6 Design and Principles of Synchrotrons and Circular Colliders
253
6.7.3 Large Piwinski Angle and Crab Waist Collision Scheme
The need for precision measurements of rare decay modes with small cross sections
at e + e − factories has driven requirements on peak luminosity to unprecedented
levels. Conventional collision schemes, see Eq. (6.63), are based on pushing up
the beam currents, lowering the β ∗
y , and increasing the beam emittance so as not to
exceed the beam-beam tune-shift limits. Passing from single to double ring colliders
allowed the number of bunches to be increased considerably. However in order
to avoid luminosity reduction due to parasitic (or long-range) bunch encounters
near the collision point, beams had to be collided with a small horizontal crossing
angle rather than head-on. However, this approach has come to a dead end since
high currents result in high power losses, beam instabilities and increased power
consumption.
Because of the parabolic variation of β y (s) = β ∗
y + s 2 /β ∗
y in the vicinity
of the interaction point (IP), the longitudinal region in which individual particle
collisions occur will include places where the effective β y (s) >> β ∗
y at the IP, and
will therefore contribute less to the luminosity. This so-called hour-glass effect
imposes a condition on the bunch-length: σ z ß y (s). Unfortunately, shortening the
bunch length is costly since it requires high voltage in the RF cavities, can excite
collective instabilities, induce higher-order mode (HOM) heating in the beam pipe,
and lead to coherent synchrotron radiation emission, which in turn deteriorates the
bunch shape. On the other hand, increasing the bunch current leads to coupled bunch
instabilities, HOM heating of the beam pipe, and higher wall-plug power.
A solution to these problems came with the idea of a new collision scheme, called
“Large Piwinski Angle and Crab Waist Sextupoles” (LPA&CW), by P. Raimondi in
2006 [62]. This scheme has two main ingredients:
1. A large horizontal crossing angle at the IP, combined with very small horizontal
beam size, resulting in a large Piwinski angle;
2. a pair of sextupoles, each placed on one side of the IP at a specific betatron phase
from it.
The Piwinski angle is defined as:
=
σ z tan (θ )
σ x
≈ θ
σ z
σ x
.
(6.64)
Consider two bunches with RMS beam size σ x and bunch length σ x , colliding
at a horizontal crossing angle 2θ . For flat beams colliding at a small crossing angle
θ 1 and large Piwinski angle 1, the luminosity L and the tune-shifts scale
as [63]:
L ∝
Nξ y
β ∗
y
(6.65)
253
6.7.3 Large Piwinski Angle and Crab Waist Collision Scheme
The need for precision measurements of rare decay modes with small cross sections
at e + e − factories has driven requirements on peak luminosity to unprecedented
levels. Conventional collision schemes, see Eq. (6.63), are based on pushing up
the beam currents, lowering the β ∗
y , and increasing the beam emittance so as not to
exceed the beam-beam tune-shift limits. Passing from single to double ring colliders
allowed the number of bunches to be increased considerably. However in order
to avoid luminosity reduction due to parasitic (or long-range) bunch encounters
near the collision point, beams had to be collided with a small horizontal crossing
angle rather than head-on. However, this approach has come to a dead end since
high currents result in high power losses, beam instabilities and increased power
consumption.
Because of the parabolic variation of β y (s) = β ∗
y + s 2 /β ∗
y in the vicinity
of the interaction point (IP), the longitudinal region in which individual particle
collisions occur will include places where the effective β y (s) >> β ∗
y at the IP, and
will therefore contribute less to the luminosity. This so-called hour-glass effect
imposes a condition on the bunch-length: σ z ß y (s). Unfortunately, shortening the
bunch length is costly since it requires high voltage in the RF cavities, can excite
collective instabilities, induce higher-order mode (HOM) heating in the beam pipe,
and lead to coherent synchrotron radiation emission, which in turn deteriorates the
bunch shape. On the other hand, increasing the bunch current leads to coupled bunch
instabilities, HOM heating of the beam pipe, and higher wall-plug power.
A solution to these problems came with the idea of a new collision scheme, called
“Large Piwinski Angle and Crab Waist Sextupoles” (LPA&CW), by P. Raimondi in
2006 [62]. This scheme has two main ingredients:
1. A large horizontal crossing angle at the IP, combined with very small horizontal
beam size, resulting in a large Piwinski angle;
2. a pair of sextupoles, each placed on one side of the IP at a specific betatron phase
from it.
The Piwinski angle is defined as:
=
σ z tan (θ )
σ x
≈ θ
σ z
σ x
.
(6.64)
Consider two bunches with RMS beam size σ x and bunch length σ x , colliding
at a horizontal crossing angle 2θ . For flat beams colliding at a small crossing angle
θ 1 and large Piwinski angle 1, the luminosity L and the tune-shifts scale
as [63]:
L ∝
Nξ y
β ∗
y
(6.65)