Let us consider experiments by shooting atomic nuclei (projectiles) with high speed onto a thin foil target, which contains a huge number of atomic nuclei. We are interested in the interaction of the projectile nuclei (which are matter waves) with the spherical target nuclei. In the picture below you can see a big target nucleus and different possible trajectories of a projectile.
Most of the projectiles pass straight through the target foil without any penetration (blue trajectories). The distance between the projectile trajectory and the target nucleus is called the impact parameter b. A large distance between the two nuclei (large b), leads to small scattering angles. Only a tiny fraction of the incident nuclei gets really close to one of the target nuclei (small b) and is repelled by its electric charge to greater scattering angles. The smaller the distance between the projectile and the target nucleus, the bigger the scattering angle is (ice blue trajectories in the picture).
If a projectile nucleus comes with a sufficiently high energy, it can overcome the electrostatic repulsion (Coulomb wall) and collide directly with the target nucleus. This collision is governed by the strong interaction, which is a force between the constituents of a nucleus, the nucleons. Due to this attractive interaction the projectile penetrates and is pulled towards the smaller scattering angles (green and yellow trajectories).
Experiments performed at the Helmholtz-Zentrum Berlin with oxygen nuclei (16O + 16O) at impact energies of 350 MeV, that corresponds to 20% of light velocity, have shown that some nuclei can pass through each other, and the projectile is deflected to negative scattering angles (orange and red trajectories).
The deviation of the projectile to the negative angles reaches a maximum for the red trajectories at an angle ΘR. For smaller values of b the deviation again becomes smaller.
The picture shows a target nucleus and different possible trajectories of the projectile. The atomic nucleus can be considered as a ball and the distance b describes well the behavior of the projectiles. The nuclear rainbow is, just like the rainbow, the enhanced intensity at the angle ΘR.
Since many impact parameters b are deflected into one rainbow angle ΘR, at ΘR a concentration of many scattering trajectories occur. We thus measure an enhanced intensity (number of detected particles) at this ΘR, which corresponds to the observation of the nuclear rainbow. Like in the case of light, the position of angle ΘR depends on the strength of the interaction as well as on the energy (or the wavelength) of the projectile nuclei. We are dealing here with particle waves and therefore the projectile masses play also an important role.
In an experiment we can't distinguish, from which side the projectile passed the target nucleus, we always get a superposition of "positive" and "negative" scattering angles in the measured intensity distribution. To be able to watch the tiny contributions to the negative scattering angles, the contributions to the positive scattering angles must be even less! This requires a very careful choice of the experimental parameters.
The attractive interaction of the particle while passing the target nucleus causes a refraction of the particle wave. For our nuclear scattering this corresponds to the refraction index of n = 2 to 3. Penetration of the two nuclei at the collision leads to an increased nuclear density by the factor 2. Therefore these collisions are also used to study the behavior of compressed nuclear matter.