!! |- | || 0.15 |- | || 0.39 |- | || 0.87 |- | || 0.95 |} While the x and y axes are generally equivalent mathematically, in horizontal rings where the x coordinate represents the plane of the ring, consideration of dispersion can be added to the equation of the emittance. Because the magnetic force of a bending magnet is dependent on the energy of the particle being bent, particles of different energies will be bent along different trajectories through the magnet, even if their initial position and angle are the same. The effect of this dispersion on the beam emittance is given by: where is the dispersion at location s, is the ideal particle momentum, and is the root mean square of the momentum difference of the particles in the beam from the ideal momentum. (This definition assumes F=0.15) Longitudinal emittance The geometrical definition of longitudinal emittance is more complex than that of transverse emittance. While the and coordinates represent deviation from a reference trajectory which remains static, the coordinate represents deviation from a reference particle, which is itself moving with a specified energy. This deviation can be expressed in terms of distance along the reference trajectory, time of flight along the reference trajectory (how "early" or "late" the particle is compared to the reference), or phase (for a specified reference frequency). In turn, the coordinate is generally not expressed as an angle. Since represents the change in z over time, it corresponds to the forward motion of the particle. This can be given in absolute terms, as a velocity, momentum, or energy, or in relative terms, as a fraction of the position, momentum, or energy of the reference particle. However, the fundamental concept of emittance is the same—the position of the particles are plotted along one axis of a phase space plot, the rate of change in that position over time is plotted on the other axis, and the emittance is a measure of the area occupied on that plot. One possible definition of longitudinal emittance is given by: where the integral is taken along a path which tightly encloses the beam particles in phase space. Here is the reference frequency and the longitudinal coordinate is the phase of the particles relative to a reference particle. Longitudinal equations such as this one often must be solved numerically, rather than analytically. RMS emittance The geometric definition of