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[转载]Grüneisen parameter

已有 1376 次阅读 2018-7-19 15:52 |个人分类:理论基础|系统分类:科研笔记| Grüneisen, 热力学 |文章来源:转载

 

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.

  • If E increases, the matter wants to increase its V due to α, because T should increase with E increase. Therefore, γth is proportional to α.

  • If KT is higher, the potential V increase that the E increase wants to causes larger Pincrease. Accordingly, γth is proportional to KT.

  • If the CV is larger, T increase becomes smaller for a given E increase, and therefore, γth is inversely proportional to CV.

In the case of MgSiO3 bridgmanite at ambient conditions, α = 2.0×10-5 K-1KT = 260×109Pa, V = 24×10-6 m3 mol-1CV 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 parameterγi, is defined as:

 

(8)

where ωi is the angular frequency of the lattice vibration of the mode i.
The Debye gammaγD, 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.


本文转账自:

https://www.katsura-highpressure-earth.com/english-home/lecture-note/gr%C3%BCneisen-parameter/




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