# Clarification of accelerator physics terminology

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• Wrichik Basu
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In summary, the terms beta-beat, tune of a beam, betatron coupling, linear coupling, amplitude detuning, dynamic aperture, difference resonance, sum resonance, fractional tunes, and resonances are all explained in detail in the lectures of the CERN Accelerator School.

#### Wrichik Basu

Gold Member
I have been reading some papers on accelerator physics recently, especially those on the LHC and the upcoming FCC-hh. As a beginner, I am not supposed to know everything, but there are a few terms that I feel I have to know in order to penetrate further into this field. Unfortunately Google search does not help, because if I do a search with any of these terms, the results are all papers, all of which contain those terms but not necessarily explain/define them. The following is a list of such terms:
1. ##\beta##-beat
2. Tune of a beam
3. Betatron coupling
4. Linear coupling
5. Amplitude detuning
6. Dynamic aperture
7. Difference resonance ##f_{1001}##, sum resonance ##f_{1010}## and fractional tunes (found in ref. [1])
One paper [2], for example, explains some of the terms:

5. Amplitude detuning is the variation of tune with single particle emittance.​
6. The dynamic aperture (DA) defines the boundary in phase space beyond which particle motion becomes unstable.​

But I still clearly don't know what this tune means.

For linear coupling (#4), I am interested in knowing the nature of the coupling.

If you do not wish to explain everything here and instead provide me with relevant papers that define/explain these terms, I will be happy to read them up.

Definitions of some of the basic terminology like emittance and beta function is given here:
https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance

References:

[1]: T. Persson & R. Tomás Phys. Rev. ST Accel. Beams 17, 051004
[2]: E. H. Maclean et al. Phys. Rev. ST Accel. Beams 17, 081002

Greg Bernhardt
Hello!
It seems that I’m a bit late, so you might already have found answers to your questions.
Most (if not all) of your questions are addressed in during the lectures of the CERN Accelerator School (CAS), either the introductory [1] or the advanced [2] one. The lectures are generally easy to follow, so you can take a look. Most relevant for your questions are the lectures on transverse linear beam dynamics and “A first taste of non-linear beam dynamics” from the introductory CAS. Anyway, I’ll try to give a quick overview so that you can ask further questions if you have any, but I’ll approach your list non-chronologically:

1) Beta-beat is the error in the beta-function w.r.t. the design beta-function due to e.g. focusing errors.
$$\frac{\beta_e-\beta_0}{\beta_0}$$,
where ##\beta_e## is the current beta-function while ##\beta_0## is the design beta-function. To achieve the design performance of the machine, one should attempt to correct beta-beating.

2) The tune of a beam is the number of transverse oscillations that a particle will do during one round trip of the machine. It is the phase-advance, ##\mu##, in one turn divided by ##2\pi##. The phase-advance in the two transverse planes are generally different, meaning that a particle will have both a horizontal- and a vertical tune. The tune is generally denoted with the symbol Q (sometimes ##\nu## is used).

The fractional tune is simply the fractional part of the tune. So if Q = 5.2, then the fractional tune is 0.2.

3) and 4): The terms linear coupling and betatron coupling are typically used interchangeably. “Coupling” means a coupling between a particles motion between two planes. Betatron coupling specifically emphasizes that the coupling is between the two transverse planes, while linear coupling can also arise between e.g. the horizontal and longitudinal plane (known as synchrobetatron coupling). However, most often people mean “linear betatron coupling” when they say either betatron coupling or linear coupling.

What does (linear/betatron) coupling mean? Assume that you have a particle that is horizontally displaced, but not vertically. If no (linear betatron) coupling is present in the machine, then all movement of the particle will remain in the horizontal plane. However, if coupling is present, then some of the horizontal motion will be transferred into the vertical plane, and the particle will start oscillating in both planes.

Where does coupling come from? Typically, the largest contribution to linear betatron coupling is through so-called skew quadrupole components. Imagine that you have a regular quadrupole magnet; it focuses or defocuses the particles in either plane, but does not couple the two. But if the quadrupole is just slightly rotated, then a horizontal displacement in the rotated quadrupole field will also give the particle a kick in the vertical direction. The skew fields can arise either from rotated quadrupoles, but also from magnet construction errors, fringe fields, and much more. In general, one would like to have good control of the coupling. This is often done by installing dedicated skew-quadrupole magnets in the machine.

7) Resonances are a big thing for synchrotrons. Actually, they are a complete research topic in itself. I don’t want to go too much into detail and introduce resonances in a general way, but here is a quick sketch. Check [3] for a very nice description and pretty plots:

A particle is on a resonance, if the tune (i.e. the number of oscillations per turn) multiplied by an integer is also an integer, such that m*Q = n, where both m and n are integers. For example, the situation, where the (horizontal or vertical) tune of the particle is 5.5. Then ##2Q = 11##, meaning that the tune is on the half-integer resonance, and the order of the resonance is 2 (this is typically an unstable, and therefore bad, situation). A 2nd order resonance is driven by a 2nd order magnet, i.e. a quadrupole. If Q = 5.2, then 5Q = 26 and the particle is on 5th order resonance (this is typically not a bad situation, in fact the MAX-IV synchrotron light source has a design horizontal tune of ##Q_x = 42.20##). Note that only the fractional tune matters for the resonance conditions.

Okay, this leads me to the coupling resonances. As the name states, these resonances arise due to the coupling of the machine. The resonance condition is now: ##mQ_x + nQ_y = p##, where m, n and p are integers.

Imagine the situation where ##Q_x = 5.2## and ##Q_y = 3.8##, that means that ##1Q_x + 1Q_y = 9##, and the particle is located on the linear sum resonance. This is generally a bad resonance; it “does not converse action (emittance)” and leads to beam blow-up.

Next up, there is the situation where ##Q_x = 5.2## and ##Q_y = 3.2##, that means that ##1Q_x – 1Q_y = 2##, and the particle is located on the linear difference resonance. This is generally a “good” resonance, since it conserves action (emittance), but also leads to a sharing of the emittance between the two transverse planes. In the extreme case where the fractional tunes are equal, one will get the round beam condition, where the emittance is equal in the two planes. ##f_{1001}## and ##f_{1010}## are the so-called resonance driving terms (RDTs) associated with the linear difference resonance and linear sum resonances, respectively. RDTs are a more advanced concept, but in the case of linear coupling, ##f_{1001}## and ##f_{1010}## describes how strongly coupled the beam is transversely.

6) To describe the dynamic aperture (and why it is of interest), it must be mentioned that other than having dipole- and quadrupole-magnets, synchrotrons are also equipped with sextupole-magnets used for so-called chromaticity correction (essentially a compensation of the energy-dependent focusing due to the finite energy spread of the particles within the beam). Sextupoles have non-linear magnetic fields which may cause chaotic motion. This means, that a particle with e.g. a large horizontal displacement will become unstable and spiral out to higher and higher amplitudes and eventually get lost. The dynamic aperture is the region of the transverse planes which is stable. I.e. a particle with coordinates x,y placed within the dynamic aperture will have a stable motion while those placed outside will be unstable. It is the job of the accelerator physicist to design (and operate) the machine with a lattice that maximizes the dynamic aperture. [3] p. 47 of has nice plots of dynamic apertures.

5) Amplitude detuning is consequence of nonlinear magnets. A particle that has a large displacement (amplitude) will be focused differently than a particle with a small amplitude. The different focusing means that the particle will have a different tune, leading to the term “amplitude detuning”. This can be bad, because a particle with a large amplitude might be detuned so much that it hits a dangerous resonance. This means that amplitude detuning might limit the dynamic aperture, and therefore it is generally preferable to avoid amplitude detuning. [3] gives a nice description and plots of amplitude detuning.

Note that amplitude detuning is sometimes introduced on purpose for beams with space charge to create Landau damping [4]. It is often done using octupole magnets.

Hope it cleared up a few things. Otherwise feel free to ask.

[1] https://cas.web.cern.ch/schools/vysoke-tatry-2019
[2] https://cas.web.cern.ch/schools/slangerup-2019
[3] https://cas.web.cern.ch/sites/cas.web.cern.ch/files/lectures/vysoke-tatry-2019/non-lineardynamicsii.pdf
[4] https://cas.web.cern.ch/sites/cas.web.cern.ch/files/lectures/slangerup-2019/kornilov2019ld1.pdf

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DrClaude, berkeman, dlgoff and 3 others
Hello!
It seems that I’m a bit late, so you might already have found answers to your questions.
No, you are not late. In fact, I was reading one paper[1] which had answers to some of the questions, but not all. Thank you for taking the time out to write such a detailed explanation. It was really very helpful. And I will surely look up the references you have given.

By the way, is their any place where CERN has listed the acronyms used in the LHC? For example, TCP and TCT are primary and tertiary collimators respectively. But TCTPV.4L1.B1 refers to some specific TCT, and I have no idea which one. Some papers clearly state which acronym stands for what, but many don't and I find it rather confusing. A list of all such acronyms would be helpful for a beginner.

[1]: S. R. Mane , Yu M. Shatunov and K. Yokoya J. Phys. G: Nucl. Part. Phys. 31 (2005) R151–R209

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The best list of acronyms for I could find online was from the LHC working group [1]. Note that these acronyms are CERN (and maybe even LHC) specific, meaning that the acronyms might differ at e.g. Fermilab, Brookhaven etc. One must probably be an LHC expert to know the naming conventions by heart to deduce where TCTPV.4L1.B1 is located. I don't know if there exists an Equipment Identifier and Locator-system (there probably does internally at CERN).

As for an accelerator physics-specific acronym list, I don't know if a "complete" one exists. Normally, I have to google a lot of acronyms and expressions when reading papers outside of my area of research.

[1] https://lhccwg.web.cern.ch/lhccwg/Bibliography/UsefulAcronyms.htm

mfb and Wrichik Basu
Two questions, again.

I am reading this paper on halo removal with hollow ##e^{-}## lens at RHIC. I haven't completed reading, but here are three things that were not clear to me after reading the first two pages:
1. As far as I can understand, by "electron lens", they mean the optics that are operating on the electron beam specifically. Is this right?
2. What is drift space perveance?
3. What is Pierce instability?

Some of the concepts of ion- (and proton-) machines aren't within my area of expertise, so unfortunately I cannot give you a whole lot of (qualified) help here. Please take my reply with a grain of salt and extra caution! ;-)

An electron lens is a setup originally developed to compensate beam-beam effects in hadron colliders. The idea was to use the electromagnetic field of the electron beam itself to compensate the effects instead of magnets. The beam-beam effect arises from the interaction between e.g. two proton beams. If you can then make the same type of interaction, but with different sign of the charge (i.e. with electrons instead of protons), then you can compensate the beam-beam effect [1]. You can consider the electron beam to be the lens.
An "electron lens" is the full setup pictured on Fig. `1, consisting of electron guns and solenoids for focusing of the electron beam. To get some background you can check out [2], a paper by some of the same authors with a throughout description of the RHIC electron lenses. The authors of the paper you are reading is not aiming to compensate beam-beam effects but rather removing the halo of the main beam.

Perveance is a new concept to me, but it seems to be quite significant, since it has its own Wiki page [3]. The electron gun perveance seems to be a parameter of the electron gun, describing the proportionality between the anode voltage applied and the electron beam current you get out (Eq.(2)).
From what I understand, there is a limit as to how high hollow-beam current can be transported called the Pierce Instability threshold. So if you know this threshold, and define the drift space perseverance (i.e. a property of the (hollow) beam sizes and the vacuum chamber) as in Eq.(4), then you know how high the energy of the electron beam must be in order to transport the current set by the Pierce instability threshold. This explanation is probably not completely correct, but maybe not too bad either.
The authors of the paper then measure the perverance of the electron gun and the drift space in Fig. 3.
For Fig 3a they simply measure the relationship between how much electron beam current they get from the gun for different anode voltages, and fit it with Eq. (2) to get the electron gun perveance. On Fig. 3b they measure the drift space perveance by changing the electron beam energy (i.e. adjusting the cathode voltage), and then recording the maximum beam current they can achieve this way. They do it for three different magnetic fields of the GS1 solenoid (first solenoid after the electron gun, far-left on Fig. 1).

[1] https://www.sciencedirect.com/science/article/pii/S0168900213016677?via=ihub
[2] X. Gu et. al, "Electron lenses for head-on beam-beam compensation in RHIC ", PRAB 20, 023501 (2017)
[3] https://en.wikipedia.org/wiki/Perveance

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Klystron, Wrichik Basu and mfb
Another important topic related to hollow electron beams is Diocotron Instability. Wikipedia[1] has a stub article on it. As far as I can understand, it was first reported[2] in 1955. Some important papers are ref. [3-5]. I haven't yet studied in detail about it, but thought it was worth mentioning here.

[1]: https://en.wikipedia.org/wiki/Diocotron_instability
[2]: H. F. Webster, Breakup of hollow electron beams, J. Appl. Phys. 26, 1386 (1955).
[3]: W. Knauer, Diocotron instability in plasmas and gas discharges, J. Appl. Phys. 37, 602 (1966).
[4]: V. V. Mikhailenko et al., Non-modal analysis of the diocotron instability for cylindrical geometry with conducting boundary, Phys. Plasmas 21, 052105 (2014).
[5]: R. H. Levy, Diocotron instability in a cylindrical geometry, Phys. Fluids 8, 1288 (1965).

@JonasKK What is luminosity burn-off ? I learned it from ref. [1], which talks about the emittance growth due to luminosity burn-off in the LHC. The abstract of the paper says,
Future hadron colliders aim at producing larger integrated luminosities by burning-off an increased fraction of the bunch particles. This burn-off removes particles unevenly in the bunch distribution generating an emittance growth.
Two questions in this regard:
1. How is this "burning" carried out experimentally?
2. How can integrated luminosity increase if particles in a bunch are removed before collision? As far as I can guess, these are not beam halo particles that they are talking about. Decreasing number of particles in a bunch would decrease the luminosity, and thereby the integrated luminosity, isn't it?
N.B.: I haven't yet finished reading the whole paper, so I don't know if any of the questions are answered later.

[1]: R. Tomás, J. Keintzel, and S. Papadopoulou, Phys. Rev. Accel. Beams 23, 031002

This is again not really my area of expertise, and I'm not completely sure about the collider-jargon, but here are my two cents:

When you have a stored beam of particles, you will always loose some particles.
The "luminosity burn-off" is the loss of particles due to the Luminosity, i.e. the collisions, and for e.g. LHC the burn-off is responsible for the main part of the particle loss. So the answer to 1) is: the burning is carried out experimentally by colliding the beams.
The paper talks about the "a moderate burn-off of about 15%". I believe the 15% refers to the reduction of number of particles due to the luminosity burn-off when the beam is dumped for refill. I.e. for a burn-off of 15%, 85% of the beam is left when the beam is dumped. That's at least what I could interpret from Fig.1 and the caption. I haven't been able to find another source using that kind of expression, and 15% sounds low to me, so I might be mistaken. It does fit pretty well with Fig. 2 in [1], which is though a plot of older date. The number 70% for HE-LHC however agree reasonably well with Fig. 2.44 in [2].

2) You're right; the burn of particles is mainly in the core of the beam, so yes decreasing the number of particles in each bunch would mean fewer collisions and i.e. lower luminosity (unless the halo particles have some kind of unwanted space-charge-like impact to the remainder of the beam, causing e.g. emittance growth, but that is just speculations from my side).

Maybe these papers can be of interest for you:
https://accelconf.web.cern.ch/p05/PAPERS/TPAP036.PDF
https://www.sciencedirect.com/science/article/pii/S0168900218308994?via=ihub

[1] http://accelconf.web.cern.ch/ICALEPCS2015/papers/wepgf066.pdf

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Wrichik Basu
Wrichik Basu said:
How is this "burning" carried out experimentally?
That's the collisions you want. More collisions, higher luminosity, a larger fraction of particles in the beam is collided, higher integrated luminosity.

At an initial ~4 billion collisions per second (2E34/(cm2s) in two experiments) the LHC starts by colliding ~1.5*10-5 of its protons every second. It's an important loss mechanism.
Wrichik Basu said:
N.B.: I haven't yet finished reading the whole paper, so I don't know if any of the questions are answered later.
You can expect papers that discuss burn-off in the abstract to explain it in more detail in the main paper.

Wrichik Basu
The "luminosity burn-off" is the loss of particles due to the Luminosity, i.e. the collisions, and for e.g. LHC the burn-off is responsible for the main part of the particle loss. So the answer to 1) is: the burning is carried out experimentally by colliding the beams.
mfb said:
That's the collisions you want.
So when they are referring to the "emittance growth from luminosity burn-off", they are basically referring to the growth of emittance after the IP. That makes things simpler. Thanks.

Wrichik Basu said:
I haven't yet finished reading the whole paper, so I don't know if any of the questions are answered later.

Wouldn't it be more respectful of others' time for you to do that before asking here?

Wrichik Basu said:
"emittance growth from luminosity burn-off", they are basically referring to the growth of emittance after the IP.

Not really. Emittance grows after the IP for several factors, e.g. beam-beam tune shift. What this paper calculates is the emittance growth due to the interactions preferentially depopulating the core of the beam.

Wouldn't it be more respectful of others' time for you to do that before asking here?
I had a quick look through the rest of the paper and found that there was no proper explanation given, and only after that did I post here.

Till now, I have advanced to the third page, and they still haven't defined what the term means. If I read the whole paper without the explanations by JonasKK and mfb, it would have been like learning QFT without a basic knowledge of QM.

In ref. [1], page 2 left column (under Fig. 1), the authors write,
During luminosity production the LHC operates with crossing angle orbit bumps in the experimental insertions.
In the same paper, page 3, left column (beside Fig. 2), it is written,
...flat orbit (a closed orbit without any orbit bumps in the experimental IRs),...
What are "orbit bumps"?

[1]: E. H. Maclean et al., New approach to LHC optics commissioning for the nonlinear era, PRAB 22, 061004 (2019)

Orbit bumps are modifications to the closed orbit by the use of corrector (steerer) magnets in a local part of the machine. Orbit bumps are typically either 2-, 3- or 4-corrector bumps: before the first corrector and after the last corrector, the orbit will be unchanged.

2-corrector bumps are only possible when the two correctors are placed with a phase-advance of ##\pi##, and are therefore rarely used.
3-corrector bumps are not restricted by phase-advance.
4-corrector bumps are the superior ones; using four magnets, you can make a bump in between corrector #2 and #3 with an arbitrary angle and displacement.

You want to shift the closed-orbit by 2mm and -2.3mrad at the IP? No problem, do a 4-corrector bump! Formulæ to calculate the corrector magnet kick-sizes based on the twiss functions at the four magnets are readily available.
That being said, the optics of the machine has to be well-known; if the twiss functions are not perfectly known at all correctors, then the orbit bump will not be closed, and the closed-orbit of the rest of the machine will be perturbed slightly.
Additionally, whenever non-linear magnetic fields (e.g. sextupoles) are present within the bump, unwanted things might happen. For example, a vertical orbit off-set in a sextupole magnet will lead to linear transverse coupling, similar to a skew quadrupole magnet (see my first comment to this thread).

Wrichik Basu
A simple question.

From this paper,
One of the main measures of a collider’s performance is its luminosity. However, from the point of view of experiments, what is most important is not the peak luminosity but, rather, the integrated luminosity. For the detection of events, it is also preferable that the luminosity remain constant for as long as possible. Therefore, luminosity levelling can be introduced. This means that the natural decay of the luminosity is pre-empted and the luminosity is spoilt initially with respect to the nominal. Then, as the luminosity decays, it is spoilt less and less in order that it remain constant for as long as possible. While doing this, it is still very much worthwhile to start with as high a luminosity as possible, as this will translate in the luminosity being constant for a longer amount of time after levelling.
Why is it necessary to keep the luminosity constant? The detector shouldn't have a problem with finding event tracks in a beam with lower luminosity if by design it can withstand a higher luminosity. Why can't we let the luminosity decay naturally with beam lifetime?

You want to collect as much data as possible, but you want your events to stay recognizable. There is an optimal luminosity and ideally the accelerator delivers this optimal luminosity the whole time. Usually the detectors can handle more than the accelerators can deliver, so the accelerator just tries to keep the luminosity as high as possible. The LHC is an exception. It quickly reached the limits of ALICE and then LHCb (as expected), but then it reached twice its design luminosity, at that point ATLAS and CMS asked to not exceed that before they can upgrade their detectors.

A more moderate initial luminosity is also better for the beam lifetime: You can keep the luminosity high for a longer time because you don't waste so many protons/so much beam quality early in the run.

Klystron and Wrichik Basu