How can the Gruneisen value be determined using the Lennard-Jones potential?

[hen]

Guys, i am frustrated!
i don't know it well, and if someone who knows it really good could expalin me then i'll be more than thankfull! i devoted too many hours for this one:
question:
" find the value of the gruneisen for a crysal inert gas. use the lennard-jones potential."

i understand that the meaning of the gru' is to show dependence of V in temp' in anaharmonic models. i know that grun'=d(lnw)/d(lnV)
but i didn't understand how they got to it (anyone?). i know you need to use the hlmoltz free energy (F=U-TS), but somewhere in the middle of the prove of that in kittle, i got lost...
also kittle has no example for this one, so i don't get it. someone?

p.s
sorry for any bad english, my primary is hebrew... the question is translated from hebrew also...

 

[Gokul43201]

I'm not sure exactly what your equation means. What is "w" in d(lnw) ? Are you talking about the frequency of a certain mode $\omega_i$ ?

The gruneisen parameter is defined as a certain quantity (I can't recall the exact form) that I vaguely recall looks like what you've written. That is by definition and hence requires no derivation.

If I recall correctly, the Grun' parameter tells you the correlation between the specific heat and the thermal expansion coefficient.

I'll have to find my Kittel or Ashcroft to do better than that.

 

[inha]

Gruneiser parameter tells you how the anharmonicity affects the volume dependence of the lattice's vibration modes. The parameter can be put into other form that's more useful in this case.

$$\gamma=-\frac {a} {6} (\frac {d^3 U} {dr^3})_{r=a} / (\frac {d^2 U} {dr^2})_{r=a}$$

where U is the potential and a is the value of r that minimizes the potential.
This expression is derived from the expression by typing the log-derivative open and doing a few manipulations.

This is an excercise in Hook&Hall btw.

 

[wangasu]

Gruneisen constant is simply a parameter to quantify the vibrational anharmonicity of materials. The introduction of anharmonicity is associated with the difference of heat capacity at constant pressure Cp and at constant volume Cv. With the numerical values, the anharmonicity of various materials can be compared easily. Its effect is relatively weak at low temperature (below Debye temperature, for example), while at high temperature, this property is quite important, especially in the solid-liquid transition. I believe that without anharmonicity, crystals will never melt. Although there are lots of vibrational modes in lattice vibration, the long wavelength vibrational modes might dominate in determining the total Gruneisen constant of materials.