be calculated using its beam matrix elements: To obtain the emittance measurement, the following procedure is employed: For each value (or value combination) of the quadrupole(s), the beam line transfer transfer matrix is calculated to determine values of and . The beam propagates through the varied beam line, and is observed at the monitor screen, where the beam size is measured. Repeat step 1 and 2 to obtain a series of values for and , fit the results with a parabola . Equate parabola fit parameters with original beam matrix elements: , , . Calculate RMS emittance of the original beam: If the length of the quadrupole is short compared to its focal length , where is the field strength of the quadrupole, its transfer matrix can be approximated by the thin lens approximation: Then the RMS emittance can be calculated by fitting a parabola to values of measured beam size versus quadrupole strength . By adding additional quadrupoles, this technique can be extended to a full 4-D reconstruction. Mask-Based Reconstruction Another fundamental method for measuring emittance is by using a predefined mask to imprint a pattern on the beam and sample the remaining beam at a screen downstream. Two such masks are pepper pots and TEM grids. A schematic of the TEM grid measurement is shown below. By using the knowledge of the spacing of the features in the mask one can extract information about the beam size at the mask plane. By measuring the spacing between the same features on the sampled beam downstream, one can extract information about the angles in the beam. The quantities of merit can be extracted as described in Marx et al. The choice of mask is generally dependent on the charge of the beam; low-charge beams are better suited to the TEM grid mask over the pepper pot, as more of the beam is transmitted. Emittance of electrons versus heavy particles To understand why the RMS emittance takes on a particular value in a storage ring, one needs to distinguish between electron storage rings and storage rings with heavier particles (such as protons). In an electron storage ring, radiation is an important effect, whereas when other particles are stored, it is typically a small effect. When radiation is important, the particles undergo radiation damping (which slowly decreases emittance turn after turn) and quantum excitation causing diffusion which leads to an equilibrium emittance.