[MUSIC] During this first model we are introducing the objects studies in particle physics, so matter, forces and space-time. In this forth video we give an example. for calculating and measuring a cross-section for the Rutherford experiment. This is the scattering process which demonstrated the existence of the atomic nucleus as we know it today. The goals are to apply the concept of cross-section, in a process that can be treated in a semi-classical manner. And to compute an interaction rate from the characteristics of an experiment. We will use a seminal experiment to demonstrate, how the cross-section serves to understand The unknow subatomic structure of a target. Geiger and Marsden measured in 1909 the angular distribution of alpha particles – fully ionized He-4 – scattering off thin foils of heavy metal like gold. The preferred model of the time for the nucleus and for the atoms was due to Thomson who had discovered the electron in 1897. This model described the atom as a homogeneous positive substance in which electrons are embedded. It thus has a very modest charge density. In this case, the alpha particles, being much smaller than atoms, should be able to penetrate the foil with only minor perturbation. If on the other hand, the atom consists of a tiny, dense, and positively charged nucleus surrounded by electrons to fill the atomic volume, as proposed by Rutherford, larger scattering angles would be possible. As we have seen in the previous video, the classical calculation of a cross section requires to find the relation between the impact parameter b and the scattering angle theta. We will calculate the relation for non relativistic velocities of the incident alpha particle. We base this calculation on three laws. The first one is Newton's law which relates the rate at which the momentum changes to the Coulomb force between projectile and target nucleus. It is quoted here using the variables indicated in the sketch on the right. The second law is the conservation of kinetic energy, characteristic for elastic scattering on a heavy target, which does not recoil when hit by the projectile. The third one is the conservation of angular momentum which introduces the angular velocity dgamma/dt. Combining these three equations we obtain the relation between the impact parameter b and the scattering angle theta, which we are looking for. If you are interested in the details of the calculation, we refer you to the video 1.4a. Inserting our result into the general relation for the classical cross-section, we obtained the differential cross-section for the Coulomb interaction with a point like heavy target. The cross section is proportional to the square products of the two charges (z Z)^2. This factor determines the order of magnitude of the cross section. The cross section is strongly peaked in the forward direction according to 1/sin(theta/2) to the fourth power, and inversely proportional to the square of the kinetic energy of the projectile. These factors are due to the propagator of the photon exchanged between the two reaction partners. These basic properties have indeed been experimentally established by Geiger and Marsden counting the impact of the scattered alpha particles on a screen covered by zink sulfate. This molecule emits a small flash of light when hit by a charged particle. The experimenters counted these flashes by watching the screen through a microscope. Is it realistic to count scatters by eye in such a setup? The small video 1.4b will convince you by actually calculating the rate. So the procedure to measure a cross section is thus clear. One must count the rate of interactions per second. For that we need a detector at a distance r covering a surface r^2 dOmega. One must then normalize the counting rate by the maximum possible interaction rate to obtain a probability. The maximum rate is obtained when every projectile interacts with the target. It is thus the product of the incident rate and the surface density of the target particles. It can also be expressed by the flux of projectiles multiplied by the number of target particles covered by the incident beam. The proportionality factor between the counted rated and the maximum counted rate is the cross section. The proportionality factor between counting rate and cross section is called luminosity. In the laboratory frame where the target is at rest it is given by the rate of the incoming beam times the surface density of target particles. If you want to use the volume density of the target you multiply by the target thickness. In the center of mass frame, where two beams collide head-on like in a collider, the luminosity is proportional to the population of the two beams, n_a and n_b, and to the frequency f at which they cross. And it is inversely proportional to the common surface S of the two beams. Counting rate and luminosity thus depend on the details of the experiment that we are conducting. Their ratio, the cross section, characterizes the physical process independent of these details. Until now we have only considered processes using the tool kit of classical physics. But we already know that this is not sufficient. In the next video we will show how quantum physics approaches scattering processes. Further on we will introduce an indispensable tool kit of particle physics, Feynman diagrams. [MUSIC]
