emittance assumes that the distribution of particles in phase space can be reasonably well characterized by an ellipse. In addition, the definitions using the root mean square of the particle distribution assume a Gaussian particle distribution. In cases where these assumptions do not hold, it is still possible to define a beam emittance using the moments of the distribution. Here, the RMS emittance () is defined to be, where is the variance of the particle's position, is the variance of the angle a particle makes with the direction of travel in the accelerator ( with along the direction of travel), and represents an angle-position correlation of particles in the beam. This definition is equivalent to the geometric emittance in the case of an elliptical particle distribution in phase space. The emittance may also be expressed as the determinant of the variance-covariance matrix of the beam's phase space coordinates where it becomes clear that quantity describes an effective area occupied by the beam in terms of its second order statistics. Depending on context, some definitions of RMS emittance will add a scaling factor to correspond to a fraction of the total distribution, to facilitate comparison with geometric emittances using the same fraction. RMS emittance in higher dimensions It is sometimes useful to talk about phase space area for either four dimensional transverse phase space (IE , , , ) or the full six dimensional phase space of particles (IE , , , , , ). The RMS emittance generalizes to full three dimensional space as shown: In the absences of correlations between different axes in the particle accelerator, most of these matrix elements become zero and we are left with a product of the emittance along each axis. Normalized emittance Although the previous definitions of emittance remain constant for linear beam transport, they do change when the particles undergo acceleration (an effect called adiabatic damping). In some applications, such as for linear accelerators, photoinjectors, and the accelerating sections of larger systems, it becomes important to compare beam quality across different energies. Normalized emittance, which is invariant under acceleration, is used for this purpose. Normalized emittance in one dimension is given by: The angle in the prior definition has been replaced with the normalized transverse momentum , where is the Lorentz factor and is the normalized transverse velocity. Normalized emittance is related to the previous definitions of emittance through and the normalized