## Anderson-Grüneisen parameter

The thermal expansivity, α, decreases by compression. The Anderson-Grüneisen parameter, δT, describes the degree of decrease in α by compression. Namely,

(1)

Or

(2)

Chopelas and Bohler [1989] suggested δT = 5.5 ± 0.5 for mantle minerals. This means that, if V decreases by 10% by compression, α decreases by 40~50%.

Anderson [1967] showed the following relation:

(3)

In the case of MgSiO3 bridgmanite at ambient conditions, α = 2.0×10-5 K-1 [Tange et al., 2012], KT = 260×109 Pa [Tange et al., 2012], (∂KT/∂T)P = -0.029×109 Pa K-1 [Aizawa et al., 2004]. These values give δT = 5.6. Katsura et al. [2010] suggested α of MgSiO3 bridgmanite at HP-HT to be δT = 5.8(2). Thus, Eq (3) agrees with the experimental data.

Here, we here prove Eq. (3). We start from the following equation,

(4)

as follows.

The left side of Eq (4) is:

(5)

The right hand of Eq. (4) is:

(6)

By using Eq. (4), Eq. (1) becomes:

(7)