Beam emittance

Samples of a bivariate normal distribution, representing particles in phase space, with position horizontal and momentum vertical.

In accelerator physics, emittance is a property of a charged particle beam. It refers to the area occupied by the beam in a position-and-momentum phase space.

Each particle in a beam can be described by its position and momentum along each of three orthoganal axes, for a total of six position and momentum coordinates. When the position and momentum for a single axis are plotted on a two dimensional graph, the average spread of the coordinates on this plot are the emittance. As such, a beam will have three emittances, one along each axis, which can be described independently. As particle momentum along an axis is usually described as an angle relative to that axis, an area on a position-momentum plot will have dimensions of length times angle (for example, millimeters per milliradian).

Emittance is important for analysis of particle beams. As long as the beam is only subjected to conservative forces, Liouville's Theorem shows that emittance is a conserved quantity. If the distribution over phase space is represented as a cloud in a plot (see figure), emittance is the area of the cloud. A variety of more exact definitions handle the fuzzy borders of the cloud and the case of a cloud that does not have an elliptical shape. In addition, the emittance along each axis is independent unless the beam passes through beamline elements (such as solenoid magnets) which correlate them.

A low-emittance particle beam is a beam where the particles are confined to a small distance and have nearly the same momentum, which is a desirable property for ensuring that the entire beam is transported to its destination. In a colliding beam accelerator, keeping the emittance small means that the likelihood of particle interactions will be greater resulting in higher luminosity. In a synchrotron light source, low emittance means that the resulting x-ray beam will be small, and result in higher brightness.

Definitions

Emittance has units of length, but is usually referred to as "length × angle", for example, "millimeter × milli-radians". It can be measured in all three spatial dimensions. The dimension parallel to the motion of the particle is called the longitudinal emittance and the other two dimensions are referred to as the transverse emittances.

Geometric Emittance

The arithmetic definition of a transverse emittance ($\varepsilon$) is:

$\varepsilon ={\frac {6\pi \left({\text{width}}^{2}-D^{2}\left({\frac {\mathrm {d} p}{p}}\right)^{2}\right)}{B}}$

Where:

• width is the width of the particle beam
• dp/p is the momentum spread of the particle beam
• D is the value of the dispersion function at the measurement point in the particle accelerator
• B is the value of the beta function at the measurement point in the particle accelerator

Since it is difficult to measure the full width of the beam, either the RMS width of the beam or the value of the width that encompasses a specific percentage of the beam (for example 95%) is measured. The emittance from these width measurements is then referred to as the "RMS emittance" or the "95% emittance", respectively.

One should distinguish the emittance of a single particle from that of the whole beam. The emittance of a single particle is the value of the invariant quantity

$\epsilon =\gamma x^{2}+2\alpha xx'+\beta x'^{2}$

where x and x′ are the position and angle of the particle respectively and $\beta ,\alpha ,\gamma$ are the Courant–Snyder (Twiss) parameters. (In the context of Hamiltonian dynamics, one should be more careful to formulate in terms of a transverse momentum instead of x′.) This is the single particle emittance.

RMS Emittance

In some particle accelerators, Twiss parameters are not commonly used and the emittance is defined by the beam's second order phase space statistics instead. Here, the RMS emittance ($\varepsilon _{\text{RMS}}$) is defined to be,

$\varepsilon _{\text{RMS}}={\sqrt {\langle x^{2}\rangle \langle x^{\prime 2}\rangle -\langle x\cdot x^{\prime }\rangle ^{2}}}$

where $\langle x^{2}\rangle$ is the variance of the particle's position, $\langle x^{\prime 2}\rangle$ is the variance of the angle a particle makes with the direction of travel in the accelerator ($x^{\prime }={\frac {\mathrm {d} x}{\mathrm {d} z}}$ with $z$ along the direction of travel), and $\langle x\cdot x^{\prime }\rangle$ represents an angle-position correlation of particles in the beam. This definition reverts to the prior listed definition of geometric emittance in the case of a periodic accelerator lattice where the Twiss parameters can be defined.

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.

$\varepsilon _{\text{RMS}}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle \\\langle x\cdot x^{\prime }\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle \end{vmatrix}}}$

Depending on context, some may also add a scaling factor in front of the equation for RMS emittance so that it will correspond to the area of uniformly filled ellipse shaped distribution in phase space.

RMS Emittance in Higher Dimensions

It is sometimes useful to talk about phase space area for either four dimensional transverse phase space (IE $x$, $x^{\prime }$, $y$, $y^{\prime }$) or the full six dimensional phase space of particles (IE $x$, $x^{\prime }$, $y$, $y^{\prime }$, $\Delta z$, $\Delta z^{\prime }$). It is now clear from the matrix definition of RMS emittance how the definition may generalize into higher dimensions.

$\varepsilon _{{\text{RMS}},6D}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle &\langle x\cdot y\rangle &\langle x\cdot y^{\prime }\rangle &\langle x\cdot z\rangle &\langle x\cdot z^{\prime }\rangle \\\langle x^{\prime }\cdot x\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle &\langle x^{\prime }\cdot y\rangle &\langle x^{\prime }\cdot y^{\prime }\rangle &\langle x^{\prime }\cdot z\rangle &\langle x^{\prime }\cdot z^{\prime }\rangle \\\langle y\cdot x\rangle &\langle y\cdot x^{\prime }\rangle &\langle y\cdot y\rangle &\langle y\cdot y^{\prime }\rangle &\langle y\cdot z\rangle &\langle y\cdot z^{\prime }\rangle \\\langle y^{\prime }\cdot x\rangle &\langle y^{\prime }\cdot x^{\prime }\rangle &\langle y^{\prime }\cdot y\rangle &\langle y^{\prime }\cdot y^{\prime }\rangle &\langle y^{\prime }\cdot z\rangle &\langle y^{\prime }\cdot z^{\prime }\rangle \\\langle z\cdot x\rangle &\langle z\cdot x^{\prime }\rangle &\langle z\cdot y\rangle &\langle z\cdot y^{\prime }\rangle &\langle z\cdot z\rangle &\langle z\cdot z^{\prime }\rangle \\\langle z^{\prime }\cdot x\rangle &\langle z^{\prime }\cdot x^{\prime }\rangle &\langle z^{\prime }\cdot y\rangle &\langle z^{\prime }\cdot y^{\prime }\rangle &\langle z^{\prime }\cdot z\rangle &\langle z^{\prime }\cdot z^{\prime }\rangle \\\end{vmatrix}}}$

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.

$\varepsilon _{{\text{RMS}},6D}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle &0&0&0&0\\\langle x^{\prime }\cdot x\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle &0&0&0&0\\0&0&\langle y\cdot y\rangle &\langle y\cdot y^{\prime }\rangle &0&0\\0&0&\langle y^{\prime }\cdot y\rangle &\langle y^{\prime }\cdot y^{\prime }\rangle &0&0\\0&0&0&0&\langle z\cdot z\rangle &\langle z\cdot z^{\prime }\rangle \\0&0&0&0&\langle z^{\prime }\cdot z\rangle &\langle z^{\prime }\cdot z^{\prime }\rangle \\\end{vmatrix}}}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle \\\langle x^{\prime }\cdot x\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle \\\end{vmatrix}}}{\sqrt {\begin{vmatrix}\langle y\cdot y\rangle &\langle y\cdot y^{\prime }\rangle \\\langle y^{\prime }\cdot y\rangle &\langle y^{\prime }\cdot y^{\prime }\rangle \\\end{vmatrix}}}{\sqrt {\begin{vmatrix}\langle z\cdot z\rangle &\langle z\cdot z^{\prime }\rangle \\\langle z^{\prime }\cdot z\rangle &\langle z^{\prime }\cdot z^{\prime }\rangle \\\end{vmatrix}}}=\varepsilon _{x}\varepsilon _{y}\varepsilon _{z}$

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. For this purpose we define normalized emittance which is invariant under acceleration.

$\varepsilon _{n}={\sqrt {\langle x^{2}\rangle \langle p_{x}^{2}\rangle -\langle x\cdot p_{x}\rangle ^{2}}}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot p_{x}\rangle \\\langle x\cdot p_{x}\rangle &\langle p_{x}\cdot p_{x}\rangle \end{vmatrix}}}$

where the angle $x^{\prime }={\frac {\mathrm {d} x}{\mathrm {d} z}}$ has been replaced with a transverse momentum $p_{x}$which does not depend on longitudinal momentum.

Normalized emittance is related to the previous definitions of emittance through the Lorentz factor ($\gamma$) and relativistic velocity in direction of the beam's travel ($\beta _{z}$).

$\epsilon _{n}=\beta _{z}\gamma \epsilon$

The normalized emittance does not change as a function of energy and so can track beam degradation if the particles are accelerated. If β is close to one then the emittance is approximately inversely proportional to the energy and so the physical width of the beam will vary inversely to 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.

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. When no radiation is present, the emittances remain constant (apart from impedance effects and intrabeam scattering). In this case, the emittance is determined by the initial particle distribution. In particular if one injects a "small" emittance, it remains small, whereas if one injects a "large" emittance, it remains large.

Acceptance

The acceptance, also called admittance, is the maximum emittance that a beam transport system or analyzing system is able to transmit. This is the size of the chamber transformed into phase space and does not suffer from the ambiguities of the definition of beam emittance.

Conservation of emittance

Lenses can focus a beam, reducing its size in one transverse dimension while increasing its angular spread, but cannot change the total emittance. This is a result of Liouville's theorem. Ways of reducing the beam emittance include radiation damping, stochastic cooling, and electron cooling.

Emittance and brightness

Emittance is also related to the brightness of the beam. In microscopy brightness is very often used, because it includes the current in the beam and most systems are circularly symmetric.

$B={\frac {{\eta }I}{{\epsilon _{x}}{\epsilon _{y}}}}$

with $\eta ={\frac {1}{8\pi ^{2}}}$

See also

• Accelerator Physics
• Thermal Emittance / Mean Transverse Energy
• Etendue