Grüneisen parameters

The Grüneisen parameter or (after Eduard Grüneisen ) describes the dependence of the frequency of lattice vibrations ( phonons ) in a crystal on the relative change in volume , which in turn depends on the temperature . It is used to describe anharmonic effects in crystals that are neither electrically conductive nor magnetic , and is used in the Mie-Grüneisen equation of state . ${\displaystyle \gamma }$ ${\displaystyle \Gamma }$

description

In a simple model one assumes that all interactions in a crystal are harmonious . However, this does not adequately describe real solids , as these e.g. B. show a volume expansion with increasing temperature , which is not taken into account by such a harmonic model. Therefore, terms of higher order are introduced into the interaction potential in the solid and new effects are obtained.

Thus, the relative change δω / ω of the oscillation frequency of a phonon with a certain impulse and in a certain phonon branch depends linearly on the relative volume expansion δ V / V :

${\displaystyle {\frac {\delta \omega }{\omega }}=-\gamma \cdot {\frac {\delta V}{V}}}$

The dimensionless Grüneisen parameter is defined as:

${\displaystyle \gamma =-{\frac {\partial (\ln \omega )}{\partial (\ln V)}}=-{\frac {V}{\omega }}\cdot {\frac {\partial \omega }{\partial V}}}$

Typical values ​​for are between 1 and 2 at room temperature (see  here ), i.e. H. the volume and the phonon frequencies change roughly equally. ${\displaystyle \gamma }$

Strictly speaking, a separate green iron parameter must be defined for each mode ; in particular, transverse and longitudinal modes can differ. However, in the Debye or Einstein model, all frequencies scale with the Debye frequency or with the Einstein frequency . Accordingly, there is only one green iron constant for all modes: ${\displaystyle \omega _{D}}$ ${\displaystyle \omega _{E}}$

${\displaystyle \gamma =-{\frac {\partial (\ln \omega _{D/E})}{\partial (\ln V)}}=-{\frac {3B_{T}\cdot \alpha }{c_{V}}}}$

With

• ${\displaystyle B_{T}}$as an isothermal compression module
• ${\displaystyle c_{V}}$as specific heat capacity at constant volume ${\displaystyle V}$
• ${\displaystyle \alpha }$the linear thermal expansion coefficient .

This is synonymous with the fact that specific heat and coefficient of expansion have a similar temperature dependence. That is why the definition of a constant Grüneisen parameter makes sense.

A thermodynamic representation of the Grüneisen parameter describes the change of the pressure  p with the internal energy  U at a constant volume  V :

${\displaystyle \gamma =V\cdot \left({\frac {\partial p}{\partial U}}\right)_{V}}$

This means that the Grüneisen parameter can be measured directly. One can increase the internal energy in one area of ​​the crystal at constant volume if one z. B. irradiates with a laser pulse . A pressure wave is generated, which is then detected on the crystal surface.

swell

• Siegfried Hunklinger : Solid State Physics . 2nd Edition. Oldenbourg Wissenschaftsverlag, Munich 2009, ISBN 978-3-486-59045-6 .