# Specific heat

# Instructions

**1) Background of the applet**

The specific heat C of a solid is a property which determines the amount of heat necessary to change the temperature T by a given value. The specific heat is a function of temperature, C(T). Here, we focus on non-magnetic insulators, where the amount of energy or heat stored in a given sample depends on the number of excited phonons (quantized lattice vibrations).

In this case, the specific heat at high temperatures approaches a constant value as predicted by classical theory (Dulong-Petit value). Upon cooling down, the phonons "freeze out" and the specific heat is suppressed. There are two simple approaches to describe this behavior, the Debye model and the Einstein model.

The Debye model assumes a contribution of acoustic phonons with omega(k) ~ k, where the number of modes is determined by the cut-off frequency omega_Debye (or, expressed as a temperature, k_B theta_Debye = hbar omega_Debye). The Einstein model assumes that the phonon energy hbar omega_Einstein is independent of the wave vector k. Again, this value can be described as a temperature, k_B theta_Einstein = hbar omega_Einstein.

For comparison, the applet also features the specific heat of a Fermi gas with Fermi temperature T_F, i.e., the specific heat of a most simple metal.

2) What does the applet show?

This applet shows the temperature dependence of the specific heat for two non-magnetic insulating solids A and B Both solids are described by a sum of one Debye model and up to two Einstein models. In these models, the user may vary parameters such as the specific temperatures discussed above or the number of atoms per cell contributing to each model. The latter determines the number of modes (or "oscillators"). Both plots show the specific heat vs. temperature (left: linear scale, right: log-log scale) and feature the asymptotic upper limit for the specific heat (grey line), which reflects the classical Dulong-Petit law which is approached at high temperature.

As stated above, there is the option to add the linear contribution of a Fermi gas of free electrons.

3) How to use the applet?

The applet allows to compare the specific heat of two independent solids A and B, which can be activated by the checkboxes 'solid A' and 'solid B' (top right). You can also add experimental data of EuCoO3 to the plots (and try to fit it with one of the solids).

Both solids allow to add up a Debye model and two Einstein models. By choosing e.g. the Debye model for solid A and one Einstein model for solid B, one can directly compare the two models. By adding up different models, one can try to simulate realistic data such as the data for EuCoO3.

The adjustable values in both solids are the following:

1) Models can be turned on/off by using the trailing checkboxes.

2) The characteristic temperatures can be adjusted by using the sliders.

3) The number of atoms per unit cell which contribute to a specific model can be modified by using the text field adjusters at the bottom.

If you move with the cursor across one of the two plots, the temperature and specific heat are indicated below the plot.