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Birch's law at elevated temperatures

Birch's law in high pressure physics postulates a linear relationship between elastic wave speed and density and one of its most well known applications is in investigations into the composition of the inner core of the Earth using the Preliminary Reference Earth Model as the primary source of constraints. However, it has never been subjected to high precision tests even at moderately elevated temperatures. Here we carry out such a test by making use of the Density Functional Theory of electronic structure calculation and the Density Functional Perturbation Theory of calculating the phonon dispersion relation. We show that a recently proposed modification to the Birch's law is consistently satisfied more accurately than its original version. This modified version states that it is the product of elastic wave speed and one-third power of density that should be a linear function of density. We have studied the cases of platinum, palladium, molybdenum and rhodium with cubic unit cell and iron with hexagonal-close-packed unit cell with temperatures up to 1500K and pressures up to about 360 GPa. We also examine the genericity of the validity of a recently proposed extension of the Birch's law according to which elastic wave speed is a linear function of temperature at a given density. Within the error bars of our calculation, we find that this is consistent with our data for the four cubic materials at temperatures up to 3300 K.