STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
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2.4 Transverse Emittance<br />
One of the most important measurements is the transverse beam emittance.<br />
The emittance is a figure of merit that can be used to characterize the extent to<br />
which beam quality is preserved. To observe the effects of energy recovery on the<br />
beam quality, the emittance of the beam in the injector, in arcs 1 and 2 and of the<br />
energy recovered beam were measured.<br />
Each particle in the machine is defined by a point in six dimensional phase space<br />
with coordinates (x, px, y, py, ℓ, δ) where x (y) is the horizontal (vertical) displace-<br />
ment from the central trajectory, px (py) is the deviation of horizontal (vertical)<br />
momentum, ℓ is the path length differential from the synchronous particle and δ<br />
is the deviation of the longitudinal momentum from the design orbit. To study<br />
the collective motion of a bunch, the ensemble of electrons is projected onto two-<br />
dimensional phase sub-spaces. That is, the horizontal, vertical and longitudinal<br />
phase spaces are the projections of the beam onto the (x, px), (y, py), and (ℓ, δ)<br />
coordinate systems, respectively. The emittance is defined as the area of the el-<br />
lipse enclosing the beam in the phase space divided by π. There are two transverse<br />
emittances, horizontal and vertical, and one longitudinal emittance. Whereas the<br />
transverse emittances are commonly used as figures of merit, the bunch length and<br />
energy spread are often used in lieu of the longitudinal emittance.<br />
For an ensemble of non-interacting particles, Liouville’s Theorem states that<br />
under the influence of conservative forces the density of particles in the phase space<br />
remains constant [37]. In the (x, px) and (y, py) phase sub-spaces, the area of the<br />
beam ellipse is defined as the normalized emittance, ɛN. An alternate definition<br />
is the geometric emittance, ɛg, which is the area of the beam ellipse in the (x, x ′ )<br />
and (y, y ′ ) coordinate systems. Here x ′ = dx/dz (y ′ = dy/dz) and is the angle the<br />
trajectory makes in the horizontal (vertical) plane. The normalized and geometric<br />
36