Is atom vibration (not speed) correlated with temperature? As far as I know every atom vibrates / oscillates / I don't know what it's called in English with a frequency, specific for its element. I don't mean the velocity of the atom, which defines temperature. Although, honestly the two might be related in a way I don't know.
For example, I know that the proof that time is a real physical dimension and not just something people made up with clocks is that you take an atomic clock at sea level and one on a high mountain and one will run behind.
I know atomic clocks don't measure time by the atoms' vibration but by a more complicated mechanism but I just thought that the other mechanism might be affected by the difference in temperature (which I am sure they account for, if needed) but still, I thought to myself if you just measure the rate of oscillation / vibration then that shouldn't be affected by temperature and then realised I am not sure about that.
tl,dr:

*

*Is the vibration / oscillation of an atom with a frequency, strictly specific to that element, different from the atom's velocity?

*Is said vibration / oscillation not affected by temperature? Difficult question to formulate, as it's more than atoms define temperature than the other way around.

 A: Atoms do not have vibrational degrees of freedom, but rather the electronic excitations. On the other hand, molecules (which consist of multiple atoms) do have vibrational and rotational degrees of freedom. As the question seems to be motivated by statistical physics, I will further make the remarks only about the latter.
The question (implicitly) refers to the equipartition theorem, which tells us that energy per degree of freedom in a gas is approximately $$\frac{k_B T}{2}$$ It is clear, according to the OP, how this is applied to the translational degrees of freedom, i.e., the velocities of the molecular center-of-mass. It is probably also clear how this applies to the molecular rotations, since these can be characterized by velocities. When it comes to oscillations of atoms in respect to each other, each degree of freedom can be modeled as a harmonic oscillator with energy:
$$
E=\frac{mv^2}{2}+\frac{m\omega^2x^2}{2},
$$
where $x$ and $v$ refer to the relative displacement of the two parts of the molecule. It is the energy of such oscillation that is given by $k_BT$. As the amplitude of the oscillations is related to their energy, one could equally express the equipartition in terms of the oscillator velocity at its minimum (at $x=0$) where we have:
$$
E=  \frac{mv^2}{2}=\frac{k_B T}{2}.
$$
Remarks:

*

*Rotational and vibrational degrees of freedom is the reason why the heat capacity of molecular gases is higher than that of monoatomic gases, see here and here.

*When treated quantum mechanically, the vibrational degrees of freedom of molecules usually have high excitation energy and are not involved at low temperatures. Thus, depending on temperature, the number of degrees of freedom that need to be taken into account changes, and the heat capacity of a molecular gas changes with temperature as well!

