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## Q. 13.10

Neutrons are used to study structures of solids and their properties. What energy (and temperature) neutrons are needed if the atomic structures are of the size 0.060 nm?

Strategy We use the de Broglie wavelength relation to match the wavelength needed with an appropriate momentum. From the momentum we determine the kinetic energy and temperature from $K=\frac{3}{2} k T$.

## Verified Solution

We determine the neutron momentum from the de Broglie wavelength relation with $\lambda=0.060$ nm.

$p=\frac{h}{\lambda}=\frac{6.63 \times 10^{-34} J \cdot s }{0.060 \times 10^{-9} m }=1.1 \times 10^{-23} kg \cdot m / s$

Because we expect the kinetic energy to be low, we use a nonrelativistic relation to determine the kinetic energy.

\begin{aligned}K &=\frac{p^{2}}{2 m}=\frac{\left(1.1 \times 10^{-23} kg \cdot m / s \right)^{2}}{2\left(1.67 \times 10^{-27} kg \right)}=3.6 \times 10^{-20} J \\&=0.23 eV\end{aligned}

Thus we see that the nonrelativistic relation is certainly adequate here. In thermal equilibrium, the temperature is found from $K=\frac{3}{2} k T$.

$T=\frac{2 K}{3 k}=\frac{2\left(3.6 \times 10^{-20} J \right)}{3\left(1.38 \times 10^{-23} J / K \right)}=1740 K$

Such energy is easily obtained by thermalizing neutrons from a nuclear reactor.