Download 6th BPPC meeting. Thursday 5th July 2012 Attendees: J. Baillie, W

6th BPPC meeting. Thursday 5th July 2012
Attendees: J. Baillie, W.Bartmann, P.Beloshitskii, O.Berrig, N.Biancacci, A.Burov, F.Butin,
C.Carli, T.Erikson, S.Hancock, R.Kersevan, M.Martini, E.Métral, M.Migliorati, S.Pasinelli, M.Pivi,
T.Rijoff, G.Tranquille
[Note from the editor: There are many places in these minutes where I did not clearly hear the
speaker. These places are marked with three question marks. It will take too much effort to
correct these places. A.Burov invites anyone who has questions about these minutes, to come and
talk with him.]
A.Burow: Review of the main ELENA parameters; cooling, diffusion and stability
P.Beloshitskii makes introduction of A.Burov
A. Burov recalls from his experience in Fermilab (FNAL). He presents a Mathcad file
(https://espace.cern.ch/ELENA-BPPC/BPPCmeetings/Document%20Library/26/ELENA.xmcd
,https://espace.cern.ch/ELENA-BPPC/BPPCmeetings/Document%20Library/26/ELENA_parameters_web.html
), with the main ELENA parameters. He invites the attendees to interact with him, while he
presents the parameters and calculations in the Mathcad file.
All the parameters in in the Gaussian unit system (unless indicated otherwise), the masses
and energies are in units of electron-volt [eV].
The equations are color-coded: the input values are highlighted in yellow, the output values
are highlighted in red, there is one disabled parameter highlighted in gray, and parameters
that have been defined earlier – but are only used now in a formula – are repeated/reminded
by highlighting in blue.
Machine parameters:
Average radius is (Machine parameters: R0 = 483.831 cm)
The average beta-function is calculated (Machine parameters: a = 279. 807m)
E.Metral: Is it the same average, both for x and y?
A. Burov: Yes, I am not taking the exact numbers. I am establishing “reasonable” numbers,
and then I check the consistency of these numbers. I only need the numbers to be precise to
within 20%.
The energy is set to 100 keV (i.e. the lowest energy. The anti-protons are ejected at 100 keV
towards the experiments) and the revolution time about 7 s.
Beam parameters:
As far as I understand, the parameters have approximately the values shown here. When I ask
different people, I get slightly different values, which is normal at this early stage of the
project. So these parameters are put in to check that the results that we get are reasonable.
It is assumed that the beam is coasting and that cooling is applied.
The beam emittance before cooling is 𝜀𝑖𝑛𝑖 = 2.5 ∙ 10−4 and after cooling it is 𝜀𝑓𝑖𝑛 = 0.7 ∙
10−4.
The average radius of the beam before cooling is 𝑎𝑖𝑛𝑖 = 0.246 cm and after cooling it is
𝑎𝑓𝑖𝑛 = 0.14 cm.
The average angle of the beam before cooling is 𝜃𝑖𝑛𝑖 = 9.452 ∙ 10−4 and after cooling it is
𝜃𝑓𝑖𝑛 = 5.002 ∙ 10−4.
All the values given here are root mean square (rms).
𝑑𝑝
The momentum spread of the beam before cooling is ( 𝑝 ) = 5 ∙ 10−4 and after cooling it
𝑖𝑛𝑖
𝑑𝑝
is ( 𝑝 )
= 2.5 ∙ 10−4 . (Again it must be emphasized that these values are rms.)
𝑓𝑖𝑛
We can now calculate the final stage “space charge tune shift”, which is 𝑑𝑄𝑠𝑐 = 0.02 . This is a
low value, which is reasonable for coasting beam. When the beam will be bunched – the
bunching factor would be roughly 10 – then the “space charge tune shift” will increase by a
factor 10 approximately.
E.Metral: Maybe the space charge tune shift would be 0.1
P.Beloshitskii: Minimum 0.1
A. Burov: 0.1 to 0.2 for bunched beam.
P.Beloshitskii: The number of antiparticles were originally set to 2.510-4, which was a bit
optimistic. It is now reduced to 1.810-4, which will bring down the “space charge tune shift” a
bit.
A. Burov: Yes, all the numbers here are rough estimates. This Mathcad file is a tool to check
that the numbers are consistent, that the numbers are within limits, compatibility and so on …
Later I will update the file, to include the parameters for bunched beam. We are still
discussing the parameters for bunched beam. It is pending. So I have left an empty space for
“Bunched beam”.
Electron Cooler:
The electron current is 2 mA; the cooling length is 90 cm; half a second of cooling time. The
eta () parameter is the ratio of the length of the cooler, divided by the length of the
circumference of ELENA – it is 0.03. ae is the radius of electron beam. The electron density of
the electron cooler beam is ne = 1.447106, and the ratio of the electron density, divided by the
anti-proton density is 0.6. This is important, because it means that the tune shift of the
electrons will be 0.6 times the tune shift of the anti-protons themselves. This tune shift of the
electrons is more restrictive than the tune shift of the anti-protons, because you have
entrance and exit from the electron-cooler, so you cannot have as high as the anti-proton
tune-shift. The anti-proton space-charge tune-shift could be about 0.1 to 0.2 as we have
discussed, but the electron tune-shift can only be around 0.02 (pessimistic, up to 0.5 big
uncertainty). If the electron tune shift is higher, then we will have beam-beam limitations. The
beam-beam effect will degrade the anti-proton beam, due to incoherent beam-beam effects.
The tune-shift of the anti-protons was 0.02 (close value to the electron tune shift) so for the
electrons, we are not far from the incoherent beam-beam limit! This tells us, that it is NOT
possible to increase the electron current significantly
E.Metral: You are referring to the beam-beam limit of cooled anti-protons
A. Burov: Yes
E.Metral: So it is really defined by ??? 395 between anti-protons …
A. Burov: … and space-charge for electrons. Electron cooler, we have borders, so you are
exciting a lot of resonances. ??? 400
P.Beloshitskii: We have a constant distribution of electron, so higher harmonics …
A. Burov: No, transverse. But longitudinally, you have a step function for anti-protons. It is
good, I realize that you have a big radius of 2.5 cm of the electron beam.
P.Beloshitskii: With uniform distribution!
A. Burov: Yes, with uniform distribution, which is a very wise decision, because this does not
– as you say – excite additional resonances.
So it is in fact a question: “How much is the maximum allowed space-charge tune-shift of the
electrons, due to lifetime requirement of the anti-protons? How far we are here, I do not
know. It is an open question for me. Maybe you can still allow a factor 2 or 4 in electron
current – from this point of view, but not from other points of view.
C.Carli: These end-effects, they happen only once or twice per revolution, but with tune-shift,
but with tune-shift, as to the electron beam only depends of the length of the electron-cooler…
A. Burov: Yes, it is here!
C.Carli: So without increasing the effect, due to the entrance end exit of the cooling, couldn’t
we decrease the electron tune shift, just by making the eta longer? So is the tune-shift the only
criteria?
A. Burov: Eta is limited
C.Carli: Yes, but if you reduce eta, then you would reduce the tune-shift, and then it would be
well below the beam-beam limit? But then one might still not improve the situation, because
we still have the entrance and exit?
A. Burov: Probably you are right. It is an open question. Still, we are probably not far from the
limit.
C.Carli: Yes, and one could also say that eta equal to 3%, is not a typical value.
A. Burov: Yes, it is not a typical value
C.Carli: Another question, where does the cooling time come from?
A. Burov: Somebody told me
100 Gauss is the magnetic field in the electron-cooler. The Larmor pitch is for electrons, it is
0.251 cm i.e. about 2 mm, and so the electrons are doing a lot of Larmor rotations, so the
phase advance is 358 radians. For the protons, you also have some Larmor phase advance
from the electron cooler, and this phase advance is 0.196 radians, or if you want to express it
as tune-shift, it is 0.031. This is a small value. It means, that if you are staying closer to the
coupling resonance than by 0.031, then your Eigen optical modes of anti-protons, will be
circular. They will surely not be planar, but circular.
P.Beloshitskii: This is much smaller than what we have in AD. In AD we have a factor 3 or 4;
which means that normal operation is close to the coupling resonance. The tune in AD is
0.45/0.43 and it is safe.
A. Burov: I am not saying that it is not safe
P.Beloshitskii: It is very strongly coupled in AD ???
A. Burov: I am in favor of that actually i.e. to stay close to the coupling resonance. I want to
point out that that if you stay close, then you should not expect your Eigen-modes for the antiproton to be planar. They will be circular both horizontally and vertically.
E.Metral: Was it predicted for AD that tunes will be close to the coupling resonances, or was it
measured?
P.Beloshitskii: I cannot really answer you, because in the AD we do not adjust the tune-split,
but only the efficiency. Generally close is better!
C.Carli: Is it foreseen lamor in one direction in the electron cooler, but in the opposite
direction of the same amount, which would remove ??? partly
A. Burov: Yes, we will remove, partly by the solenoid – placed in the proper place.
E.Metral: Why do you want to remove it?
P.Beloshitskii: We know that careful compensation of coupling gives 7% more intensity.
C.Carli: Why don’t we compensate coupling in LHC
A. Burov: Sometimes it is beneficial if the coupling is circular modes.
C.Carli: With additional solenoids, it can be compensated quite easily.
Two-beam transverse instability:
The electron current not only gives a potential limitation of the lifetime of the anti-protons,
but it is also a source of dynamical effects, which can result in a coherent perturbation of the
anti-proton beam. It works as a sort of beam-impedance, which can provoke a transverse
instability. A transverse instability is more dangerous than a longitudinal instability. The
reason is that the electrons are magnetized, and if you have – at the entrance of the electron
cooler – some vertical offset of the anti-protons, then the electron-beam will respond
horizontally. So a vertical offset of anti-protons, will give a horizontal response of the
electrons. So if you are uncoupled for the anti-protons, as a first order approximation, you will
not have any interaction between electrons and anti-protons, because the response is
orthogonal to the perturbation. However, if we have coupled modes for the anti-protons – we
are returning to the previous questions – then this drift response of the electrons is not
orthogonal to the original perturbation, and then we might have a collective instability, which
is caused by a very local interaction between electrons and anti-protons. So the growth rate of
this instability would be constant for all the wave-lengths up to upper ??? . I considered this
problem about six years ago. (Two beam instability in electron cooling A.Burov)
E.Metral: The two-beams instability was also studied by B.Zotter and Zenkevich in the 1970’s
(see: http://care-hhh.web.cern.ch/care-hhh/hhh-2004/Talks Session 6/Giov_CARE.ppt), but
in your case you have coupling.
A. Burov: Yes, I took coupling into account, because without coupling this instability is much
bigger. So this instability is really very sensitive to coupling.
E.Metral: Is this a problem of coupled oscillators, coupled by the beam-beam interaction?
A. Burov: There are two sorts of coupling. One type of coupling is an optical coupling of antiproton modes – it has nothing to do with electrons, but the electron cooler might contribute to
this coupling. The other type of coupling is the dynamical coupling due to the coherent
interaction with the electron beam. And to have two-beam instability, we need both these two
types of coupling, both: optical – and dynamical coupling.
E.Metral: In the past B.Zotter and Zenkevich studied only electron coupling, which led to
instability.
A. Burov: Yes, but a much bigger instability. Here I assume that the solenoid will compensate
the coupling. If you are going to compensate coupling, then the growth rate will be much
smaller – I will show this.
E.Metral: So it is better to compensate the coupling?
A. Burov: Yes.
The growth time is approximately 0.01 seconds. I would say it is the STRONGEST instability of
all the instabilities. Later on, I will show you my estimates for resistive-wall instability, which
is much smaller.
E.Metral: How long time do we cool?
C.Carli: Before, Alexey showed, that the cooling time is half a second.
A. Burov: The cooling time is ½ a second, but the default cooling time is this number – they
are sort of comparable (my estimation). But, in my opinion, we should not rely on cooling to
damp instabilities. This means that WE NEED A DAMPER. Later I will show my estimation of
the beam-beam space-charge tune-shift, which is small compared to space-charge. So in terms
of the stability diagram, it means that we are sitting at the far tails of stability diagram. The
reason is that this growth rate (Λ 𝑝𝑒 ) is much smaller than the space-charge tune-shift, which
means that we are sitting at the far tails of the stability diagram.
Ions:
The main question is: “How dangerous are ions trapped inside the electron beam?”
My assumptions are: The vacuum pressure is 310-12 Torr. Now I am populating the vacuum
with ions and I assume that the cross-section is 0.7    aB2 (0.7 time PI times Bohr radius
square). This corresponds to a hydrogen cross-section, which is a conservative estimation.
More or less all the other ions that we will find inside the electron beam are all in that range of
cross-section. For our specific energy range 100 keV all the other ions have a smaller crosssection. The cross-section of hydrogen is also bigger than the cross-section of the anti-protons
for any energy level; however they are about at the same level. I do not believe that we will
have a real limitation because of the ions.
The number of ions produced during the ½ second cooling time, divided by the number of
anti-protons is 1.3110-3. It means that any instability related to ions, is about a factor 4
smaller than the instability related to space-charge tune-shift. If we calculate, you will see that
the maximum growth rate due to this ion related beam-beam impedance, is about 1/5 (FLD
~0.2) of this number from space-charge tune-shift, so it is smaller than the drift instability. So
the ions will not make a difference in stability. We could have 10 times more ions without
changing the situation significantly. So, I do not believe that we are limited by the vacuum.
E.Metral: By the vacuum you mean the ions?
A. Burov: Yes, this vacuum is so good that not many ions created by collision with the
electron beam.
E.Metral: Do you think we can achieve this good vacuum (to Roberto)?
R.Kersevan: Yes
Number of ions produced by electrons is: 𝑛𝑖𝑒 = 1.895 ⋅ 103 and this should be compared to
𝑛
the number of anti-protons, i.e. : 𝑛𝑖𝑒 = 0.028 and as this is much less than ~0.6 so this is not
𝑝
a problem.
Impedance and stability:
I am only analyzing the resistive wall instability, because it is the only one that I can estimate.
Impedances from changes in the size of the vacuum chambers, I can not analyze, as I do not
know the final layout.
For the resistive wall instability I assume that the vacuum chamber is made of stainless steel;
that the radius is 3.1 cm – however I was later informed by Gerard that it is really 4 cm – but I
kept the 3.1 cm as it gives a higher beam-impedance (resistive wall beam impedance). This is
the usual formula for the resistive wall beam impedance, where I assume that the vacuum
chamber is thicker that the skin depth. The skin depth is 1.32 mm or thinner. Therefore the
vacuum chamber should not be thinner than 1 mm. So the formula for Zx(w) is correct, and by
using the formula for the growth rate (Λ 𝑥 ), we get growth rates that are very long, in the order
of half an hour. ESSENTIALLY NOTHING! So resistive wall instability is essentially nonexistent, because the number of particles (Np=2.5107) is so low. Also the resistive wall
impedance is reduced by the low relativistic beta.
E.Metral: Can there be another beam impedance, which is important?
A. Burov: Yes, the two-beam instability, which I described before, is important – it has a
growth time of about 10 ms. So the two-beam instability is the dominant instability in the
machine. It can be suppressed if we compensate with the solenoid, but we are not obliged to
do this, and therefor we should have damper. I am very much in favor of a damper.
Now I will calculate the Landau damping. If the frequency spread, from the Landau damping,
is large enough, then we do not need the damper. If at high frequencies, your eta
(approximately 1 at high frequencies) combined with dp/p are comparable with the spacecharge, then we have enough Landau damping. Then we do not care about this slow
instability, because the Landau damping will suppress it. Actually, this threshold, since we are
sitting in the tails of the stability diagram, is not dQsc divided by dp/p, but it is also inversely
proportional to a form factor, which means that the bandwidth should be smaller. So in this
case, we will not use this cumbersome formula, but set FLD=8, which results in a bandwidth of
the damper of fZth ~ 1.5 MHz. We should be able to build a damper, without too much trouble,
with this bandwidth. In Fermilab we always worked with the damper, and our digital damper
had a 75 MHz bandwidth, and our ability to cool was limited by the damper. We only used our
cooler to 10% of its power – it was always underused – because if we cooled more, we would
have instability! SO IN FERMILAB, OUR ABILITY TO COOL WAS LIMITED BY THE DAMPER.
We started to work with a damper, with a bandwidth of 35 MHz, and later we doubled the
bandwidth of the damper to 75MHz. This resulted in an increase of 50% of the transverse and
longitudinal phase space density.
For ELENA, you will need a damper with a bandwidth of 1.5 MHz to damp instabilities,
maybe even more. Therefore I recommend that if/when you build a damper; foresee that you
might have a higher bandwidth in the future.
E.Metral: Maybe 10 MHz
A. Burov: Yes, 10 MHz looks perfect.
Another equipment that ELENA might also need, and which is closely related to the damper, is
a “noiser”. In order to regulate the ratio of the longitudinal and transverse temperatures of the
final beam, I think you will need a “noiser”. It should be a transverse noiser. This will give you
a good regulation of the transverse emittances. Do not let the beam be overcooled, or be
stopped by space-charge. These problems will reduce the lifetime of the beam, so I think that
it will be good to foresee a regulation that can prevent transverse overcooling of the antiproton beam. For that you need a “noiser”, which would be rather narrowband compared to
the damper, but its frequency should be high enough (compared to the previous one for the
damper), it is determined by the space-charge tune-shift – because you will act on the
incoherent part. So the mode number is about 40, noiser 5.8 MHz [Note the noiser
parameters are not in the Mathcad file I received] . So we need two devices: a transverse
damper and a transverse noiser. These two instruments could be built together into the same
equipment.
The next question is the longitudinal instability. Can longitudinal instability reign-in???
dp/p.
E.Metral: Do we have a damper in AD?
T.Eriksson: Yes we have a damper in AD, but it is not needed.
E.Metral: And in LEIR?
C.Carli: Yes, and it works and it is required!
A. Burov: What is the bandwidth?
In Fermilab we have the same type of damper in the main injector and in the recycler ring.
E.Metral: Because you did not compensate the coupling?
A. Burov: Even if we do not compensate the coupling, we do not need the damper, for
instabilities.
E.Metral: In fermilab?
A. Burov: In Fermilab we need the dampers, because the resistive wall impedance is much
higher. This is because the number of anti-protons is much higher than in ELENA, in Fermilab
it is 61012.
E.Metral: In LEIR, do we have a damper because of the resistive wall impedance?
C.Carli: LEIR has a significantly higher intensity than ELENA.
The next question is longitudinal instability. For the longitudinal instability, we will see how
far we are from the threshold. You have to compare the phase-velocity of the longitudinal
modes, which is proportional to space-charge density of the anti-proton beam. The phasevelocity is this formula, and you have to compare it with the usual dp/p. Actually with the
final dp/p. And when I compare, I see that the ration is just 0.763. For stability, it needs to be
smaller than 6 to 10, so this is an order of magnitude below the threshold. NO PROBLEM.
Gas scattering:
There are many logarithms in this program, and one is the logarithm of the residual gas
pressure, which is equal to 10. I assume that the residual gas is hydrogen.
The time-constant for the growth rate of the emittance, resulting from multiple scattering of
the beam against the residual gas, is 46 seconds. THIS IS ESSENTIALLY NOTHING, compared
with the half a second cooling time. However, we must pay attention of Z i.e. the charge of
nuclei, which enters the formula as Z squared in the denominator. So, f.ex. if the restgas is
oxygen, which has Z=16, then the time-constant for the growth rate will be a factor 162 = 256
smaller i.e. 46/256 = 0.18 seconds which is less than the 0.5 seconds cooling time. So we must
keep in mind that heavier atoms could be problem.
Another parameter is the single-scattering halo. The question is: “How many percent (%) of
the anti-protons are outside the beam envelope?” It is essentially the same formula, except
that the factor: “2LogGas” is removed and instead the factor cool has been added. cool is the
electron-cooling time (equal to half a second). So if I would like to see how many anti-protons
are in the halo – which means they are outside 3 sigma i.e. the rms size of the beam – after ½
second cooling time, then it is 1.35710-4 which is very little. However, one must again be
careful about the factor Z2.
E.Metral: So dnn() is the number of particles outside the halo created by single-scattering?
A. Burov: Yes.
C.Carli: I think that the assumption of only the hydrogen is optimistic and that LogGas=10 is
pessimistic in this case, because the blow-up is not really a multiple-scattering process,
because many of the anti-protons does not interact with the rest gas molecules, and among
those that interacts, a relatively large proportion are scattered outside the envelope
immediately. It makes not too much sense to include them in the blow-up
A. Burov: This we agree! For 45 seconds it is nothing. Strictly speaking, you will still have
emittance growth, but it will be very small.
C.Carli: Still, the use of LogGas is pessimistic, it should be smaller.
A. Burov: No, it is not pessimistic. For multiple scattering, this is the conventional formula, it
is correct.
It was agreed to look into the scattering again by C.Cali and A.Burov Agree , should be 1. !!!!
Intra Beam Scattering:
After long discussions with Christian we have agreed on the following formula for the intrabeam-scattering. The longitudinal and transverse temperatures are significantly different –
about an order of magnitude. The longitudinal temperature is smaller. This means that the
dominating intra-beam-scattering effect will be heat transfer from the transverse degree of
freedom to the longitudinal. To calculate this heat transfer I am using the Ichimaru and
Rosenbluth formula for infinite plasma, where the transverse temperatures are different from
the longitudinal temperatures. When this formula is used in our context, we assume that the
plasma density equals the average density of the beam. The average density of the beam we
set to ½ of the maximum density of the beam. Then we get the result for IBS (intra-beamscattering) with this simple formula, and we do not need to apply the cumbersome BjorkenMtingwa formulation. The simple formula that I use is accurate within a factor of 2, and I do
not believe that we need to be better than a factor of two at this stage of the project.
So, we use this Ichimaru and Rosenbluth formula – which I can recommend – for twotemperature plasma, where the longitudinal and transverse temperatures are different; and
which gives very good agreement with Bjorken-Mtingwa (which is used by M.Martini; while
C.Carli use Piwinski). There is a reference to Ichimaru and Rosenbluth in Martin Reiser’s book
in eq. 6.154: Theory and Design of Charged Particle Beams published in 1994 and 1996 (Wiley
and Sons. http://www.ipr.umd.edu/ireap/personnel/Reiser/Reiser.htm ).
By the way, The formula 6.152 in M.Reiser’s book, assumes that the longitudinal temperature
is zero. So, you should not use formula 6.152, but instead formula 6.151, which takes into
account the difference of the two temperatures. This difference in temperature is part of this
form factor. I have a factor of 3 disagreement with C.Carli and M.Martini).
P.Beloshitskii: Are you only referring to coasting beam?
A. Burov: Yes
IBS is the e-fold IBS cooling time of the transverse degree of freedom i.e. the transfer of
energy from the transverse to the longitudinal planes. And the longitudinal e-fold heating, it is
the ratio of the longitudinal temperature divided by the transverse temperature.
So, all in all, the longitudinal e-fold heating time for dp/p, due to IBS temperature exchange,
gives me around 200 ms. The 200 ms should be compared with the electron cooling e-fold
time.
E.Metral: What if they are the same?
A. Burov: Later I will come to electron cooling e-fold !
Electron cooling:
The electrons, in the electron beam, should have an energy of about 50 eV.
Tc = 0.1 eV is the temperature of ???. From this we calculate the transverse thermal angle of
the electrons eT , which is 30 mrad. It is a bigger angle ??? and since it is so big, it means that
the electron beam will need to be magnetized. And only magnetized scattering ??? count,
because in electron cooling we distinguish between magnetized and non-magnetized
electron-cooling. It all depends on the ratio of “the impact parameter” – for two-particle
scattering – and the Lamor radius of the electron.
There are three cases:
 If there is no magnetic field in the cooler, then minor effects.
 If we do have magnetic fields, but at the same time you have very small impact
parameters (for scattering of anti-protons on electrons), so that the impact parameter
will be smaller than the Lamor radius. For such a small impact parameters, the
magnetic field does not matter and this is again minor effect.
 If we have magnetic fields, and longer impact parameters i.e. longer than the Lamor
radius, then we have magnetized cooling, where there are closed ??? temperature of
the electron transverse ???, because if the electron doesn’t matter, because anti-proton
is scattered on the Lamor radius, transverse degree of freedom is averaged out. Since
the electron-cooling rate is inversely proportional to ??? effective ??? of U. This
envelope ??? makes electron cooling look impossible ??? because this factor ??? so it
means that very few is needed in the cooler, so only magnetized cooling will help.
𝑛
The electron-cooling rate formula (𝑟𝑒𝑐 = θ3 𝑒 ; rec=the electron cooling rate; ne= the number of
𝑒𝑓𝑓
electrons; eff= the effective scattering angle) is a function of the effective scattering angle.
2
2
2
2
2
2
𝜃𝑒𝑓𝑓 = √𝜃𝑒𝑙𝑡
+ 𝜃𝑒𝑘𝑖𝑛
+ 𝜃𝑒𝑆
+ 𝜃𝑒𝑆𝐶
+ 𝜃𝑒𝑑
+ 𝜃𝑒𝑖𝑛𝑖
 the effective angle of the electrons
Several effects are contributing to this effective angle. The first contribution is the angle of the
anti-protons. It has it’s own transverse and longitudinal velocity, i.e. the relative velocity of
the electrons and anti-protons, and then there are several contributions to the longitudinal
velocity of the electrons. These contributions are only longitudinal, because the transverse
velocity is averaged out by the Lamor rotations – remember that we are only considering
magnetized beam.
The longitudinal contributions are:
1. ekin . Due to the acceleration of the electron beam, there is a kinematic transformation
of longitudinal density. It is inversely proportional to the kinetic energy of the electron
beam. The numerical value is 6.44710-4, so it is a lot smaller than the initial thermal
angle, and is therefore not significant. This is a general principle; if the electron beam is
accelerated enough, then the kinematic transformation is small.
2. eS . Schottky energy of the schottky noise of the electron beam could contribute to
longitudinal electron temperatures. Again the numerical value is small, so it will not
contribute significantly to the effective scattering angle.
3. eS . Now the famous space-charge contribution. Due to the space-charge of the
electrons, the longitudinal velocity of the electrons, has an excess of electron beam, and
at some pulse ???, are different, because the total energy of the electrons are the same.
Since the ??? of the energy at the exit and the entrance, therefore the kinetic energy is
different, and this gives a parabolic behavior of longitudinal velocity of the electron
beam, and if I ??? at the rms offset of the anti-proton beam, I have 1.15910-4. Again this
is an insignificantly number. However, if you look at particles that sits very far from the
core, i.e. at the far end of the transverse tails of the anti-proton distribution, then we
should not use the rms offset, but the real offset of the particles that are far from the
core. So some “halo” particles will not get cooled. But, they are cooled anyhow, because
they have a big transverse angle by themselves.
Electron IBS contribution:
The electron beam by itself also has IBS (intra-beam-scattering). The IBS will make an energy
transfer, from the high transverse temperatures, to the low longitudinal temperatures. Again I
am using the same Ichimaru / Rosenbluth formula, but here I have only ??? tized for ???
scattering, because I you put larger circles ??? then you cannot scatter, because the transverse
degree of freedom is ??? itched off for electrons, they are detached, they should have impact
parameters smaller than the Lamor radius, then the magnetic field doesn’t matter for the
scattering, so the coulomb law is determined by the ratio, of the Lamor radius and the impact
parameter. To get the maximum value of this ratio, we use the maximum Lamor radius and
and the minimum impact parameter (depending on the minimum distance). The logarithm of
this ratio is again around 10 (exact: 8.578). Now we calculate the time constant for heating of
the longitudinal degree of freedom from the transverse: ee. I have to divide this time, by the
time of flight of electrons, from the cathode to the center of cooling, which is the time of this
heating. The entire time-of-flight is small (3.52310-7).
E.Metral: The distance from the cathode to the …
A.Burov: The distance from the cathode to the entrance of the cooler. And the total distance,
during which the electrons are suffering from this heat transfer (from transverse to
longitudinal degree of freedom), is the total time, which is half the electron-cooler length plus
this distance (=the distance from the cathode to the electron-cooler entrance),
W.Bartmann: We are only considering the electron beam?
A.Burov: Yes, only the electron beam by itself. This is the intra-beam-scattering in the
electron beam; I am trying to estimate the longitudinal temperature in the electron beam, due
to IBS in the electron beam. The transverse temperature of the electron beam is huge, while
the longitudinal temperature is much smaller, so the question is: What will be the
contribution of the IBS? And the answer is that the contribution to the scattering angle is
1.06310-3 for the electron-electron IBS. IT IS THE BIGGEST CONTRIBUTION! to the
longitudinal velocity of the electrons so far.
Now, I will calculate the ratio of effective scattering angle in the cooler – and this effective
scattering angle is again the probabilistic combinations of all the scattering angles, from the
different sources: Thermal, kinematic, schottky, space-charge, anti-protons and initial;
expressed as the square root of the squares of the angles – it is important for calculating the
cooling rate. The number is 𝜃𝑒𝑐𝑜𝑜𝑙_𝑓𝑖𝑛 = 1.459 ∙ 10−3 – actually this number will inform you of
the tolerances of alignment of the magnetic field in the electron cooler – so it means that the
magnetic field is misaligned, but also that ??? here, but we will say factor of two smaller than
ecool , so this misalignment is tolerable. So misalignment should be factor 2 lower, to neglect
it, than this number. So, say 0.6 mrad, this is the destination ??? of misalignment of magnetic
field in the cooler, which follows from this result. Otherwise, if the misalignment is
comparable to that, we also have an additional factor of ecool square. In the electron cooling
𝑛
2
rate formula: 𝑟𝑒𝑐𝑜𝑜𝑙 = 𝜃3𝑒 , ne is roughly proportional to is 𝜃𝑒𝑓𝑓
This is important! In the
𝑒𝑓𝑓
3
denominator, we have 𝜃𝑒𝑓𝑓
this means our cooling rate is reduced. So the total ecool_fin
scattering angle is important. It is about a factor 3 higher than the final rms angle of the antiproton beam ( fin ).
CCarli: How can that fit? You say that you can have a misalignment between the field lines and
the circulating anti-proton beam, which is three times the rms angle of the anti-proton beam?
A.Burov: Yes, because … Actually, the cooling rate is determined, NOT by the rms angle of the
anti-protons, but the effective angle between the anti-protons and the leptons. So here you see
that we are in a range of parameters, where … you see that fin = 510-4 is the rms angle of
anti-protons, and the effective angle for electron-cooling is, say, a factor of three bigger. In this
case we are already in the situation, that the cooling rate determined, NOT by the rms angle of
the anti-protons, but by the longitudinal temperature of the electrons.
Another very important parameter, is to compare this total angle of the electron-cooler 𝜃𝑒𝑐𝑜𝑜𝑙
with the same angle 𝜃𝑒𝑐𝑜𝑜𝑙_𝑛𝑜_𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛_𝑓𝑟𝑜𝑚_𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛_𝑐𝑢𝑟𝑟𝑒𝑛𝑡 , but with the contribution of
electron current is removed (i.e. zero electron current, or with other words, current
independent). We devise the formula:
𝑥𝑥 =
2
𝜃𝑒𝑐𝑜𝑜𝑙
2
𝜃𝑒𝑐𝑜𝑜𝑙_𝑛𝑜_𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛_𝑓𝑟𝑜𝑚_𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛_𝑐𝑢𝑟𝑟𝑒𝑛𝑡
− 1,
also written as:
𝑥𝑥 =
2
𝜃𝑒𝑐𝑜𝑜𝑙
2
2
2
𝜃𝑒𝑘𝑖𝑛 +𝜃𝑒_𝑎𝑛𝑡𝑖_𝑝𝑟𝑜𝑡𝑜𝑛 +𝜃𝑒𝑆
− 1,
which is then an expression of the electron-current.
By making a plot of the cooling rate ( Fcool(xx)), as a function of this formula, we can find the
electron-current that gives the best cooling
This plot shows that the optimum value of xx is between 1.5 and 3. Since our xx, for the start
and end of the cooling, are both very close to the optimum, we have selected a very good
electron current in our design. It also shows that if we increase the electron current, our
cooling rate will be decreased.
W.Bartmann: Because of the IBS?
A.Burov: Yes, because IBS increases the longitudinal temperature, and the longitudinal
temperature is proportional to the electron current (ne), and the effective angle squared is
𝑛
also proportional to the electron current, so according to the formula: 𝑟𝑒𝑐𝑜𝑜𝑙 = 𝜃3𝑒 , the
𝑒𝑓𝑓
cooling rate will go down.
E.Metral: How will you define XX in words?
2
A.Burov: It is the current dependent contribution to the 𝜃𝑒𝑐𝑜𝑜𝑙
, so it is an expression of the
density of the electron beam. You see that xxfin is 1.6 so we are really very close to the
optimum!
E.Metral: Was the electron current optimized?
A.Burov: I do not know, but I agree with the value!
C.Carli: It is very nice
A.Burov: Yes, and since this optimum is very flat, the difference between, say, 1.4 or 0.7 is not
big difference. So the maximum is flat, so if you are close to the maximum, you may increase
or decrease the electron current by a factor of 2, and still get the same cooling rate.
E.Metral: In the TEVATRON, you had to increase the damper bandwidth in order to increase
the cooling.
A.Burov: Yes, in the TEVATRON, we had a limitation by the damper, so we could not increase
the electron current.
E.Metral: How far from the maximum were you?
A.Burov: We stayed here at low density.
One more interesting thing, it is comparing electron cooling with IBS. For this parameter the
electron cooling is much powerful … 20 times faster than IBS! It means that we may cool
deeper, so in ELENA we are not limited, I do not see any limits for the cooling, the beam could
be cooled deeper.
C.Carli: This seems to be contradictory to Beta-cool calculations?
A.Burov: We need to check the parameters that were used for the Beta-cool calculations.
effective envelope. Beta-cool uses the Parachuck ??? model, what is your effective angle, we
see that the main parameter of the electron-cooling rate is the effective angle. F.ex. here I do
not take any misalignment into account, or we can say that I assume that the misalignment is
below 0.5 mrad. I also does not take the ripple of the power supplies into account ( the
tolerances of the power supplies has been set to
𝑑𝑉𝑜𝑙𝑡𝑎𝑔𝑒
𝑉𝑜𝑙𝑡𝑎𝑔𝑒
=110-3), and I also not the
uncertainty of longitudinal velocity of the electrons i.e. energy. So I assume that the electron
energy that we assume in this file, has a better precision than 10-3, so it means that the
electron energy is 50 eV with a precision better than  a few 100 meV. This is the
requirement of the electron energy!
So how good can we build the electron cooler, to fulfill these requirements? If we cannot fulfill
the requirements, we will be limited by other parameters, say, fluctuations in the electron
energy or misalignment of the magnetic field, and again the parameters that we enter into the
Beta-cool calculations; i.e. all the parameters that will be unavoidable e.g. IBS in the electron
beam, the electron current,
E.Metral: What are the results that you gain from the Beta-cool calculations?
G.Tranquille: What we obtain from the IBS simulations is that we cannot cool much further
than the dp/p that we have here.
P.Beloshitskii: We are limited by the IBS of the anti-protons
A.Burov: There are two IBS, for anti-protons and electrons. But in my calculations, I do not
see that ELENA is limited! With these parameters, you are pretty far from the limit – you may
cool much better.
C.Carli: We have to add misalignments to these calculations, i.e. between the magnetic field
lines and the circulating beam.
P.Beloshitskii: I did put limits to the misalignment, in the order of 10-4 .
C.Carli: Yes, but between the perfect alignment and the perfect cooler it is maybe 2.510-4 rms.
P.Beloshitskii: Maybe it is smaller, but not much. Within the precision of this model –
definitely.
A.Burov: Also keep in mind that the electron-cooling rate is very sensitive to this angle. If you
have a factor of 2 in the angle, then you will have an order of magnitude in the electroncooling rate.
P.Beloshitskii: The beta-function in the cooler is 2 instead of 3.
A.Burov: I do not think this is so important.
Bunched beam:
There are no calculations here; I would like to present some ideas about bunched beam.
Imagine that you cool, as deep as you want, with the coasting beam, but then you want to
bunch the beam. If you bunch the beam, but without cooling, then the longitudinal
temperature will increase because the phase-space volume is preserved, but then we will
have different relation between longitudinal and transverse temperatures; and if you are
staying sufficiently long time in the bunched state, then you will have some exchange of
temperature between longitudinal and transverse (and in fact this would change your
longitudinal and transverse phase-space).
If we cool the bunched state, so you cool in coasting, and then, up to some ultimate parameter,
not necessarily those that you see here, and then you bunch the beam, and then you continue
to cool in the bunched state. We can then make the bunch shorter or longer, and this will
control the cooling rate. So this is one more knob to control the cooling, and is a parameter for
optimization. Having one more knob to control, means that we will do better.
Let us consider sigma_z, in the final cooled bunch state. But sigma_z is not necessarily the
same as the length required by the trap. After cooling in the bunched state, we can still change
the bunch length. We can squeeze the bunch or make it longer; and then transfer the bunch to
the experiment. I think that we can use this extra knob, but it is a topic for discussion. My
purpose here is to inform about this extra parameter, which allows us to optimize the cooling
process and have better parameters for the final beam.
So how is the sigma_z parameter important for the optimization of the cooling? Due to the
space-charge, the emittance in the final stage, is proportional to space-charge. And spacecharge is proportional to the number of anti-protons divided by the bunch length. And one
more important thing, for the cooler IBS elimitation, it is very beneficial to stay in thermal
equilibrium, while the ??? damper ???. Then you do not have this problem of energy transfer
from longitudinal to transverse or vice-versa. You still have IBS, but, say, an order of
magnitude or more bigger than this one, because on average the temperatures are the same,
and you have emittance growth due to beta-function variations …
C.Carli: But then, for a beam with electron cooling, it is difficult to get a smaller longitudinal
temperature, because longitudinal cooling is faster. How do you want to avoid thermal IBS, for
the circulating beam?
A.Burov: For this set of parameters, it is not clear that longitudinal is faster – let’s discuss it. If
you have longitudinal temperatures that are smaller than transverse, then cooling rate for
longitudinal and transverse are not too different.
C.Carli: Yes, you are right. That was also the outcome of the BETACOOL simulations that the
difference is not too large.
A.Burov: Yes, that’s what I said. Then you may stay ??? cooling, in thermal equilibrium, semi
thermal equilibrium, and then you will not fight against IBS, as much, i.e. IBS will be much
smaller, because the IBS will be due to only beta-function variation, the IBS rate will be an
order of magnitude smaller, and then you may cool much deeper. And then later … you may
squeeze your bunch as much as you can. So there is a parameter of optimizations, this is my
main message. The bunch length, during the thermal equilibrium stage while cooling, and then
you can expand of squeeze the bunch, as you want, and now I assume that the sigma_z trap,
required bunch length, for the experiment, that sigma is the bunch length while you are
cooling, then the transvers damper temperature, is inversely proportional to sigma_z, ???
look at smaller transverse temperatures, and the longitudinal temperature, is proportional to
sigma_z, but ??? so. But you have some ratio between longitudinal and transverse
temperatures, in the final state, which you might discuss with the experiments what ratio do
they want, adjust, so one more place where we can optimize!!! – cooling in the bunched
state!!! – that’s what I want to be my message.
C.Carli: We have discussed to bunch the beam during cooling. So if it is a problem for the
experiments?
A.Burov: So It could be shorter or longer. So ??? this your voltage, RF voltage, higher or lower,
then ???
Conclusions.
Note: “To reach minimal available dp/p, the broad-band transverse noiser is required.” So in
order to avoid being limited by the space-charge, transverse emittance, as big or small as you
want, it could be above the space-charge ???
Tx = transverse temperature, Tz = longitudinal temperature.
C.Carli: If we look for possible limitations of the electron cooler, what about the azimuthal
drift of the electrons i.e. what happens to the azimuthal drift of the electrons due to the ???
magnetic and electric fields, in particular the ??? due to beta-function of 3m and the ??? due to
IBS?
A.Burov: I overlooked, it should be checked! Due to space-charge of anti-protons and ???
???ects
C.Carli: Mainly space-charge of the electrons plus the longitudinal magnetic field, ??? drift
velocity. This electron velocity ??? compared to the velocity of ??? I am afraid