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Suppose we have a gapped system i.e. no gapless excitation is possible. If the thermal energy is insufficient to excite atoms from ground state to excited state of any kind (of a single atom or of a collection of atoms) i.e. $k T \ll \Delta{E} = E_e - E_g$, can the system store any thermal energy? If yes in what form is the thermal energy stored?

Alternative statement of my question: is zero heat capacity possible without violating the third law of thermodynamics i.e. at nonzero temperature?

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    $\begingroup$ Phonons, excitons, polarons, magnons, molecular rotation and vibration, molecular kinetic energy in gases... lot's of good ways to stash away energy. An atom can, by the way, be in a mixed electronic state, and in a thermal environment they are, the idea that atoms can only exist in well defined energy eigenstates is just a simplification for the non-thermodynamic state of matter. $\endgroup$
    – CuriousOne
    Commented Jun 7, 2016 at 22:22
  • $\begingroup$ @CuriousOne: thanks for the answer but I have assumed that excited state of all kinds including phonons, excitons, polarons, magnons, molecular rotation and vibration etc. is beyond the reach of $kT$ $\endgroup$
    – PhysicsMath
    Commented Jun 7, 2016 at 22:25
  • $\begingroup$ If there is no way to store the energy — it will not be absorbed. Basically, the phenomenon of superfluidity is based on a particular disability to excite phonons in some cases. However, it is impossible to remove all possible excitations in a general sense that you imply — many kinds of excitations do not have a lower energy bound (position shift, momentum increase, emission of a photon) $\endgroup$
    – Andrii Magalich
    Commented Jun 7, 2016 at 22:41
  • $\begingroup$ @AndriiMagalich: let's say we have a gapped system i.e. no gapless excitation is possible. Do you mean that in such cases the temperature is actually zero since there is not at all thermal energy stored in the system? $\endgroup$
    – PhysicsMath
    Commented Jun 7, 2016 at 22:47
  • $\begingroup$ Phonon modes are excited all the way to cryogenic temperatures. Can you freeze it all out, eventually. Sure, but that wasn't the question, was it? The temperature is never zero per 3rd law of thermodynamics, but you would be correct to say that heat capacity drops like a stone at sufficiently low temperatures. It just can't drop to zero. $\endgroup$
    – CuriousOne
    Commented Jun 7, 2016 at 23:12

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Mostly kinetic energy.

The kinetic energy of a free particle is not quantized. It becomes so when the particle is closed in a box. But even in this case the energy levels are often so closely spaced that the spectrum is almost continuous.

In fact, if you solve the Schroedinger equation for a particle in a 1D infinite square well you will find the following energy levels:

$$E_n = \frac{( \pi \hbar)^2}{2 m L^2} n^2 $$

where $L$ is the length of the box and $n=1,2,3,\dots$.

Let's put some numbers in the above formula. If $m$ is the mass of an hydrogen atom ($\sim 10^{-27}$ kg) and $L=1$cm, we will get

$$E_n \simeq (2.0 \cdot 10^{-18} \text{eV}\ ) \ n^2$$

So the difference in energy between the ground state ($n=1$) and the first excited state ($n=2$) is

$$E_2-E_1 = 3 \cdot (2.0 \cdot 10^{-18} \text{eV}\ ) = 6.0 \cdot 10^{-18} \text{eV}\ $$ At ambient temperature, $T\simeq300$K, we have

$$kT \simeq 2.6 \cdot 10^{-2} \text{eV}$$

You can see how this energy is enormous with respect to $E_2-E_1$:

$$\frac{kT}{E_2-E_1} = 4.33 \cdot 10^{15}$$

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  • $\begingroup$ many thanks for your detailed answer but I am considering a thought experiment in which I assume the temperature is sufficiently low that $kT$ cannot reach any excited state, including the translational kinetic energy (I am considering a condensed matter with finite size so the energy is quantized). $\endgroup$
    – PhysicsMath
    Commented Jun 8, 2016 at 0:13
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    $\begingroup$ Ok, I understand. But be careful: if you talk about the system having "thermal energy", you are implying that the system has a nonzero temperature that we can measure, and hence must have kinetic energy of some kind (translational, rotational or vibrational). In this case we already are in some excited state. If on the contrary the system has no thermal energy and we are giving it some energy of the order $kT$ coming from a source at temperature $T$, then we have to discuss the mechanism with which this energy is transferred (radiation? contact between the two bodies?) $\endgroup$
    – valerio
    Commented Jun 8, 2016 at 8:41
  • $\begingroup$ I see. So you mean if there is no kinetic energy of any kind the temperature of the system is absolute zero i.e. zero heat capacity necessarily implies zero temperature? $\endgroup$
    – PhysicsMath
    Commented Jun 9, 2016 at 1:32

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