[MUSIC] During this fourth module, we go into details about the properties of electromagnetic interaction. In this third video, we discussed the phenomenon of spin and its consequences for electromagnetic interactions. After following this video, you will know spin as a typical quantum observable, and its measurable consequences and the functional principle of a Penning trap. We have seen in video 1.1 that the matter particles known up to now are all fermions with spin one-half. And the particles carrying the forces are spin one bosons. The only elementary scalar boson observed to date is the Higgs boson. It is therefore inevitable that we discuss date spin in more depths. We will not do that with a formal approach, instead we discuss only the consequences of spin. Spin is a typical quantum observable which has no classical equivalent. It is an intrinsic degree of freedom corresponding to an angular momentum manifesting itself by a magnetic moment. There's no movement associated to either of these moments, let alone a rotation. This is evident because spin is a property of pointlike particles. We will nevertheless use a classical analogy laid out on this slide. For a particle with charge q and mass m, the magnetic moment µ is related to the angular momentum L by µ = q/(2m) L. This is easy to see for a particle traveling on a circle of radius r. Its magnetic moment is the product of the current caused by the motion and the surface included in the path. The current can be expressed using charge and angular velocity, the magnetic moment is thus proportionate to the angular momentum. The proportionality factor is simply q/(2m). It is plausible that an intrinsic angular momentum would cause an analogous magnetic moment. A new proportionality constant comes in, it is called the gyromagnetic ratio g. It's value depends on the internal structure of the particle. For a pointlike fermion, such as the electron, muon or tau, the spin is one-half, the charge is -e, so µ is equal to -ge/(4m). A QED calculation to first order gives g simply equal to two. The classical experiment to measure g uses an electron trapped in an electromagnetic bottle called a Penning Trap. A very good description is available in the article by Ekstrom and Wineland, in Scientific American, that is quoted on the bottom of this page. The system resembles a macroscopic atom where the nucleus is replaced by an external field. The electron circulates in a magnetic field which is pretty much homogeneous except for the focusing component generated by a ferromagnetic nickel ring in the equatorial plane. An electric field counteracts excursions along the axis of the magnetic field. It is generated by a ring electrode and two cap electrodes. We calculate the energy levels of such a system in video 4.3a. The result which is given here, neglecting the focusing effects of the ring the caps, is relatively simple. The first term in blue corresponds to a free movement in the z direction. This movement is constraint by the focusing components we have neglected. And they caused the axial oscillation. The two terms in red correspond to harmonic oscillator in the equatorial plane with energy eigenvalues which are omega times (n+1/2) for n equal to zero, one, two and so on. Adding the magnetic moment of the spin µ_s, the last term in green is obtained. The two directions of spin, s_z equal plus one-half and minus one-half, are therefore separated by the same energy difference omega as the main levels, provided that g is exactly equal to two. The energy levels for spin parallel and anti-parallel to z would then be displaced by exactly one step omega. This degeneracy does not exist if the value of g differs from two. The levels are then slightly displaced by a frequency shift delta omega equal to omega times (g-2)/2, which can be measured with impressive accuracy using a Penning trap. The principle of the measurement is shown here by its electrical analogue. In an empty trap, the three electrodes form a capacitance network that transmits the AC signal generated on top to the detector on the bottom. The trapped electron introduces an additional capacitance-inductance in the circuit. The amplitude of the transmitting signal as a function of frequency thus indicates the number of electrons in the device. And the frequency shift delta omega which determines the deviation of g from two. The figure on the left shows the detected signal as a function of time at the well adjusted frequency. Each major step corresponds to the loss of one electron in the trap. The right figure shows the frequency shift delta omega as a function of time. It's too minima indicate the two directions of spin. Also indicated are the much larger shifts due to a change in orbit. The precidion of the experimental result is amazing. You see the value obtained in blue on the bottom of the page. The value in bracket indicates the measurement error which applies to the 13th and 14th decimal places. Comparison of this value to electromagnetic theory requires a calculation of the same quantity to 10th order in the pertubation expansion. One finds good agreement between measurement and theory even at this unrivalled precision. The measurement also gives us one of the most precise values of the fine structure constant alpha equals e^2/(4π). In the next video, Mercedes will discuss Compton scattering, the elastic scattering of a photon off an electron. [SOUND]