Beam emittance

It refers to the area occupied by the beam in a position-and-momentum phase space.

In addition, the emittance along each axis is independent unless the beam passes through beamline elements (such as solenoid magnets) which correlate them.

In a colliding beam accelerator, keeping the emittance small means that the likelihood of particle interactions will be greater resulting in higher luminosity.

[4] The coordinate system used to describe the motion of particles in an accelerator has three orthogonal axes, but rather than being centered on a fixed point in space, they are oriented with respect to the trajectory of an "ideal" particle moving through the accelerator with no deviation from the intended speed, position, or direction.

When a particle moves through a circular accelerator or storage ring, the position

is shown on the right of the equation, and would often be included in the units of emittance, rather than being multiplied in to the computed value.

will depend on the application and the author, and a number of different choices exist in the literature.

Some common choices and their equivalent definition of emittance are: [1]: 83 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.

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.

represents the change in z over time, it corresponds to the forward motion of the particle.

[3]: 218 The geometric definition of emittance assumes that the distribution of particles in phase space can be reasonably well characterized by an ellipse.

In cases where these assumptions do not hold, it is still possible to define a beam emittance using the moments of the distribution.

is the variance of the angle a particle makes with the direction of travel in the accelerator (

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.

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.

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).

):[6] 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.

In this case, the physical width of the beam will vary inversely with the square root of the energy.

Then the RMS emittance can be calculated by fitting a parabola to values of measured beam size

By adding additional quadrupoles, this technique can be extended to a full 4-D reconstruction.

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.

[11] When no radiation is present, the emittances remain constant (apart from impedance effects and intrabeam scattering).

This is the size of the chamber transformed into phase space and does not suffer from the ambiguities of the definition of beam emittance.

Lenses can focus a beam, reducing its size in one transverse dimension while increasing its angular spread, but cannot change the total emittance.

In microscopy brightness is very often used, because it includes the current in the beam and most systems are circularly symmetric[clarification needed].

, describing a normal distribution in the initial phase space, is diffused by the emittance introduced by aberrations