Cohen-Tannoudji Beam of Neutrons incident on a linear chain of atomic nuclei I have been self studying Cohen-Tannoudji - Quantum Mechanics, Vol.1 and trying to finish the exercises for the first chapter.
I was stuck on the very first problem about beam of neutrons incident on a linear chain of atomic nuclei. I have read the materials of chapter 1 for four/five times but still can't get really get how the physics work in this scenario.
For a very low energy E, the neutrons should just work like classical particles and the neutron detectors will only detect neutrons for the exact direction? But on the other hand, how could that possibly create any resonance?
Below is the question:
A beam of neutrons of constant velociy, mass Mn and energy E, is incident on a linear chain of atomic nuclei, arranged in a regular fashion as shown in the figure (these nuclei could be, for example, those of a long linear molecule). We call l the distance between two consecutive nuclei, and d, their size (d << l). A neutron detector D is placed far away, in a direction which makes an angle of $\theta$ with the direction of the incident neutrons.
a) Describe qualitatively the phenomena observed at D when the energy E of the incident neutrons is varied.
b) The counting rate, as a function of E, presents a resonance about $E=E_1$. Know that there are no other resonances for E < $E_1$., show that one can determine l. Calculate l for $\theta=30^o$ and $E_1=1.3 * 10^{-20}$ joule
 A: Your problem starts with the assumption that low energy means the neutron behave like classical particles.
It does mean that they are non-relativistic, so that:
$$E = \frac{p^2}{2m_n} $$
holds. Quantum mechanically, that means:
$$ \sqrt{2m_nE} = p =\hbar k = \frac{h}{\lambda}$$
where $\lambda$ is the de Broglie wavelength.
So you have a wave:
$$ \psi_i(x)=Ae^{ikx-\omega t} $$
incident on a lattice of scattering centers.
The scattering is coherent, so to get to a point described by $\theta$ at the detector, you do not have "which way" information.
You need to add the amplitudes for scattering from each of the $n$ scattering centers:
$$\psi_f(\theta)=[\psi_1(\theta_1)+\psi_2(\theta_2)+\ldots+\psi_n(\theta_n)]$$
and then square that to get a probability:
$$\frac{d\sigma(\theta)}{d\theta}\propto \psi_f(\theta)^*\psi_f(\theta)$$
The key is that the path length from each scattering center to a point on the detector is different, and when those differences are multiple of $\lambda$, there will be constructive interference.
You can look up light scattering from a diffraction grating to get a hint at the math (trig).
