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As a followup question to this question: The relation between energy quanta of an Einstein solid and the equipartition value of heat capacity and this answer https://physics.stackexchange.com/a/133053/55779:

Apparently energy quanta and the equipartition values are related somehow.

I'd like to approach the problem without the formula, since the problem the question is asked in is before the problem where that formula is derived.

How can I know that value of $x\approx3$ without the formula ? I can't even tell that I should be looking for the half maximum. Or can I?

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I'm not sure how one can know that the half maximum corresponds to $kT/\epsilon \approx 1/3$ without resorting to the formula for the heat capacity.

Still, notice that there are only two energy scales, i.e., $\epsilon$ and $kT$, in the problem. Then, whatever (dimensionless) number that determines whether the equipartition holds or fails has to be the ratio $x\equiv kT/\epsilon$. The limits $x \rightarrow 0$ and $x\rightarrow \infty$ should correspond to the two cases, and $x$ would be of order one in the crossover region. This consideration at least lets you make an order-of-magnitude estimation of $\epsilon$.

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  • $\begingroup$ You can find out the half maximum by looking at the graph, which reaches half of its maximum value for approximately one third of the unit of $kT/\epsilon$ (granted you plotted the graph that way). I'm just not sure how to know that I should be looking for half the value of $C$ to find the value for $\epsilon$. $\endgroup$ – 1010011010 Aug 29 '14 at 10:36
  • $\begingroup$ Surely you'll find that the half-maximum amounts to $kT/\epsilon \approx 1/3$ by looking at the plot, but only if you know the value of $\epsilon$ beforehand. What I described above is how you can estimate $\epsilon$ from the $C$ vs. $T$ plot. The point is that $kT/\epsilon = a$ at the crossover region, where $a$ is an arbitrary number of order one. Of course, there is freedom in choosing the exact value of $a$ and in defining what we exactly mean by the crossover point, but taking $a=1$ and the half-maximum would be a reasonable thing to do. $\endgroup$ – higgsss Aug 29 '14 at 10:55
  • $\begingroup$ But I don't think one can get $a\approx 1/3$ without looking at the actual formula. $\endgroup$ – higgsss Aug 29 '14 at 10:56
  • $\begingroup$ The R code from my linked question shows a plot of $C/Nk$ as a function of $kT/\epsilon$. A screenshot can be found in this link: i.imgur.com/t4V87CC.png - As you can see, for $N=50$, when $C/Nk\approx 1/2$, $kT/\epsilon\approx1/3$. :-) $\endgroup$ – 1010011010 Aug 29 '14 at 10:59
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For visibility I will post this as an answer as well.

The problem in my text book wanted me to fit the numerical/dimensionless approximation to the graph in the text book, to estimate at what value of $T$ the graph would reach half its maximum value (in Figure 1.14 in Schroeder's Introduction to Thermal Physics that's $C_V=3R$).

The estimation of $\epsilon$ can be done too for $C/Nk=1/4$, but you will need to fill in different (higher) values of $T$, but $\epsilon$ should be the same for those calculations as well.

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