Question 3.9: Suppose that the Maxwell-Boltzmann distribution, Eq. (2.34),......
Suppose that the Maxwell-Boltzmann distribution, Eq. (2.34), represents the neutron density in Eqs. (3.43) and (3.44):
a. Find the value of \overline{v}.
b. If we define \overline{E} ≡ 1/2 m\overline{v}^2 , show that \overline{E} = 1.273 kT .
c. Why is your result different from the average energy of 3/2 kT given by Eq. (2.33)?
For part a, we substitute Eq. ( 2,34) into Eq. (3.44), along with \displaystyle\mathbf{v}={\sqrt{2E/m}}~.
\overline{v}=\int_0^{\infty}\sqrt{2E\,/m}M(E)d E \Bigg/\int_{0}^{\infty}M(E)d E and Eq. (2.36) sets the denominator equal to one. Thus \overline{v}=\int_{0}^{\infty}{\sqrt{2E/m}}M(E)d E ={\sqrt{2/m}}{\frac{2\pi}{(\pi k T)^{3/2}}}{{\int_0^{\infty}}}\ E\exp(-E/k T)d E
Let x = E/ kT Then
\overline{v}=={\sqrt{2/m}}\,{\frac{2(k T)^{1/2}}{\pi^{1/2}}}\int_{0}^{\infty}\,x\exp(-x)d x = ={\sqrt{2k T/\pi m}}
For part b, we have \overline{{{{E}}}}\equiv1/2 \ m\Bigl(2\sqrt{2k T/\pi m}\Bigr)^{2}=\frac{4}{\pi}k T=1.273\ k T
For c, we note that \overline{{{{E}}}}\equiv1/2 \ m \overline{v}^2 holds only if all of the neutrons have the same speed and energy.