## 1. Introduction

[2] The amount of heat flowing across the core-mantle boundary (CMB) critically determines the thermal evolution of both the mantle and the core. By determining the fraction of the total heat budget not produced in the mantle, it sets key constraints for the required budget of heat producing elements in the mantle, and is likely the main source of heat for the formation of mantle plumes. It constrains the amount of energy required to drive the geodynamo, sets the age of the inner core, and therefore also the required budget of radiogenic isotopes in the core. By limiting the rate at which heat is conducted across the thermal boundary layer at the very base of the mantle, often associated with the seismic D″, the thermal conductivity () of the lower-most mantle is the key physical property in constraining geodynamical models of CMB heat flux from geophysical observations.

[3] Estimates for in the lowermost mantle are very poorly constrained, ranging between 4–16 W/m K [*Goncharov et al.*, 2009; *Lay et al.*, 2008; *Hofmeister*, 1999; *Brown and McQueen*, 1986]. for MgSiO_{3} perovskite is not well known (see below), while its pressure and compositional dependence has yet to be investigated. Extrapolation to the extreme pressure conditions at the base of the mantle (136 GPa) is therefore a very uncertain exercise. The importance of heat transport by radiation remains an additional source of debate. Though some studies have suggested that MgSiO_{3} perovskite will have a high radiative conductivity in the lower mantle [*Keppler et al.*, 2008], this is unlikely to be the case for a polycrystalline aggregate of (Mg, Fe)O periclase and (Mg, Fe)SiO_{3} perovskite [*Goncharov et al.*, 2008]. Another factor contributing to uncertainty is the post-perovskite phase, which is likely present in the colder parts of the D″ [e.g., *Catalli et al.*, 2009; *Tsuchiya et al.*, 2004]. Of course, even if its conductive properties differ strongly from that of perovskite, this will be of secondary importance to the heat flux from the core, since the mantle temperature directly adjacent to the core is likely outside the post-perovskite stability field.

[4] When considering the large degrees of compression relevant to Earth's mantle, the change of lattice with compression is usually expressed relative to some reference value at *ρ*_{ref} and *T*_{ref} through a power law relation

with different relations for the exponent *g* = ∂ ln /∂ ln *ρ*, all of which are based on approximations to the expression for a kinetic gas of phonons

where *v* is the sound speed and *τ* the mean phonon lifetime (Table S1 of Text S1 in the auxiliary material gives a glossary of mathematical symbols used). The most frequently used relation, often referred to as Debye theory, follows by applying the Debye approximation to the theoretically predicted form of *τ* [*de Koker*, 2010; *Ross et al.*, 1984; *Roufosse and Klemens*, 1973], to get

where *γ* is the Grüneisen parameter and *q* = −∂ ln *γ*/∂ ln *ρ*. If one further relates *v* to rather than to the Debye frequency [*Manga and Jeanloz*, 1997], one can show that

If the density dependence of *τ* is neglected, *g* can also be represented as [*Hofmeister*, 2007]

[5] With only sparse high pressure data available with which to test how best to perform extrapolations, there is very little consensus on which level of approximation is sufficient. Using first-principles molecular dynamics calculations, *de Koker* [2010] showed that *g*(*γ*) (equation (3)) works extremely well for MgO periclase, finding that is strongly dominated by acoustic phonons, indicating that the relation should perform reasonably well for more complex solids as well.

[6] Direct measurement of in MgSiO_{3} perovskite is very challenging, and has only been performed in a supercooled sample at ambient pressures [*Osako and Ito*, 1991]. Obtaining experimental constraints on the pressure dependence of in this mineral is therefore not currently possible. Analogue materials have proven invaluable in elucidating the trends to be expected in the physical properties of mantle phases. As lattice conductivity is primarily controlled by the nature of the crystal lattice [e.g., *Slack*, 1979], analogue materials should give a good indication of the pressure dependence of in deep mantle phases. CaGeO_{3} perovskite is an ideal analogue phase for MgSiO_{3} perovskite, because they have the same space group (*Pbnm*), similar ratios among unit cell parameters, and similar radius ratios between the cations. CaGeO_{3} is stable above 6 GPa, making it an ideal analogue to study the pressure dependence of thermal conductivity in mantle perovskite.