Elasticity of calcite: thermal evolution

Abstract

The elastic properties of calcite have been determined by Brillouin spectroscopy for temperatures up to 600 °C. The results reveal that the variations of the aggregate bulk (K _VRH) and shear (G _VRH) moduli of calcite with respect to temperature can be approximately expressed as follows:

$$\begin{aligned} K_{{{\text{VRH}}}} ({\text{GPa}}) & = 79.57-0.0230\;T\, (T\;{\text{in}}\;^{^\circ } {\text{C}}) \\G_{{{\text{VRH}}}} ({\text{GPa}}) & = 32.23 - 0.0097\;T. \\\end{aligned}$$

This indicates a nearly constant Poisson’s ratio (0.322) for calcite from 22 to 600 °C. A further analysis shows that the compressibility along the c axis (β _||) and that perpendicular to the c axis have comparable contributions to the volume compressibility of calcite, although the contribution of β _|| decreases with an increase in the temperature.

Appendix: Estimation of the minimum uncertainty for acoustic velocity data

The cleavage of calcite is generally described as $\left\{ { 10\bar{1}1} \right\}$ on the basis of the cleavage rhombohedral cell. This plane of a form is $\left\{ { 10\bar{1}4} \right\}$ according to the X-ray smallest hexagonal cell. The lattice parameters of the calcite used in this study at room temperature are as follows: a = 4.984 Å and c = 17.039 Å. By taking this data set into account, we calculated the angle between the optic axis (i.e., the c axis) and the normal of the $\left( { 10\bar{1}4} \right)$ cleavage face of calcite to be 44.622°. Figure 5 shows the relationship among the normal of the $\left( { 10\bar{1}4} \right)$ cleavage face, the optic axis, and the incident ray, and the cone formed by rotating obliquely the optic axis around the normal of the cleavage face. Here, ϕ, α, and θs denote the rotation angle of the optic axis, the angle between the optic axis and the incident ray, and the scattering angle, respectively. The ϕ angle is related to the χ angle of the Eulerian cradle because the rotation of the optical axis is caused by the rotation of the specimen about the normal of the specimen’s face (but the ϕ angle may or may not be equal to the χ angle). For the 60°/60° scattering geometry, the angle between the optic axis and the incident ray is 14.622° as shown in Fig. 5. Let ϕ = 0° when the normal of the specimen’s face, the optic axis, and the incident ray are coplanar. Then, the angle α can be obtained by using the following formula:

$$ \alpha = \cos^{ - 1} \left[ ({p^{2} - q^{2} + 1} )/2p \right], $$

(3)

where $q = \left[ {(nm - nc\cos \phi )^{2} + (nc\sin \phi )^{2} } \right]^{1/2}$, $p = \left[ {\left( {1 + \tan^{2} \theta i} \right)\cos^{2} \gamma } \right]^{1/2}$, and nm = cosγ tanθi. p and q are the distances of the om and c*m segments, respectively in Fig. 5b. θi is the incident angle, and γ is the angle between the normal of the specimen’s face and the optic axis. Let the distance oc in Fig. 5 be one unit, then $nc = \sin\gamma$. Equation (3) is a general formula for a uniaxial crystal under symmetric scattering geometry. In the present case (calcite with 60°/60° geometry), γ and θi are 44.622° and 30°, respectively. Hence, p = 0.822, nm = 0.411, and nc = 0.702 for the cleavage specimen. For the specimen with two faces being ideally parallel, the angle between the scattered ray and the optic axis can be calculated by Eq. (3) using $q = \left[ {(nm + nc\cos \phi )^{2} + (nc\sin \phi )^{2} } \right]^{1/2}$.

Fig. 5

Relationship among the normal of the $\left( { 10\bar{1}4} \right)$ cleavage face, the optic axis, and the incident ray (a), and the cone formed by rotating the optic axis around the normal of cleavage face (b). ϕ is the rotation angle of optic axis, and α is the angle between the optic axis and the incident ray. θs is the scattering angle. Here, ϕ = 0° when the normal of cleavage face, the optic axis, and the incident ray are coplanar

The refractive index of the e ray in a uniaxial crystal can be calculated by using the following equation (Pascal et al. 2011):

$$ 1/\varepsilon^{\prime} = \cos^{2} \alpha /\omega + \sin^{2} \alpha /\varepsilon , $$

(4)

where ε′ denotes the refractive index of the e ray inside the crystal when the angle between the incident (or scattered) ray and the optic axis is α. ω and ε are the two axes of the optical indicatrix corresponding to the o and e rays in the crystal. When the optic axis rotates obliquely at an angle ϕ around the normal of the $\left( { 10\bar{1}4} \right)$ face as shown in Fig. 5b, the corresponding refractive indices for both the o and e rays in calcite under ambient conditions are displayed in Fig. 6 (for 60°/60° scattering geometry with ω = 1.658 and ε = 1.486 for calcite). The blue circle indicates that the refractive index of the o ray, ω, is independent of the orientation. The red and green data in Fig. 6 represent, respectively, the refractive indices of the e ray corresponding to the refracted and the scattered rays at the same rotation angle ϕ.

Fig. 6

Refractive indices of both o (blue, ω = 1.658) and e (red and green, ε′) rays in calcite at ambient condition and 30° of incident angle against the rotation angles (ϕ) of the optic axis (60°/60° scattering geometry and $\left( { 10\bar{1}4} \right)$ cleavage face). Red and green data represent the refractive indices of the two e rays corresponding to the refracted and the scattered rays at the same ϕ angle, respectively

Double refraction adds an extra uncertainty to the estimation of acoustic velocity and elasticity. At any χ angle, we cannot say that the scattering data related to the o ray are superior to those related to the e ray and vice versa because birefringence is a property of the crystal. To estimate the acoustic velocity at a given χ angle, the averaging of the scattering data caused by both the o and e rays is inevitable and reasonable, although this may be just an approximation. This case also indicates the drawback of Brillouin spectroscopy with respect to the determination of the elastic data of an anisotropic crystal with large birefringence. Therefore, to calculate the acoustic velocity, one needs to use both Eqs. (1) and (2). The scattering angle (θs) is an essential data element for adopting Eq. (1) in the calculation. Unlike that of the o ray, which is isotropic inside calcite, the propagation of the e ray in an anisotropic crystal does not follow Snell’s law, although the direction of the wave normal of the e ray can be predicted by Snell’s law. A theoretical prediction or in situ measurement to the ε′–refraction angle relationship of the e ray at any ϕ (or χ) angle is necessary for calculating the scattering angle of the e ray. However, this information is not available at present. A simplified way for the estimation of θs is to assume that the e ray also follows Snell’s law. This simplification does not exclude the influence of birefringence on the calculation of the acoustic velocity because the refractive index of the e ray varies with the ϕ (or χ) angle (see Fig. 6). As shown in Fig. 6, except the ϕ angle at 90° and 270°, the refractive index of the e ray corresponding to the refracted and that to the scattered beams (which will go to the detector) are different at the same ϕ (or χ) angle, even if Snell’s law is assumed to be valid. This indicates that θs of the e ray will vary with the ϕ angle. Accordingly, the percentage difference [Δ %, relative to the data obtained by using Eq. (2)] between the acoustic velocities related to both the e rays calculated by Eqs. (1) (anisotropic) and (2) (isotropic) ranges from 0 (ϕ = 90° and 270°) to 4.7 % (ϕ = 0° and 180°) depending on the ϕ angle. The Δ % is independent of the frequency of the scattering signal, but it shows a reverse ϕ angle (or refractive index) dependence for the two e rays (refracted and scattered). The average of the acoustic velocities related to the two e rays is almost equal to that calculated using Eq. (2). However, the e ray entering the detector is the scattered. Hence, the range of 0–4.7 % is more reasonable for Δ %. It is known that the deviation in the refraction angle of the e ray from that of the o ray is no smaller than that predicted by Snell’s law. Therefore, 0–4.7 % is the minimum uncertainty for the acoustic velocity related to the e ray under ambient conditions and the 60°/60° scattering geometry. For the current study, the velocity data adopted for solving elastic constants are the averages of the two velocities related to the o and e rays. This treatment will decrease the net uncertainty if the uncertainty of the velocity related to the o ray is relatively low. The uncertainty may be relatively high when the true path of the refracted ray is considered. Hence, if not impossible, an exact correction of the velocity data is difficult because it is complicated by several factors: the variations of the refractive index and the refraction angle of the e ray with the χ angle, the polarizability of the e ray involved, the parallelism of the specimen, the quality of the scattering signal, and the intrinsic error of the apparatus. To find θs of the e ray, the relationship between the refractive index and the refraction angle for the e ray under the 60°/60° geometry needs to be measured in the future. A theoretical study on the polarizability of calcite at the orientation involved is also a requisite for an accurate correction of the acoustic velocity data at all χ angles.

When the temperature rises, the c axis of calcite elongates, while the a axis shortens as given in Table 1. In addition to an increase in the refractive indices of calcite (Li et al. 2004), the angle between the optic axis and the incident ray increases with an increase in the temperature. These changes result in a decrease in θs. Therefore, according to Eq. (1), the uncertainty of the acoustic velocity contributed from birefringence may remain almost the same even at a relatively high temperature because of the decrease in θs and the increase in the refractive index. The argument about a possible temperature-independent uncertainty may be valid in the calculation of C _ij s and elastic moduli because the modulus-T plots of calcite obtained by Brillouin spectroscopy in the current study and those by ultrasonic methods are close to each other as shown in Fig. 3.