Excitation by neutron impact
Task number: 4417
A neutron moving at 34 km·s−1 collides with a hydrogen atom in its ground state, initially at rest. Demonstrate that this collision must be elastic, i.e., kinetic energy is conserved.
Instruction: Show that the atom cannot be excited as a result of this collision.
Hint
If the hydrogen atom were to be excited, this would constitute an inelastic collision, in which part of the mechanical energy is converted into another form of energy — in this case, the internal energy of the excited hydrogen atom.
Determine the kinetic energy of the incoming neutron and the energy required to excite the hydrogen atom.
Analysis
In an elastic collision, both the law of conservation of momentum and the law of conservation of mechanical (kinetic) energy apply. In an inelastic collision, momentum is still conserved, but part of the mechanical (kinetic) energy is transformed into another form — typically some form of internal energy. In our example, in the case of an inelastic collision, part of the neutron’s energy would be used to excite the electron in the hydrogen atom.
To excite the electron in a hydrogen atom initially in its ground state, it is necessary to supply at least the energy corresponding to the difference between the first two energy levels (ground state and first excited state). For excitation to a higher level, the required energy would be correspondingly greater.
We therefore compare the kinetic energy of the neutron with the energy required to excite the hydrogen atom.
Known quantities
v = 34 km·s−1 = 34·103 m·s−1 neutron speed mn = 1,67·10−27 kg neutron mass E1 = −13,6 eV energy of the electron in the hydrogen atom in the ground state Solution
First, we determine the kinetic energy of the incoming neutron
\[E_k=\frac{1}{2}m_{n}v^2=\frac{1}{2}\cdot1{,}67{\cdot}10^{-27}\cdot\left(34{\cdot}10^3\right)^2\,\mathrm{J}=9{,}6{\cdot}10^{-19}\,\mathrm{J}=6{,}0\,\mathrm{eV}.\]In the calculation, we used the conversion 1 eV = 1,6·10−19 J.
Next, we determine the energy required to excite a hydrogen atom from the ground state. This energy equals the difference between the electron energy in the first level (ground state) E1 and the second level (first excited state) E2. We use the formula for the energy of the n‑th level in a hydrogen atom
\[E_n =\frac{E_1}{n^2}\,.\]The excitation energy is then
\[E_{excit.}=E_2-E_1=\frac{E_1}{4}-E_1=-\frac{3}{4}E_1=-\frac{3}{4}\cdot\left(13{,}6\right)\,\mathrm{eV}=10{,}2\,\mathrm{eV}\,.\]We see that the kinetic energy of the incoming neutron is smaller than the energy required to excite the hydrogen atom.
Answer
We see that the kinetic energy of the incoming neutron is smaller than the energy required to excite the hydrogen atom. This means that the hydrogen atom cannot absorb this energy (any part of it), and the collision between the neutron and the hydrogen atom will therefore be elastic.
