2 Beam Dynamics
39
We may find the relationship between canonical momentum and divergence from
the substitution:
p y = m 0
dy
dt
γ = m 0
ds
dt
dy
ds
γ = mc (βγ ) y
.
(2.50)
Writing Liouville’s Theorem expressed in canonical coordinates we can use the
above expression to define a conserved quantity and relate it to the area in (y, y )
space
p y dy = m 0 c (βγ )
y
dy = p 0
y
dy
(2.51)
where p 0 is the momentum in the direction of motion of the particle.
This invariant is the emittance, ε, of our transverse phase space multiplied by
the relativistic βγ which is proportional to momentum. Accelerator physicists often
call this the invariant or ‘normalised’ emittance:
ε
∗
= βγ ε [π mm · mrad]
(2.52)
This normalised emittance, ε ∗ , is conserved as acceleration proceeds in a
synchrotron and the physical emittance within the right-hand side of the equation
must fall inversely with momentum if the whole term is to be conserved. Close to
the velocity of light this implies that it is inversely proportional to energy.
Emittance = πε =
y
dy = πε
∗ / (βγ ) ∝ 1/p 0 .
(2.53)
We therefore expect the beam dimensions to shrink as (Fig. 2.19) a phenomenon
called ‘adiabatic damping’.
2.3.1 Chains of Accelerators
As a consequence of the adiabatic shrinking, the beam emittance is largest at low
energy, and so is the beam dimension. Proton accelerators need their full aperture
at injection and it is then that their design is most critical. For this reason it is
economic to split a single large ring into a chain of accelerators—the smaller
radius rings having a large aperture while the higher energy rings with large radius
can have smaller apertures. In these chains of proton accelerators, such as the
CERN accelerator complex, Linac – Booster – PS – SPS, the invariant emittance,
determined by the parameters of the beam as it leaves the ion source at the beginning
of the linac, may be conserved to several hundred GeV. Of course one must
guard against mismatches between machines or non-linear fields which dilate the
emittance.
39
We may find the relationship between canonical momentum and divergence from
the substitution:
p y = m 0
dy
dt
γ = m 0
ds
dt
dy
ds
γ = mc (βγ ) y
.
(2.50)
Writing Liouville’s Theorem expressed in canonical coordinates we can use the
above expression to define a conserved quantity and relate it to the area in (y, y )
space
p y dy = m 0 c (βγ )
y
dy = p 0
y
dy
(2.51)
where p 0 is the momentum in the direction of motion of the particle.
This invariant is the emittance, ε, of our transverse phase space multiplied by
the relativistic βγ which is proportional to momentum. Accelerator physicists often
call this the invariant or ‘normalised’ emittance:
ε
∗
= βγ ε [π mm · mrad]
(2.52)
This normalised emittance, ε ∗ , is conserved as acceleration proceeds in a
synchrotron and the physical emittance within the right-hand side of the equation
must fall inversely with momentum if the whole term is to be conserved. Close to
the velocity of light this implies that it is inversely proportional to energy.
Emittance = πε =
y
dy = πε
∗ / (βγ ) ∝ 1/p 0 .
(2.53)
We therefore expect the beam dimensions to shrink as (Fig. 2.19) a phenomenon
called ‘adiabatic damping’.
2.3.1 Chains of Accelerators
As a consequence of the adiabatic shrinking, the beam emittance is largest at low
energy, and so is the beam dimension. Proton accelerators need their full aperture
at injection and it is then that their design is most critical. For this reason it is
economic to split a single large ring into a chain of accelerators—the smaller
radius rings having a large aperture while the higher energy rings with large radius
can have smaller apertures. In these chains of proton accelerators, such as the
CERN accelerator complex, Linac – Booster – PS – SPS, the invariant emittance,
determined by the parameters of the beam as it leaves the ion source at the beginning
of the linac, may be conserved to several hundred GeV. Of course one must
guard against mismatches between machines or non-linear fields which dilate the
emittance.