Question

derive the debye law explain the difference between Einstein and debye model of spefic heat of solids?​

Answer 1

Answer:

Explanation:

Both of these models are agreeing well at high temperature limit as both of these model are able to recover Dulong-Petit Law (lattice heat capacity is constant at high temperature). They contradicts at low temperature limit as experimentally, materials (e.g Diamond) are found to have its heat capacity proportional to T3 .

Einstein model considered that there was only one type of mode of vibration exist within the crystal lattice that contributes to the lattice heat capacity. This mode is known as Optical mode. Moreover, he assumed that the energy of this mode was quantized (phonon) and has no dependence on momentum space (k space). This model predicts that the heat capacity will drops exponentially at low temperature. Though it is incorrect in very low temperature, but it is still a good approximation for studying optical phonon spectrum. From Einstein model, you can also say that at low temperature, it is very difficult to excite an optical phonon.

Debye model considered all the mode of vibrations in the crystal lattice have linear dispersion relation (E vs k is linear). Moreover, Debye introduced a cutoff frequencyℏωD (known as Debye frequency) which is the maximum frequency that the mode of vibration can have (Why Debye frequency is important? Because this means that you have excited all different mode of vibrations within a crystal, continue heating the crystal will only increase the amount of modes but will no longer excite any new mode of vibrations, this means that we do not get any new physics). This model is used to approximate accoustic phonon spectrum.

There are a lot of good notes in the internet.