velocity in the direction of the beam's travel (): The normalized emittance does not change as a function of energy and so can be used to indicate beam degradation if the particles are accelerated. For speeds close to the speed of light, where is close to one, the emittance is approximately inversely proportional to the energy. In this case, the physical width of the beam will vary inversely with the square root of the energy. Higher dimensional versions of the normalized emittance can be defined in analogy to the RMS version by replacing all angles with their corresponding momenta. Measurement of Emittance Quadrupole Scan Technique One of the most fundamental methods of measuring beam emittance is the quadrupole scan method. The emittance of the beam for a particular plane of interest (i.e., horizontal or vertical) can be obtained by varying the field strength of a quadrupole (or quadrupoles) upstream of a monitor (i.e., a wire or a screen). The properties of a beam can be described as the following beam matrix. where is the derivative of x with respect to the longitudinal coordinate. The forces experienced by the beam as it travels down the beam line and passes through the quadrupole(s) are described using the transfer matrix (referenced to transfer maps page) of the beam line, including the quadrupole(s) and other beam line components such as drifts: Here is the transfer matrix between the original beam position and the quadrupole(s), is the transfer matrix of the quadrupole(s), and is the transfer matrix between the quadrupole(s) and the monitor screen. During the quadrupole scan process, and stay constant, and changes with the field strength of the quadrupole(s). The final beam when it reaches the monitor screen at distance s from its original position can be described as another beam matrix : The final beam matrix can be calculated from the original beam matrix by doing matrix multiplications with the beam line transfer matrix : Where is the transpose of . Now, focusing on the (1,1) element of the final beam matrix throughout the matrix multiplications, we get the equation: Here the middle term has a factor of 2 because . Now divide both sides of the above equation by , the equation becomes: Which is a quadratic equation of the variable . Since the RMS emittance RMS is defined to be the following. The RMS emittance of the original beam can