dependence of elastic constants, we have selected five mantle materials, viz. CaSiO₃ perovskite, hydrous ringwoodite, Mg₂SiO₄ spinel, MgSiO₃ ilmenite, and MgSiO₃ perovskite because of the fact that various sets of theoretical as well as experimental data are available [22–26] so that the results obtained in the present work may be discussed in the light of earlier investigations. The input data are all listed in Table 1. The results obtained with (13) are plotted in Figures 1–5. The simple method developed in the present work is free from the theory of potentials, and the results obtained are in good agreement with the corresponding experimental data and the other theoretical results. However, there are still a few exceptions at high pressure such as C₁₂ of MgSiO₃ perovskite.

Figure 1:

Calculated elastic constants of CaSiO₃ perovskite as a function of pressure. First principles results from Karki and Crain [22] are shown as open symbols.

Figure 2:

Calculated elastic constants of hydrous ringwoodite as a function of pressure. The experimental data from Wang et al. [23] are shown as open symbols.

Figure 3:

Calculated elastic constants of Mg₂SiO₄ spinel as a function of pressure. The calculated results based on plane-wave pseudopotential method are shown as open symbols [24].

Figure 4:

Calculated elastic constants of MgSiO₃ ilmenite as a function of pressure. Ab initio results from Da Silva et al. [25] are shown as open symbols.

Figure 5:

Calculated elastic constants of MgSiO₃ perovskite as a function of pressure. First principles results from Karki et al. [26] are shown as open symbols.

CaSiO₃ perovskite is generally expected to be present in the lower mantle with abundance (6–12% by volume) next to (Mg,Fe)SiO₃ perovskite and magnesiowüstite [22]. Hydrous ringwoodite is probably the single largest potential water reservoir in the depth range from 520 to 660 km [23]. Mg₂SiO₄ spinel is thought to be the most abundant mineral in the lower part of the transition zone (520–660 km depth) [24]. Their elastic properties will play an important role for our understanding of the composition of this region. In the case of CaSiO₃ perovskite, hydrous ringwoodite, and Mg₂SiO₄ spinel, there are only three elastic constants, viz. C₁₁, C₁₂, and C₄₄. C₄₄ has a relatively small pressure dependence when compared with the other elastic constants. The elastic constant C₁₂ increases with pressure about 1.5–4 times faster than C₄₄, and these two constants intersect and become equal at a certain pressure. At the same time, we find that C₁₁ varies largely under the effect of pressure when compared with the variations in C₁₂ and C₄₄. The constant C₁₁ represents elasticity in length, and the constants C₁₂ and C₄₄ are related to the elasticity in shape. A longitudinal strain produces a change in volume, and a transverse strain or shearing causes a change in shape. So, C₁₂ and C₄₄ are less sensitive of pressure.

Equation (13) has also been extended to calculate the elastic constants of more complicated minerals, viz. MgSiO₃ ilmenite and MgSiO₃ perovskite. For MgSiO₃ ilmenite, there are seven elastic constants, and nine for MgSiO₃ perovskite. MgSiO₃ ilmenite is a high-pressure polymorph of enstatite and occurs only in cold subduction environments near the bottom of the transition zone [25]. Its unusual elastic properties may play an important role in the interpretation of three-dimensional seismic structure. The elastic constant C₁₁ remains much larger than C₃₃ throughout the pressure regime studied indicating that the c-axis is more compressible than the a-axis. Magnesium silicate perovskite has been considered as the major constituent of the earth’s lower mantle, the largest single region of the planet making up about 55% of its volume [26]. The effect of pressure on all the elastic constants is large. The longitudinal moduli vary by more than a factor of two over the mantle pressure regime, off-diagonal moduli vary by more than a factor of three, and the effect of pressure is weakest for the shear elastic constants.

In conclusion, we have thus presented a simple and straightforward method to predict the elastic properties of solids at high pressure. The results obtained are encouraging. Due to the simplicity of the method, it can be applied to the more complicated solids, like minerals of geophysical importance and applications.