Beam emittance

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

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

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

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

where x and x′ are the position and angle of the particle respectively and ${\displaystyle \beta ,\alpha ,\gamma }$ are the 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.

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 (${\displaystyle \varepsilon _{\text{RMS}}}$) is defined to be,[1]

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

where ${\displaystyle \langle x^{2}\rangle }$ is the variance of the particle's position, ${\displaystyle \langle x^{\prime 2}\rangle }$ is the variance of the angle a particle makes with the direction of travel in the accelerator (${\displaystyle x^{\prime }={\frac {\mathrm {d} x}{\mathrm {d} z}}}$ with ${\displaystyle z}$ along the direction of travel), and ${\displaystyle \langle x\cdot x^{\prime }\rangle }$ 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.

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

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.

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

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

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.

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

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.

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

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

Normalized emittance is related to the previous definitions of emittance through the Lorentz factor (${\displaystyle \gamma }$) and relativistic velocity in direction of the beam's travel (${\displaystyle \beta _{z}}$).[2]

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

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.