Raman-based geobarometry of ultrahigh-pressure metamorphic rocks: applications, problems, and perspectives

Abstract

Raman-based geobarometry has recently become increasingly popular because it is an elegant way to obtain information on peak metamorphic conditions or the entire pressure-temperature-time (P-T-t) path of metamorphic rocks, especially those formed under ultrahigh-pressure (UHP) conditions. However, several problems need to be solved to get reliable estimates of metamorphic conditions. In this paper we present some examples of difficulties which can arise during the Raman spectroscopy study of solid inclusions from ultrahigh-pressure metamorphic rocks.

Appendix

Multi-shell elastic model

We assume that a system consisting of j ₀ layers is formed at the pressure P ₀ and temperature T ₀. The first substance occupies a sphere of the radius r ₁, the second occupies the shell r ₁ < r < r ₂, the j-th substance occupies the shell r _j−1 < r < r _j . We also assume that all processes are slow, so that the temperature will be uniform and the body will maintain in mechanical equilibrium. The temperature of the formation is high and the stress tensor σ _ik relaxes to an isotropic form: ${\sigma _{rr}} = {\sigma _{\theta \theta }} = {\sigma _{\varphi \varphi }} = - {P_0}$. The radial displacement in initial stage is expressed by the formula

$$ {u_0} = {A_{0j}}r,\quad {A_{0j}} = - {{{P_0}} \mathord{\left/{\vphantom {{{P_0}} {\left( {3{K_{0j}}} \right)}}} \right.} {\left( {3{K_{0j}}} \right)}} $$

where K _0j is the bulk modulus in j-th substance in initial state. Let the system change to an environment with the pressure P and temperature T. We assume that during this change the system behaves as an elastic body. Below we indicate the component σ _rr of stress tensor as σ _r and ${\sigma _{\theta \theta }} = {\sigma _{\varphi \varphi }}$ as σ _t . It is well known [36] that the conditions of equilibrium give the expression for radial displacement $u(r) = {C_1}r + {C_2}/{r_2}$ in each layer. In our case it is more convenient to write for j-th substance

$$ u = \left( {{A_j} + {B_j}\frac{{r_{j - 1}^3}}{{{r^3}}}} \right) r $$

In this notation the total displacement is u _tot = u + u ₀. For the stress tensor we have

$$ \begin{array}{*{20}{c}} {{\sigma_r} = {3}{K_j}\left( {{A_{0j}} + {A_j} - {\varepsilon_j}} \right) - 4{G_j}{B_j}\frac{{r_{j - 1}^3}}{{{r^3}}}} \hfill \\{{\sigma_t} = {3}{K_j}\left( {{A_{0j}} + {A_j} - {\varepsilon_j}} \right) + {2}{G_j}{B_j}\frac{{r_{j - 1}^3}}{{{r^3}}}} \hfill \\\end{array} $$

where ε _j is the coefficient of linear expansion due to temperature change and phase transformation, A, B, and C are three constants and G is shear modulus. Note, that ε _j can be written as ε _j  = (ρ _0j /ρ _j )^1/3 − 1, where ρ _0j is the substance density at initial temperature T ₀ and zero pressure and ρ _j is the substance density at final temperature T and (may be) another phase state also for zero pressure.

The conditions of finiteness of displacement in the center, continuity of radial displacement u, and normal stress tensor component σ _r at the boundary between layers and the condition on the outer boundary of the body may be expressed thus:

$$ \begin{array}{*{20}{c}} {{B_1} = 0} \\{{A_j} + {B_j}{w_j} = {A_{j + 1}} + {B_{j + 1}},j = 1,...,j0 - 1} \\{3{K_j}\left( {{A_j} + {A_{0j}} - {\varepsilon_j}} \right) - 4{G_j}{B_j}{w_j} = 3{K_{j + 1}}\left( {{A_{j + 1}} + {A_{0j + 1}} - {\varepsilon_{j + 1}}} \right) - 4{G_{j + 1}}{B_{j + 1}},{\hbox{J}} = 1,...{,_{j0}} - 1} \\{3{K_{j0}}\left( {{A_{j0}} + {A_{{0_{j0}}}} - {\varepsilon_{j0}}} \right) - 4{G_{j0}}{B_{j0}}{w_{j0}} = - P} \\\end{array} $$

where ${{{w_j} = r{{_j^3}_{ - 1}}} \mathord{\left/{\vphantom {{{w_j} = r{{_j^3}_{ - 1}}} {r_j^3}}} \right.\kern-\nulldelimiterspace} {r_j^3}}$ We solve this system numerically.