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
