Problem: Validity of the semiclassical approximation
Cold and thermal neutrons are considered as classical particles at the instrument level, saving the wave description to the component level. We will now look closer at the soundness of this semiclassical approximation.
Question 1
In a typical high-angular-resolution experiment to measure diffraction from single crystals, the neutron direction is determined within \(10'\) (10 arc minutes) in the horizontal direction. Consider a set-up with a low neutron energy of 3.7 meV, where the beam is limited in space by a slit (or diaphragm) with a width (horizontally) of 1 mm. How does this set-up agree with the uncertainty relations? And how small a beam would still be consistent with the present value of collimation?
The momentum of a neutron with \(E = 3.7\) meV is
\( p=\sqrt{2mE}=\sqrt{2 \cdot 1.675\cdot 10^{-27} \cdot 0.0037 \cdot 1.602\cdot 10^{-19}}=1.409\cdot 10^{-24} \,\text{kg m/s}.\)
The horizontal uncertainty of the momentum due to divergence \(\eta\) is \(\delta p_x \approx |\bar{p}|\cdot \eta\). The divergence in in radians \( \frac{\pi\cdot 10}{180\cdot60}=2.909\cdot 10^{-3} \). The spacial horizontal limitation is \(dx = 1\) mm and since
\( \delta p_x \text{d}x =1.409\cdot 10^{-24} \cdot 2.909\cdot 10^{-3}\cdot 0.001 \dfrac{\text{kg}\cdot\text{m}^2}{\text{s}}= 4.1 \cdot 10^{-30}\text{J}\cdot\text{s} > \dfrac{\hbar}{2} ,\)
Heisenberg's uncertainty relation is valid with the set-up.
Question 2
At the high-energy-resolution neutron scattering instrument BASIS at SNS, the neutron energy can be measured with an accuracy of 2.2 \(\mu\)eV, using backscattering from Si analyzer crystals, while the neutron energy itself is \(E=2.08\) meV. Since SNS is a pulsed neutron source, the energy resolution of the incoming neutrons is determined by the pulse length, which is around 40 \(\mu\)s. How does that match the Heisenberg uncertainty relations? (Similar instruments operate at reactor sources, e.g. the instrument SPHERES at FRM-2 (Munich), with an energy resolution of 0.6 \(\mu\)eV.)
\( \delta E \delta t =2.2\cdot 10^{-6} \cdot 1.602\cdot 10^{-19}\text{J} \cdot 4\cdot 10^{-5}\text{s} = 140.98\cdot 10^{-31}\text{J}\cdot\text{s}> \dfrac{\hbar}{2} .\)
The energy resolution is limited at a pulsed source by the pulse length.