In the previous piece we treated a solid as 3N oscillators. That is Einstein's model. It gave Dulong and Petit's 3R at high temperature and a heat capacity that falls at low temperature. So far it agreed. The falling, though, was steeper than what is measured.
The reason it is too steep lies in the assumption: that all 3N oscillators carry the same frequency. That is the upper row of the figure, where every atom swings on its own.
Real atoms, however, are tied to their neighbours by springs. Move one and the next is pulled after it. That is the lower row, where the motion is not one atom at a time but a wave running along the whole line. A different wavelength means a different frequency, so one frequency could never have been enough.
Once we treat them as waves, the counting is the same as for light in a cavity. A line with both ends held still admits only certain waves, those in which a whole number of half wavelengths fits. Each one of those is a mode.
The figure raises the number of the wave one step at a time. The higher the number, the shorter the wavelength and the higher the frequency. The vertical bars at the ends are where the line is held, and it is that holding which decides what can stand.
Where the counting is the same, so is the count. The number of modes with frequency between \omega and \omega + d\omega is proportional to \omega^2, exactly the reckoning we did in the piece on black-body radiation.
There is one difference from light. The modes of light go up without end; the modes of a solid do not.
The reason is a count. There are N atoms, each able to move in 3 directions, so there are only 3N independent ways to move at all. The modes therefore stop at 3N. The shortest wavelength is the wave in which neighbouring atoms move opposite ways, and nothing finer than that fits on a line of atoms.
The shaded region of the figure is those 3N modes. The frequency where it stops is the Debye frequency \omega_{\mathrm{D}}. The faint curve is what it would look like carried on without stopping, which is the case for light. No ultraviolet catastrophe happens in a solid because the high modes are simply not there.
Now the low temperature end. The spacing of the steps differs from mode to mode, and the higher the frequency the wider it is. As in the previous piece, a mode whose steps kT cannot reach is frozen out.
So the only modes that can move are those with \omega below \dfrac{kT}{\hbar}. The shaded region of the figure is that range, and raising the temperature widens it.
The number of modes grows as \omega^2, so the number inside that range grows as the 3rd power of its width, which is the 3rd power of T. Each of them carries something like kT, so the energy goes as the 4th power of T, and the heat capacity, its derivative, as the 3rd. That is Debye's T^{3} law.
The figure sets Debye beside Einstein. At high temperature both lie on Dulong and Petit's 3R, because once kT exceeds the step spacing of every mode the difference in the counting stops showing.
The difference appears at low temperature. Einstein has only one kind of oscillator, so the moment its steps freeze out the curve falls exponentially. Debye still has low frequency modes left over without end, so the fall is gentle and goes as T^{3}. It is Debye that matches measurement.
All that mattered here was one thing: whether the atoms are looked at separately or as things tied together. How the partition function is used, and why a degree of freedom freezes, are unchanged from the previous piece. It is an example of reaching the measurement by correcting nothing but the counting. In the next chapter we let the number of particles move as well, and go on to quantum statistics.