ILC School Lecture #2 Homework and Solutions

5/20/2006

ILC School Lecture #2 Homework and Solutions

Problems:

Assuming round beams at the IP, express the luminosity in terms of the beam power, the beamstrahlung energy loss, the bunch charge and the bunch length. What does this imply?
What beam parameters have to be modified to reduce the disruption while holding the AC power consumption and beamstrahlung energy loss constant?
Beamstrahlung photons degrade the luminosity spectrum. Express the luminosity at the center-of-mass in terms of the average number of photons radiated per e+/e- (this is just Poisson statistics). Compare these numbers against the expected luminosity within 1% and 5% of the center-of-mass as listed below.

Nominal / Low Q / Large Y / Low P / High Lum
Ecms / 500 / 500 / 500 / 500 / 500
gamma / 4.89E+05 / 4.89E+05 / 4.89E+05 / 4.89E+05 / 4.89E+05
N / 2.00E+10 / 1.00E+10 / 2.00E+10 / 2.00E+10 / 2.00E+10
Dx / 1.62E-01 / 7.08E-02 / 5.59E-01 / 2.26E-01 / 1.70E-01
Dy / 1.85E+01 / 1.00E+01 / 2.80E+01 / 2.70E+01 / 2.19E+01
Theta0 / 3.53E-04 / 2.34E-04 / 5.05E-04 / 5.11E-04 / 5.12E-04
xp_max_out / 2.70E-04 / 1.79E-04 / 3.87E-04 / 3.91E-04 / 3.91E-04
yp_max_out / 7.60E-05 / 8.40E-05 / 7.72E-05 / 8.03E-05 / 9.57E-05
Uave / 0.046 / 0.061 / 0.039 / 0.100 / 0.133
delta_B / 0.022 / 0.018 / 0.028 / 0.057 / 0.070
P_Beamstrahlung [W] / 2.48E+05 / 2.05E+05 / 3.14E+05 / 3.06E+05 / 7.90E+05
ngamma / 1.257 / 0.823 / 1.811 / 1.756 / 1.725
Lum within 1% Ecms / 73% / 79% / 64% / 55% / 53%
Lum within 5% Ecms / 94% / 95% / 91% / 81% / 78%

What would be the rf system efficiency improvement if ILC operated at 1 Hz instead of 5 Hz assuming a constant luminosity and ignoring the technical limitations associate with long pulse operation? The cavity fill time is roughly 500 us while the rf pulse length is roughly 1 ms in the 5 Hz parameters. What might be the technical difficulties associated with such operation?

Solutions:

Problem 1.

With round beams the luminosity and beamstrahlung are – see pages 11 and 26 of the lecture:

;

where Y<1 was assumed. This can be re-written in terms of the beamstrahlung loss, the bunch charge, and the bunch length for round and flat beams as:

where there is an additional factor in the flat beam case that depends on the ratio of the horizontal to vertical beam size – this can be as much as a factor of 100. There are different ways to write the different expressions but the main point is that for flat beams the luminosity is proportional to N/sxsy and the beamstrahlung energy loss scales as (N/sx)2 while for round beams the luminosity scales as N/sr2 while the beamstrahlung scales (N/sr)2. In both cases, one can increase the ratio of luminosity to beamstrahlung by decreasing the single bunch charge however, in the flat beam case, one can also increase the luminosity without impacting the beamstrahlung by decreasing sy. This extra knob is very important to achieve the desired luminosity while maintaining a reasonable level of beamstrahlung.

Problem 2.

In general, it is always possible to reduce the disruption and the beamstrahlung to develop cleaner interactions at the expense of the luminosity – such an optimization might be desirable to do precise measurements while scanning a narrow resonance and is most effectively done by increasing the horizontal spot size. It is more difficult to reduce the disruption while maintaining the luminosity although in many cases this may be desired.

Going back to the flat beam expression, the luminosity expression can be written in terms of the disruption parameter – see page 34 of the lecture:

To maintain the luminosity while decreasing the disruption parameter, the bunch length must be decreased in proportion to the disruption, ie. the ratio Dy/sz is constant. Now to understand the scaling of the beamstrahlung energy loss, plug in Dy/sz = const and we are left with the energy loss is proportional to sy2/sz. Thus to keep the beamstrahlung energy loss constant as well the luminosity, the vertical spot size must be decreased in proportion to the square root of the bunch length.

Of course, these parameters are not all independent. To see how this works, let’s take a specific example. Starting from the ‘nom’ ILC parameters – see page 15 of the lecture – with Dy =20, N=2e10, sz=300mm, sx =550nm, and sy =6nm. Now to get Dy =10 with the same luminosity, decrease the bunch length to 150mm, decrease the vertical spot to 4.2 nm and decrease N to 1.4e10. To maintain the same beam power, the number of bunches or the rep rate must be increased. To keep the rf system parameters constant, the average current should be maintained and thus the number of bunches should be increased by sqrt(2) while the bunch space is decreased by the same factor.

The ILC ‘lowN’ parameters – see page 15 of the lecture – are an example of this optimization which was taken a bit further where the beamstrahlung energy loss is actually decreased along with the disruption parameter. To do this, the bunch charge is decreased by a factor of 2 while the number of bunches is increased by a factor of 2. To maintain the luminosity, the vertical spot size is also decreased by nearly a factor of 2 and the horizontal spot is also decreased slightly.

Problem 3.

To calculate the fraction of the luminosity at the full center-of-mass energy, we need to calculate the probability of radiating a beamstrahlung photon – any beamstrahlung will reduce the particle energy from the design. Photon emission is a statistical process described by Poisson statistics. The parameter ng is the average number of photons emitted by a particle during the collision and thus the probability of not emitting any photons is given by:

P0 = e-ng (remember Poisson statistics: Pm = ngm e-ng / m!)

Now to calculate the fraction of luminosity without any beamstrahlung, we need to integrate through the collision. For example, a slice of the beam at head of one beam is always colliding with particles that have not yet had a chance to radiate while the particles at the tail of a beam are always colliding with particles that have had the greatest probability of radiating.

To do the calculation, assume uniform longitudinal distributions with length l. In this case, the probability of not radiating after passing through s / l of the opposing beam is

P0(s / l) = e-s ng / l

The desired integral is:

Where L100% is the full luminosity and L0% is the fraction that is within 0% of the full center-of-mass energy. Filling out the Excel table:

Nominal / Low Q / Large Y / Low P / High Lum
Ecms / 500 / 500 / 500 / 500 / 500
gamma / 4.89E+05 / 4.89E+05 / 4.89E+05 / 4.89E+05 / 4.89E+05
N / 2.00E+10 / 1.00E+10 / 2.00E+10 / 2.00E+10 / 2.00E+10
Dx / 1.62E-01 / 7.08E-02 / 5.59E-01 / 2.26E-01 / 1.70E-01
Dy / 1.85E+01 / 1.00E+01 / 2.80E+01 / 2.70E+01 / 2.19E+01
Theta0 / 3.53E-04 / 2.34E-04 / 5.05E-04 / 5.11E-04 / 5.12E-04
xp_max_out / 2.70E-04 / 1.79E-04 / 3.87E-04 / 3.91E-04 / 3.91E-04
yp_max_out / 7.60E-05 / 8.40E-05 / 7.72E-05 / 8.03E-05 / 9.57E-05
Uave / 0.046 / 0.061 / 0.039 / 0.100 / 0.133
delta_B / 0.022 / 0.018 / 0.028 / 0.057 / 0.070
P_Beamstrahlung [W] / 2.48E+05 / 2.05E+05 / 3.14E+05 / 3.06E+05 / 7.90E+05
ngamma / 1.257 / 0.823 / 1.811 / 1.756 / 1.725
Lum at Ecms / 32% / 46% / 21% / 22% / 23%
Lum within 1% Ecms / 73% / 79% / 64% / 55% / 53%
Lum within 5% Ecms / 94% / 95% / 91% / 81% / 78%

Note that the luminosity at the full center-of-mass is much lower than that within the top 1% or the top 5%. For most measurements, these later values are more reflective of the useful luminosity however for some measurements having a very clean spectrum may prove important. In such a case, it may be desirable to optimize the collider for a cleaner spectrum even with some loss in the total luminosity – see Problem 2.

Problem 4.

The AC power required to operate the 500 GeV ILC is roughly 200 MW. This is a large amount of power and begs the question if there is a way to improve the collider operating efficiency. The AC à Pbeam efficiency can be separated into two portions: the efficiency of the generation of the rf power, and the efficiency of the transfer of the rf power to the beam. The efficiency of the generation of the rf depends on the detailed technology and is subject to extensive technical optimization.

Is there a way to optimize the transfer of the rf power to the beam? The superconducting rf cavities have very small losses into the cavities themselves but still the rfàbeam efficiency is only about 66% because rf power must be used to fill the rf cavities before the beam arrives – this power is not recovered after the beam pulse is accelerated. A schematic of the rf and beam pulse through a cavity is shown on page 13 of the lecture.

Thus, in a superconducting cavity, the rfàbeam efficiency is given by: h = Tb / (Tf + Tb) where Tb is the beam pulse length and Tf is the cavity fill time. For the nominal ILC parameters, the cavity fill time is roughly 0.5 ms while the beam pulse length is roughly 1 ms. Thus, the rfàbeam efficiency is 67%. If the collider repetition rate were to be decreased to 1 Hz while the beam pulse length is increased by a factor of 5 to maintain the same luminosity, this efficiency would increase to 91%, a significant improvement.

However, such a change would be quire hard to implement. First, the rf technology would become more difficult because the modulators, klystrons, and rf distribution systems would be required to deliver much more energy per pulse. Second, the rf cavities may have difficulty working at much longer pulse lengths because transient heating within the long pulse could lower the useful gradient. Finally, the damping ring would have to store 5 times as many bunches – because the damping ring size is determined by the number of bunches and the fastest injection/extraction kicker rise/fall times, this would imply a ring that is 5 times as large or >30km in circumference which would be a bit bigger than the LEP/LHC rings.