The Grüneisen parameter [Wiki], γth, is defined as a change in P by a change in density of E at constant V. Namely,
(1)
To put it plainly, γth is the ratio of thermal pressure to thermal energy [Wiki]. Or, it describes how much P increases when the matter is heated.
It is expressed by:
(2)
If γth is defined by this equation, it is called the thermodynamic Grüneisen parameter.
To put it plainly, what Eq (2) means is as follows.
In the case of MgSiO3 bridgmanite at ambient conditions, α = 2.0×10-5 K-1, KT = 260×109 Pa, V = 24×10-6 m3 mol-1, CV is 80 J K-1 mol-1, and therefore, γth = 1.1.
Note that γth is a dimensionless parameter.
Eq. (2) is obtained from Eq. (1) by the following way.
(3)
The numerator of Eq (3) is:
(4)
The denominator of Eq (3) is:
(5)
Using Eqs (3)-(5), Eq. (2) is obtained from Eq. (1) as:
(6)
The definition of the Grüneisen parameter, Eq. (1), is the Mie-Grüneisen EOS [Wiki] itself:
(7)
There are other ways to define the Grüneisen parameter.
For example, the mode Grüneisen parameter, γi, is defined as:
(8)
where ωi is the angular frequency of the lattice vibration of the mode i.
The Debye gamma, γD, is defined as:
(9)
where ωD is the Debye cut-off (angular) frequency [Wiki], and ΘD is the Debye temperature [Wiki].
We will discuss the mode Grüneisen parameter and Debye gamma elsewhere.