In the derivation of the equilibrium concentration of vacancies by statistical mechanics method, I was stumped by this procedure (marked by "?").

$\textbf{Physical Model}:$

1.Solid viewed as a collection of $N$ atomic sites;

2.Each site may or may not be occupied, and assume now that $N_o$ sites are occupied and $N_v$ sites are vacant;

3.If a site is not occupied then the system has an additional energy, namely the formal energy $E_v$;


1.multiplicity function

$$\Omega = C_N^{N_o} = \dfrac{N!}{N_o!N_v!}$$


$$S=k_B\ln \Omega = -N k_B (c\ln c+(1-c)\ln(1-c)) \qquad (c=\dfrac{N_v}{N} \quad ;\quad (1-c) = \dfrac{N_o}{N})$$

3.the internal energy ($\textbf{?}$)

$$U = N c E_v$$

(Why we don't consider the internal energy of the whole system rather than the vacancies ? )

4.The Helmholtz free energy

$$F = U-TS = N(cE_v + k_B T (c \ln c + (1-c) \ln (1-c)))$$

and taking $c \ll 1 $

$$\dfrac{F}{N} = c E_v + k_B T c \ln c$$

5.equilibrium concentration (by minimizing the Helmholtz free energy.)

$$ c \rightarrow e^{-\dfrac{E_v}{k_B T}}$$

So what's the missing points to understand the marked procedure above?

  • $\begingroup$ Surely you mean $NcE_v$ rather than $NcN_v$ right below the question mark. I'm assuming that that's a typo rather than the source of your confusion. $\endgroup$ – David Hammen Feb 10 '17 at 12:11
  • $\begingroup$ It's just a typo. $\endgroup$ – Jack Feb 10 '17 at 12:46

Why we don't consider the internal energy of the whole system rather than the vacancies?

Every atom being exactly at a crystalline site represents the minimum energy configuration. (Aside: This configuration only happens at absolute zero temperature.) What this energy is is irrelevant; it's some constant. You might as well treat it as zero.

  • $\begingroup$ Why the temperature is 0 K? $\endgroup$ – Jack Feb 10 '17 at 13:00
  • $\begingroup$ @Jack -- Third law of thermodynamics. $\endgroup$ – David Hammen Feb 10 '17 at 13:19
  • $\begingroup$ To derive the concentration the constant energy (due to on-site atom) has no influence with the final minimization procedure? $\endgroup$ – Jack Feb 10 '17 at 13:38

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