During acceleration, individual particles can be defined by their positions X and momenta P, each specified by their Cartesian projections x, y, z, Px, Py, Pz. Within a beam, one can define distribution probabilities for the individual particle coordinates. These distributions are characterized by their variances:

^x = <(x -<x})2} (HL40a)

<2 = <(y -<y})2} (Kb)

<2 =<(z -<z})2} (HL40c)

<4 = <(Px -<Px})2} (III.40d)

4y = ((Py -<Py})2} (m.40e)

<Pz = ((Pz -<Pz})2} (III.40f)

the beam direction is taken to be Oz. It is assumed in the following that Pz ^ Px, Py. Hence the angles of the particles with respect to the beam axis are defined by their two projections dx = Px/Pz and dy = Py/Pz.