In introductory physics, we learned that a particle (ideal gas) in thermal equilibrium with its surroundings has a kinetic energy of 3kT/2. Calculate the de Broglie wavelength for (a) a neutron at room temperature (300 K) and (b) a “cold” neutron at 77 K (liquid nitrogen). Strategy In both of these

Question

In introductory physics, we learned that a particle (ideal gas) in thermal equilibrium with its surroundings has a kinetic energy of 3kT/2. Calculate the de Broglie wavelength for (a) a neutron at room temperature (300 K) and (b) a “cold” neutron at 77 K (liquid nitrogen).

Strategy In both of these cases we will use Equation (5.2) to find the de Broglie wavelength. First, we will need to determine the momentum, and we note in both cases the energies of the particles will be so low that we can perform a nonrelativistic calculation. Neutrons have a rest energy of almost 1000 MeV, and their kinetic energies at these temperatures will be quite low (0.026 eV at 300 K).

[latex]\lambda=\frac{h}{p}[/latex] (5.2)

Accepted Answer

We begin by finding the de Broglie wavelength of the neutron in terms of the temperature.

[latex]\frac{p^{2}}{2 m}= K . E .=\frac{3}{2} k T[/latex] (5.8)

[latex]p=\sqrt{3 m k T}[/latex]

[latex]\begin{aligned}\lambda &=\frac{h}{p}=\frac{h}{\sqrt{3 m k T}}=\frac{h c}{\sqrt{3\left(m c^{2} ight) k T}} \&=\frac{1}{T^{1 / 2}} \frac{1240 eV \cdot nm }{\sqrt{3\left(938 imes 10^{6} eV ight)\left(8.62 imes 10^{-5} eV / K ight)}}\end{aligned}[/latex]

It again has been convenient to use eV units.

[latex]\lambda=\frac{2.52}{T^{1 / 2}} nm \cdot K ^{1 / 2}[/latex]

[latex]\lambda(300 K )=\frac{2.52 nm \cdot K ^{1 / 2}}{\sqrt{300 K }}=0.145 nm[/latex] (5.9)

[latex]\lambda(77 K )=\frac{2.52 nm \cdot K ^{1 / 2}}{\sqrt{77 K }}=0.287 nm[/latex]

These wavelengths are thus suitable for diffraction by crystals. “Supercold” neutrons, used to produce even larger wavelengths, are useful because extraneous electric and magnetic fields do not affect neutrons nearly as much as electrons.

Question 5.4: In introductory physics, we learned that a particle (ideal g...

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