[MUSIC] During this first module, we are introducing the objects studied by particle physics namely matter, forces and space-time but also, of course, reactions between particles. In this third video, we present the concept of probability, to characterize the strength of fundamental forces. We'll also define one of the most important quantities in particle physics, which is called the cross section. This quantity allows to calculate and to measure the strength of subatomic reactions. After watching this video, you will be able to define the cross section in terms of incident flux and number of target particles. You will also be able to relate the cross section to the probability, and the rate of a particle reaction. In quantum physics the motion of fields and their way of interacting is described by probability amplitudes and probabilities themselves. They are defined in a frequentist way by multiple measurements. For the movement of a particular the wave function psi, describes the probability amplitude as a function of space-time, and the four-momentum of the particle, namely as a function of t and x and of energy E and momentum p. The probability density to find the particle at time t and at location x, is given by the square of the wave function psi* psi. Integrating this density over a finite volume gives us the probability that the particle is in this volume. The probability amplitude M for a reaction is the joint probability amplitude that two current densities j_1 and j_2 meet and exchange a virtual photon of four-momentum q. The probability for the reaction is then given by the cross section sigma, which is proportional to the square of the amplitude |M|^2. The physics of particle motion and particle reactions is thus expressed via statistical probabilities. Mathematically speaking, a probability is a real number between 0 and 1. In a frequentist approach, the probability is defined as the ratio between the number of desired outcomes of an experiment, we might call successes, and the number of trials it took to obtain them. In terms of physics, this is the ratio between the number of times an event happens, and the number of times it could have happened. An event can for example be the localization of a particle inside a given volume. The ratio can also be the number of times a reaction takes place, divided by the number of times it could have taken place. This means in particular that reactions do not always happen. In fact, we will find that they happen rather rarely. Probabilities follow their own relatively simple algebra. For our purposes it suffices to know our two simple rules. The joint probability of two independent events to happen is the product of their individual probabilities. The probability that at least one of two incompatible events happen, is the sum of the two properties. The two probabilities, sorry. Incompatible means that the two events cannot be realized at the same time. And all of these rules applied to the probability amplitudes that we have introduces earlier. The two limiting cases for a probability are p=0 for impossible events and p=1 for certain events. In general, we will find that probabilities for particle reactions are rather small numbers. Processes which follow probabilistic laws are not certain individually, but can still be predicted collectively. This little simulation shows what is called a Galton board as an example. The small balls are introduced in the middle at the top of the board. The obstacles scatter them either left or right with 50% probability. The multiple scatters are independent of each other, the result does not depend on a previous one. At the bottom of the board, the balls show, as you can see, a Gaussian distribution around the middle position. For those of you who know a little statistics, this is an example of how chains of multiple events with an arbitrary individual distribution – Boolean in this case but not necessarily Boolean – asymptotically produce a Gaussian distribution, the famous bell-shaped distribution, which is perfectly calculable and characterized by just two parameters, the mean and the width of the distribution. But let us come back to particle reaction. Particle physics explores matter by means of scattering experiments. For this purpose a beam of particles is directed towards a target. One counts the number of scatters which take place, and measures the scattered particle direction and energy. This sketch shows a fictitious situation where each target particle is represented by a small gray surface. Let us consider a simple model where a reaction takes place if and only if, a particle hits a gray surface. Let us make the assumption that the distance between target particles is large compared to their size such that the gray surfaces do not overlap. The interaction thus stays rare. In real life this corresponds to a target which is made by a thin foil or a rarified gas. The number of possible scatters is evidently proportional to the flux of incoming particles, of incoming projectiles. That is to say, their number per unit time which pass through a perpendicular surface. The number of actual scatters is proportional to the number of target particles per square meter. That is to say, the surface density n and the individual surface sigma. If fact, the probability to interact is simply the product of the surface density and the surface size sigma. A small part dI of the initial flux I will thus be scattered. The rest of the flux is just transmitted. The small fraction of the scattered flux -dI/I is thus equal to the interaction probably n times sigma. The density of targets can also be expressed as the volume density rho, if one multiplies by the infinitesimal target thickness, dx. Integrating over the thickness up to ∆x, one obtains an exponential law for the attenuation of the incident flux. We call sigma the cross section for the reaction because it has the dimension of a surface. It has nothing to with the geometrical size of the target particle, it rather represents the probability of interaction between an individual projectile and an individual target particle. The cross-section is often measured in enormous units which are one « barn » ten to the minus 28 square meters, which is a very large one. The cross-section is, thus, a sort of probability expressed by strange units, one must admit. But the rules for probability calculation apply. If two processes are independent of each other, the probabilities just simply add up. One distinguishes, for example, processes of elastic scattering, where the kinetic energy of the projectile does not change, just its direction changes, inelastic scattering, where the projector looses energy and the target particle recoils and/or changes mass, and absorption where the projectile simply disappears inside the target. For these mutually exclusive processes, the cross-sections can just be added to form what is called a total cross section, like in the bottom equation on this slide. The cross-section must obviously be a positive real number. It has a maximum, which corresponds to a reaction which always takes place. That is called the unitarity limit. An application example is calculated in video 1.3a, it deals with the attenuation of photons by a sheet of lead. The result will probably surprise you, and encourage you hopefully, to be very careful the next time you expose yourself to a beam of energetic photons. It is clear that the properties of the outgoing particles are also important to understand the reaction. For example, the angle by which the projectile is scattered gives useful information about the structure of the target, and the properties of the interaction as we'll see with many examples in the course of these lectures. Keeping in mind that the cross-section has nothing to do with the physical size of the target, let us continue to imagine the cross-section as an effective area for the interaction. If the projectile hits it a reaction takes place, otherwise the projectile passes without perturbation. The scattering into an infinitesimal element dOmega of solid angle, centered on the polar angle theta and the azimuth angle phi, then comes from an element dsigma of that surface in our geometrical model. The fraction of the incoming beams scattered into the solid element is proportional to the differential cross section, dsigma dOmega. The probability that a projectile is scattered into a solid angle element is not normally uniform. It depends on the angles theta and phi, the scattering and the azimuth angle, and allows one to look into the target in a literal sense. The total cross section is obtained integrating the differential cross section dsigma dOmega over the solid angle dOmega. Let us consider a system with cylindrical symmetry. The scattering probability is then independent of the azimuthal angle phi and only depends on the scattering angle theta. An example from classical physics is the scattering by a central force. It shows a fixed relation between the impact parameter b and the polar scattering angle theta. All projectiles with an impact parameter between b and b plus db and an initial azimuthal angle between phi and phi plus dphi, are scattered into an element dOmega around the direction (theta,phi). The fraction of the incoming beam which falls into such a region is dI/I equal rho times ∆x times b db dphi. Integrating over the azimuthal angle on which the force does not depend we will just obtain a factor of 2π, and the results defines the cross-section in terms of impact parameter, and the scheduling angle. Taking the absolute value guarantees that the cross section, is always a positive definite real number. The scattering angle theta depends on the impact parameter b, in a specific manner, which is determined, in fact, by the distance law of the central force or its equivalent potential. Let us take a really trivial example: the classical shock between two rigid bodies. The geometry and the laws of elastic reflection relate b to thetha, in the simple geometrical way sketched in this little drawing. The differential cross-section is just constant, it does not depend on the scattering angle. The angular distribution of the scattered particles is isotropic. The total cross section is found to be sigma times πR^2, equal to the geometrical surface of the target and we're not surprised to find this result which we expected in the first place. In the next video, Mercedes will give you a much more interesting example of a realistic electromagnetic interaction. In this case, the interaction of a He-4 nucleus also known as an alpha particle with a heavy nucleus like gold. This process is called Rutherford scattering, its experimental observation has revealed the existence of the atomic nucleus. [MUSIC]