Academic literature on the topic 'Combined eqns'

Abstract Beam polarization gives information on the electron’s inclination to behave with aberration. In the preceding chapters, we expounded a convenient way to calculate helicity amplitudes for various processes at high energies. In this chapter, we show how those helicity amplitudes must be combined for the description of processes in which arbitrary spin polarizations occur. We first treat the case of fermion polarization in detail and then analyse the polarization effects for photons and gluons. Throughout this chapter, we again neglect the finite mass effects for simplicity. The positive helicity states for fermions, introduced in Chapter 3 [eqns (3.1)], describe particles with spin polarization along the direction of motion, where as negative helicity states have their spin polarization opposite to the direction of motion. This can be seen as follows.

Abstract Just as deviations form the ideal orbit lead to betatron oscillations, so also deviations from the ideal energy lead to energy oscillations which are sometimes called synchrotron oscillations. These oscillations contribute to the size of the electron beam in the horizontal plane because energy deviations lead to deviations in position and direction of the electrons through the dispersion function fsee eqn ( 11.9)]. Because the dispersion function depends on the value of the s-coordinate around the ring, the contribution of the energy oscillations to beam size also depends on D(s). Furthermore, because the dispersion function tends to follow the contour of the beta function [see eqn ( I 1.11)], the effects of the betatron and synchrotron oscillations tend to combine to increase the beam size instead of cancelling each other out. If the energy distribution of the electrons follows a Gaussian function with standard deviation σ(E), then from eqn (11.9) the contribution of the energy oscillations to the total beam width is (σ (E)/ Eo) × D(s). In any practical storage ring, Q is a long way from resonance so we can assume that the two types of oscillations are uncorrelated and add the contributions from the betatron [eqn ( 10.29)] and synchrotron oscillation in quadrature.