A I A. Introduction A. Simple derivation of the total impedance of the structure A. Impedance of the single vertical collimator A. Case of two collimators with identical Twiss beta functions A. Case of two collimators with di ferent Twiss beta functions A. Conclusion B B B. Parameters for simulations with an inductive impedance model B. Parameters for TMCI simulations with the LHC impedance model B. Comparison of collimator gaps for the TMCI measurement B. Collimator gaps and beam parameters for the HL-LHC TMCI simulations B. Collimator gaps and beam parameters for the HE-LHC impedance model and stability simulations at injection energy B. Collimator gaps and beam parameters for the HE-LHC impedance model and stability simulations at injection energy C R C. Results of TCP measurements C. Results of TCSG measurements D E

PyHEADTAIL

. After the proton source, a linear accelerator, the Linac , regroups the protons in bunches and accelerates them up to a kinetic energy of MeV.

. The Proton Synchrotron Booster accelerates the proton beams from MeV to . GeV kinetic energy. It is also in the Booster that the proton bunches intensities and transverse sizes are de ned.

. The Proton Synchrotron accelerates the beams from . GeV to GeV. The PS is also a key accelerator: thanks to its various RF systems, one bunch can be split in multiples bunches, they can be merged or shortened to provide nely tuned beams for the SPS and the LHC.

where f 0 is the is the beam revolution frequency and σ x and σ y are the RMS transverse beam size of the bunches, assuming those have a Gaussian pro le. The transverse beam sizes at the collision point are determined by the focusing strength applied to the bunches and by the particle distribution divergence.

A reduction factor F is applied because the beams do not collide head-on. Instead they collide with a small crossing angle which reduces the overlap of the two bunches at the interaction point. In turn the number of events which could happen is reduced during the crossing. The LHC design peak luminosity in ATLAS and CMS is cm -s -.

.

CERN and the CERN accelerator complex

In HL-LHC the top energy of the beam will also be increased from . TeV to TeV after an extensive training campaign for the superconducting magnets. Most of the hardware implementation for HL-LHC will occur during the Long Shutdown in and . To obtain these smaller and more intense beams, the LHC injector chain must also be upgraded. In the framework of the LHC Injectors Upgrade (LIU), major hardware changes to CERN accelerators will occur during the Long Shutdown in and :

. The Linac will be replaced by the Linac which will accelerate H -ions up to MeV kinetic energy.

. In the Proton Synchrotron Booster the H -electrons will be stripped out and the proton beams will be accelerated from MeV to GeV.

. The Proton Synchrotron will accelerate the beams from GeV to GeV.

. The SPS injection and top energy remain unchanged but the accelerating RF cavities power will be increased to cope with the more intense beams.

This increase in performance of the LHC poses many challenges in di ferent elds. Among those challenges we can cite:

• Machine protection: each LHC beam has a stored energy of MJ, which will rise to MJ in HL-LHC. Collimators are used to protect the superconducting magnets from proton losses. New materials will be deployed for some of them to increase their robustness against beam losses. The collimation and protection layout close to the ATLAS and CMS interaction points will also be redesigned to withstand the damages caused by high energy collisions debris.

• Beam induced heat-load which a fects the surrounding environment of the beam when it circulates in the accelerator. It is caused by synchrotron radiation, impedance as well as electron-cloud, a parasitic avalanche e fect generated during the passage of the beam. The cryogenics plants having a limited cooling capacity, heating must be contained within nominal values.

• Coherent beam instabilities: the charged particle bunches interact with their environment, creating electromagnetic wakes after their passage. These wakes can then perturb the following bunches or the bunch itself. As the bunch intensity will increase by a factor of two in HL-LHC, these elds will become stronger. These perturbations have thus to be reduced through hardware changes or machine parameters optimisations.

Coherent beam instabilities are indeed one of the major concern for the LHC and HL-LHC as they can lead to beam quality degradation, beam losses and equipment damage. Their study is therefore of importance to reach and go beyond the luminosity target. We will now introduce elements of accelerator physics useful to understand how an accelerator works. This will be followed by an introduction to the concept of wake elds and impedances needed to study coherent beam instabilities.

1 Bρ ∂B y ∂x x , ( . )

This assumption is made for the derivation of the single particle motion. We will see that electromagnetic elds are generated by the beam current when considering a collection of particles. These elds can then perturb the beam motion

A

The High-Luminosity upgrade of the CERN LHC will increase the performance of the accelerator and the potential physics discoveries. The beam intensity will be multiplied by two to increase the collider luminosity. With such high intensities, collective e fects and in particular beam coupling impedance are a possible performance limitation for the accelerator.

After an introduction to accelerator physics and wake-elds, two collective e fects codes will be detailed:

PyHEADTAIL, a time-domain macro-particle code, and DELPHI, a Vlasov equation solver. Both are important to estimate coherent beam stability margins in the CERN accelerator complex, therefore a detailed comparison for di ferent wakes and impedances, including the LHC model, will be presented.

The current LHC stability limits will then be investigated with DELPHI simulations. In particular the Transverse Mode Coupling Instability, a fast instability occurring for high intensity beams with chromaticities close to zero, will be studied. The results will then be compared to measurements performed in the accelerator. Beam based measurements of several collimators will also be presented and compared to predictions from the impedance model. Combining these measurements and their comparison to simulations, we will estimate the uncertainty on the LHC impedance model.

To cope with the increased beam intensity, the impedance of the High Luminosity upgrade of the LHC will be reduced. The impact of di ferent upgrade scenarios will be studied from the Transverse Mode Coupling Instability perspective. The potential bene ts of an impedance reduction will be demonstrated through measurements in the LHC. Measurements performed in the LHC on a low impedance prototype collimator, the key component to the impedance reduction, will also be presented. These simulations and measurements will con rm the increase in the mode coupling threshold towards a value three times higher than the nominal bunch intensity. R Le projet de LHC Haute Luminosité sera une amélioration majeure de l'accélérateur visant à accroître ses performances et son potentiel de découvertes. L'intensité du faisceau sera multipliée par deux pour augmenter la luminosité du collisionneur. À de telles intensités, les e fets collectifs et en particulier l'impédance de couplage du faisceau sont une limitation possible des performances du LHC.

Après une introduction à la physique des accélérateurs et aux e fets des champs électromagnétiques induits, deux codes de simulation des e fets cohérents du faisceau seront détaillés : PyHEADTAIL, un code macro-particulaire temporel, et DELPHI, un solveur de l'équation de Vlasov. Ces derniers sont importants pour estimer les marges de stabilité dans les accélérateurs du CERN, c'est pourquoi une comparaison détaillée pour di férentes impédances, y compris celle du LHC, sera présentée.

Les limites de stabilité actuelles du LHC seront ensuite étudiées à l'aide de simulations DELPHI. En particulier, l'instabilité de couplage des modes transversaux, une instabilité rapide se produisant dans les faisceaux de haute intensité pour des chromaticités proches de zéro, sera étudiée. Les résultats seront ensuite comparés à des mesures réalisées dans l'accélérateur. Des mesures de plusieurs collimateurs réalisées avec le faisceau seront également présentées et comparées aux prédictions du modèle d'impédance. L'association de ces di férentes mesures et leur comparaison avec les simulations nous permettra d'estimer l'incertitude sur le modèle d'impédance du LHC.

Pour faire face à l'augmentation de l'intensité du faisceau, une réduction d'impédance est prévue pour le projet de LHC Haute Luminosité. L'impact de di férents scénarios sera étudié du point de vue de l'instabilité de couplage des modes transversaux. L'e fet béné que d'une réduction d'impédance sera démontré par des mesures sur le LHC. Les mesures e fectuées dans le LHC sur un prototype de collimateur d'impédance réduite, un élément clé du projet, seront également présentées. Ces simulations et mesures con rmeront l'augmentation du seuil de couplage des modes à une valeur trois fois supérieure à l'intensité nominale des paquets.

I

.

CERN CERN

The European Organization for Nuclear Research was founded in by European states. It had the purpose to rebuild the European nuclear physics landscape which had been shattered by the Second World War, and to share the increasing cost of research instruments among its members. The rst accelerator, the Synchro-Cyclotron (SC), accelerated its rst beam in . A second accelerator, the Proton Synchrotron (PS) was already under construction. In , its rst proton beam was accelerated. While the SC was shutdown in , the PS still runs after years of operation and delivers beams to xed target experiments as well as to the Super Proton Synchrotron (SPS) and the Large Hadron Collider (LHC).

As high energy physics required larger instruments to reach higher energies, a km long accelerator was designed. In , the Super Proton Synchrotron was inaugurated. Five years later, it was converted to a proton-antiproton collider, leading to the discovery of the W an Z bosons. More than forty years after its start, the SPS continues to deliver protons and heavy-ions beams to xed targets and to the LHC.

As the understanding of the standard model progressed further, the energy reached by colliders became insu cient to explore rare phenomena. In consequence, CERN designed in the s the km long Large Electron Positron (LEP) collider. The LEP helped further investigate the properties of the W and Z bosons discovered beforehand in the SPS.

Meanwhile the rst ideas of the Large Hadron Collider (LHC) came to life in the late s. At the beginning of the s, the LEP was dismantled to make room for the LHC. From the beginning the LHC design foresaw the use of cutting-edge technologies: T superconducting magnets to guide the beam along the km ring. The two counter rotating beams are kept separate from one another, requiring twin-aperture magnets and they are accelerated by superconducting radiofrequency cavities. The magnets cold bore are maintained at ≈ K with a ow of super-uid helium created by large cryogenics plants. Of course the LHC is the last element of the vast complex of CERN accelerators: . The Super Proton Synchrotron is the last stage before the LHC: the bunches are accelerated from GeV to GeV.

These accelerators also have their own physics program, serving xed target experiments or antimatter production. Operation with ion beams is also possible and involves additional accelerators, the Linac and the Low Energy Ion Ring (LEIR) at the beginning of the chain. The beams are then accelerated in the PS and SPS before reaching the LHC where they can be collided. Other experiments require antiproton or heavy-ions beams. The full accelerator complex operated at CERN is shown in Fig. The bunches are injected in the accelerator from the SPS in multiples trains, each containing multiples batches of proton bunches coming from the PS and PSB before it. Figure . schematizes the injection process of the LHC. Once all the trains have been injected, the two LHC beams are accelerated from GeV to . TeV, ramping the magnets from . T to . T. After reaching their top energy, the two beams are brought into collision in the four experiments, ATLAS, ALICE, CMS and LHCb. The . CERN and the CERN accelerator complex beams keep circulating and colliding for ∼ h for the experiments to acquire data. After this period, if the beam quality is too poor, they are dumped and the magnets elds are ramped down. A new injection can then start. Figure . summarises the supercycle for particle physics operations.

A gure of merit for a particle collider is the instantaneous luminosity L. It links the number of physics events per second dR dt to the cross-section of this event σ event dR dt = Lσ event .

( . )

Assuming that the two LHC beams have the same number of bunches M and that each bunch has the same number of protons N b and the same transverse size, the luminosity can be written Before this ramp, the machine is con gured for injection and then the trains of bunches are injected from the SPS. Once the ramp is nished, the beams are focused to reduce their size at the experiment points and are then adjusted in collision. After ∼ h of collisions, the beams are dumped and the magnets are ramped down to start a new cycle.

L = f 0 M N 2 b 4πσ x σ y F , (
In , despite a number of bunches reduced to compared to the design value of , the LHC peak luminosity could reach up to × cm -s -. This performance was made possible by the smaller beam sizes at the collision points, as well as the increased bunch intensities, and thus could not have been possible without the great exibility of the CERN accelerators preparing the beam for the LHC.

An upgrade of the LHC, the High-Luminosity LHC, aims at increasing the peak luminosity as well as the integrated luminosity, i.e the total number of events recorded by the experiments. This will allow to further investigate the properties of the Higgs boson, as well as other rare physics processes. Equation . shows that to increase the luminosity, di ferent parameters can be improved:

• The number of bunches can be increased: in HL-LHC it will reach bunches for the standard beam where the bunches are separated by ns [ ].

• The bunch intensity can also be increased: in HL-LHC it will be increased to . × p.p.b. at injection, more than two times the design value.

• The transverse beam sizes at the interaction points can also be reduced, which will be achieved in HL-LHC by using stronger focusing magnets before the interaction points and by reducing the beam divergence.

• The factor F coming from the crossing angle of the bunches can be optimized. In HL-LHC speci c radiofrequency cavities called crab cavities will de ect the beam before the ATLAS and CMS detectors so that the two bunches fully overlap at the center of the experiments.

Introduction

. P

We will now derive the equations of motion in the transverse and longitudinal planes for a single particle.

Several parameters essential to the study of collective e fects such as transverse and synchrotron tune or chromaticity will be introduced. The full details of the derivations can be found in [ , , , , , , ]. These references also provide further insight on single particle dynamics in the presence of machine imperfections, a critical aspect of beam dynamics.

. . E Let us consider a charged particle travelling at a speed v = βc along a circular orbit of radius ρ. We also consider an arbitrary position along the orbit as a reference for the curvilinear position s of the particle at an instant t, we then have s = vt. This reference for the curvilinear position is also used as the zero for the particle azimuthal angle θ. The reference frame is pictured in Fig. Let us now consider an other particle which has a position o fset from the reference particle. Its coordinates with respect to the reference particle are detailed in Fig. . . The particle position vector with respect to the accelerator center #r can be written

#-r = (ρ + x) #-u x + y #-u y + z #-u s , ( . )
and its momentum #p is decomposed as #p = p x #u x + p y #u y + p z #u s .

We now assume that the reference particle stays on the design orbit and energy. Its coordinates are (x = 0, y = 0, z = 0) and its momentum is (p x = 0, p y = 0, p z = p 0 ). Here p 0 = γm 0 v is the particle momentum at the design energy, where γ = 1 √ 1-β 2 and m 0 is the proton rest mass. This particle will be referred to as the synchronous particle.

The position o fset of the other particle is assumed to be small compared to the machine radius: x, y ρ. The transverse momenta are also small compared to the total momentum: p x , p y p 0 . The particle has a longitudinal o fset z. Its total momentum p is assumed to be slightly di ferent from the design one and is written p = (1 + δ)p 0 where δ 1.

. Principles of accelerator physics

x y #-u s z #-u x #-u y Figure .
: Detail of the coordinate system used to derive the equation of single particle motion. The gray circle represents the machine circumference and the the black dot the reference particle. The mobile coordinate system is de ned with respect to this particle. A blue dot shows a particle with an o fset from the reference particle. It circulates on the blue path called the closed orbit.

The particle state can then be represented by the D phase space vector (x, x , y, y , z, δ) with

x = dx ds = dx dt dt ds = v x v z = p x p z ≈ p x p , ( . ) y
= dy ds = dy dt dt ds = v y v z = p y p z ≈ p y p , ( . ) δ
= p -p 0 p 0 . ( .
)
We will now concentrate on the transverse motion of the particle. It is assumed that there is no coupling between the longitudinal and transverse plane: the coordinate z with respect to the reference particle will be taken equal to zero, as well as the momentum deviation δ. We then have #r = (ρ + x) #u x + y #u y = r #u x + y #u y .

For the particle submitted to a magnetic eld #-B only we have

d #-p dt = #- F Lorentz = q #-v × #- B , ( . )
with q the particle charge. We will assume in the following that q = +e where e is the elementary positive charge.

The right hand side term can be written . ) assuming that the magnetic eld has only transverse components, and can therefore be written as

e #-v × #- B = e(-v z B y #-u x + v s B x #-u y + (v x B y -v y B x ) #-u s ) , (
#- B = B x #-u x + B y #-u y .
For the left hand side we have #p = γm 0 #v , and assuming that the particle energy does not change, its γ remains constant in time. Then the left hand side of the equation becomes

d #-p dt = γm 0 d #-v dt = γm 0 #-r , ( . )
where a dot means taking the derivative with respect to time.

The rst derivative of #r can be written

# - r = ṙ #-u x + r #- u x + ẏ #-u y ( . ) = ṙ #-u x + r θ #-u s + ẏ #-u y ( . )
where θ = v s /r. The second derivative is then

#-r = r #-u x + r θ + 2 ṙ θ #-u s + r θ #- u s + ÿ #-u y ( . )
= r -r θ2 #u x + r θ + 2 ṙ θ #u s + ÿ #u y .

( . )

Combining Eq. . and . we obtain for the projection on the #u x axis r -r θ2 = -ev s B y γm 0 , ( . )

It was assumed that p x , p y p 0 , and since #p 0 = #p x + #p y + #p z we get p z ≈ p 0 , leading in turn to γm 0 v z ≈ p 0 . The former equation becomes

d 2 r dt 2 -r dθ dt 2 = - ev 2 s B y p 0 . ( .
)
At order zero, we must keep the proton on its design orbit r = ρ which requires to have r = 0. Since x ρ we obtain r = ρ + x ≈ ρ. Equation . then becomes

-ρ v s ρ 2 = - ev 2 s B 0 y p 0 , ( . )
which can be written

B 0 y ρ = p 0 e , ( . )
where B 0 y is the constant component of the vertical magnetic eld. Equation . is the basic equation to design a magnet based circular accelerator. It states that to increase the particle energy E = p 2 0 c 2 + m 2 0 c 4 , either the magnetic eld or the machine radius has to be increased. We assume that the accelerator is planar and thus only the constant vertical magnetic eld is used to guide the particles i.e B 0 x = 0. The quantity B 0 y ρ is often considered as a single value called the magnetic rigidity of the accelerator and it will be noted (Bρ). The constant vertical magnetic eld is provided by dipole magnets, thus its name of dipolar eld.

. Principles of accelerator physics

Because the accelerator must host other types of magnets such as quadrupoles or sextupoles, the radiofrequency cavities to accelerate the beam, the kicker magnets to inject and extract it as well as the particle detectors, the bending radius ρ is in practice smaller than the accelerator radius itself. We will have an e fective bending radius ρ = η dipole Lacc 2π where L acc is the total accelerator length and η dipole is the fraction of it occupied by dipole magnets. Table . details the magnetic rigidity and the energy reached by various accelerators operated at CERN, as well as for projected ones [ , , ]. -). During Run I ( -), the eld was rst limited to . T for an energy reach of . TeV per beam. The eld was then slightly increased to . T and the beam energy could reach

TeV. The beam kinetic energy assumed here is GeV/c . It will be the energy used for LHC beams extracted to the PS after the LIU upgrade of the machine.

We must now switch from t to s for the independent variable, which leads to and assuming d 2 s dt 2 = 0 i.e that the particle is neither accelerated nor decelerated we obtain

d 2 dt 2 = ds dt 2 d 2 ds 2 = ρv s r 2 d 2 ds 2 . ( . )
The equation of motion . is now written

ρv s r 2 d 2 r ds 2 -r v s r 2 = - ev 2 s B y p 0 , ( . )
and substituting r by r = ρ + x it can be simpli ed to

d 2 x ds 2 - ρ + x ρ 2 = - eB y p 0 ρ + x ρ 2 . ( .
)
The derivation is similar for the vertical plane motion, and yields

d 2 y ds 2 = eB x p 0 ρ + y ρ 2 . ( . )
These equations of a single particle motion are valid for any kind of magnetic eld #-B at a position s along the accelerator:

#-B = B x (x, y, s) #u x + B y (x, y, s) #u y . They are generally non-linear. The magnetic eld can be expanded in a Taylor series

B x (x, y) = B x (0, 0) + ∞ i=0 ∞ j=0 x i y j i! j! ∂ i+j B x (x, y) ∂x i ∂y i 0,0 , ( . ) B y (x, y) = B y (0, 0) + ∞ i=0 ∞ j=0 x i y j i! j! ∂ i+j B y (x, y) ∂x i ∂y i 0,0 . ( . )
It is assumed in these equations that the magnetic elements do not generate a eld which depends on the curvilinear position s.

We will now keep only the linear terms in x and y for the magnetic eld, as well as for the equations of motion. The eld components can then be written

B x = B x (0, 0) + ∂B x (x, y) ∂x x + ∂B x (x, y) ∂y y , ( . )
B y = B y (0, 0) + ∂B y (x, y) ∂x x + ∂B y (x, y) ∂y y .

( . )

.

Principles of accelerator physics

The terms ∂By ∂y and ∂Bx ∂x are assumed to be zero. This is to ensure that there is no coupling between the two planes of motion: only magnetic elds along the #u y axis will act on the horizontal motion and vice-versa. The B y (0, 0) component is the one entering into the magnetic rigidity formula

(Bρ) = B 0 y ρ = B y (0, 0)ρ = p 0 e , ( .
)
and it is assumed to be independent of the curvilinear position s.

Maxwell-Ampere's law in vacuum states that

# - rot #- B = µ 0 #- j + µ 0 ε 0 ∂ #- E ∂t ( . )
and in our case no time varying electric eld #-E is present, and the current #j induced by the single particle is negligible compared to the elds induced by the accelerator magnets . Thus we have

# - rot #- B = ∂B y ∂x - ∂B x ∂y #-u s = #- 0 . ( . )
Therefore ∂By ∂x = ∂Bx ∂y and the elds expressions . and . can then be simpli ed to

B x = ∂B y ∂x y , ( . )
B y = B 0 y + ∂B y ∂x x , ( . )
which are in turn inserted in the equations of motion . and . , yielding for the horizontal plane one

x - x ρ 2 - 1 ρ = - e p 0 B 0 y 1 + x ρ 2 - e p 0 ∂B y ∂x x 1 + x ρ 2 , ( . )
Using (Bρ) = B 0 y ρ = p 0 e it can be simpli ed to

x - x ρ 2 - 1 ρ = - 1 ρ 1 + x ρ 2 - 1 (Bρ) ∂B y ∂x x 1 + x ρ 2 . ( . )
We only keep linear terms, thus 1 + x ρ 2 ≈ 1 + 2 x ρ and the equation becomes

x = x ρ 2 + 1 ρ - 1 ρ 1 + 2 x ρ - 1 Bρ ∂B y ∂x x 1 + 2 x ρ ( . ) = - 1 ρ 2 +
where the term in x 2 has been neglected to obtain the second line.

A similar derivation for the vertical plane equation . completes the set of two equations:

x + 1 ρ 2 + 1 (Bρ) ∂B y ∂x x = 0 , ( . ) y - 1 (Bρ) ∂B y ∂x y = 0 . ( . )
Equations . and . are called Hill's equations and are similar to those of an harmonic oscillator. The di ference with respect to the harmonic oscillator is that the spring constant depends on the curvilinear position s along the ring. These spring constants will be noted as K x for the horizontal plane and K y for the vertical plane

K x (s) = 1 ρ 2 + 1 (Bρ) ∂B y ∂x , ( . ) K y (s) = - 1 (Bρ) ∂B y ∂x . ( . )
For the horizontal plane, two terms enter the spring constant: the 1 ρ 2 term is a weak focusing coming from the beam bending alone. The other term

1 (Bρ) ∂B y ∂x , ( . )
is stronger than the rst one especially in large accelerators. These two components are usually created by di ferent kind of magnets: dipole magnets to bend the trajectory of the beam, quadrupole magnets to focus it. When writing the expressions of the magnetic elds in Eq. . and . , coupling and non linear terms were neglected. In a real machine such elds can be created by magnet misalignments, manufacturing imperfections or roll angles among others. Higher order magnets such as sextupoles, octupoles or decapoles and skewed magnets are thus needed to compensate these errors.

In our case the spring constant K is also periodic which means that a distance L exists such that

K(s + L) = K(s) ( . )
The distance L can be the circumference of the accelerator or a fraction of it. Studied by George Hill in the th century, the solutions for this class of di ferential equations resemble the ones of a harmonic oscillator [ ]. The results for the horizontal plane and the vertical plane are similar. The solution for the equation of motion in the horizontal plane can be written as ) where J x and µ x,0 are integration constants determined by the initial conditions, α x (s) = -1 2 β x (s) and where β x (s) is a periodic function determined by the magnetic elements assembly. The function µ x (s) is the phase advance from the s = 0 position to s and is equal to

x(s) = β x (s)J x cos(µ x (s) + µ x,0 ) , ( . ) x (s) = - J x β x (s) (α x (s) cos(µ x (s) + µ x,0 ) + sin(µ x (s) + µ x,0 )) , ( .
µ x (s) = s 0 1 β x (s)
ds .

( . )

The transverse tune Q x0 is de ned as the number of oscillations per turn and derived from the phase advance

Q x0 = 1 2π 1 β x (s)
ds .

( . )

From Eq. . we get cos(µ(s

) + µ x,0 ) = x(s) √ βx(s)Jx
which, when inserted in Eq. . provides

J x = γ x (s)x(s) 2 + 2α x (s)x(s)x (s) + β x (s)x (s) 2 , ( .
)
where γ x (s) = 1+αx(s) 2 βx(s) . This equation is a parametric representation of an ellipse in the (x, x ) phase-space. The action J x is the Courant-Snyder invariant. The three parameters α x , β x and γ x are called the Twiss parameters and they de ne the ellipse shape and orientation. They are a function of the curvilinear position s and derive from the magnetic elements con guration. At di ferent locations along the accelerator, the ellipse will di fer in shape and orientation but its area will remain the same. This area will be noted as A and it results that A = πJ x .

( . )

Figure . represents the phase space ellipse at one location of the accelerator and its remarkable points. The particle ellipse Eq. . can also be written in a matrix form

X = x x ( . ) Ω = β x -α x -α x γ x , ( . )
X T ΩX = J x . ( . )
The Ω matrix is called the Twiss matrix.

The derivation was made for a single particle, however a real beam is composed at least of several millions of protons. We can therefore consider a distribution of particles in the (x, x ) phase space. The covariance matrix Σ of such a distribution is written

Σ = x 2 xx xx x 2 . ( . ) -α A/β -α A/γ √ Aγ √ Aβ x x Figure .
: Ellipse in the (x, x ) phase space described by a particle at a certain location in the accelerator.

The geometric emittance is de ned as the square root of the covariance matrix determinant

ε geom = √ detΣ = x 2 x 2 -( xx ) 2 . ( .
)
The covariance matrix Σ can also be expressed in terms of the Twiss matrix Ω as

Σ = ε geom Ω . ( . )
A Gaussian beam distribution in x and x is often encountered in lepton machines when the beam has reached an equilibrium state. It is also a good approximation for the LHC hadron beams at high energy [ , ]. For these beams, an other de nition of the beam emittance can be used: ε F is de ned as the area in the (x, x ) phase space which contains a certain fraction F of all the particles. For a bi-Gaussian beam, the emittance is [ ]

ε F = - 2πσ x β x ln(1 -F ) . ( .
)
where ln is the natural logarithm and σ x is the standard deviation of the particle distribution. If the fraction F = 15 %, then the Gaussian beam emittance ε F =0.15 equals the general de nition of the emittance ε geom from Eq. . . It can be shown that the betatron oscillations amplitude reduces when the particle energy increases [ , p. ]. Thus the emittance de ned previously in Eq. . is not an invariant of motion if the beams are accelerated. The normalised emittance ε n can be used instead

ε n = βγε . ( . )
The normalised transverse emittance is the one which is preserved throughout the LHC acceleration chain and we will refer to this de nition. We now have the equation of motion for the ideal synchronous particle and its solution. We also de ned a quantity to characterize an ensemble of particles and found an invariant of motion. We can now investigate the behavior of a particle with a small o fset from the synchronous particle parameters.

. . E

We derived fundamental tools to study the unperturbed particle motion when the two transverse planes are uncoupled. The study was assuming that the longitudinal momentum of the particle was equal to the one of the reference particle. We will now highlight the e fect a small momentum o fset can have on the horizontal motion of the particle.

For this we start from Eq. . , the equation of motion which included the linear terms of the magnetic eld. In this case the particle had an momentum p equal to the reference particle momentum p 0 . We now assume that its momentum is slightly di ferent from the reference particle p = p 0 (1 + δ) with δ 1. Equation . is now written as

x - x ρ 2 - 1 ρ = - e p 0 (1 + δ) B 0 y 1 + x ρ 2 - e p 0 (1 + δ) ∂B y ∂x x 1 + x ρ 2 . ( . )
The term e p 0 1 1+δ can be approximated as

e p 0 (1 + δ) ≈ e p 0 (1 -δ) , ( . )
and can in turn be expressed as a function of the magnetic rigidity (Bρ) = p 0 e of the reference particle:

e p 0 (1 + δ) ≈ 1 (Bρ) - 1 (Bρ) δ . ( . )
Equation . then becomes

x - x ρ 2 - 1 ρ = - 1 (Bρ) B 0 y 1 + x ρ 2 - 1 (Bρ) ∂B y ∂x x 1 + x ρ 2 + δ 1 (Bρ) B 0 y 1 + x ρ 2 + δ 1 (Bρ) ∂B y ∂x x 1 + x ρ 2 . ( .
)
This equation is similar to Eq. . except for the last two terms given by the momentum o fset δ. The second or higher order terms are neglected, as well as the terms in xδ for now, leading to

x + 1 ρ 2 + 1 (Bρ) ∂B y ∂x x = 1 ρ δ , ( . )
which is in turn similar to Eq. . and can also be written

x + K x (s)x = 1 ρ δ . ( . )
This equation is the inhomogeneous Hill's equation. Its solutions are the sum of the deviation from the closed orbit caused by betatron oscillation x β (s) and the deviation caused by the momentum o fset x D (s)

x(s) = x β (s) + x D (s) .
( . )

The term x β (s) is the solution of the homogeneous Hill's equation derived previously. We write x D (s) = D(s)δ where D(s) is called the dispersion function which is the solution of equation

D + K x (s)D(s) = 1 ρ . ( . )
If the mean value of the machine dispersion is positive, a particle with a positive momentum o fset will have a larger bending radius than the reference particle and will therefore travel a longer distance along the machine. The di ference between the path length C = C 0 + ∆C of an o f-momentum particle and the one of the reference particle is characterised by the momentum compaction factor α p

∆C C 0 = α p δ . ( . )
It can be shown that [ ]

α p = 1 C 0 D(s) ρ ds ≈ 1 Q 2 x0 . ( . )
In Eq. . the terms in δx were neglected. We will now include them in the equation. Since ) and Eq. . can be written

1 ρ 2 1 (Bρ) ∂By ∂x we have 2 ρ 2 + 1 (Bρ) ∂B y ∂x ≈ 1 ρ 2 + 1 (Bρ) ∂B y ∂x δ = K x (s)δ , ( .
x + (K x (s) -δK x (s))x = 1 ρ δ , ( . )
This equation shows that a particle with a momentum o fset will have a di ferent focusing strength. This di ference in focusing strength leads to a change of the betatron tune Q x0 by an amount ∆Q x0 . The chromaticity ξ characterises the variation of the betatron tune with respect to the particle momentum o fset

∆Q x0 Q x0 = ξδ . ( . )
The particle tune is then written

Q x = Q x0 + ξQ x0 δ.
The quantity Q x = ξQ x0 will also be called chromaticity and we will also use the chromatic angular frequency ω ξ = Q x ω 0 , where ω 0 is the particle angular revolution frequency.

. Principles of accelerator physics

. . S

For the study of beam motion perturbed by intensity e fects, we will use the smooth approximation: the focusing term in Hill's equation will be considered constant. The average value of the betatron functions β x,y will be used. The wavelength of the particle betatron oscillation is 2π β x and the tune Q x0 simpli es to

Q x0 = 1 2π β x ds = C 0 2π β x = R 0 β x . ( . )
We derived fundamental concepts of transverse linear motion in a particle accelerator. Notions such as the Twiss functions, dispersion and chromaticity were introduced. We must now complete the picture with the motion in the longitudinal plane before introducing the concepts of impedance and collective e fects.

. . P

The Lorentz force acting on the particle shows that the magnetic eld components result in a de ecting force perpendicular to the particle velocity. The acceleration of the proton beam can therefore only be performed with electric elds. The Lorentz force acting on a particle submitted to a eld #-E is written

#- F Lorentz = e #- E . ( . )
The electric eld can be electrostatic or time varying. In high energy accelerators, the eld is time varying and is created by a set of radiofrequency (RF) cavities through which the particle passes every turn. To accelerate the particles, the electric eld must be in the #u s direction:

#-

F Lorentz = eE s #-u s .
The time varying electric eld E s felt by a particle travelling in a cavity of length g can be written

E s = E 0 sin(φ RF (t) + φ 0 ) = E 0 sin(hω 0 t + φ 0 ) , ( . )
where E 0 is the electric eld amplitude, φ RF (t) is the phase of the eld in the cavity and φ 0 is the phase angle of the synchronous particle with respect to the RF wave and ω 0 is the particle angular revolution frequency. In the RF phase φ RF (t) = hω 0 t, h an integer called the harmonic number.

The energy ∆E 0 gained by the synchronous particle during its passage in the cavity is

∆E 0 = e +g/2 -g/2 E 0 sin(hω 0 t + φ 0 ) d(βct) ( . ) = eE 0 βc +g/2βc -g/2βc sin(hω 0 t + φ 0 ) dt ( . ) = eE 0 g sin(hω 0 g/2βc) (hω 0 g/2βc) sin(φ 0 ) . ( . )
To go from the rst to the second line, it was assumed that the particle velocity increase per turn in the cavity gap is small compared to its longitudinal speed, hence ds = βcdt. Because the particle travels through the cavity at a nite speed, the electric eld it feels is not constant: it will experience an average electric eld V = E 0 gT . We call transit time factor T the quantity

T = sin(hω 0 g/2βc) (hω 0 g/2βc) . ( . )
The transit time factor can be kept close to one by having a small cavity gap g. At each passage in the cavity, the synchronous particle will gain an energy ∆E 0 = eV sin(φ 0 ). Its energy change rate is therefore Ė0 = ω 0 2π eV sin(φ 0 ), assuming no other device or phenomena a fect the particle energy during its revolution. A particle with a momentum o fset will be non-synchronous. It will have a RF phase φ instead of φ 0 . Its energy gain will be ∆E = eV sin(φ) and its energy change rate Ė = ω 0 2π eV sin(φ). We saw that the momentum compaction factor α p relates the change of the particle path length to its momentum o fset with respect to the reference particle. The slip factor η is an other parameter which links the change of the non-synchronous particle revolution frequency ω to its momentum o fset:

ω -ω 0 ω 0 = -ηδ . ( . )
The momentum compaction factor and the slip factor are related: taking the logarithm-di ferentiate of the expression ω = 2πv/C = 2πβc/C gives

dω ω = dβ β - dC C 0 ( . ) = 1 γ 2 dp p 0 - dC C 0 ( . ) = 1 γ 2 -α p dp p 0 , ( .
)
where we used p = βm 0 c/ 1 -β 2 which gives dβ β = 1 γ 2 dp p . The slip factor can then be written

η = α p - 1 γ 2 . ( . )
A particle with a positive momentum o fset will have a higher velocity than the synchronous particle. However its bending radius is also increased and thus it will travel a longer distance compared to the synchronous particle. These two e fects compete to either increase or decrease the particle revolution frequency. One can see that the slippage factor η changes sign at a certain energy γ t = 1/α p . Three di ferent situations can be distinguished according to the particle energy γ:

. If γ < γ t then η < 0. A momentum increase will lead to an increase of the particle revolution frequency.

. If γ = γ t then η = 0. At rst order, a momentum increase or decrease will not change the particle revolution frequency.

. If γ > γ t then η > 0. A momentum increase will lead to a decrease of the particle revolution frequency.

The last case can seem counter-intuitive: a momentum, and thus a velocity increase leads to a lower revolution frequency. This is because the path length of the particle has been increased by an amount that the particle velocity increase can not compensate. The quantity γ t is called the transition energy and it is determined by the optics design, in particular by the dispersion function. For a regular lattice design based on alternating focusing and defocusing elements, γ t ≈ Q x0 . The transition crossing, i.e acceleration from γ < γ t to γ > γ t requires special measures to avoid beam quality deterioration. In the LHC the transition energy is at γ t ≈ 53 and is therefore never crossed since the injection occurs at γ = 480. Instead the transition crossing occurs in the PS: the beams are injected at γ = 2.5, extracted at γ = 27.7 and the transition occurs at γ ≈ 6. To mitigate instabilities close to this energy, a transition crossing scheme was put in place [ ].

E -

We now have all the elements to obtain the equation of motion in the longitudinal plane. We will establish a rst relation between the particle energy and the RF phase it experiences inside the RF cavity.

Similarly to the derivation done for the transverse motion, we assume that the particle parameters have a small deviation from the synchronous particle parameters which are denoted with a zero subscript

θ = θ 0 + ∆θ , R = R 0 + ∆R , E = E 0 + ∆E , p = p 0 + ∆p , ω = ω 0 + ∆ω , φ = φ 0 + ∆φ , ( . )
with respectively the azimuthal angle, the closed orbit radius R = C/2π, the energy, the momentum, the angular revolution frequency and the synchrotron phase. We have ∆θ = -ω 0 ∆t where ∆t = t-t 0 is the time delay between the o f-momentum particle and the synchronous particle. Therefore the delayed proton arrives at the center of the RF cavity with a phase delay ∆φ = hω 0 ∆t. Combining these two relations gives ∆φ = -h∆θ .

( . )

In turn we have

∆ω = d dt ∆θ = - 1 h d dt ∆φ = - 1 h dφ dt , ( .
)
where it is assumed that the synchronous particle phase φ 0 varies slowly in time with respect to φ. Since

∆ω ω 0 = -η ∆p p 0 we obtain ∆p = p 0 hηω 0 dφ dt . ( . )
The total particle energy is E 2 = E 2 rest + (pc) 2 so dE = vdp . The energy deviation ∆E is thus ∆E = v∆p = ω 0 R 0 ∆p. Combining this relation with Eq. . we obtain the rst energy-phase equation:

∆E ω 0 = R 0 p 0 hηω 0 dφ dt . ( . )
To obtain the second energy-phase relation, we start from the energy change rate for the synchronous and o f-momentum particles: ) which can be simpli ed to ) if we assume that the particle velocity increase per turn is small. These two equations lead to

Ė = ω 2π sin(φ) , ( . ) Ė0 = ω 0 2π sin(φ 0 ) , ( .
R dp dt = 1 2π sin(φ) , ( . ) R 0 dp 0 dt = 1 2π sin(φ 0 ) , ( .
2π R dp dt -R 0 dp 0 dt = eV (sin(φ) -sin(φ 0 )) . ( . )
The left hand side can be expanded and simpli ed to the rst order

R dp dt -R 0 dp 0 dt = (R 0 + ∆R) dp 0 dt + d∆p dt -R 0 dp 0 dt ( . ) ≈ R 0 d∆p dt + ∆R dp 0 dt . ( . )
Assuming the parameter o fsets are small, we have ∆R ≈ dR 0 dp 0 ∆p and then

R 0 d∆p dt + ∆R dp 0 dt = R 0 d∆p dt + ∆p dR 0 dt = d(R 0 ∆p) dt , ( . )
and using R 0 ∆p = ∆E ω 0 , Eq. . yields the second energy-phase equation

2π d(∆E/ω 0 ) dt = eV (sin(φ) -sin(φ 0 )) . ( . ) Since d E 2 = 2EdE on one hand and d E 2 = d E 2 rest + d (pc) 2 = 2c 2 pdp = 2cβEdp
on the other hand, we obtain 2EdE = 2cβEdp.

. Principles of accelerator physics E Combining Eq. . and .

nally provides

2π d dt R 0 p 0 hηω 0 dφ dt = eV (sin(φ) -sin(φ 0 )) . ( . )
This is a non linear equation in φ. The synchronous particle parameters inside the left hand term are also time dependent. However if ones assumes that those parameters are slowly varying in time, the equation simpli es to

2π R 0 p 0 hηω 0 d 2 φ dt 2 = eV (sin(φ) -sin(φ 0 )) . ( . )
This equation has analytic solutions if one further assumes that the oscillations have a small amplitude. Since φ = φ 0 + ∆φ, the term (sin(φ) -sin(φ 0 )) can be expanded to the rst order in ∆φ (sin(φ) -sin(φ 0 )) = sin(φ 0 ) cos(∆φ) + sin(∆φ) cos(φ 0 ) -sin(φ 0 ) ( . )

≈ ∆φ cos(φ 0 ) , ( . )
and the equation of motion . simpli es to

d 2 ∆φ dt 2 - eV hηω 0 cos(φ 0 ) 2πR 0 p 0 ∆φ = 0 , ( . )
where it was assumed that the synchronous particle phase φ 0 varies slowly with time so that φ0 ≈ 0.

The motion is stable and the equation corresponds to an harmonic oscillator only if η cos(φ 0 ) < 0.

The angular revolution frequency of the oscillations in the longitudinal plane is called the synchrotron frequency ω s

ω s = eV hω 0 |η cos(φ 0 )| 2πR 0 p 0 , ( . )
and the synchrotron tune Q s is de ned as

Q s = ω s ω 0 = eV h|η cos(φ 0 )| 2πv 0 p 0 . ( . )
This equation was derived by assuming that the phase o fset of the particle is small and that the synchronous particle parameters vary slowly in time. For a larger energy o fset or for quick changes in the longitudinal beam parameters, Eq. . must be solved numerically. The treatment can of course lead to important di ferences in the beam physics results. As we now have seen the elements needed to understand single particle motion in an circular accelerator, we can investigate the impact of collective e fects on the beam dynamics.

I

We will now study in more detail the e fect of the electromagnetic elds generated by the beam itself on the particles motion. A rst part will detail the concept of beam coupling impedance and showcase simple examples which can be encountered in circular accelerators. The second part will make use of these examples to study two di ferent beam instability regimes: the head-tail instability which is a chromaticity dependent e fect and the Transverse Mode Coupling Instability, a stronger instability observed for chromaticities close or equal to zero.

. N . . W Until now we studied the unperturbed particle motion in the longitudinal and transverse planes. The study focused on a single particle and we assumed that only the external magnetic and electric elds used to guide and bunch the particles were present. However the beam is not composed of a single particle but of several billions of them. It also travels in an environment which is not perfectly conducting, thus the image currents induced on the beam pipe boundaries will generate parasitic electromagnetic elds. These elds will in turn perturb the other particles present in the bunch itself or the following bunches if present. The more particles the bunch contains, the stronger these elds get and thus the stronger are the perturbations. These perturbations can be included in the equations of motion derived previously to study their impact on the beam dynamic.

We will now introduce the concepts of wake elds and impedances which are used to describe these beam induced electromagnetic elds. Detailed explanations of the concept of wake-elds and derivations for di ferent structures can be found in [ , , , , , , , ].

We saw that for the beam to circulate inside the accelerator, a set of external electromagnetic elds is needed. These elds generate a Lorentz force #-F ext which writes

#- F ext = e #- E ext + #-v × #- B ext . ( .
)
This relation is valid for a single particle travelling inside the machine, and the equations of motion were previously derived in this framework. However the charged particle beam interacts with its environment, creating images charges and currents in the accelerator components. They generate in turn electromagnetic elds which will act back on the beam. These elds are called wake elds. We then have a force #-F wake acting on the beam

#- F wake = e #- E wake + #-v × #- B wake . ( . )

Impedance induced instabilities

Source (xs, ys, ss) Let us consider an accelerator equipment through which two charged particles are travelling. Figure .  shows the situation under study: a source particle with charge q s enters rst the structure. We assume that it travels with a speed #v = βc #u s parallel to the structure axis. It also has a transverse o fset in both horizontal and vertical planes #r s = x s #u x + y s #u y . During its passage in the structure, the particle will generate electromagnetic elds and lose some of its energy. Now a test particle with charge q t enters the structure after the source particle. We assume that its velocity and direction are the same as the source particle and its o fset with respect to the structure axis is #r t = x t #u x + y t #u y . In the gure, the positions along the ring of the two particles are denoted s s and s t .

Test (x t , y t , s t ) z #-u s #-u x
We will make a second assumption: we assume that the wake eld perturbation is small enough so that it does not a fect the motion of the beam during its passage in the structure. In this rigid beam approximation, the distance between the two particles stays constant and is therefore noted z. As we assumed that the two particles travel at the same speed #v = βc #u s , we can write s s = βct and s t = βct -z. If z > 0, then the test particle is behind the source particle.

The Lorentz force #-F wake created by the source particle and acting on the test particle is thus a function of the two particles o fsets #r s and #r t , of their positions s s and s t and of time t

#- F wake = #- F wake ( #-r s , #-r t , s s , s t ) = q t #- E wake ( #-r s , #-r t , s s , s t ) + βc #-u s × #- B wake ( #-r s , #-r t , s s , s t ) . ( . )
This force will create a momentum change ∆ #p t for the test particle which can be written

∆ #- p t ( #-r s , #-r t , z) = +∞ -∞ #- F wake ( #-r s , #-r t , s s = βct, s t )| st=ss-z dt , ( . )
and can be projected on the longitudinal plane ∆p and on the transverse planes

∆ #-p ⊥ ∆p ( #-r s , #-r t , z) = q t +∞ -∞ E wake, ( #-r s , #-r t , s s = βct, s t ) st=ss-z dt , ( . ) ∆ #-p ⊥ ( #-r s , #-r t , z) = q t +∞ -∞ #- E wake,⊥ ( #-r s , #-r t , s s = βct, s t ) + βc #-u s × #- B wake,⊥ ( #-r s , #-r t , s s = βct, s t ) st=ss-z dt , ( . )
.

Notions and examples of beam coupling impedance

The Lorentz force components have been projected on the longitudinal and transverse planes which for the electric eld gives #-E wake = E wake, #u s + #-E wake,⊥ , the magnetic eld decomposition being similar. The wake functions are de ned for the longitudinal and transverse planes as

w ( #-r s , #-r t , z) = - βc q s q t ∆p ( #-r s , #-r t , z) , ( . ) #- w ⊥ ( #-r s , #-r t , z) = - βc q s q t ∆ #-p ⊥ ( #-r s , #-r t , z) . ( . )
The wake functions w , #w ⊥ can be Taylor expanded if the source and test particles o fsets are small. For the longitudinal wake w , the dependence on a transverse o fset is neglected and the wake is expanded to zero order, yielding

w ( #-r s , #-r t , z) = w #- 0 , #- 0 , z + O( #-r s ) + O( #-r t ) . ( . )
The w #-0 , #-0 , z function will be called the longitudinal wake.

For the transverse wake functions #w ⊥ = w ⊥,x #u x + w ⊥,y #u y , the expansion can be made to the rst order, yielding for example in the horizontal plane

w ⊥,x ( #-r s , #-r t , z) ≈w ⊥,x #- 0 , #- 0 , z + ∂w ⊥,x ∂x s #-0 , #-0 x s + ∂w ⊥,x ∂y s #-0 , #-0 y s + ∂w ⊥,x ∂x t #-0 , #-0 x t + ∂w ⊥,x ∂y t #-0 , #-0 y t .
( . )

In this decomposition, the rst term is a constant which is equal to zero if the structure is symmetric. If not it will cause a shift of the closed orbit path of the particles as they travel through the device [ ].

The second and third terms are the dipolar or driving wake elds. These terms correspond to an o fset of the source particle while the test particle remains on axis. For these terms the force experienced by the test particle is independent of its transverse position, like in a dipole magnet. The fourth and fth terms are the quadrupolar or detuning wake elds. These terms correspond to an o fset of the test particle while the source particle remains on axis. In this case the force experienced by the test particle is proportional to its transverse o fset, as in a quadrupole magnet. This force would therefore change the particle tune, hence its name of detuning wake.

We will use the notations

w x,dip = ∂w ⊥,x ∂x s #-0 , #-0 , w xy,dip = ∂w ⊥,x ∂y s #-0 , #-0 , ( . ) w x,quad = ∂w ⊥,x ∂x t #-0 , #-0 , w xy,quad = ∂w ⊥,x ∂y t #-0 , #-0 , ( . )
Impedance induced instabilities for the di ferent terms of the Taylor expansion. The cross terms w xy,dip and w xy,quad will be neglected as well since they only become important in asymmetric structures and with large displacements of the source and test particles [ ]. By de nition, the wake functions are the response of the device to an impulse excitation, thus it is equivalent to a Green function. The wake potential # -W created by a line charge density λ(z) travelling inside the structure is therefore the convolution of the distribution with the wake function

#- w( #-r s , #-r t , z) = w , w ⊥,x , w ⊥,y # - W ( #-r s , #-r t , z) = +∞ -∞ #- w #-r s , #-r t , z -z λ z dz . ( . )
For some element geometries, the wake functions can be computed analytically. Exact derivations are often limited to simple geometries such as circular or at beam pipes. For more complex structures such as cavities or corrugations, approximate models can be found. The wake functions for various elements are derived in detail in [ ] and an overview can be found in [ ]. For complex devices installed in accelerators, the wake functions can also be obtained with simulations codes such as CST Particle Studio [ ]. In this case the source charge can not be a pure impulse: a line charge density is used and the wake function is computed by deconvoluting the simulation results. The simulation set-up must be carefully thought of to obtain accurate results in a reasonable computation time.

The beam coupling impedance is de ned as the Fourier transform of the wake elds

Z ( #-r s , #-r t , ω) = 1 βc +∞ -∞
w ( #r s , #r t , z)e jωz/βc dz , ( . )

Z ⊥,x ( #-r s , #-r t , ω) = -j 1 βc +∞ -∞
w ⊥,x ( #r s , #r t , z)e jωz/βc dz , ( . )

Z ⊥,y ( #-r s , #-r t , ω) = -j 1 βc +∞ -∞
w ⊥,y ( #r s , #r t , z)e jωz/βc dz .

( . )

They are de ned for negative and positive frequencies and they follow the relations [ ] ) where Z * denotes the complex conjugate of the beam coupling impedance function. The beam coupling impedance, which will be referred to as impedance, is more adapted to circular machines where the periodicity can help simplify the expressions. Similarly to the wake functions, the impedance can be Taylor expanded and the same notations apply. Again the main terms considered are the longitudinal impedance Z , the transverse dipolar impedances Z x,dip and Z y,dip and the transverse quadrupolar impedances Z x,quad and Z y,quad .

Z (-ω) = Z * (ω) , ( . ) Z ⊥ (-ω) = -Z * ⊥ (ω) , ( .
The longitudinal wake is in units of V C -, the transverse dipolar and quadrupolar wakes in V C -m -, the longitudinal impedance in Ω and the transverse dipolar and quadrupolar impedances in Ω m -.

We saw that the wake is a space or time domain quantity. Physically, the larger the wake functions are, the stronger the kicks to the beam will be. If the wake is short-ranged, the particles closer to the . Notions and examples of beam coupling impedance source will be a fected. In that case the wake will mainly act on the bunch itself. On the contrary if the wake is long-ranged, then the following bunches will also be a fected. A very long ranged wake can have a multi-turn e fect: the wake generated during a rst passage will a fect the bunch after it has performed a full revolution of the machine.

Since the impedance is the Fourier transform of the wake, a short ranges wake will give a broadband impedance spanning over a large frequency range. On the contrary a long ranged wake will give a sharply peaked impedance.

We now have an overview of the wake eld and beam coupling impedance concepts. Their impact on the beam dynamics will be further explored in the following parts. Before we will show the wake and impedance functions for two speci c cases, the resistive wall and the broadband resonator. We will then focus on the LHC transverse impedance model which will be the basis for beam stability studies.

. . R An accelerator requires a vacuum chamber to keep the beam circulating. Depending on the materials used, the dimensions and the geometry of the chamber, the main source of wake elds can be caused by this vacuum chamber. Maxwell's equations are used to derive analytically the impedance generated by such element. Details on the derivation are given in [ , , , ]. The longitudinal and transverse dipolar resistive wall impedances for a circular pipe can be written [ ]

Z ,RW (ω) = (1 + j sgn(ω)) L 2πbσ c δ skin , ( . )
Z ⊥,RW (ω) = (sgn(ω) + j)µcδ skin L 2πb 3 , ( . )

and the longitudinal and transverse wakes are written

w (z) = - βc 4πb µc πσ c L |z| 3/2 , ( . ) w ⊥ (z) = - βc πb 3 µc πσ c L |z| 1/2 , ( .
)
where b is the beam pipe radius and L its length, σ c is the beam pipe material conductivity and sgn(ω)

means the sign of the angular frequency ω. δ skin = 2 |ω|µσc is the skin depth and µ the magnetic permeability of the beam chamber material. These relations are an exact expression of the chamber impedance only in a certain range of frequencies [ , ]

χc b ω χ -1/3 c b , χ = 1 Z 0 σ c b , ( . )

Impedance induced instabilities

where Z 0 is the free space impedance. For the wake function the previous condition becomes

2πbχ 1/3 z 2πb χ . ( . )
Figure . a shows the transverse wake as a function of the distance between the source and test particle. Time is used here as a variable since t = z/βc. The corresponding impedance as a function of frequency f = ω/2π is plotted in gure . b. The vacuum chamber properties used to obtain these functions are given in Tab. . . Following the conditions given in Eq. . and . the resistive wall model will be valid in the Hz to . THz range, equivalent to a wake ranging between µm and × m. This speci c resistive wall impedance model will later be used in the benchmark of two instability codes 10 -11 10 -7 10 -3 10 1 10 5

Time / s 10 -7

10 -5 10 -3 10 -1 . Notions and examples of beam coupling impedance . . B

10 1 10 3 |W dip x |/V C -1 m -1 |W dip x | (a) Wake eld -2 -1 0 1 2 Frequency / GHz -20 -10 0 10 20 Z dip x /M Ωm -1 (Z dip x ) (Z dip x ) (b) Impedance
A second simple wake and impedance model is the broadband resonator. It can be used as a rst approximation to account for the various cross-section changes occurring in the vacuum chamber. A broadband impedance means a quickly decayed wake eld. The wake will mainly a fect the bunch itself and not the following ones, leading to so-called single bunch instabilities. The longitudinal and transverse broadband impedance as a function of angular frequency can be written

Z ,BB (ω) = R s 1 -jQ ωr ω -ω ωr , ( . ) Z ⊥,BB (ω) = c ω R s 1 -jQ ωr ω -ω ωr , ( .
)
where ω r is the angular resonance frequency of the resonator, R s the shunt impedance, Q the quality factor. For the wake function, the formulas for the longitudinal and transverse components are [ ]

W ,BB (z) = R s, ω r Q exp - αz c cos |ω 2 r -α 2 |z c - α 2Q sin |ω 2 r -α 2 |z c , W ⊥,BB (z) = R s, ω 2 r Q |ω 2 r -α 2 | exp - αz c sin |ω 2 r -α 2 |z c , α = ω r 2Q .
( . ) Figure . a shows the wake function as a function of the distance between the source and test particles. The parameters for this wake are ω r = 2π × 2 × 10 9 GHz, R s = 25 MΩ m -1 and Q = 1. The time scale in ns highlights that for the broadband resonator the wake quickly decays. This decay time is comparable to the RMS bunch length of LHC beams: . cm, equivalent to . ns. The impedance function is pictured in gure . b. As highlighted before a short range wake implies a large frequency range with an impedance extending up to GHz.

The broadband resonator model presented here will be used to study single bunch e fects and in particular study and benchmark two codes in the Transverse Mode Coupling Instability regime. We will now detail the LHC impedance and the elements that compose it. Only the main elements with a strong impact on the machine impedance will be detailed.

. . T LHC

The LHC impedance model was developed in [ ] and has been completed and improved over the years. It was also extended to cope with the new or upgraded equipments of the High-Luminosity upgrade of the LHC and now includes a large part of the di ferent elements present in the machine [ ]. The contributions are frequency dependent and are also changing according to the accelerator con guration. Figure . shows the transverse impedance as a function of frequency at the injection energy of GeV and the top energy of . TeV. The impedance at top energy is larger than at injection energy by a factor of ∼ .

Impedance induced instabilities

Z dip x /M Ωm -1 (Z dip x ) (Z dip x ) (b) Impedance
Figure . : Left plot represents the horizontal dipolar wake function of a broadband resonator versus time after the source particle. The horizontal axis is in ns, highlighting the quick decay of the wake for these resonator parameters. The right plot represents the corresponding beam coupling impedance as a function of frequency. For frequencies below GHz, the impedance is mainly inductive i.e

Z dip x ≈ 0 and Z dip x ≈ constant.
The real part and imaginary part of the LHC impedance at top energy are detailed in gures . a and . b. These plots show the impedance functions of the various elements included in the model. They highlight the major contribution of the collimators and the beam screen on a large portion of the frequency range. The models used for these two parts will now be further detailed.

T

The cold beam screens are the tubes shielding the magnets cold bore from synchrotron radiation and beam induced heating from image currents [ ]. Figures . b and . a show the assembly of the beam screen in the cold bore and their location in a dipole magnet. The beam screens are stainless steel tubes with a µm copper lamination, cooled between K and K [ ]. Thanks to the copper coating and the low operating temperature, the impedance per unit length of the cold beam screens is small. But because they cover km out of the km circumference of the LHC [ ], their contribution to the impedance becomes sizeable at injection and top-energy.

Oblong perforations are made on the top and bottom parts of the cold beam screens, as pictured in They cover % of the beam screen surface, and their position and length was semi-randomised to minimise their impact on longitudinal impedance [ ]. Their impedance depends on the perforations geometry and the beam pipe radius [ , ]. Their relative contribution at injection energy is high, in particular for the imaginary part of the impedance. At top energy, the total impedance of the machine increases because of the collimators and therefore the pumping holes have a smaller relative contribution. Figure . : The beam screen (left) is a racetrack shaped tube inserted in the magnet cold bore (right). It is cooled between K and K and serves as a heating shield for the cold bore, cooled to K.

. Notions and examples of beam coupling impedance

10 -1 Z dip x / M Ωm -1 (Z dip x ), inj. energy (Z dip x ), inj. energy (Z dip x ),
T In a single LHC beam had a stored energy of MJ ( bunches of . × protons at . TeV). In HL-LHC this energy will reach MJ ( bunches of . × protons at TeV) [ ]. Particles at the fringes of the beam core form a halo that may be lost in the superconducting magnet coils. High loss levels can lead to an unwanted quench, and might damage equipment. The collimators are beam cleaning devices required to mitigate particle losses. Figure . shows the assembly of a single collimator. The two jaws are positioned close to the beam and must be made of robust materials such as Carbon Fiber-reinforced Carbon (CFC) [ , p. ].

To improve the cleaning e ciency, of these devices are installed for each of the LHC beam [ , ]. The system is multi-staged [ ]: primary collimators (TCP ) will intercept the main halo, secondary collimators (TCSG ) located downstream will intercept the secondary halo and the particle showers created by the interaction of high energy particles with the primary collimators. The principle is repeated with tertiary collimators (TCT) and particle showers absorbers (TCL and TCLA ). The collimator names follow this convention: T stands for target, the general category for the equipment, C for collimator, the subcategory, P for primary, the type of collimator. For a TCSG, S stands for secondary and G for Graphite The L designates auxiliary collimators and the A designates absorbers. TCL collimators are located close to the interaction points and TCLA are located in the betatron and dispersion cleaning regions. . The collimators apertures are set to intercept the fringes particles without intercepting the beam core. The physical gap g of each collimator in m is computed from the RMS beam size at the collimator. The beam size σ i in the plane i ∈ (x, y) is given by

Notions and examples of beam coupling impedance

σ i = β i ε βγ + (D i δ) 2 , ( .
)
where β i and D i are the Twiss beta and the dispersion values at the collimator position. Only the betatron part is kept to compute the horizontal and vertical beam sizes in the collimator [ ]

σ i,coll = β i ε βγ . ( . )
The collimators are also oriented with an angle θ with respect to the horizontal plane. An e fective beam size in the collimation plane is then computed with the beam sizes from . and the collimator angle θ [ , ]

σ coll = (σ x cos(θ)) 2 + (σ y sin(θ)) 2 . ( . )
Moreover, the normalised beam emittance used to compute the collimators gaps is xed to . µm rad for the LHC [ ] and will be xed to . µm rad for HL-LHC [ ].

Finally, the collimator jaws are set at a certain number of the e fective beam σ coll . Since the collimators settings are symmetric, this setting is the collimator half-gap g/2 g = 2n σ σ coll .

( . )

The various collimators families (primary, secondary, tertiary and absorbers) will have di ferent gaps which are then reported in number of e fective beam size n σ . Equations . and . show that as the beam energy increases, the beam size at the collimator decreases. Since the collimators settings are de ned for a certain normalized emittance, their gaps must follow the beam size and therefore are reduced as the energy increases. The collimators then become the main contributors to the impedance at top energy, as shown in Fig. . .

. A simplified approach to beam instabilities: the two-particle model

The jaws materials are also a key element to explain their predominance on the beam impedance. Most of them are made of graphite based materials, resulting from mechanical and vacuum compatibility considerations [ , p. ]. Their electric conductivity is rather poor: . MS m -for CFC [ ] against

. MS m -for copper at room temperature [ , ]. All-in-all the beam coupling impedance in LHC is higher at top energy than at injection energy, as can be seen in Fig. . . Because of their tight gaps, as shown in Table . , each collimator has a measurable impact on the impedance. Methods were developed to measure their individual contribution which can then be compared to predictions from the LHC impedance model. The results will be detailed in part . .

Because they are the main impedance source at top energy, the betatron cleaning collimators located in IR will be upgraded in the framework of the HL-LHC project. The scope and the impact of this impedance reduction on the mode coupling instability will be investigated in chapter .

The simple impedance models presented at the beginning of the section and the LHC impedance model will now be used to study transverse beam instabilities. But before benchmarking two beam instability simulation codes with those impedance models, we will rst give a qualitative overview of beam instabilities for a model with two particles.

. A : -

The simplest way to study beam instabilities is to use a two-particle model: the bunch is divided in two macroparticles each containing half the bunch intensity. This approach also enables to derive approximate formulas which can then be compared to more involved models. This model was suggested by R.Kohaupt [ ] and R.Talman [ ] and detailed derivations can be found in [ , ]. We will rst investigate the beam instability qualitatively before giving the main results from the aforementioned references.

The study starts by dividing the bunch in two macroparticles, one at the head of the bunch and one at the tail. Their charge is N b /2 where N b is the total number of protons in the bunch . The particle at the head of the bunch generates a wake eld which perturbs the particle at the tail of the bunch. This initial situation is depicted in Fig. . a. The particles are initially located at z 1 = 75 mm, z 2 = -75 mm, x 1 = 0.3 µm and x 2 = 0.3 µm. Their individual intensity is . × p.p.b. The summary of the parameters for these simulation is given in Table . . Macroparticle (MP ) at the head of the bunch creates a wake eld depicted by the solid line. Initially the wake felt by the macroparticle (MP ) trailing behind is small and thus has little impact on its transverse motion.

The two particles execute synchrotron oscillations: they periodically exchange their longitudinal position, the period being the synchrotron tune Q s . Figure . shows the longitudinal position of the two macroparticles as a function of the turn number. For this simple example the synchrotron tune has been chosen so that Q s = 1/500 i.e the particles execute a full synchrotron oscillation in turns. Because of the synchrotron oscillations, the trailing particle which was a fected by the wake will become the leading particle after 1/(4Q s ) turns. This is depicted in Figs. . b and . c which show the position of the macroparticles at turn and at turn , before and after 1/(4Q s ). At turn , MP at the tail is strongly a fected by the wake created by MP at the head, as can be seen in Resonator impedance with resonance frequency f res = 10 GHz, shunt impedance R s = 25 MΩ m -1 and quality factor Q = 1. Only the horizontal plane is used. This tune value is chosen so that a particle executes a full revolution in the (x, x ) phase space in ve turns (1/0.2 = 5). That is after ve turns the transverse position of the particle is the same if the motion is unperturbed. This is impossible to set in a real machine because magnet imperfections would create a resonance at this tune value. generating the wake. In turn this wake strongly a fects MP transverse position. After 1/(2Q s ) turns, MP is at the tail of the bunch and MP at the head, as showed in Fig. . d. The transverse position of both macroparticles has been a fected by the wake eld. While they both started with a transverse o fset of . µm, they end at ∼ µm after half a synchrotron oscillation. In this case the kicks to the trailing particle are piling up and create an exponentially growing transverse oscillation. Figure . e shows the situation after one full synchrotron oscillation. The two macroparticles are far from their original position: ± µm, a factor ∼ 7 compared to the . µm original position. This simple description highlights the important role of the synchrotron tune in the instability mechanism. Indeed if the particles exchange their position quickly enough, the trailing particle is a fected for a shorter time. The kicks to the transverse position do not have the time to accumulate and the beam remains stable.

W dip x /V C -1 m -1 ×10 18 W dip x (a) Turn -100 0 100 z / mm -2 0 2 x / µm MP MP -0.5 0.0 0.5 W dip x /V C -1 m -1 ×10 18 W dip x (b) Turn -100 0 100 z / mm -2 0 2 x / µm MP MP -0.5 0.0 0.5 W dip x /V C -1 m -1 ×10 18 W dip x (c) Turn -100 0 100 z / mm -2 0 2 x / µm MP MP -0.5 0.0 0.5 W dip x /V C -1 m -1 ×10 18 W dip x (d) Turn -100 0 100 z / mm -2 0 2 x / µm MP MP -0.5 0.0 0.5 W dip x /V C -1 m -1 ×10 18 W dip x (e) Turn
An other important parameter is of course the bunch intensity. Since the wake strength is proportional to the head particle intensity, the instability is more likely to appear for high bunch intensities. Figure . shows the same snapshots of the particles motion at di ferent periods of the synchrotron oscillation but for a bunch intensity reduced to p.p.b. The kicks to the trailing particle do not accumulate and the motion remains stable. The transverse position of the two macroparticles during turns are plotted in Figs. . and . , for both the unstable and the stable case. In the unstable case, the transverse excursions of the particles are clearly increasing because of the wake eld kicks. In the stable case, these kicks are not strong enough and the transverse motion remains regular.

Impedance induced instabilities

The two-particle model can easily be put in equations as detailed in [ , , ], and approximations can be made using it. The situation presented before is further simpli ed:

• The two macroparticles are assumed to be separated by a distance z for the rst half of the synchrotron period T s = 1/Q s . MP is at the head of the bunch, MP at the tail. At T s /2, their longitudinal position are instantaneously switched. MP is now at the head of the bunch and MP at the tail.

• The wake force W dip x (z) acting on the tail macroparticle located at position z behind the head particle is assumed to be constant.

• The machine is assumed to be smooth i.e transverse focusing is constant along the accelerator.

The transverse tune

Q x0 is then equal to Q x0 = C 0 /2π β x .
Using these assumptions, the equations of motion of the head and the tail particles are written [ ]

ẍ1 + (ω βx ) 2 x 1 = 0 , ( . ) ẍ2 + (ω βx ) 2 x 2 = e 2 N b W dip x 2C 0 γm 0 x 1 , ( . )
where

ω βx = Q x0 ω 0 .
x 1 corresponds to the transverse position of the macroparticle which starts at the head of the bunch. x 2 is the transverse position of the macroparticle which starts at the tail of the bunch. In the equations, the tail particle MP motion is perturbed by a force generated by the wake eld of the head particle MP . This force is also proportional to the charge of the particle.

. A simplified approach to beam instabilities: the two-particle model . The e fect of the wake eld is visible when the two particles are close longitudinally, at , , and turns. At these time periods, the wake is strong enough to create a displacement of the trailing particle which accumulates with time. . A simplified approach to beam instabilities: the two-particle model After a time T s /2, the positions of the two particles are switched, and so are the equations

W dip x /V C -1 m -1 ×10 18 W dip x (a) Turn
W dip x /V C -1 m -1 ×10 18 W dip x (b) Turn
W dip x /V C -1 m -1 ×10 18 W dip x (
W dip x /V C -1 m -1 ×10 18 W dip x (d) Turn
ẍ1 + (ω βx ) 2 x 1 = e 2 N b W dip x 2C 0 γm 0 x 2 , ( . ) ẍ2 + (ω βx ) 2 x 2 = 0 . ( . )
In the rst half of the synchrotron period, equations . and . govern the particles motion. Introducing the phasors of the two particles [ ] ) allows to write the system of equations in the matrix form

x1,2 = x 1,2 -j 1 ω βx ẋ1,2 ( .
x1 (T s /2) x2 (T s /2) = e jω βx Ts/2 A x1 (0) x2 (0) . ( . )
During the second half of the synchrotron period, MP and MP are simply switched. The corresponding equations . and . lead to the matrix system

x1 (T s ) x2 (T s ) = e jω βx Ts/2 B x1 (T s /2) x2 (T s /2) . ( . )
Combining the two former systems of equations allows to nd the transverse positions of the two particles after one full synchrotron period

x1 (T s ) x2 (T s ) = e jω βx Ts M x1 (0) x2 (0) . ( .
)
where the matrices A, B and M are written

A = 1 0 -jΥ 1 B = 1 -jΥ 0 1 M = B • A Υ = e 2 N b T s W dip x 8ω βx C 0 γm 0 . ( . )
The system is stable if the matrix trace

|Tr(M)| = |2 -Υ 2 | < 2. This condition leads in turn to Υ < 2 also written e 2 N b T s W dip x 8ω βx C 0 γm 0 < 2 . ( . )

Impedance induced instabilities

This condition con rms the qualitative observations which were made before. Decreasing the synchrotron period and thus exchanging more often the positions of the head and tail particles helps stabilize the beam. A higher beam energy also contributes to the stabilization: if the beam is more rigid, the perturbation caused by the head particle has less impact on the tail. On the other hand an increased beam intensity or a stronger wake will reduce the beam stability. This result can be applied to the simple case presented beforehand. The beam parameters entering in the stability criterion are

W dip x ≈ 8 × 10 17 V C -1 m -1 , Q s = 1/500 , Q x0 = 50.2 , ω 0 = 2π × 11 250 Hz , γ = 6930 , m 0 = 1.67 × 10 -27 kg . ( . )
Two bunch intensities were considered, N b = 10 10 p.p.b. and N b = 5 × 10 11 p.p.b. which led respectively to a stable motion and an unstable motion. In the rst case Υ ≈ 1 and in the second case Υ ≈ 51. The stability criteria predicts indeed an intensity threshold for the instability at Υ = 2 which gives to N b ≈ 1.9 × 10 10 p.p.b..

We have seen that a two-particle model of the bunch allows to derive a stability criteria which depends on the bunch intensity, the synchrotron and betatron tunes, the wake eld strength and the beam energy. This model is valid for the zero chromaticity case where the betatron and synchrotron motions are independent. Chromaticity can however be introduced in the model as showed in [ ]. In this case a two-particle beam can have two oscillation modes: a σ mode where the head and the tail of the bunch oscillate in phase and a π mode where the head and the tail have opposite phase. The sum of the modes growth rates is equal to zero so one of the two mode is always unstable [ ].

The two-particle model is however limited when more than two degrees of freedom are needed to describe the bunch behavior. A N-particle model can then be used to simulate the beam dynamics. We will now brie y introduce the physics underlying one of such code, PyHEADTAIL. The Vlasov formalism, which is an other approach to treat beam instabilities, will then also be introduced.

. T

We saw that the perturbation to the beam transverse motion caused by a wake eld can be treated analytically using a simple two-particle model. However analytic models are more di cult to obtain when more than two degrees of freedom are required. The two-particle model method has therefore to be extended to a multi-particle problem where the particles of the bunch are considered. The movement of each particle is tracked along time, using the single particle equations of motion. The kicks induced by the wake elds are then applied at every turn to each particle. With this method a computational limit is quickly encountered: it is impossible to track particles for enough turns to see the instability rising. The solution is to regroup them: instead of tracking particles of charge e, the code will track for example macroparticles of charge e. Despite the code optimisation and progress made on computing power, simulation time remains the main limitation of this method.

Studying di ferent machine parameters for many turns can be resource-intensive and thus the simulations parameters have to be carefully chosen. Macroparticle codes can also be extended to include many other e fects such as RF manipulations of the bunches (acceleration, splitting, rotation), direct space-charge, electron cloud or synchrotron radiation among others. This versatility makes macroparticle codes powerful tools to study the beam dynamics in accelerators. The code PyHEADTAIL [ ] is used at CERN to study coherent instabilities in the presence of wake elds, space-charge or feedbacks among others. Its implementation in the case of wake eld induced instabilities will be brie y presented in part . . . With the macroparticle approach, tracking macroparticles implies that modes of motion will be studied for each transverse and longitudinal coordinate. The tracking is unconcerned by the fact that many of these modes describe microscopic motions, thus irrelevant to coherent beam instabilities. It is the macroscopic motion, i.e modes with wavelengths in the order of the beam pipe or the bunch length dimensions that are of interest. So instead of considering the beam as a collection of macroparticles in time domain, the bunch can be studied as a whole and its modes of oscillation can be studied in the frequency domain. This approach relies on Vlasov equation [ ] and was rst applied to bunched beams by F.Sacherer [ , , , ]. Di ferent methods to solve the equation were developed along the years [ , , , , , , , , ]. We will present the method used in the code DELPHI [ , ] in part . . .

. . M

We will brie y introduce the underlying mechanisms of PyHEADTAIL interesting for the study of impedance induced instabilities. Further details on PyHEADTAIL implementation and functionalities can be found in [ , ]. A simulation follows these main steps:

. Machine initialization: the accelerator parameters (circumference, particle used, Twiss and dispersion values) are loaded. The ring is divided into K segments each separated by an interaction point IP.

. Beam initialization: a beam made of N macroparticles is generated at interaction point IP . Typically macroparticles are used. The initial transverse and longitudinal distributions of the macroparticles can be speci ed. A macroparticle with index i ∈ [1, N ] now has a set of six coordinates (x i , x i , y i , y i , z i , δ i ), a charge q i and a mass m i .

. Linear tracking: using Hill's equation solution detailed in part . . , the macroparticle transverse coordinates are transported from IP to IP . This transport for macroparticle i can be written in a matrix form [ , , ]

x i x i 1 = M x i x i 0 , M = √ β 1 0 -α 1 √ β 1 1 √ β 1 cos(∆µ 0→1 ) sin(∆µ 0→1 ) -sin(∆µ 0→1 ) cos(∆µ 0→1 ) 0 -α 0 √ β 0 √ β 0 , ( . )
Impedance induced instabilities where α 0,1 and β 0,1 are the Twiss functions values at the IP and ∆µ 0→1 the phase advance from IP to IP . The smooth approximation is used in PyHEADTAIL linear tracking so the phase advance ∆µ 0→1 = Q x0 L 0→1 /C where L 0→1 is the rst segment length.

. Chromaticity and detuning e fects: these e fects are implemented as a change of each macroparticle phase advance. Amplitude detuning is the dependence of the particle tune with its Courant-Snyder invariant 2J x,y . It is an important e fect for beam stabilization through Landau damping [ , ]. The phase advance ∆µ 0→1 for macroparticle i is derived from the phase advance ∆µ 0→1 which was used for all macroparticles

∆µ i,0→1 = ∆µ 0→1 + (ξ x δ i + a xx J x,i + a xy J y,i ) ∆µ 0→1 2πQ x0 ( . )
where a xx and a xy are the detuning coe cients, J x,i and J y,i the transverse actions of the particle.

. Synchrotron motion: the longitudinal coordinates (z i , δ i ) of the macroparticle are updated once every turn using the equations of longitudinal motion. The motion can be linear or not and complex RF systems can be modeled, allowing for bunch acceleration, splitting or rotation among others.

. Collective e fects: they are applied at each interaction point. PyHEADTAIL implements wake elds e fects but also feedbacks and space charge. For wake elds, we saw that the particles in front generate a wake which provides a momentum kick to the following particles. The dipolar wake kick ∆x i on macroparticle i caused by the other macroparticles j is thus

∆x i = - e 2 β 2 E 0 N j=1 W dip x (z i -z j )∆x j . ( . )
If the beam is made of macroparticles, applying this formula for each particle, at multiple interaction points along the ring and for several thousands of turns would be computationally impossible. Instead the bunch is sliced longitudinally as schematized in Fig. For thin enough slices, the wake can be considered constant inside the slice. The wake kick ∆x i generated by all the slices j located upstream of slice i are then computed and summed to obtain the wake kick acting on slice i

∆x i = - e 2 β 2 E 0 N slices j=1 x j W dip x (i -j) , ( . )
where x j is the mean value of all the transverse o fsets of macroparticles in slice j.

This whole process is then repeated for several thousands of turns depending on the beam revolution frequency. For example in the LHC the revolution frequency is ∼ kHz, so an instability with a rise time of s would be seen after × turns of simulation. On the other hand in the PSB the revolution frequency at injection energy is ∼ MHz so an instability with the same rise-time of s would require × turns of simulation. The simulation length must therefore be chosen carefully to ensure that eventual instabilities can develop properly. The wake slicing is also crucial since it assumes that the wake is constant within a slice. If the wake has sharp peaks, a very thin slicing has to be used to resolve it correctly.

We saw that PyHEADTAIL is a exible simulation code implementing the particle equations of motion and adding the chromaticity and the wake eld induced e fects on top of it. However computing power remains a limit of such code. Performing large parameters scans and studying slow instabilities can be time consuming. We will now see an other way to treat the instability problems which relies on Vlasov's equation.

. . V '

In the macroparticle approach the beam is treated as a collection of individual particles interacting with each other and transported along the ring. The limit case would be to consider a beam made of an in nite number of particles and apply the results to a subsystem of particles. In this approach the beam is described as a superposition of modes rather than a collection of particles. The basic tool is Vlasov's equation which describes the evolution of an arbitrary distribution of particles along time [ , ]. The code DELPHI [ ] implements this approach which will now be detailed.

Let us consider two conjugate variables q and p, where q is the coordinate and p the momentum. (q, p) forms the phase space, longitudinal or transverse. Vlasov's equation states that if we follow a local particle distribution Ψ(q, p, t) along time, the phase space area will not change. It can be written as

dΨ dt = 0 . ( . )
We now consider that the distribution Ψ is a function of the horizontal coordinate x and momentum p x , the longitudinal coordinate z and momentum o fset δ and of the curvilinear position along the accelerator s which encloses the time dependence as s = vt Ψ = Ψ(x, p x , z, δ, s) .

( . )

Impedance induced instabilities

Equation . can then be rewritten [ ]

∂Ψ ∂s + x ∂Ψ ∂x + p x ∂Ψ ∂p x + z ∂Ψ ∂z + δ ∂Ψ ∂δ = 0 . ( . )
The transverse plane coordinates can be changed from the position and momentum variables (x, p x ) to the action and angle variables (J x , θ x ) ) where Q x0 refers to the transverse betatron tune from single particle dynamics without momentum o fset. Vlasov's equation . becomes

x = 2J x R Q x0 cos(θ x ) , p x = 2J y Q x0 R sin(θ x ) , ( .
∂Ψ ∂s + J x ∂Ψ ∂J x + θ x ∂Ψ ∂θ x + z ∂Ψ ∂z + δ ∂Ψ ∂δ = 0 . ( . )
The Hamiltonian H describing the motion of a single particle, including the e fect of wake elds, can be written [ , ]

H = Q x R J x - 1 2η ω s v 2 z 2 - η 2 δ 2 - xF dip x (z, s) E 0 . ( . )
The rst term describes the transverse betatron motion of the beam in the smooth approximation. The e fect of chromaticity ξ is accounted for in

Q x = Q x0 (1 + ξ x δ).
The second and third terms of Eq. . describe the linear synchrotron motion of the beam. The last term corresponds to the force created by the dipolar wake eld at the position s along the accelerator acting on the particle at longitudinal coordinate z. The derivatives with respect to s can be expressed

J x = - ∂H ∂θ x = ∂x ∂θ x F dip x (z, s) E 0 , ( .
)

θ x = ∂H ∂J x = Q x R - ∂x ∂J x F dip x (z, s) E 0 , ( . ) z = ∂H ∂δ = -ηδ , ( . ) δ = - ∂H ∂z = ω s v 2 z η , ( . )
. Treatment of coherent beam instabilities and Vlasov's equation . is now written

∂Ψ ∂s + F dip x (z, s) E 0 ∂x ∂θ x ∂Ψ ∂J x + Q x R - F dip x (z, s) E 0 ∂x ∂J x ∂Ψ ∂θ x -ηδ ∂Ψ ∂z + ω s v 2 z η ∂Ψ ∂δ = 0 .
( . )

To write the particle Hamiltonian, di ferent assumptions were made. The longitudinal motion is assumed to be linear, thus the formalism can not handle particles with large synchrotron amplitudes, accelerated beams or complex beam manipulations. Also only the transverse dipolar wake is taken into account.

P

The perturbation formalism will now be used to solve the di ferential equation. We assume that the phase space distribution can be decomposed in a sum of an unperturbed distribution ψ 0 and a small perturbation ψ 1 . This perturbation develops itself at a complex frequency ω c = Q c ω 0 where Q c is the perturbed tune value associated with the mode. The total distribution can thus be written [ , ] ψ(s, J

x , θ x , z, δ) = f 0 (J x )g 0 (r) unperturbed distribution + f 1 (J x , θ x )g 1 (z, δ) exp jωcs v perturbation to be found , ( . )
where r = z 2 + ηvδ ωs 2

. Using polar coordinates z = r cos φ and δ = ωs ηv r sin φ, Vlasov's equation . simpli es to

f 1 g 1 jω c s v + Q x R g 1 ∂f 1 ∂θ x + ω s v f 1 ∂g 1 ∂φ exp jω c s v = sin θ x E 2J x R Q x0 F dip x (z, s)g 0 (r)f 0 (J x ) .
( . )

The function f 1 (J x , θ x ) can be Fourier expanded as

f 1 (J x , θ x ) = k=+∞ k=-∞ f k 1 (J x )e -jkθx , ( . ) but only the term f k=1 1 term is non-zero [ ]. Therefore f 1 (J x , θ x ) is expressed as f (J x ) exp(-jθ x ). The term g 1 (r, φ) is also Fourier expanded as g 1 (r, φ) = exp - jQ x z ηR m=+∞ m=-∞ R m (r)e -jmφ , ( . )
where R m (r) is the azimuthal mode m function. This leads to

m=+∞ m=-∞ R m (r)e -jmφ   f (J x )(Q c -Q x0 -mQ s ) f 0 (J x ) 2J x R Q x0   = R 2E F dip x (z, s) exp -j Q c s R exp -j Q x z ηR .
( . )

The wake force F dip x (z, s) for a dipolar impedance Z dip x is proportional to

F dip x (z, s) ∝ exp j Q c s R +∞ m=-∞ j -m +∞ p=-∞ exp -j(Q c + p) z R Z dip x (-ω 0 (Q c + p)) ∞ 0 rR m (r)J m (ω ξ -ω 0 (Q c + p)) r v , dr .
( . )

The transverse damper can be treated as a single turn wake acting on the dipole motion of the bunch [ ].

A damper force term can be added to the wake force term

F dip x (z, s) ∝ exp j Q c s R +∞ m=-∞ j -m ∞ 0 rR m (r)J m ω ξ r v , dr . ( . )
Combining equations . , . and . , integrating over φ and taking τ = r v yields

(ω c -Q x0 ω 0 -mω s )R m (τ ) = -κg 0 (τ ) ∞ m =-∞ j m -m ∞ 0 d , τ τ R m τ damper term µ ω 0 J m (-ω ξ τ )J m -ω ξ τ + ∞ p=-∞ Z dip x (ω p )J m ((ω ξ -ω p )τ )J m (ω ξ -ω p )τ impedance term . (
R m (τ ) = τ τ b |l| e -bτ 2 ∞ n=0 c m,n L |m| n aτ 2 , ( . ) g 0 (τ ) = e -bτ 2 n 0 k=0 g k L 0 k aτ 2 , ( .
)
which are then substituted in Eq. . . An integration over the variable τ then leads to an eigenvalue problem [ , ] (

ω c -Q x0 ω 0 )c m,n = ∞ m =-∞ ∞ n =0 c m ,n δ mm δ nn mω s + M m,n,m ,n , ( . )
where M is the combined impedance and damper matrix. The eigenvalues problem presents a two fold in nity with l the azimuthal mode number and n the radial mode number. When the problem is numerically solved, the matrix is truncated and only some modes are taken into account.

S

The eigenvalue (ω c -Q x0 ω 0 ) associated with the mode (m, n) is the complex frequency shift of this mode. The real part (ω c -Q x0 ω 0 ) will provide the coherent betatron frequency shift of the mode caused by the impedance. The imaginary part (ω c ) will give the rise or damping time of the oscillation mode. If (ω c ) < 0, the mode is unstable and the time needed for the oscillation amplitude to grow by a factor e 1 is 1 (ωc) . If (ω c ) ≥ 0 the mode remains stable. The eigenvector associated to mode (m, n) can be used to retrieve the mode frequency spectrum. After a Fourier transformation, the spectrum gives the signal observable at a xed location in the accelerator [ , ]. This signal represents the bunch shape when it is disturbed by the impedance. The bunch pro le can be obtained from the eigenvectors by reconstructing the transverse perturbation g 1 (r, φ). From [ ] the distribution spectrum λ(ω ) can be written as

λ ω = ω s ηc r=+∞ r=0 φ=2π φ=0 r exp j ω r cos φ c g 1 (r, φ) , dr , dφ , ( . )
where ω = pω 0 + ω c and p ∈ Z. Inserting Eq. . in Eq. . then yields

λ ω = ω s ηc +∞ m=-∞ r=+∞ r=0 rR m (r) , dr φ=2π φ=0 exp -jmφ + j ω r c - Q x r ηR cos φ , dφ . ( . )
With the relations

1 2π 2π 0 exp(jmφ -jx cos φ) , dφ = j -m J m (x) , ( . ) J -m (x) = J m (-x) , ( . )
we obtain

λ ω = 2πω s ηc +∞ m=-∞ j m +∞ 0 rR m (r)J m ω r c - Q x r ηR , dr . ( . )
This function is the spectrum of the unstable mode. A Fourier transform then allows to obtain the time domain pro le of the bunch. The signal can be reconstructed from DELPHI's output following these steps:

. An eigenvalue ω c is selected and its corresponding eigenvector is retrieved.

. From equation . R m τ = r c is reconstructed for each azimuthal mode m. The retrieved eigenvector gives the coe cient c m,n of the decomposition.

. Equation . is computed for a range of ω .

The signals obtained by simulations can be compared to those measured in an accelerator. In the SPS and the LHC, the head-tail monitor system is used to acquire these beam pro les [ ]. It employs a set of strip-line bunch position monitors to record the bunch longitudinal pro le. The baseline pro le is then subtracted to obtain the oscillation pattern of the bunch. We will see in parts . and . examples of pro les for di ferent oscillation modes and instability types.

Vlasov formalism allows for a fast treatment of the instability problem: it is reduced to an eigenvalue problem, which is computationally interesting. This allows to perform simulation scans over a large range of beam parameters such as bunch intensity, transverse and synchrotron tunes, chromaticity or impedance model. However many assumptions have been made during the problem derivation: for instance only linear synchrotron and betatron motions are kept, the transverse motions are assumed to be decoupled, and only the dipolar term of the impedance is used. These assumptions will matter when the results from PyHEADTAIL and DELPHI will be compared.

We saw that DELPHI fully solves Sacherer's integral equation, including azimuthal and radial modes. However some of the modes can be very weak and neglected as a rst approximation. We will now give the results of a simpler treatment of Sacherer's integral which allows to estimate the mode frequency shifts and growth rates.

. Treatment of coherent beam instabilities

-6 2τ b -5 2τ b -4 2τ b -3 2τ b -2 2τ b -1 2τ b 1 2τ b 2 2τ b 3 2τ b 4 2τ b 5 2τ b 6 2τ b
Frequency / Hz

. . A S '

In DELPHI, many of the computed matrix elements will correspond to weak modes with very slow growth rates and frequencies far from the unperturbed tune. In a rst approximation, the treatment of Sacherer's integral equation . can be further simpli ed. Only the modes with the same azimuthal and radial number are kept. The longitudinal beam distribution can also be considered constant

g 0 (τ ) = 4 π 2 τ 2 b
where τ is the longitudinal coordinate expressed in time unit and τ b is the full bunch length i.e τ b = 4σ z /βc. These assumptions allow to get a simple expression for the complex frequency shift ω c,m of azimuthal mode m for a low intensity bunch and outside the mode coupling regime [ , ] (ω c,m -

Q x0 ω 0 -mQ s ω 0 ) = 1 |m| + 1 je 2 N b 4πγm 0 cQ x0 τ b +∞ k=-∞ Z x (ω k )h m (ω k -ω ξ ) +∞ k=-∞ h m (ω k -ω ξ ) ( . )
where Z x is the transverse dipolar impedance and

ω k = (k + Q x0 )ω 0 + mω s . The mode m spectrum h m is h m (ω k -ω ξ ) = (|m| + 1) 2 τ 2 b 2π 4 1 + (-1) |m| cos((ω k -ω ξ )τ b ) τ 2 b π 2 (ω k -ω ξ ) 2 -(|m| + 1) 2 2 ( . )
Figure . represents the spectra of di ferent azimuthal modes. It is the interaction of these mode spectra with the impedance which determines the coherent frequency shift of the modes.

Impedance induced instabilities

A quantity called the e fective impedance appears in Sacherer's formula

Z x,ef f = +∞ k=-∞ Z x (ω k )h m (ω k -ω ξ ) +∞ k=-∞ h m (ω k -ω ξ ) . ( . )
This value represents the interaction of each mode with the beam coupling impedance. It can be computed for di ferent modes, chromaticities or bunch lengths and can be used to compare di ferent machines and their con gurations. Sacherer's formula . is therefore written

(ω c,m -Q x0 ω 0 -mQ s ω 0 ) = 1 |m| + 1 je 2 N b 4πγm 0 cQ x0 τ b Z x,ef f . ( . )
A few observations can be drawn from Eq. . and .

• The mode shifts and growth rates are proportional to the bunch intensity and inversely proportional to the beam energy.

• Azimuthal modes di ferent from zero have a smaller amplitude and an instability is more di cult to drive.

• The complex mode frequency shift is proportional to jZ x,ef f . In consequence the imaginary part (inductive or capacitive) of the impedance will a fect the mode frequency shift whereas the real part (resistive) of the impedance will a fect the mode growth rate. An instability can only appear if the real part of the impedance is non-zero.

Sacherer's formula . is useful to obtain a rst estimate of the mode frequency shifts for a given impedance. However, e fects such as mode coupling are not taken into account. We will compare the complex mode frequencies as a function of intensity obtained with DELPHI and Sacherer's formula to highlight their di ferences. We will also compare them to PyHEADTAIL results. However a post-processing of PyHEADTAIL output is necessary: the code yields time domain data whereas DELPHI and Sacherer's formula directly give the mode complex frequencies. We will now detail these post-processing steps.

. . P -

PyHEADTAIL

PyHEADTAIL being a tracking code, it records the turn-by-turn positions and momentas for the six phase space coordinates of all macropaticles. This data must be post-processed to allow comparisons with DELPHI output An example of an instability obtained with PyHEADTAIL is plotted in Fig. . . The horizontal mean position of the bunch is plotted versus the turn number. To obtain the instability growth rate, one can t an exponential function to the position signal envelope as shown in the plot. The simulation was performed with a resistive wall impedance for a chromaticity Q = +14 and a bunch intensity of 6 × 10 11 p.p.b.. This type of instability will be detailed in part . . We will see that for this chromaticity the most unstable mode has an oscillation frequency shifted by 2Q s from the unperturbed tune.

To obtain the frequencies of the beam oscillation modes, a fast Fourier transform of the signal can be performed. normalised to the maximum amplitude value A 0 which corresponds to the most unstable mode. One can see that the oscillation mode with the largest amplitude develops at a frequency corresponding to Q x0 -2Q s . Other modes with smaller amplitudes are present at

Q x0 -3Q s , Q x0 -Q s , Q x0 and Q x0 + Q s .
The mode spectrum can also be obtained with the SUSSIX algorithm [ ] or the Harpy code [ ]. These tools can provide more accurate results than the FFT for shorter signal lengths, which will be useful during measurements with the LHC beam for example.

We now have the elements to compare DELPHI and PyHEADTAIL results for di ferent impedance models and beam parameters. Instability growth rates, modes frequency shifts and head-tail pro les will be investigated. The rst instability studied will be the head-tail one.

Impedance induced instabilities

Q x0 -3Qs Q x0 -2Qs Q x0 -1Qs Q x0 Q x0 + 1Qs
Frequency

10 -3 10 -2
10 -1 10 0 ln(A/A 0 + 1) .

H -T

We saw in the previous part that two main approaches are possible to study beam instabilities: the macroparticle formalism and the mode formalism. The mode formalism and its implementation in the solver DELPHI have been detailed. We will now compare DELPHI and PyHEADTAIL for two types of bunched beam instabilities: the head-tail and the Transverse Mode Coupling Instability. We start with the head-tail instability: it is a chromaticity dependent instability which was encountered early in the history of circular accelerators [ , , ]. Indeed we will see that this instability appears for any bunch intensity and can be quite strong for some chromaticity values. We will rst investigate the case of a purely inductive impedance model. Then in parts . . and . . the head-tail instability will be studied with a resistive wall impedance model and the LHC impedance model.

. . M

The Sacherer integral equation . written before holds for any bunch intensity and any impedance function. Approximations can however be made in the low intensity case as shown in [ ]. In this case the di ferent oscillation modes are decoupled. Solving the eigenvalue problem will provide a set of complex frequency shifts ω c . We saw in part . . that the imaginary part of the beam coupling impedance is responsible for the frequency shift whereas the real part is responsible for the growth rate. As a result, if the impedance is purely inductive, the oscillation modes only have a frequency shift and no instability appears.

Let us take a transverse dipolar impedance Z dip x (ω) constant and purely inductive at j25 MΩ m -1 . The code DELPHI was used to obtain the complex frequencies ω c of the oscillation modes, with a beam intensity of p.p.b. The other relevant beam parameters are detailed in Table B. reported in appendix B. Since the beam intensity is low, the oscillation modes are independent. The Gaussian bunch power spectra as a function of frequency for several azimuthal modes are plotted in Fig. . with the . Head-Tail instability

-6 2τ b -5 2τ b -4 2τ b -3 2τ b -2 2τ b -1 2τ b 1 2τ b 2 2τ b 3 2τ b 4 2τ b 5 2τ b 6 2τ b Frequency / Hz 0 1 Mode spectra / arb. units m = 0 m = 1 m = 2 m = 3 0 25 Z dip x / MΩm -1 (Z ⊥ )
Figure . : Azimuthal mode power spectra for a low intensity bunch in a purely inductive impedance, computed with DELPHI. Several modes are represented, they all have been normalized to their maxima. The imaginary part of the impedance function is plotted alongside as a dashed line. Each mode frequency shift and growth rate result from the convolution of its respective spectra with the impedance function.

constant inductive impedance plotted alongside. All spectra have been normalized to their respective maxima. This case is for a chromaticity Q x = 0: all the spectra are centered on f = 0 GHz. The mode spectrum is peaked at zero frequency while the higher order modes have extrema at higher frequencies. The bunch longitudinal particle distribution is Gaussian with σ z = 7.5 cm bunch length. We will also refer to the full bunch length τ b which corresponds to the ±2σ z extent of the bunch expressed in seconds. In this case τ b = 4σz βc = 1 ns and we can see in Fig.

. -6

2τ b -5 2τ b -4 2τ b -3 2τ b -2 2τ b -1 2τ b 1 2τ b 2 2τ b 3 2τ b 4 2τ b 5 2τ b 6 2τ b Frequency / Hz 0 1
Mode spectra / arb. units

m = 0 m = 1 m = 2 m = 3 0 25 Z dip x / MΩm -1 (Z ⊥ )
Figure . : Azimuthal modes power spectrum for a low intensity bunch in a purely inductive impedance at Q = -10. All the mode spectra are shifted towards negative frequency values and they interact with a di ferent frequency range of the impedance function.

. Head-Tail instability

-1 0 1
Bunch length / ns obtained when Q = 0. The same observation can be made for mode : the head and the tail are not in phase opposition anymore at a speci c turn but superimposing multiple signals reveals the single-node structure observed in the Q = 0 case.

These results are similar to those obtained in the past with analytic derivations or measured in the PS and PSB [ , ]. In J.-L. Laclare and J.-P.Garnier analytic approach [ , ] the modes at low intensity are used as a basis for the high intensity stability eigenvalue problem. This approach was implemented in the Valsov solver GALACTIC [ ].

We will now study with DELPHI the head-tail instability for a range of chromaticities. As we saw the chromaticity shifts the mode spectra, changing their interaction with the impedance. For a certain chromaticity value, some oscillation modes will interact more strongly with the impedance while others can be stabilised.

. . H -

We start the study of the head-tail instability with the resistive wall impedance model. As seen in part . . , both impedance and wake functions can be written exactly given the beam chamber properties. This model is valid for an intermediate frequency regime speci ed in part . . . For the study we will take the same cylindrical copper beam pipe as in part . . . Its radius is b = 10 mm, its length L = 27 km and copper resistivity at K is 1/σ c = 17 nΩ m. The impedance and wake functions were plotted in Fig. . b and. a. The resistive wall impedance is larger at frequencies close to zero: the mode spectrum sampling this frequency region will drive a strong instability. Changing the chromaticity will shift the mode spectra and therefore changes the most unstable mode.

The beam and machine parameters for DELPHI and PyHEADTAIL simulations are detailed in Table . . The PyHEADTAIL and DELPHI convergence parameters are detailed in Table . . They correspond to a modi ed version of the LHC top energy beam parameters. Both codes use the smooth approximation for the Twiss beta functions. The average beta functions in the machine are

β x = R Q x0 = 26658.8832 2π × 62.31 ≈ 68.1 m , ( .
) Resistive wall impedance and wake computed analytically assuming a km long cylindrical beam pipe of radius r = 10 mm, made of copper at room temperature. An intensity of × p.p.b. was used for simulations with the LHC impedance model, whereas the intensity was increased to × p.p.b. for the resistive wall impedance model.

β y = R Q y0 = 26658.8832 2π × 60.32 ≈ 70.3 m . ( . )
We rst investigate the head-tail instability when the transverse damper is deactivated. The study is made for a xed bunch intensity, scanning the horizontal chromaticity from Q = -50 to Q = 50. These corresponds to chromatic frequencies of f ξ = ±1.6 GHz. The RMS bunch length used is σ z = 7.5 cm which corresponds to a full bunch length of τ b = 1 ns. The wake eld and the impedance in the vertical plane are set to zero so that the instability develops only in the horizontal plane.

Figure . represents the real part of the mode complex frequency shifts as a function of chromaticity.

DELPHI output is represented with blue dots. The mode with the largest negative imaginary part is highlighted in the plots. PyHEADTAIL results are plotted in a yellow-purple gradient, where purple

. Head-Tail instability

(ω i )-(ω i-1 ) (ω i-1 )
| < crit, the computation is stopped. Otherwise the matrix size is increased by computing new azimuthal and radial modes.

indicates the modes which have the largest spectral amplitude. The same color scale will be used for similar plots within the chapter. The modes were obtained using Harpy for the post-treatment [ ].

Figure . shows the growth rates of the modes -(ω c,x ) as a function of chromaticity. For DELPHI simulations only the most unstable mode growth rate is represented. For PyHEADTAIL we saw in part . . that the growth rate is directly obtained with the turn-by-turn transverse position data of the bunch. The positive chromaticities part is magni ed in Fig. . . An excellent agreement between the two codes over the whole chromaticity range is achieved.

Figure . shows that for negative chromaticities a quick instability develops. Associated with the most unstable mode in Fig. . , we can deduce that it is the mode which is the most unstable at negative frequencies. For negative chromaticities and above the transition energy, the modes spectra are shifted towards negative frequencies. The mode spectra will therefore sample an impedance Z x which has (Z x ) < 0 and (Z x ) > 0 as Eq. . and Fig. . b show. From Sacherer's formula . , we saw that the mode complex shift is proportional to jZ x so the mode will have a large imaginary component, hence the quick instability.

For positive chromaticities, mode samples an impedance which has (Z) > 0 and (Z) > 0. The mode frequency will now have a positive imaginary part and therefore will be stable. The higher order modes however will become unstable as they are now sampling more of the negative frequency region of the impedance. When the chromaticity becomes higher, the most unstable mode changes and the instability growth rate becomes smaller.

The head-tail pro le of all modes can be obtained with DELPHI and PyHEADTAIL for the di ferent simulated chromaticities. Figure . shows a selection of signals for the most unstable mode obtained with DELPHI. We can see that the pro le is modi ed by chromaticity:

• At Q = -3, mode is the most unstable mode and the pro le has no node as seen in Fig. . a. We will now study the e fect of the transverse damper on beam stability. The transverse damper is a feedback system: it detects the transverse position of a bunch which is becoming unstable, then computes and applies an electromagnetic kick to restore its transverse position [ , ]. In an ideal feedback the kick would be applied right after the position detection and would immediately restore the bunch position. But because of the hardware limitations, the feedback system will apply small kicks over several turns after a certain delay. The damper provides therefore a momentum kick x d proportional to the bunch dipole moment

Impedance induced instabilities

x d = g x sin φ d β d ( . )
where g is damper gain, x is the mean position of the bunch, φ d is the damper phase and β d is the Twiss beta function value at the damper location. A damper gain g = 0 corresponds to the absence of a transverse damper whereas a gain g = 1 corresponds to an instantaneous damping. The damping time τ D computed in unit of turns is the inverse of the damping gain τ D = 1/g. In the LHC the transverse damper (ADT) is a bunch-by-bunch feedback: each bunch in the beam will receive a correction kick. The system bandwidth of MHz [ ] guarantees that the kick a fects

Impedance induced instabilities

-1 0 1 Bunch length / ns -1 0 1
HT sig. / arb. units

×10 -3 (a) Q = -3 -1 0 1
Bunch length / ns

-3 0 3
HT sig. / arb. units

×10 -4 (b) Q = 5 -1 0 1 Bunch length / ns -1 0 1
HT sig. / arb. units

×10 -4 (c) Q = 25 -1 0 1 Bunch length / ns -1 0 1
HT sig. / arb. units . Head-Tail instability

×10 -4 (d) Q = 35
-1 0 1
Bunch length / ns only the selected bunch. The ADT was designed to damp injection oscillations and coupled bunch instabilities [ ] but is now routinely used in operation to stabilize both coupled and single bunch instabilities [ ]. It can also be used to excite the beam for measurement purposes and its beam position monitors can be used to record the turn-by-turn position of all bunches for several thousands of turns [ ].

In the tracking code PyHEADTAIL the behavior of the damper is implemented following the previous formula. More complex cases are also available: the feedback phase can be changed so that the ADT can a fect the bunch tune or drive an instability.

In DELPHI, the transverse damper is implemented as a delta function impedance [ ], corresponding to a constant wake. Because of the limited bandwidth of the ADT hardware, the damper is more e cient at damping mode , a rigid bunch oscillation when Q = 0, than higher order modes of oscillation.

Figure . shows the real part of the mode frequency shifts as a function of chromaticity when the transverse damper has been activated. The agreement between DELPHI and PyHEADTAIL is again excellent over the full chromaticity range. One can see that the transverse damper strongly a fects the beam instabilities by acting on the dipolar motion of the bunch. For Q ∈ [-20, 0], mode is not the most unstable one anymore. For positive chromaticities, the damper also a fects the higher-order modes due to the dipole moment produced by chromaticity.

Looking at the instability growth rate in Figs. . and . , we can see that they are reduced compared to the case without damper. For Q ∈ [-20, 0] the mode growth rate is reduced to almost zero. For chromaticities Q < -20, a positive growth rate is still present and comparable to the case without damper. At positive chromaticities, the growth rates of higher order modes are reduced by a factor ∼ 2 compared to the case without damper, as can be observed by comparing Figs. . and. .

We can examine the head-tail pro les obtained with DELPHI and compare them to those obtained with PyHEADTAIL. Figure . shows the pro les for the same chromaticities as the case without damper. We can see that for the negative chromaticity case in HT sig. / arb. units

×10 -4 (b) Q = 5 -1 0 1
Bunch length / ns HT sig. / arb. units

(c) Q = 25 -1 0 1
Bunch length / ns 0 HT sig. / arb. units -1 0 1

×10 -3 (d) Q = 35
Bunch length / ns We have seen with the simple resistive wall impedance model that the two approaches, macro-particle tracking and Vlasov's equation solver, provide consistent results both without and with the damper activated. We can now use the more complex and realistic impedance model of the LHC.

. . H -LHC

As for the previous case we will start the analysis with the damper deactivated. Figure . compares the real parts of the modes. The results of a third code, Nested Head-Tail Vlasov Solver (NHTVS) [ ] are also plotted in green triangles. This Vlasov solver uses a di ferent approach from DELPHI to solve Sacherer's integral equation. The longitudinal distribution is assumed to be a sequence of equally charged concentric rings. A basis of head-tail modes is computed for each of these rings and grouped

. in an eigenvalue problem. Figures . and. show the most unstable mode growth rate obtained with the codes. Both plots con rm the excellent agreement which was obtained in the resistive wall impedance case. Minor discrepancies on the mode growth rate between DELPHI and PyHEADTAIL are present for chromaticities Q ∼ 45. These are potentially caused by DELPHI convergence scheme: the convergence criteria checks the most unstable mode imaginary part. To reach the convergence on this value, DELPHI increases both the number of azimuthal and radial modes to be computed. If the computation is stopped because of too large matrices, the modes remain potentially non-converged.

We can see that bunch behavior is similar to the resistive wall impedance case, with the shift of head-tail modes at positive chromaticities and the mode dominating the negative chromaticities. The head-tail pro les are also very similar, and the two codes results are again consistent. Activating the transverse damper with the same damping time of turns as done previously results in a behavior similar to the resistive-wall case. Results are plotted in Figs. . , . and . . At negative chromaticities the mode is damped by the feedback for Q ∈ [-30, 0], as can be seen in Fig. . . But when the chromaticity is set to a large negative value, mode becomes unstable again. Figure . shows the signal reconstructed with DELPHI for Q = -35 and Q = -45 The signals envelope corresponds to a mode instability and the head-tail pro les show a large phase shift because of the chromaticity value.

Similarly to the resistive wall impedance case, some minor discrepancies remain between the two codes. In the Q ∈ [-30, -10] and in the Q ∈ [40, 50] ranges, the growth rate obtained with DELPHI are di ferent from the ones obtained with PyHEADTAIL. The discrepancy for these chromaticities can also be observed on the real part of the modes frequencies in Fig. . . The source of these discrepancies was con rmed to be DELPHI convergence scheme [ ]. Overall the agreement between DELPHI and PyHEADTAIL is excellent for the LHC impedance model for both cases without and with the transverse damper.

Impedance induced instabilities

E --

We saw in part . . that DELPHI assumes that the bunch synchrotron motion is linear. In reality synchrotron motion is non-linear and those e fects can be simulated with PyHEADTAIL. The e fect of non-linear synchrotron motion will be investigated with PyHEADTAIL for the LHC impedance without damper and compared to DELPHI results. Figure . shows the real part of the modes frequency for the same beam parameters and chromaticity range as before. The imaginary parts of the modes are plotted in Figs. . and. .

The e fect of non-linear synchrotron motion is noticeable for chromaticities Q ≥ 18. In Fig.

. we can observe with DELPHI a shift of the most unstable mode from m = -1 to m = -2 at Q = 18. With PyHEADTAIL this shift occurs at Q = 20. The e fect is even more visible for the m = -2 to m = -3 shift: while DELPHI predicts it at Q = 26, PyHEADTAIL predicts it at Q ≈ 50. In Fig. . , for Q = 10, we see that the growth rate obtained with PyHEADTAIL is reduced by ∼ % compared to DELPHI simulations. The e fect of non-linear synchrotron motion is therefore visible, even though the LHC bunch is rather short ( ns full bunch length in a . ns long RF bucket) and thus is less sensitive to the non-linear synchrotron motion. Since the synchrotron motion a fects the longitudinal phase, and thus the bunch length, the modes spectrum are a fected and their interaction with the impedance a fects in turn the modes frequencies and growth rates. This e fect could be more important when the bunch is lling up the RF bucket, for example in the PSB beams [ ].

We saw with the head-tail instability a rst important type of coherent beam instability. This is in fact the most commonly encountered in accelerators since the chromaticity is usually di ferent from zero. For a machine operating above the transition energy, the chromaticity is often chosen to be slightly positive. Then it is a mode l = -1 or l = -2 which is unstable. These modes can be damped by a transverse damper as we saw or by the accelerator non-linearities which creates a spread of the particle tunes.

. Head-Tail instability

Impedance induced instabilities

The head-tail instability occurs for any bunch intensity as Sacherer's formula . shows. In this approximation the growth rate and mode frequency shifts are linear with the bunch intensity and the e fective impedance. In the case studied previously, we saw that for Q = 0 the instability is much weaker than for Q = 0 and therefore easier to mitigate. However at Q = 0, when the bunch intensity is increased, a fast instability can develop. The two codes DELPHI and PyHEADTAIL will now be compared in this con guration.

. T M C I

We saw that the head-tail instability is chromaticity dependent but occurs for any bunch intensity.

The instability was however weaker for chromaticities close to zero. For Q = 0 we will see that this observation is valid for a certain intensity range and that above a certain threshold a fast instability, the Transverse Mode Coupling Instability, appears. We will start the study with the simple broadband resonator model, comparing the results of PyHEADTAIL and DELPHI without and with the transverse damper. We will then use this model to investigate the very high intensity behavior where the instability rise-time becomes comparable to the synchrotron period. Finally we will perform the study with the LHC impedance model.

. . TMCI

The rst comparison between DELPHI and PyHEADTAIL is made for the case of a broadband resonator impedance. This impedance model was described in . . and shown in Fig. . b, and can be used to study single bunch instabilities. The low quality factor used for the resonator ensures that the wake elds remaining after the bunch passage are negligible. The beam parameters are the same as those used for the head-tail instability study found in Table . . Table . only lists the parameters which have been modi ed for this study. The parameters speci c to DELPHI and PyHEADTAIL are listed in Table . . Resonator impedance with resonance frequency f res = 2 GHz, shunt impedance R s = 25 MΩ m -1 and quality factor Q = 1. We saw with the two particle model in part . that the head and the tail of the bunch regularly exchange their position, mitigating the destabilizing e fect of the impedance. However when the bunch intensity is too high or the wake eld too strong, the bunch becomes unstable. Figure . shows the real part of the complex mode frequency shifts obtained with PyHEADTAIL and DELPHI as a function of bunch intensity. We can see that the mode is shifted down as intensity increases. On the other hand the mode -stays almost constant with intensity. When the bunch intensity reaches × p.p.b., the two modes have an identical oscillation frequency, they couple and a fast instability arises.

. Transverse Mode Coupling Instability

The instability growth rate in Fig. . con rms this threshold at × p.p.b.. A growth rate of s -corresponds to a rise-time of ms, whereas the revolution period is T 0 = 1 f 0 = 89 µs. The instability therefore develops in approximately turns. The growth rate increases with the beam intensity, and at . × p.p.b., a second step is present. A close examination of the real part of the modes frequency in Fig. . shows that a second mode coupling occurs at this intensity, between modes -and -.

The TMCI intensity threshold can be approximated using Sacherer's formula . . With this model the mode frequency shift is linear with intensity and proportional to the e fective impedance Z ef f,0 . Using Sacherer's mode expression from Eq. . , for a chromaticity Q = 0 we obtain an e fective impedance Z ef f,0 = 24.9j MΩ m -1 . The mode frequency needs to be shifted by -ω s to reach and couple with the mode -frequency. Using Sacherer's formula . this leads to the condition

N thresh b = -j 1 Z ef f,0 4πγm 0 cQ x0 τ b ω s e 2 . ( .
)
This simple formula gives for this rst case a threshold intensity of . × p.p.b., close to the value of × p.p.b. obtained with DELPHI and PyHEADTAIL. The di ference is mainly caused by the fact that the mode shift is not linear anymore when the intensity is close to the TMCI threshold and that mode -also has a small shift as can be seen in . appearing as a signal with no node and with a standing wave structure turn after turn. As the bunch intensity increases and becomes close to the TMCI threshold, a travelling wave pattern starts to appear, as shown in Fig. . b. Above the TMCI threshold, this travelling wave pattern is clearly present as can be observed in Figs. . c and . d. Below the instability the threshold, the two modes are decoupled and have an independent standing-wave pattern. When the intensity is increased, the two modes start to in uence each other and the travelling wave appears. Above the mode coupling threshold, the beam pattern is the superposition of the standing-wave patterns of mode and mode -.

The signals obtained with the two codes can now be compared. Because below the TMCI threshold the motion is stable, the oscillation signal obtained with PyHEADTAIL is within the numerical noise. The two codes have to be compared for intensities above the TMCI threshold.

Figure . shows the pro le for a bunch intensity of . × p.p.b., just above the TMCI threshold, whereas gure . shows the pro les for a bunch intensity of . × p.p.b. Again the two codes give results in good agreement for this impedance model. These simulations were also extended to a bunch intensity of × p.p.b.. For such high intensities, the instability rise time is comparable or faster than the synchrotron period. The mode coupling instability is then similar to a beam-break-up instability encountered in linear accelerators in which there is no synchrotron whereas the growth rate found with DELPHI and PyHEADTAIL at this intensity is s -. Figure . compares the results obtained with the scaling law to those obtain with DELPHI and PyHEADTAIL Some of the longitudinal beam pro le obtained with DELPHI for these high intensities are shown in gure . . As the intensity is increased, the oscillation amplitude becomes larger towards the tail of the bunch. This is an other indication that the bunch entered a break-up like regime.

We saw that without damper the bunch motion is stable below a well de ned intensity threshold, found at × p.p.b.. Activating the transverse damper will however introduce a slow instability of the mode -below the TMCI threshold. This instability is created by a coupling of the two modes through the interaction with the damper. This result was found with the Vlasov solver GALACTIC [ ] and the results could be reproduced with DELPHI [ , ] and NHTVS [ ].

Figures . and . reached in the mode coupling regime. Combined with Fig. . , we can see that it is mode -which is unstable while mode is damped by the transverse feedback. The mode and -coupling is also suppressed by the feedback. A stronger instability still appears at . × p.p.b. when the modesand -couple, the transverse damper being less e cient at damping these modes.

An excellent agreement between PyHEADTAIL and DELPHI is again obtained with the mode coupling instability for the simple broadband resonator model, both without and with the transverse damper active. We can now perform the same study using the LHC impedance model.

. . TMCI LHC

For the study with the LHC impedance model, the beam and machine parameters remain the same as those presented in Tables . and. . The instability will again be studied for the case without and with damper. Furthermore, the e fect of the horizontal quadrupolar impedance on tune shifts and beam stability will also be investigated, and compared to Sacherer's formula.

For the rst case of the LHC impedance without damper, the comparison results are shown in Figs. . and. . The TMCI threshold is found at × p.p.b.. A second coupling of modesand -is found at N b = 9 × 10 11 p.p.b. with DELPHI but is not observed with PyHEADTAIL.

Agreement between the two codes is good for intensities up to × p.p.b.. For larger intensity values, a discrepancy which was not seen in the previous broadband resonator case is present. This discrepancy was addressed by changing then slicing method used for PyHEADTAIL simulations from an UniformBinSlicer to a UniformChargeSlicer. More details on the di ferent slicing methods and results of the convergence study can be found in appendix D.

For the bunch pro le, the agreement between the codes remains satisfactory, as can be seen in the LHC after the High-Luminosity upgrade will not exceed . × p.p.b., the discrepancies found for higher bunch intensity are not critical.

The transverse damper will now be activated with a damping time of turns. Figures . and . show the comparison of the real and the imaginary part of the modes shifts. As in the case without damper, the agreement between the two codes is good below an intensity of × p.p.b. Beyond this value, both real and imaginary parts of the modes diverge, as observed already in the previous case without damper.

The e fect of the transverse damper is clear: the mode is no longer the most unstable below the TMCI threshold previously found. As with the broadband resonator case, there is no sharp threshold for the mode coupling instability. The bunch is instead unstable at all intensities, and the mode -is the one driving the instability.

The bunch pro le obtained for the intensities of × p.p.b., × p.p.b. and × p.p.b. are shown in Fig. . . For the latter, there is a clear di ference with respect to the case without damper pictured in . b: a node structure remains in the beam pro le. This con rms that it is mode -which is the most unstable at all intensities, while the coupling with mode is reduced by the action of the transverse feedback. We can also see that a travelling wave pattern starts to develop in the bunch for an intensity of × p.p.b., con rming that a weak coupling of modes and -occurs at this intensity because of the transverse damper.

In the cases presented until now, only the dipolar part of the beam impedance was taken into account. The quadrupolar part of the impedance has a detuning e fect, which can increase or reduce the bunch tune shift among other e fects. As seen previously, the code DELPHI only includes the e fect of the dipolar impedance. PyHEADTAIL can however account for the e fect of any component of the wake eld. A comparison of PyHEADTAIL, including the quadrupolar wake of the LHC model, and DELPHI was thus made. This allows to estimate the error made on the tune shift when simulations are performed with DELPHI. The simulations were performed without the transverse damper, and also compared to Sacherer's formula results which in one case will account only for the dipolar tune shift and in the other for both dipolar and quadrupolar tune shifts.

Impedance induced instabilities

Figure . compares the modes frequency shifts obtained with the two codes. Including the quadrupolar impedance in PyHEADTAIL (right plot) shifts the mode slightly upward for intensities below the TMCI threshold compared to the case without the quadrupolar impedance (left plot). For intensities beyond this threshold, the /-coupled mode is also shifted upward. This shows that in the LHC the quadrupolar tune shift partially compensates the dipolar one in the horizontal plane. This comes from the combination of the collimators impedance. Because of their di ferent orientations and Twiss beta functions, their dipolar and quadrupolar impedance can add or compensate each other. An illustrative example is given in appendix A.

The e fect on the modes growth rates can be seen in gure . . In this case the TMCI is not much a fected by the horizontal quadrupolar impedance. The growth rates also remain similar in the three cases for intensities up to × p.p.b.. Above this intensity DELPHI and PyHEADTAIL results start to diverge, as seen previously. However the quadrupolar impedance only a fects the growth rates for intensities above × p.p.b.. Sacherer's formula allows to estimate the mode frequency shifts by computing the e fective impedance of the machine for the given beam parameters. The formula can also be used to estimate the detuning e fect of the quadrupolar impedance. Applying it to the LHC impedance model, one obtains the horizontal dipolar Z dip ef f and quadrupolar Z quad ef f e fective impedances

Z dip ef f = 37j MΩ m -1 , ( . ) Z quad ef f = -5.8j MΩ m -1 . ( . ) ( . )
Impedance induced instabilities

-1 0 1
Bunch length / ns

Impedance induced instabilities

The estimated tune shift ∆ω c,0 for mode induced by the dipolar and quadrupolar impedance is then proportional to the sum of the e fective impedances [ ]

∆ω c,0 ∝ jN b Z dip ef f + Z quad ef f . ( . )
The left plot of Fig. . shows the case without quadrupolar impedance. For intensities below p.p.b., the tune shifts are the same for the three methods. However as the intensity increases, the mode starts to interact with other modes and its frequency shift diverges from the linear model of Sacherer's formula.

The right plot shows the case where the quadrupolar wake is included in PyHEADTAIL and Sacherer's formula sums the e fective impedances. Mode frequency is shifted upwards because of the quadrupolar impedance contribution. PyHEADTAIL and Sacherer's formula results remain in good agreement for intensities below p.p.b.. The tune shift obtained with DELPHI is now overestimated by ∼ %, the ratio of the horizontal quadrupolar and dipolar e fective impedances. This ratio obtained with Sacherer's formula can therefore be used as a correction for DELPHI simulations.

We studied the head-tail instability and the transverse mode coupling instability for di ferent impedance models and compared the results of two simulations codes. We saw that they both are in good agreement except for some speci c cases. DELPHI now also permits to reconstruct the beam pro le which would be observed at a xed location in the accelerator. The signals were cross-checked with those obtained with PyHEADTAIL for the di ferent studies, validating the implementation. For the head-tail instability we also investigated with PyHEADTAIL the e fect non-linear synchrotron motion can have on beam instabilities and compared the results with DELPHI. Some di ferences were found, notably for large positive chromaticity values where the head-tail mode change was a fected by the non-linear motion. For the transverse mode coupling instability, we studied the very high intensity regime where the instability growth rate becomes linear with the bunch intensity and observed the beam-break-up like behavior of the bunch. In this regime, the chromaticity is assumed to be perfectly corrected to . The e fect of quadrupolar impedance on the mode frequency shifts was also investigated. We could see that in the LHC impedance case the horizontal quadrupolar component compensates the mode dipolar tune shift by ∼ %. Moreover PyHEADTAIL results are close to those obtained with Sacherer's formula for low intensities. Correction factors can therefore be computed using Sacherer's approximation and applied to the mode shift obtained with DELPHI.

We will now investigate the impact of impedance on the beam dynamics in the LHC. In particular the e fect of slightly positive chromaticity on the mode coupling regime will be investigated through simulations. We will compare the results of these simulations with beam based measurements performed in the accelerator.

T T M C I LHC

We investigated two types of coherent beam instabilities: the head-tail and the transverse mode coupling.

We saw that the two simulation codes used, PyHEADTAIL and DELPHI, are in good agreement and we investigated their limits. We will now use DELPHI simulations to simulate beam stability margins in the LHC. They will rely on the accelerator impedance model described in part . . . The comparison between measurements and simulations will allow to estimate the uncertainty on the impedance model.

A rst part will be dedicated to a general overview of stability limits encountered during the machine Run II ( -). We will then investigate the Transverse Mode Coupling Instability in the LHC with simulations and measurements in parts . and . . Beam based measurements of individual collimators and head-tail instability measurements at injection energy will also be presented in part . .

. M

We saw in the previous chapter that the transverse damper and chromaticity can be e fective ways to reduce the instability growth rate. However in the LHC the beam con guration process takes several minutes. The energy ramp takes ∼ min, the at-top phase during which the transverse tunes are modi ed takes ∼ min, the squeeze and adjust process during which the tranverse size of the bunches at the interaction points are reduced and the beams brought into collision takes ∼ min [ ]. The beams are then stored for several hours, colliding at the experimental points. The beam stability needs to be ensured at all stages in order to preserve the beam quality.

A mechanism called Landau damping helps keeping the beam stable [ , ]. A spread in the individual particle oscillation frequencies i.e the transverse tunes is introduced. When a coherent excitation of the bunch is applied, the particles cannot organize themselves to provide a coherent response. Over time their average response is zero and no instability develops.

The frequency spread is introduced by the machine non-linearities resulting from the magnets misalignment and mechanical errors. But the spread they provide is often not su cient to keep the beam stable. Dedicated octupole magnets called Landau octupoles are therefore installed in the machine. They provide a frequency spread which is function of the particles betatron oscillations amplitude [ , , ]. In the LHC, of these magnets are installed [ ]. The nominal current they can reach is A. Because their number and the current they can sustain are limited, they can only stabilize the beam up to a certain threshold.

The frequency spread needed from the octupoles can be predicted from the coherent mode frequency shifts [ ]. This frequency spread corresponds to a certain current in the octupole magnets. This simulated current can be compared to the value set during machine operation or the values found during dedicated measurements [ , , ].

The Transverse Mode Coupling Instability in the LHC

Figure . compares the results of instability threshold measurements and simulations in terms of

Landau octupoles current as a function of chromaticity. The results are for the horizontal plane of beam in the year . They show that a systematically higher octupole current than predicted from the impedance model was required to stabilize the beam. The results for a case without damper (in blue) and with the transverse damper activated with a damping time of turns (in orange) are presented. Those are compared to measurements for di ferent chromaticities, represented in the same color code. A dot denotes a measurement made at the end of squeeze, i.e when the beams are ready to be collided. A cross represents a measurement made a at-top, i.e after the end of the energy ramp and before the tune change. All results have been normalized to a bunch intensity of p.p.b.

Measurements during Run II ( -) and daily monitoring of transverse beam instabilities in the LHC during the year have shown that a factor two on the octupole current is required to ensure the beam stability [ ]. Figure . shows the octupole current used in LHC operation during Run II. It is compared to the one predicted with DELPHI simulations. The octupole current margin could be reduced over the years as the understanding of various instability mechanisms improved [ , ].

The margin remains however tight to cope with the bunch intensity increase foreseen in HL-LHC. The octupole current scales linearly with the bunch intensity, therefore a factor two on the bunch intensity would bring the operational scenario to the limit of available octupole current. No margin would be left in case new instability phenomena appear.

The factor two existing between simulations and measurements needs therefore to be investigated. Impedance e fects, including the mode coupling instability mechanism, are possible candidates to explain a part of this factor. We will now focus on impedance related simulations and measurements performed in the LHC. This will allow us to estimate the uncertainty on the impedance model. Operation at small positive chromaticities would however be preferred to optimize the beam lifetime and reduce the Landau octupole current. We will now investigate the mode coupling instability in the LHC with DELPHI simulations for di ferent chromaticities and collimator settings.

. . I ,

The TMCI simulations were performed with the LHC impedance model. The year indication is essential since the collimators gaps can be modi ed during the run to match the beam size reduction at the interaction points. The collimator gaps for the speci ed year were provided by the LHC Collimation team [ ]. The simulation parameters are similar to those presented in part . , only the beam parameters are di ferent to reproduce the real machine con guration. The parameters correspond to the at-top phase, before the tune change and are summarised in Table . .

The TMCI simulations presented in the previous chapter assumed a chromaticity corrected to units. However because of the uncertainty over the parameters set in the accelerator, the chromaticity has an error range of ± units [ ]. To guarantee that the beam remains stable, the chromaticity should be kept at a positive value. Therefore a measurement on the LHC should be performed with a chromaticity set to ∼ units to keep the beam stable. For the same purpose the transverse damper is also kept activated. The e fect of these two elements, chromaticity and damper, on the TMCI must be simulated for the LHC in order to plan a machine measurement. .

The Transverse Mode Coupling Instability in the LHC

. S

The rst simulations cover the case of Q = 0 without the transverse damper. Figure . a shows the mode frequency shifts and growth rates as a function of bunch intensity. One can see that the TMCI threshold is reached at an intensity of . × p.p.b.. Figure . b shows the same simulation but with the transverse damper activated with a damping time of turns. One can see that the mode frequency shift of the mode remains una fected by the damper for low intensities. At intensities closer to the TMCI threshold found before, modes andin uence each other but no strong coupling is found. These observations are in agreement with the ones made for the comparison of DELPHI and PyHEADTAIL in part . . These results were computed for the case of a chromaticity perfectly corrected to zero units. Since an uncertainty always remains on the operationally set value of chromaticity, we will now look at the e fect of a positive chromaticity on the TMCI threshold. . . R

. Simulations of the Transverse Mode Coupling in the LHC

To understand the e fect of chromaticity on the TMCI, the same simulations as in the previous part were performed with Q = 5, both without and with damper.

The Transverse Mode Coupling Instability in the LHC

We saw that a small positive value for the chromaticity doesn't change drastically the mode coupling behaviour at high bunch intensity. We now need to check the e fect of the transverse damper: The transverse damper, combined with the positive chromaticity, e fectively removes the head-tail instability. The mode frequency remains marginally a fected by the presence of the damper and the positive chromaticity compared to the reference case.

From the di ferent simulations presented, one can conclude that measuring the tune-shift as a function of intensity is a way to infer the TMCI threshold. Indeed the slope of the mode shift versus bunch intensity is minimally a fected by the inclusion of a positive chromaticity and the transverse damper: the di ference is within % which remains within the uncertainties of a tune-shift measurement in the LHC. Simulations also show that the machine currently operates with bunch intensities largely below the TMCI threshold. If one wants to try to reach the TMCI threshold, either the beam parameters have to be modi ed or the machine impedance should be increased.

From the beam parameters side, the synchrotron tune Q s can be reduced to decrease the TMCI threshold. We saw with the two particle model that the TMCI threshold value is proportional to

Q s N thres b = 8E 0 Cω β ω s πe 2 W 0 v 2 . ( . )
A reduction of Q s can be obtained by reducing the RF cavities voltage V RF . As Q s ∝ √ V RF , a reduction of the RF by a factor of would result in a TMCI threshold reduction of only %. However a sharp reduction of the RF voltage could a fect in turn the bunch length and increase non-linear e fects from the RF bucket. The measurement could then be a fected by this reduction of the RF voltage.

The previous equation shows that a simple way to decrease the TMCI threshold is to increase the machine impedance, here represented by the wake W 0 . This increase can be obtained by tightening the IR collimators gaps. As the resistive wall impedance of a at structure is proportional to g -3 where g is the structure gap, decreasing the collimator gaps by ≈ % would increase their impedance by a factor . The next part will detail the simulations results obtained for tighter settings of the LHC collimators.

. . R Simulations with tighter collimators gaps are based on the LHC impedance model with nominal gaps. The gaps settings are given in number of transverse beam size σ coll in the plane of collimation. Since the transverse beam size is di ferent at each collimator because of di ferent Twiss beta functions, the physical collimator gaps change from one collimator to the other. Figure . shows the gaps for three of the LHC collimators and for the di ferent scenarios investigated. The nominal LHC con guration in is displayed in blue, whereas the other colors show tighter collimators settings. In the tightest setting presented here the collimator gaps are reduced by ≈ %. The picture for all collimators gaps is reported in part B.

The LHC impedance simulations were then performed with these tighter gap settings. The resulting horizontal dipolar impedance as a function of frequency is shown in Fig. . . The collimators have a strong impact on the real part of the dipolar impedance in the MHz to GHz frequency range. For the imaginary part the impact is seen in an even broader frequency range, from kHz to GHz.

In this frequency range the impedance can be increased by up to a factor of by closing the primary and secondary collimators gaps to 4.0σ and 5.0σ. Thus the TMCI intensity threshold could be reduced by the same factor. To con rm the impact on transverse beam stability, DELPHI simulations were performed for these sets of tighter collimators gaps. Figure . b shows the results for the 4.0σ and 5.0σ con guration in the TCP and TCSG collimators. To obtain a clear TMCI e fect, the chromaticity is taken equal to Q = 0 and the damper is deactivated. It is again compared to the nominal reference case with nominal collimators gaps in Fig. . a.

The impact of the impedance increase caused by the tighter gaps in the collimators is clearly visible. In this tightest con guration the TMCI threshold is reduced to . × p.p.b.. This bunch intensity becomes reachable in the accelerator. The mode shift is also clearly a fected. A measurement of the The Transverse Mode Coupling Instability in the LHC LHC tune shift versus intensity for di ferent collimators settings could therefore provide more insight on the accuracy of the impedance model.

As for the previous study with nominal collimator gaps, a positive chromaticity is required because of operational uncertainties. The chromaticity is now at units and the transverse damper has a damping time of turns. As seen previously in the case with nominal collimators gaps, the tune shift is only slightly a fected for intensities below the new TMCI threshold of . × p.p.b.. The presence of both chromaticity and damper a fects however the mode coupling behavior. The fast instability characteristic to mode coupling disappears even for intensities above the threshold.

We saw that because of operational constraints and limitations, the machine setup used for TMCI simulations can not be exactly reproduced in measurements. However an observable can help infer the TMCI threshold value: the tune shift as a function of intensity. Measuring it can help to quantify the accuracy of the impedance model. This tune shift can also be increased by tightening the collimators settings, providing more data for an identical set of bunch intensities. A scenario with tight enough collimators gaps could also bring the TMCI threshold within the intensity reach of the machine. The next part will detail the measurement performed in the LHC, using the simulations detailed before. The Transverse Mode Coupling Instability in the LHC

. Simulations of the Transverse Mode Coupling in the LHC

. A LHC . . M TMCI LHC

We saw with simulations that the TMCI threshold in the LHC is predicted to be at an intensity of . × p.p.b. However this bunch intensity can not be reached in the machine. The measurements will therefore aim to infer the TMCI threshold. Sacherer's formula developed in part . . showed that at zero chromaticity, outside the mode coupling regime, the mode zero complex frequency shift ∆ω c,0 is proportional to the e fective impedance Z ⊥,ef f and the bunch intensity

N b ∆ω c,0 ∝ jZ ⊥,ef f N b . ( . )
The imaginary part of the mode complex shift will determine its growth rate and can be measured when the beam becomes unstable. The real part of the mode corresponds to the change of betatron frequency caused by the impedance. We saw in part . that this frequency change is small since it is in the order of Q s ω 0 . It can however be measured precisely by coherently kicking the bunch and recording its turn-by-turn oscillations. With a properly controlled kick, these oscillations will then be damped by the transverse feedback. The transverse tune can be found by performing a Fourier transform or applying the Harpy algorithm to the bunch transverse position, in the same way as for the processing of PyHEADTAIL data. In many machines the impedance at a given energy is xed and the induced tune shift can be measured by injecting bunches of di ferent intensities, as was done for example in the CERN SPS [ ]. If a bunch becomes unstable and its quality degrade, the quick turn-over time allows to re-inject a new bunch and perform a new measurement.

However, in the LHC, the turn-over time is in the order of the hour. Moreover the time for a measurement is usually limited to h since many di ferent studies have to be performed in the accelerator during the year. In this limited time, two energy ramps and their consecutive measurements can be accommodated. Measurement repeatability is therefore limited and the experiment has to be carefully set-up.

As seen in part . , the collimators have a strong impact on the transverse impedance. As they are movable devices, they can then be approached or moved away from the beam to increase or reduce the impedance. The collimators and bunch intensity parameters chosen for the experiment were based on the simulations detailed in the previous part and the machine limitations.

Moving the collimators allows to repeat the same measurement for di ferent machine impedances, however it limits the number of bunches which can be injected in the machine. The total beam intensity must remain below × p.p.b. if the collimators are moved from their nominal position at top energy. This constraint results from machine protection requirements and failing to keep the beam intensity within this limit would trigger a beam dump [ ].

An experiment to infer the TMCI threshold for di ferent collimator con gurations was proposed and executed in the LHC during the MD block of year [ ], in the night of the th of September. The steps taken during the MD were the following:

. Three single bunches of di ferent intensities were injected in both rings. The bunch buckets were chosen so as no beam-beam e fects would occur between the beam and beam bunches. The bunch intensities were . × , . × and . × p.p.b..

. The energy was ramped-up to . TeV, keeping the Landau octupoles at maximum current to provide a su cient tune-spread ensuring coherent stability.

. The beam parameters were set-up: chromaticity was reduced to Q ∼ 5. This value has been chosen to stay close to Q = 0 while remaining at a positive chromaticity. The transverse fractional tunes were kept at their injection values of . and . in the horizontal and vertical planes respectively.

. The bunches were coherently kicked in both planes. The turn-by-turn position of all bunches is then recorded and the data post-processed to obtain the tune [ ].

. The kicks were then repeated for the con gurations with tighter collimator gaps.

The time allocated for the measurement allowed to perform a second ramp for which the procedure was kept identical. Only the beam intensities used were di ferent, with two bunches of . × and . × p.p.b. instead of three bunches. Four di ferent collimator con gurations were used to better probe the impedance model despite a limited number of bunches available. The collimators parameters are given in Table . . ( . / )

The rst con guration presented in Table . used larger collimator gaps than the nominal con guration. The gaps were chosen to reproduce the equivalent impedance reduction foreseen for HL-LHC. The results obtained in this con guration will be further detailed in part .

We highlighted in the previous parts that the collimators have a strong impact on the impedance. It is therefore important to obtain the gaps which were present in the machine. Those are logged in the CERN Accelerator Logging System (CALS) [ ] during machine operation. Once retrieved, the machine impedance can be computed for these real gaps. In Fig. . the gaps retrieved from the CALS are compared to the ideal ones resulting from the scaling of the physical collimator gaps with the number of sigmas used for the con guration. The di ference between the predicted gaps and measured gaps is small but visible. The measurement results will therefore be compared to the results of stability simulations performed with the impedance models derived from the real machine collimator gaps. Collimator name

The Transverse Mode Coupling Instability in the LHC

(5/6.5), predicted gaps

(5/6.5), measured gaps

(5/6.0), predicted gaps

(5/6.0), measured gaps (4.5/6.0), predicted gaps (4.5/6.0), measured gaps Figure . : Physical gaps of three of the LHC IR collimators in mm for the con gurations given in Table . . The collimator gaps measured in the machine are compared to the one resulting from a scaling with n σ,coll of the gaps and which were used for instability predictions described in the previous part.

The di ference between the gaps measured in the machine and the predicted ones are of the order of %. The picture for all collimators gaps is reported in part B.

Once the top energy was reached, the tune measurement relied on the transverse damper (ADT). The bunches were coherently kicked with the ADT. The excitation strength was chosen to obtain clear bunch oscillations without provoking intensity losses caused by scrapping on the collimators. The damper gain was also reduced to obtain a longer decoherence time. An excitation was applied to both horizontal and vertical planes, with a delay of turns between the two, and all the bunches were coherently kicked at the same time. Their oscillation signal was recorded using the ADTObsBox [ ], a system which can acquire the bunch-by-bunch and turn-by-turn transverse position of the beams using the ADT stripline pick-ups. An example of a set of acquired signals is given in Fig. . where the vertical axis unit is proportional to the bunch transverse position. The kicks were then repeated multiple times for each beam, spaced by seconds. The transverse tune is then computed for each kick signal using a procedure similar to the one described in part . . on the post-processing of PyHEADTAIL data.

Figure . shows ring collimators gaps and beam parameters as well as the three bunches intensities, full lengths and vertical emittances over time. The times at which the kicks were sent are highlighted as well. The procedure started with the ADT setup, then a rst set of kicks was applied. The collimators gaps were then tightened for a second measurement set, and again for a third set of kicks. The beam intensity losses, emittance growth and bunch lengthening were small during the tune measurements thanks to the controlled kick strength. A beam instability was induced at the end of the measurement by closing the gaps to tighter settings, leading to intensity losses and bunch lengthening.

Figure . shows the data processing steps made once the tune has been computed for each ADT signal. In the top plot all the vertical tune signals for the three bunches are represented. For each collimator con guration the values are then averaged. Knowing the individual bunch intensity over time, the tune shift as a function of intensity can be computed.

Figures . and . comparison of the two lls data and the di ferent collimators con gurations. The points represent the average tune values whereas the lines show the tune shifts predicted from the impedance model.

The tune shifts as a function of intensity are reported in Table . . They have been normalized to the accelerator synchrotron tune Q s = 1.838 × 10 -3 .

We saw in the TMCI study with PyHEADTAIL and DELPHI done in part . . that the quadrupolar impedance a fects the tune shift. As DELPHI models only the dipolar impedance contribution, a correction should to be used to account for the quadrupolar impedance e fect. The correction is based on Sacherer's formula. First the tune shifts induced by the dipolar and quadrupolar impedances, noted ∆Q dip and ∆Q quad , are computed separately. The ratio

∆Q dip +∆Q quad ∆Q dip
is then computed and applied to DELPHI simulations. The results with this correction factor are also reported in Table . . These measurements can then be used to infer the TMCI threshold. As we saw in the simulations from part . this instability results from the coupling of mode and -. The mode -frequency is weakly perturbed, therefore the TMCI threshold is approximately the intensity at which the real part of mode and mode -complex frequencies cross. From the normalised tune shift versus intensity ∆Q/Q s , the TMCI intensity threshold N thres b is approximately found at

N thres b ≈ - 1 ∆Q/Q s . ( . )
The thresholds inferred from the measurements presented in Table . are reported in Table . . The third column shows the TMCI threshold obtained with DELPHI simulations by reading the intensity value at which the growth rate becomes non-zero. The fourth and fth columns give the threshold computed from Eq. . and using the tune shift obtained with DELPHI and the tune shift corrected for the quadrupolar impedance contribution. The last column shows the TMCI threshold inferred from the measurements.

The average ratio between measurements and simulations is calculated for the two beams and planes. The simulation results are from DELPHI with the quadrupolar tune shift e fect taken into account via the correction factors. The results are reported in Table . and show that the tune shifts at top energy are underestimated by ∼ % to %.

. Assessment of the LHC impedance and stability limits with beam based measurements The bottom plot shows the average tune for each bunch in the three collimators con gurations.

Table . : Measured and simulated tune shifts as a function of beam intensity for the LHC nominal collimator settings and tighter con gurations. The third column reports DELPHI simulations results. The fourth column is the correction factor to account for the quadrupolar impedance e fect and the fth column is the simulation results with the correction factor applied.

Tune shift / 10 12 p.p.b.

× Q s -1
Conf. Sim. Correction / % Sim. w/ quad. Measured B H ( / . ) -.

-.

-.

-. ( / . ) -.

-.

-.

-. B V ( / . ) -.

-.

-.

-. ( / . ) -.

-.

-.

-. B H ( / . ) -.

-.

-.

-. ( / . ) -.

-.

-.

-. ( . / . ) -.

-.

-.

-. B V ( / . ) -.

-.

-.

-. ( / . ) -.

-.

-.

-. ( . / . ) -.

-.

-.

-.

The Transverse Mode Coupling Instability in the LHC The Transverse Mode Coupling Instability in the LHC

. . M

The measurement of tune shifts versus intensity detailed beforehand allows to compare the total machine impedance to the model. The e fect of the collimators on the tune shift was clearly shown by closing further their gaps. It is therefore interesting to study the individual collimator contribution to the overall machine impedance budget. As seen previously, the LHC collimators can be moved from their nominal position, allowing to change the machine impedance at will. The position setting of a single collimator can be modi ed and its impact on the machine transverse tune measured. Multiple machine development sessions were carried out during the years , and to measure primary, secondary and tertiary collimators [ , ]. The measurement principle is similar to the one carried out during the TMCI measurement described in part . . . For some cases in , the tune kicker (MKQA) was used [ ] instead of using the ADT to excite the beams. The collimator to be measured is further closed, then a series of kicks is applied. The collimator gap is then opened and a new series of kicks is made. This process is repeated one more time. Figure . shows the TCSG measurement process during the dedicated machine development. In this case seven di ferent secondary collimators were measured. The same process was used to measure primary and some of the tertiary collimators. Table . reports the names, location, and collimation plane of the collimators which were measured during Run II. Figure . also shows the beam parameters during one of the machine developments. Signi cant bunch intensity losses caused by too strong kicks a fected the measurement. The full bunch length shrinks down and the tune drifts upwards over time. However the e fect of the collimator impedance on the tune can still be observed, as detailed in Fig. . . The tune is changed when opening and closing the collimator jaws. Similarly to the TMCI measurement data processing, an average tune value is computed for each period of time when the gaps were opened and close. The di ference between the closed position and opened position can then be calculated and compared to simulations results.. The measurements and simulations results for the collimators impedance measurements are reported in appendix C. Figure . shows the ratios between measurement and simulation of the single collimator . Assessment of the LHC impedance and stability limits with beam based measurements induced tune shifts. The ratios are within a factor two for most of the collimators, which is consistent with the results of the full machine tune shift measurements. .

The Transverse Mode Coupling Instability in the LHC

The Transverse Mode Coupling Instability in the LHC

. M

The measurements presented beforehand investigated the machine when it is in the most critical phase stability-wise. These measurements are time consuming and the limited periods dedicated to machine studies in the LHC reduce the measurement repeatability. The injection set-up, the energy ramp, the beam parameters and equipment con guration once the top energy is reached reduce considerably the quantity of data which can be acquired. The number and intensity of bunches present in the machine must also remain below a certain limit to protect the superconducting magnets and machine equipment. On the other hand experiments performed at injection energy can pro t from a quick re lling time either to reproduce a measurement or to scan a certain parameter space.

A measurement of the single bunch instability growth rate as a function of chromaticity was planned and executed. The measurement took place during the MD block of , on the afternoon of the th of September. With nominal bunches at the injection energy, the horizontal chromaticity was trimmed in the negative range, keeping the vertical chromaticity positive. The ADT was then switched o f to let an instability develop. The turn-by-turn position signal was then recorded over turns with the ADTObsBox. From these signals the instability rise time was found by tting an exponential function.

The experiment was repeated for both beams and planes, and the results compared to DELPHI simulations using the LHC impedance model at injection. Figures . and . show the results for the horizontal and vertical plane of both beams. The impedance model for beam and beam being similar, only one set of DELPHI simulation is presented. The results are within a factor two from the predictions for both beams and planes, with the exception of the vertical plane of beam . In this case at large negative chromaticities the rise time is larger than

The Transverse Mode Coupling Instability in the LHC

. . S ,

The results of the various stability measurements performed in the LHC during Run II can be summarized by computing the ratio between measurements and predictions from the impedance model. Those were shown for the TMCI measurements, the individual collimators measurements and the instability growth rate at injection energy. Thanks to the di ferent measurements methods, the impedance was investigated from di ferent sides: individual elements and full machine, imaginary part and real part, injection and top energy. From

. the factor between measurements and simulations can be estimated to be between % and %. This value can be compared to the factor two on the Landau octupoles current required to stabilize the beam: missing elements to the the impedance model need to be found. However the impedance is not the only contributor to the discrepancy. Studies are currently ongoing to investigate potential e fects detrimental to beam stability: external noise a fecting the transverse distribution [ ], transverse damper [ ] and its interplay with Landau damping [ ] among others.

T T M C I

We investigated in the previous chapter the mode coupling instability in the LHC. Measurements performed in the machine also allowed to estimate the uncertainty on the accelerator impedance. We concluded from these measurements that the impedance is approximately % to % higher than predicted with the current model.

As the High-Luminosity upgrade of the LHC will use bunches with an intensity increased by a factor of two compared to the current value, impedance will be one of the critical aspects of the project. The mode coupling instability in the HL-LHC will be investigated and the impact of the impedance reduction assessed. We will rst present the scope of the collimation upgrade and the simulations results for di ferent scenarios in part . . Measurements performed in the LHC to showcase the bene ts of the impedance reduction will then be presented in part . . One of the measurement focused on the full machine tune shift whereas the other characterised the impedance of a prototype collimator for the collimation upgrade.

We will conclude the chapter with an overview of the impedance and stability estimates realised for the High-Energy LHC study. The HE-LHC is a proposed collider which could succeed the HL-LHC and re-use the existing accelerator infrastructures. Part . will describe the project and the impedance and beam stability studies performed.

. S T M C I H L LHC

The impedance reduction for HL-LHC will target the IR betatron cleaning collimators since they are the main contributors to the impedance at top energy as we saw in part . . . Selected collimators among the three primaries and eleven secondaries will be replaced. The overall design of the collimators will remain similar, but the jaws materials will be changed. During the Long Shutdown in and , two primary collimators (TCP) per ring will be replaced in the framework of the LHC consolidation project. Four secondary collimators (TCSG) per ring will also be replaced by low impedance ones in the framework of the HL-LHC collimation upgrade [ , ]. The remaining seven TCSGs per ring will be replaced during the Long Shutdown in and . The new collimators will use a molybdenum-graphite composite [ , ] instead of carbon-ber reinforced carbon for the jaw materials. The upgraded primary collimators, the TCPPM , will use the molybdenum-graphite (MoGr) blocks over their cm length. The new secondary collimators, the Here the M stands for a metallic material used for the jaws and the second P indicates that a beam position monitor is embedded in the collimator for orbit control.

The Transverse Mode Coupling Instability in future machines TCSPM, will use molybdenum-graphite for the jaws materials and the blocks will also receive a µm coating of pure molybdenum (Mo) [ ].

These new materials were chosen to meet the challenges presented by both the impedance reduction and the machine protection. Their resistivities are reported in Table . . The MoGr bulk material has a resistivity reduced by a factor compared to the current jaw materials and the additional Mo coating would provide an other factor ∼ reduction for the jaw resistivities. We saw in part . . that the resistive wall transverse impedance Z ⊥,RW scales as ) with δ skin the skin depth at the considered frequency. This simpli es in turn to ) where ρ c = 1/σ c is the electrical resistivity. Therefore the use of MoGr will reduce by a factor . the transverse resistive wall impedance of the collimators. The Mo coating would provide a factor of ∼ reduction on the resistive wall impedance compared to the current collimator design. Further data on these materials can be found in [ , ] TiN stands for titanium nitride. This material is a possible alternative to the Mo coating for the MoGr blocks.

Z ⊥,RW (ω) ∝ δ skin (ω) , ( .
Z ⊥,RW (ω) ∝ √ ρ c , ( .
The Landau octupole current is a limiting factor for coherent beam stability as we saw in part . . The impedance reduction will allow to preserve stability margins in HL-LHC with the higher intensity beams [ , ]. We will now study the impact of the collimation upgrade on the transverse mode coupling instability. Table . shows the di ferent scenarios which were studied. Two additional cases in which the Mo coating of the secondary collimators is not present were also studied. The HL-LHC scenarios are compared to two scenarios with the LHC impedance model: the rst one with the nominal collimator gaps and the second one with the secondary collimators opened to 14σ coll to simulate an impedance reduction.

The collimator gaps used in the di ferent simulation scenarios are reported in appendix B. . The horizontal dipolar impedance as a function of frequency is plotted in Fig. . for the di ferent cases. A clear impedance reduction with respect to the LHC nominal case can be observed. The nal upgrade with all the TCSGs made of molybdenum coated molybdenum-graphite provides a reduction by a factor ∼ over the kHz to GHz frequency range. The LHC scenario with the TCSGs opened The scenarios were used to study the transverse mode coupling instability in HL-LHC. The beam parameters used for the simulations are similar to those used for the TMCI simulations in the LHC and are therefore reported in appendix B. . The results for the horizontal plane of beam will be reported since it is the most critical plane from the stability point of view [ , ]. Figure . shows DELPHI results obtained for the LHC scenario and the HL-LHC nal scenario with molybdenum coating. For the latter case the coupling of modes and -still occurs but at an intensity of . × p.p.b. instead of . × p.p.b.

To facilitate the comparison of the di ferent scenarios, the mode shift is linearly tted in the low intensity part as showed in Fig. . . The TMCI intensity threshold can be obtained with the instability growth rate. Figure . shows the linear ts of the mode real part and the instability growth rate for the di ferent scenarios.

From these linear ts the tune shifts versus intensity can be computed. They are reported in Table . . The rst stage implementation of the collimation upgrade during the LS increases the TMCI intensity threshold by a factor . , from . × p.p.b. to . × p.p.b. The nal stage of the upgrade with the eleven secondary collimators coated in molybdenum increases the TMCI threshold by a factor . compared to the current LHC case. For all these scenarios the TMCI threshold reaches a value higher than the maximum bunch intensity of . × p.p.b. planned for HL-LHC. The molybdenum coating used for the secondary collimators helps to increase the TMCI threshold by % for the LS upgrade and by % for the nal upgrade with respect to the uncoated versions of the collimators. The tune shifts versus intensity are similarly decreased for the di ferent investigated cases.

We saw with simulations that the collimation upgrade of the LHC will increase the TMCI threshold and reduce the machine tune shift. These will provide margins to operate the machine at lower chromaticity. The bene ts of the impedance reduction will now be investigated with beam based measurements performed in the LHC. . Study of the collimation upgrade impact with beam measurements . S

. . M

We saw in the previous part that the impedance reduction planned for HL-LHC will have a visible impact on the coherent beam tune shift. To demonstrate the bene ts of the impedance reduction targeted at the collimators, a measurement was performed in the LHC by opening the secondary collimators gaps. Simulations detailed in the the previous part showed that opening the secondary collimators of IR to 14σ coll reduces the machine impedance to an intermediate level between the nominal con guration and the LS upgrade. This con guration was measured during the TMCI machine development detailed in part . . . The measurement procedure and the data post-treatment method are therefore identical. Figure . shows the results for both planes of beam and Fig. . the results for beam . The mode frequency shift predicted by DELPHI has been corrected for the quadrupolar impedance contribution. In all cases the tune-shift is reduced compared to the nominal machine con guration.

The resulting tune shifts as a function of intensity are reported in Table . and the corresponding TMCI intensity thresholds are reported in Table . . The tune shift is reduced by a factor of ∼ . in the horizontal plane and by a factor of ∼ in the vertical plane. The TMCI threshold inferred from these measurements is increased by the same factors compared to the nominal LHC case. -.

-.

-. ( / . ) -.

-.

-.

-. B V ( / ) -.

-.

-.

-. ( / . ) -.

-.

-.

-. B H ( / ) -.

-.

-.

-. ( / . ) -.

-.

-.

-. B V ( / ) -.

-.

-.

-. ( / . ) -.

-.

-.

-.

This tune shift versus intensity measurement highlights the predominant role of the collimators in the impedance model. They support the impedance reduction strategy for the HL-LHC collimation upgrade and the bene cial e fect it will have for coherent beam stability. .

The Transverse Mode Coupling Instability in future machines

(∆Q/Q s ) DELPHI simulation ( σ/ σ) ( σ/ . σ) (a)

. M

To further validate the choice of low impedance collimators for the HL-LHC collimation upgrade, a prototype collimator was installed in the LHC during the -winter shutdown [ ]. It was positioned in a spare slot of beam adjacent to the TCSG.D R [ ]. This collimator, the TCSPM.D R , is a vertical collimator. Unlike the production version, this prototype does not have fully molybdenum coated jaws. As showed in Fig. . , the jaws have three di ferent materials: a molybdenum coating, the molybdenum-graphite bulk and a titanium-nitride coating. A motor can shift horizontally the jaws and expose a di ferent stripe to the beam.

This prototype collimator was designed to validate with beam measurements the choice of materials for the collimation upgrade. The goal was to measure the tune shifts induced by the three di ferent materials and compare the results to predictions from the impedance model. The adjacent collimator, the TCSG.D R was also measured during the procedure.

A machine development session took place on the th of June and st of July . The measurement procedure applied was identical to the one described in part . . on the single collimator tune shift measurements. The beam was kicked with the transverse damper and the oscillations recorded with the ADTObsBox. As showed in Fig. . a a series of kicks was performed for ∼ min and the collimator gap was then changed. The procedure was repeated for a few cycles of the collimator gap. However the machine tune is also jittering with time. In the TCSPM case the tune jitter is comparable to the impedance induced tune shift. The measurement method was therefore slightly modi ed to bypass the tune jitter e fects and reach the desired precision.

For this second attempt the collimator gap was changed in quick successions as showed in Fig. . b. The beam was kicked in quick series with the ADT, decorrelating the kicks from the collimator gap. The tune jitter is still present as can be seen in Fig. . b but two distinct lines appear as well in the time evolution of the tune. The upper one corresponds to kicks which were sent when the collimator gap was wider, the lower one to kicks applied when the gap was tighter.

The tune jitter e fect could be removed and the tune shift induced by the TCSPM di ferent materials computed [ ]. This improved method allowed to reach a tune shift measurement precision in the order of a few × -. The procedure was repeated for di ferent lower positions of the collimation gap: 3.5σ coll , 4σ coll , 4.5σ coll and 6σ coll . The subsequent data treatment is further detailed in [ , ]. Figure . shows the tune shift as a function of the collimator gap for the di ferent considered materials. From the resistive wall impedance model the tune shifts should scale in 1/n 3 σ . The measured tune shifts include both the resistive wall impedance and the geometric impedance from the collimator elements. The contribution of the geometric impedance was therefore subtracted from the measured tune shifts [ , ]. The measurements are compared to simulations results.. For carbon-ber reinforced carbon, molybdenum-graphite and titanium-nitride, measurements are lower than predicted whereas for the molybdenum coating measurements are two times higher than the prediction.

Despite the discrepancy found between the model and measurements, the molybdenum coating on molybdenum-graphite jaws still provides the largest impedance reduction of all the investigated materials. Measurements of the electrical properties of molybdenum coatings are ongoing to systematically check and validate the production series of molybdenum coated blocks [ , , , , ]. .

The Transverse Mode Coupling Instability in future machines

I H E LHC

We will now investigate the impedance model of the High-Energy LHC (HE-LHC) which was derived from the HL-LHC one. A quick overview of the Future Circular Collider Study and the HE-LHC impedance model at injection and top energy will be detailed in part . . . We will then use this model to study the mode coupling instability in part . . .

. . I HE-LHC

The High-Luminosity LHC is planned to operate until [ ]. To follow it, two collider design studies have been launched at CERN: the Compact Linear Collider (CLIC), a multi-TeV linear electronpositron collider [ ] and the Future Circular Collider (FCC). In the FCC study three di ferent machines have been considered:

• The FCC-ee, a km long electron-positron accelerator with collision energies ranging from GeV to

GeV [ ].

• The FCC-hh, a TeV hadron collider ( TeV per beam) which could follow the lepton collider, re-using the same infrastructures. High eld magnets would be needed to reach the target energy [ ].

• The High Energy LHC, a TeV hadron collider which would re-use the LHC infrastructures with the high eld magnets of FCC-hh to reach the collision energy [ ].

The hadron colliders rely on high eld magnets to obtain the desired energies. These magnets could reach a magnetic eld of T with NbSn 3 superconductors instead of the NbTi technology used in the LHC [ ]. The machine protection requirements will become even more stringent as the beam energy increases. These requirements will in turn pose challenges for coherent beam stability from the very beginning of the machine design [ , ].

The HE-LHC transverse impedance was derived from the HL-LHC impedance model [ ]. The same optics function as in HL-LHC were used but the beam screen was replaced by the FCC-hh one, pictured in Fig. . . This new beam screen design must cope with an increased synchrotron radiation power and the subsequent vacuum and cooling constraints [ ] while keeping its impedance as low as possible.

Three injection energies options are considered for HE-LHC. The rst at GeV would allow to reuse the SPS as an injector without further modi cations. The second and third options at GeV and . TeV would require to replace the existing SPS by a superconducting machine. These two options are however more favorable in terms of available aperture for the beam injection [ ]. For the impedance study the GeV and the . TeV injection energies will be investigated. For the collimators, the HL-LHC collimation layout was used. The collimator materials were assumed to be those of the fully upgraded HL-LHC which was detailed in part . . The collimator gaps have been scaled with the beam energy and reference emitttance. Because of the higher energy, the transverse beam size will be reduced compared to LHC and HL-LHC and therefore the collimator gaps will also be smaller. The gap settings, their physical gaps and the beam parameters for stability simulations are reported in appendix B. and B. . GeV option, even if the beam size is unchanged compared to HL-LHC, the impedance is higher because the collimators must be set closer to the beam to protect the magnets.

The Transverse Mode Coupling Instability in future machines

At top energy the impedance increases by a factor compared to HL-LHC in the kHz to GHz frequency range. With a single bunch intensity for HE-LHC similar to the HL-LHC one, . × p.p.b., the stability margins in HE-LHC must be investigated and mitigation techniques proposed.

. Impedance and beam stability considerations for the High Energy LHC study

10 -1 Z dip x / M Ωm -1

HL-LHC nal injection LHC injection HE-LHC injection

GeV HE-LHC injection . TeV 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10

Frequency / Hz

10 -1 Z dip y / M Ωm -1 (Z dip y ) (Z dip y )

. . M HE-LHC

The impedance models were used to estimate the stability limits at the di ferent energies. We will highlight here the results for the mode coupling instability at Q = 0. Additional results on the Landau damping requirements and multi-bunch stability can be found in [ , , , ].

The horizontal instability growth rate as a function of intensity for the di ferent considered energies are presented in Fig. The top energy is the most constraining case from the beam stability point of view. The TMCI threshold is found at × p.p.b. in the vertical plane, a value below the nominal bunch intensity. Using the transverse damper mitigates the mode coupling instability but as we saw in part . . , a slow instability then appears below the threshold. Chromaticity and Landau damping from octupole magnets can be used to stabilize the beam and mitigate these single bunch instabilities.

However the beam stabilization with Landau octupoles become more challenging as the beam energy increases [ ]. The e fective frequency spread provided by the octupoles scales as 1/γ 2 . An HL-LHC type octupole would therefore be ∼ times less e cient with an HE-LHC beam. Novel instability mitigation techniques are therefore being considered. For example an electron lens would create a frequency spread from the electromagnetic interaction of the proton beam with a high brightness low energy electron beam [ , , ]. An RF quadrupole is an other option to increase Landau damping [ , ]. While the octupole magnets create a frequency spread dependent on the transverse actions of the beam J x and J y , an RF quadrupole generates a spread dependent on the longitudinal action J z . For higher energies the transverse actions are much smaller than the longitudinal one: for example in HL-LHC there is a factor ∼ between the two [ ].

We saw with the High-Energy LHC study that beam coupling impedance poses a challenge for beam stability in future high energy machines. New and robust mitigation techniques will be needed to maintain stability margins and allow for optimal machine operation.

The Transverse Mode Coupling Instability in future machines

C

This work presented advancements on the understanding of transverse impedance induced instabilities in the Large Hadron Collider. In particular the Transverse Mode Coupling Instability was investigated for the current and future large colliders at CERN.

We rst validated the simulation codes for the study of coherent beam instability in the LHC and highlighted their limitations. This study was performed for two types of instabilities, the head-tail and the mode coupling ones. The e fect of the transverse damper was also included in the simulations. Di ferent impedance models were used, and in particular the LHC impedance model. The results showed good agreement between the PyHEADTAIL tracking simulations and the Vlasov solver DELPHI.

The mode coupling instability was then investigated for the LHC with simulations and measurements. Simulations for di ferent machine con gurations highlighted the predominant role of the collimators on the machine impedance and the mode coupling instability. The TMCI intensity threshold was predicted to be found at × protons per bunch, more than two times the single bunch intensity used during Run II ( -). Measurements of the tune shift as a function of intensity were performed at the LHC top energy. Di ferent collimator con gurations were tested to modify the machine impedance. From these measurements the impedance induced tune shift was found to be higher than predicted from simulations by % to % depending on the beam and plane. Despite the larger tune shifts with respect to predictions, the TMCI intensity threshold was still higher than the used bunch intensities.

The impedance at top energy of some single collimators were also measured at multiple occasions during Run II. This measurement method and the subsequent data processing allowed to measure the impedance induced tune shifts to a level of a few 10 -5 . The results for the di ferent collimators are within a factor of two from the impedance model predictions.

Measurements at top energy in the LHC pose time and machine protection constraints. A measurement of the head-tail instability was therefore performed at the injection energy to scan a larger chromaticity range. The results obtained with this method are also within a factor of two from predictions.

We concluded from the combination of the di ferent measurements that the LHC model underestimates the machine impedance by a factor of . to . . During Run II, the LHC was operated with at least two times more Landau octupole current than predicted from impedance and instability simulations. Missing impedance sources can therefore explain a part of this factor, but other destabilizing mechanisms need also to be investigated.

For the High-Luminosity LHC project, the impedance will be a critical aspect. The beams will be two times more intense than the design value. To cope with this brightness increase and maintain margins for coherent beam stability, an impedance reduction of the collimators is planned. We saw with simulations that in the nal stage of the HL-LHC upgrade the mode coupling instability threshold is at . × protons per bunch, a value four times higher than the projected single bunch intensity. Intermediate scenarios corresponding to the staged implementation of the upgrade and to di ferent Conclusion collimator jaws materials were also simulated. They all showed an increase of the TMCI threshold, but still smaller than the one obtained with the full upgrade.

The bene cial e fect of the collimators impedance reduction was also demonstrated with measurements in the LHC. A tune shift measurement with a set of opened collimators was performed. The machine impedance was reduced to a level similar to the one that will be reached during Run III ( -). We saw that the tune shift versus intensity could be reduced with this con guration. Moreover the impedance of a single collimator prototype was also measured with beam in the LHC. This prototype was installed to con rm the choice of new low resistivity metallic materials for the collimation upgrade. The bene cial e fects of these low resistivity materials compared to the current carbon based collimators has been shown.

Finally, impedance and mode coupling instability simulations were performed for the High-Energy LHC, a proposed future collider. We saw that because of the machine protection constraints, the impedance can increase by a factor up to ten over a large frequency range compared to current colliders. This results from the tight collimator gaps required to protect the machine. In consequence coherent stability margins will be much lower than in the LHC and HL-LHC. Novel beam stabilization techniques should therefore be investigated.

A I A. I

The LHC makes use of many collimators to protect the cold magnets from particle losses. Because of their tight gaps, they contribute to a sizeable portion of the beam coupling impedance at top energy. These collimators also have di ferent orientations, gaps and Twiss beta functions.

Because of these di ferences in optics and gaps, the quadrupolar impedance might increase or decrease the tune shifts, assuming that those can be derived from the sum of the dipolar and quadrupolar components of the impedance.

We will derive expressions for the impedance of a combination of two at structures, the rst one horizontal and the second one vertical. This structure approximate the case of two LHC collimators. We will rely on simple geometric considerations and on Yokoya factors [ ] for at structures to derive these simple expressions.

A. S

Let S be a source particle of coordinates (x S , y S ) and T a test particle with coordinates x T , y T as was pictured in Fig. . . For a at, multilayered, axisymmetric structure, the wall impedance can be written

Z x = Z dip x x S + Z quad x
x T (A. )

Z y = Z dip y y S + Z quad y y T (A. )

We now assume that we dispose of two identical at structures, the rst with horizontal plates and the second with vertical plates. They will be referred as 1 and 2 throughout the paper.

The second structure impedance components are then

Z dip x,2 = Z dip y,1 , (A. )
Z quad x,2 = Z quad y,1 , (A. )

Z dip y,2 = Z dip x,1 , (A. )

Z quad y,2 = Z quad x,1 . (A. ) (A. )

A Impedance of two collimators with di erent orientations Moreover the structure being at, we have the following relations between dipolar and quadrupolar impedances [ , ] Z dip

x,1 = -Z quad x,1 , (A. )

Z dip y,1 = 2Z quad y,1 , (A. )

Z dip x,1 = Z quad y,1 , (A. ) (A. )

We can now sum the impedances of structures 1 and 2

Z dip x,tot = β x,1 β x Z dip x,1 + β x,2 β x Z dip x,

A. C T

We now take the case of a combination of the two identical collimators described in part A. . We will assume that the average horizontal and vertical betatron function β x and β y are equal to 1 in all the following examples. In this rst example we put the β functions equal to 1 for both planes and in both collimators. The quadrupolar impedance is now zero for both planes and the dipolar impedances are three times the single vertical collimator of Sec. A. .

A. C T

We now take the case of di ferent β functions in both collimators. Figures A. and A. show two di ferent cases. In the rst case, the horizontal quadrupolar impedance has an opposite sign to the dipolar impedance. In consequence the tune shift induced by the dipolar impedance will be reduced if we take the quadrupolar impedance into account. This is can also be seen in the vertical impedance.

Changing the β functions of the second collimator can lead to a very di ferent situation. . for the LHC impedance and wake model showed a discrepancy at high bunch intensity between the codes DELPHI and PyHEADTAIL. The convergence of the simulations was therefore checked with the two approaches.

D. S

PyHEADTAIL

As explained in part . . , in PyHEADTAIL the bunch is divided in several thousand of macroparticles.

When applying the e fect of wake elds, the macroparticles are grouped in longitudinal slices. All macroparticles within a slice receive the same kick from the wake eld. The default slicer used for stability simulations with PyHEADTAIL is the UniformBinSlicer. With this function all slices have the same length and the number of macroparticles per slice varies according to the bunch charge distribution. This absence of macroparticles and the charge di ference between slices can introduce numerical noise in the simulations. Because of the Gaussian shape of the longitudinal distribution, increasing the number of slices has little e fect on the number of macroparticle in the edge slices.

The PyHEADTAIL code also includes a UniformChargeSicer module. In this case each slice will have a di ferent length but they will all contain the same number of macroparticles. Figure D. shows the slicing of the same bunch with this slicer. The top plot shows that indeed all slices have the same number of macroparticles. However slices at the center of the distribution are much shorter that those located at the edges of the distribution to ensure the uniform density.

D E ect of beam distribution slicing in PyHEADTAIL simulations

D. E LHC

To try understand the discrepancy found between DELPHI and PyHEADTAIL simulations showed in part . . , a convergence study was made rst using the UniformBinSlicer in PyHEADTAIL. The number of bins used for the longitudinal slicing was scanned from to with few steps. Figure ?? shows the results for this rst scan: the most unstable mode growth rate is plotted as a function of bunch intensity. The curves correspond to the di ferent number of slices used.

The results of the convergence study with the UniformBinSlicer don't show any improvement compared to the bin case. The UniformChargeSlicer was then used in PyHEADTAIL. Figures D. to D. show the resulting mode frequency shifts and most unstable mode growth-rate for to bins. The results are compared to DELPHI simulation results.

When the number of slices is increased, the results start to converge towards DELPHI simulation results. Using more than bins with this slicer type appears to ensure the results convergence.

D. E ect of slicing on the LHC stability simulations

Figure . :

 . Figure . : The CERN accelerator complex in August [ ]. The circumference and rst year of operation of each accelerator are indicated, as well the particle species which can be accelerated.

  . The LHC was designed to host bunches, with . × protons per bunch (p.p.b.)[ ].

Figure

  Figure . : The LHC beam injection process. Batches of proton bunches are rst accelerated in the PSB and ejected towards the PS. Once the PS contains two of these batches, they are accelerated and transferred to the SPS. This process is repeated until the SPS contains four of these PS trains of bunches. These four trains contain in total bunches which are then accelerated to GeV and injected in one of the two LHC rings. Picture from [ ].

  Figure . : The LHC cycle as a function of time. The start of the energy ramp is taken as the time reference.Before this ramp, the machine is con gured for injection and then the trains of bunches are injected from the SPS. Once the ramp is nished, the beams are focused to reduce their size at the experiment points and are then adjusted in collision. After ∼ h of collisions, the beams are dumped and the magnets are ramped down to start a new cycle.

  .

  Figure . : Coordinate system used to derive the equation of single particle motion. The gray circle represents the machine circumference on which the reference particle circulates. The coordinate system is de ned with respect to this particle.

Figure . :

 . Figure . : Coordinate system used for the source and test particles traveling through an accelerator element.

Figure . :

 . Figure . : Left plot represents the resistive wall horizontal dipolar wake function versus time after the source particle, obtained for a cylindrical copper beam pipe. The right plot represents the corresponding beam coupling impedance as a function of frequency.

  Fig. . b. They allow reaching equivalent pressures in the -Pa to -Pa range, ensuring a beam lifetime of ∼ h [ ].

Figure . :

 . Figure . : Transverse dipolar beam coupling impedance of the LHC as a function of frequency. The top plot shows the horizontal impedance and the bottom plot the vertical impedance. The impedance at injection energy is plotted with light colors and the one at top energy with dark colors.

Figure . :

 . Figure . : Contribution to the horizontal dipolar impedance of the di ferent accelerator elements at top energy versus frequency. The top plot shows the real part of the impedance whereas the bottom plot shows the imaginary part.

  Figure . schematizes the multistage cleaning principle used in the LHC [ ], and Fig. . shows the location along the two beams of the collimators.

Figure . :

 . Figure . : Top view of an LHC collimator during assembly. The beam trajectory in the device is represented with the blue arrow. In this picture only the left jaw has been installed. It is made of a CFC block mounted on stainless steel frame. Once the two jaws are assembled, the beam passes in a tight gap and the particles in the transverse beam halo are intercepted by the jaws. Picture from [ ].

Figure . :

 . Figure . : Principle of the multi-stage beam cleaning used in the LHC [ , ].

Figure . :

 . Figure . : Layout and names of the collimators installed in the LHC, for both ring (blue) and ring (red).

  Fig. . b.It is kicked by the wake as long as it remains behind MP . At turn , the two macroparticles have just exchanged their longitudinal position.

  Figure . c shows that MP is now at the head and is the one

Figure . :

 . Figure . : Transverse and longitudinal position x and z of the two macroparticles at di ferent periods of the synchrotron oscillation cycle. They perform a full synchrotron oscillation in turns.The total bunch intensity in this case is × p.p.b., each particle has half the bunch intensity . × p.p.b. The kick felt by the trailing particle is caused by the wake generated by the head particle. It is depicted here in a solid line the same colour as the macroparticle generating it.

  Figure . : Longitudinal position of the two macroparticles as a function of turn number. The synchrotron tune is Q s = 0.002 and dashed lines at 1/(4Q s ) multiples are plotted. As the synchrotron period is turns, each macroparticle comes back to its initial longitudinal position after this period.

Figure . :

 . Figure . : Transverse and longitudinal position of the two macroparticles MP and MP at di ferent periods of the synchrotron oscillation cycle. The total bunch intensity in this case is now p.p.b., each particle has half the bunch intensity . × p.p.b. The kicks do not accumulate and the motion remains stable: the two macroparticles recover their initial position after a full synchrotron period.

  Figure . : Transverse position of the two macroparticles as a function of turn number. The total bunch intensity is reduced to p.p.b. The wake is therefore weaker and the transverse displacements do not accumulate. The transverse position of the particles remains in a bounded region and the beam stays stable.

  .

  (a) Bunch before slicing. (b) Bunch after slicing.

Figure . :

 . Figure . : Principle of beam slicing for in PyHEADTAIL. On the left the bunch is represented as a collection of macroparticles. The e fect of the wake eld generated by all the macroparticles on the red one must be evaluated. To decrease computation time, the bunch is instead sliced as pictured on the right. Each slice contains several thousands of macroparticles, and it is the e fect of all slices on the red one which is computed. Pictures courtesy of M.Schenk [ ].

Figure

  Figure . : Mode spectra obtained with Sacherer's formula treatment. The modes are represented here for zero chromaticity. Mode is peaked at f = 0 GHz whereas the other modes are peaked at f ≈ (|m|+1) 2τ b . Chromaticity would shift all the modes by f ξ .

  Figure . : Example of an instability signal obtained with PyHEADTAIL. The mean value of the bunch horizontal position is plotted against the turn number in blue. The exponential t of the signal envelope is plotted in orange.

Figure . :

 . Figure . : Mode spectrum amplitude as a function of the transverse tune frequency. The amplitude A is normalized to the maximum amplitude A 0 . The unperturbed transverse tune is Q x0 . Positive and negative synchrotron side-bands are plotted as dashed lines. The most unstable mode is at the second negative sideband Q x0 -2Q s . Other modes are present at the other side-bands.

Figure

  Figure . : Head-tail pro les obtained with DELPHI at Q x = 0 in an inductive impedance model. The pro les corresponding to azimuthal modes |m| = 0, 1, 2, 3 are represented.

Figure

  Figure . : Head-tail pro les obtained with DELPHI for Q = -10 and an inductive impedance model. The pro les corresponding to azimuthal modes |m| = 0, 1 are represented.

Figure

  Figure . : Real part of the complex mode frequency shifts as a function of chromaticity, comparing PyHEADTAIL and DELPHI simulations. The resistive wall impedance is used and the damper is deactivated.

Figure . :

 . Figure . : Head-tail pro les obtained with DELPHI for di ferent chromaticities, at a xed bunch intensity of × p.p.b. The resistive wall impedance is used and the damper is deactivated.

  Figure . : Head-tail pro les obtained with PyHEADTAIL (left plot) and DELPHI (right plot) for a chromaticity of Q = 26. The resistive wall impedance is used and the damper is deactivated.

  Figure . : Real part of the complex mode frequency shifts as a function of chromaticity, comparing PyHEADTAIL and DELPHI simulations. The resistive wall impedance is used and the damper is activated with a damping time of turns.

Figure . :

 . Figure . : Head-tail pro les obtained with DELPHI for di ferent chromaticities, at a xed bunch intensity of × p.p.b. The resistive wall impedance is used and the damper is activated with a damping time of turns.

  Figure . : Head-tail pro les obtained with PyHEADTAIL (left plot) and DELPHI (right plot) for a chromaticity of Q = 26. The resistive wall impedance is used and the damper is activated with a damping time of turns.

  Figure . : Real part of the complex mode frequency shifts as a function of chromaticity, comparing PyHEADTAIL and DELPHI simulations. The LHC impedance is used and the damper is deactivated.

  Figure . shows DELPHI and PyHEADTAIL reconstructions of the pro les for Q = -3, and Fig. . for Q = 20.

  Figure . : Instability growth rate as a function of chromaticity, comparing PyHEADTAIL and DELPHI simulations for the LHC impedance without transverse damper. Only the most unstable mode is represented.The LHC impedance is used and the damper is deactivated.

  Figure . : Instability growth rate as a function of chromaticity, comparing PyHEADTAIL and DELPHI simulations.The LHC impedance is used and the damper is deactivated. Non-linear longitudinal motion is activated in PyHEADTAIL.

  Figure . : Real part of the complex mode frequency shifts as a function of bunch intensity, comparing PyHEAD-TAIL and DELPHI simulations. The broadband impedance is used and the damper is deactivated.

Figure . :

 . Figure . : Head-tail pro les obtained with DELPHI for di ferent bunch intensities below and above the TMCI threshold. The broadband impedance is used and the damper is deactivated.

  Figure . : Head-tail pro les obtained with PyHEADTAIL (left plot) and DELPHI (right plot) for a bunch intensity of . × p.p.b. The broadband impedance is used and the damper is deactivated.

  Figure . : Instability growth rate as a function of bunch intensity, comparing PyHEADTAIL and DELPHI simulations. Only the most unstable mode is compared in this case. The broadband impedance is used and the damper is deactivated. The dashed black line corresponds to the growth rate at N b = 60 × 10 11 p.p.b., the dashed grey line represents the intensity value corresponding to a s -growth rate.

.

  Fig. . for bunch intensities of × p.p.b. and × p.p.b.. Since the single bunch intensity reached in

Figure . :

 . Figure . : Head-tail pro les obtained with DELPHI for high bunch intensities, above the TMCI threshold. The broadband impedance is used and the damper is deactivated..

Figure . :

 . Figure . : Real part of the complex mode frequency shift as a function of bunch intensity, for PyHEADTAIL and DELPHI simulations. The broadband impedance is used and the damper is activated with a damping time of turns. Compared to Fig. . , mode is now stable at all intensities. Mode -is the most unstable, and it is coupling with mode -at an intensity of . × p.p.b.

Figure . :

 . Figure . : Real part of the complex mode frequency shifts as a function of bunch intensity, comparing PyHEAD-TAIL and DELPHI simulations. The LHC impedance is used and the damper is deactivated.

Figure . :

 . Figure . : Pro les obtained with PyHEADTAIL and DELPHI for bunch intensities of × p.p.b. . × p.p.b. The LHC impedance is used and the damper is deactivated.

Figure . :

 . Figure . : Head-tail pro les obtained with PyHEADTAIL and DELPHI for bunch intensities of × p.p.b., . × p.p.b. and × p.p.b. The LHC impedance is used and the damper is activated with a damping time of turns.

Figure . :

 . Figure . : Real part of the complex mode frequency shifts as a function of bunch intensity, comparing PyHEAD-TAIL and DELPHI simulations. The LHC impedance is used and the damper is deactivated. On the left, PyHEADTAIL simulations include only the dipolar wake and Sacherer's tune shift is computed for the e fective dipolar impedance only. On the right, PyHEADTAIL simulations include both dipolar and quadrupolar wakes and Sacherer's tune shift is computed by adding the dipolar and quadrupolar e fective impedances. In both plots DELPHI only includes the dipolar impedance e fect.

Figure . :

 . Figure . : Instability growth rate as a function of bunch intensity, comparing PyHEADTAIL and DELPHI simulations. Only the most unstable mode is compared in this case. The LHC impedance is used and the damper is deactivated.

  Figure . : Current in the Landau octupoles magnets required to stabilize the horizontal plane of beam (solid lines) as a function of chromaticity during the year .The results for a case without damper (in blue) and with the transverse damper activated with a damping time of turns (in orange) are presented. Those are compared to measurements for di ferent chromaticities, represented in the same color code. A dot denotes a measurement made at the end of squeeze, i.e when the beams are ready to be collided. A cross represents a measurement made a at-top, i.e after the end of the energy ramp and before the tune change. All results have been normalized to a bunch intensity of p.p.b.

  Figure . : Real part of the mode frequency shifts (top plots) and instability growth rate (bottom plots) as a function of bunch intensity. On the left the damper is deactivated, on the right it is activated with a damping time of turns. In both cases the chromaticity is corrected to Q = 0.

  Figure . : Real part of the mode frequency shifts (top plots) and instability growth rate (bottom plots) as a function of bunch intensity. Both left and right plots are for a deactivated damper. On the left the chromaticity is corrected to Q = 0 and on the right it is positive at Q = 5.

  Figure . : Real part of the mode frequency shifts (top plots) and instability growth rate (bottom plots) as a function of bunch intensity. On the left the chromaticity is corrected to Q = 0 and the damper is deactivated, on the right the chromaticity is positive at Q = 5 and the damper is activated with a damping time of turns.

  Figure . : Physical gaps of three of the LHC IR collimators in mm for the di ferent scenarios investigated.The nominal LHC con guration in is displayed in blue, whereas the other colors show tighter collimators settings. In the tightest setting presented here the collimator gaps are reduced by ≈ %. The picture for all collimators gaps is reported in part B.

  x )/M Ωm -1 TCP 5σ/ TCSG 6.5σ TCP 5σ/ TCSG 6σ TCP 4.5σ/ TCSG 5.5σ TCP 4σ/ TCSG 5σ

Figure . :

 . Figure . : Real and imaginary part of the LHC horizontal beam coupling impedance for di ferent collimator settings.

  Figure . b shows the simulations results for this setup. They are compared to the results with a corrected chromaticity and no damper for the same tight collimator gaps in Fig. . a.

  Figure . : Real part of the mode frequency shifts (top plots) and instability growth rate (bottom plots) as a function of bunch intensity. Both left and right plots are for a deactivated damper and a chromaticity corrected to Q = 0. On the left the collimator gaps are the nominal ones for whereas on the right they are tighter to increase the machine impedance.

  Figure . : Real part of the mode frequency shifts (top plots) and instability growth rate (bottom plots) as a function of bunch intensity. Now in both left and right plot the collimators gaps are tighter at 5σ/6σ in the TCP /TCSG . In the left plot the chromaticity is at Q = 0 and the damper is deactivated. In the right plot chromaticity is at Q = 5 and the damper is activated with a damping time of turns.

  Figure . : Example of turn-by-turn transverse beam position signals acquired with the ADTObsBox after anADT kick. The three bunches present in beam during the rst ll are represented. For each bunch, the horizontal position is plotted with a darker color than the vertical position. All three bunches are kicked at the same time, rst in the horizontal plane and after turns in the vertical plane. The bunch oscillations decay in 500 turns.

Figure . :

 . Figure . : Overview of the beam parameters in ring during the rst ramp for the tune shift measurement at top energy on the th and th of September . The top plot shows the gap of a primary (TCP.D R ) and a secondary collimator (TCSG.D R ). The second, third and fourth plot show the individual bunch intensities, full bunch lengths and the vertical emittances. The time at which the ADT kicks were sent are marked with dashed lines.

  Figure . : Tune signals obtained after the post-processing of the ADT kicks signals with SUSSIX (top plot). The data represented is for beam vertical plane and the corresponding beam parameters shown in Fig. . . Three collimators con guration, separated by a dashed line, were measured with this beam. The left part of the plot corresponds to the nominal collimators con guration ( / . ), the central part to the tighter TCSG setting ( / ) and the right part to the tighter TCP setting ( . / ).The bottom plot shows the average tune for each bunch in the three collimators con gurations.

  Figure . : Measured tune shift as a function of bunch intensity for beam , compared to DELPHI simulations corrected for the quadrupolar tune shift (dashed lines). The collimator settings measured during the MD are represented by di ferent line and points colors.

  Figure . : Measured tune shift as a function of bunch intensity for beam , compared to DELPHI simulations corrected for the quadrupolar tune shift (dashed lines). The collimator settings measured during the MD are represented by di ferent line and points colors.

  Figure . : Overview of the secondary collimators measurements performed on the night of th to th of October . The top plot shows the horizontal tune obtained from the ADT kicks (dots) correlated with the collimators gaps (solid lines) along time. The center and bottom plots show the full bunch length and the bunch intensity evolution.

  Figure . : Detail of Fig. . showing the tune variation during the measurement of two collimators. Despite the overall upwards tune drift, the e fect of the collimator gaps opening and closing remains visible and the induced tune shift can be computed.

  Figure . : Ratios between measured and predicted tune shifts induced by individual collimators of ring a and ring at top energy. The measurement years are indicated by di ferent colors.

  Figure . : Instability rise time as a function of chromaticity measured in the LHC at injection energy, for the horizontal plane. The measurements were made for beam (in blue) and beam (in red). They are compared to DELPHI simulations represented with a solid grey line. The dashed grey line corresponds to the simulated rise times with a factor two applied.

  Figure . : Instability rise time as a function of chromaticity measured in the LHC at injection energy, for the vertical plane. The measurements were made for beam (in blue) and beam (in red). They are compared to DELPHI simulations represented with a solid grey line. The dashed grey line corresponds to the same simulated rise times with a factor two applied.

Figure . :

 . Figure . : Horizontal dipolar impedance as a function of frequency for the di ferent scenarios considered. The top plot shows the real part of the impedance, the bottom plot the imaginary part.

Figure . :

 . Figure . : Horizontal mode frequency shifts as a function of bunch intensity for two of the HL-LHC scenarios considered. The modes are computed with DELPHI (points) and the mode is then tted in the low intensity part (solid lines).

  Figure . : Measured tune shift as a function of bunch intensity for beam , compared to DELPHI simulationscorrected for the quadrupolar tune shift (dashed lines). The nominal LHC con guration is represented in blue, the low impedance con guration in red.

Figure . :

 . Figure . : Top view of the prototype TCSPM jaw. Three stripes are visible: the top one is the molybdenum coating, the central one the molybdenum-graphite bulk and the bottom one the titanium-nitride coating. One the left a button beam position monitor is embedded in the transition taper. Picture from [ ].

Figure . :

 . Figure . : Tune shifts induced by the TCSPM as a function of collimator gap for the three di ferent materials. Predictions from the impedance model are represented in dashed lines and measurements are represented with the 1σ error bar. The geometric impedance of the collimators has been removed of the measured values. Results courtesy of S.Antipov [ ].

Figure . :

 . Figure . : Picture of a beam screen prototype for the FCC-hh, and cross-section of the current design. Two slits are present to allow synchrotron radiation to escape from the main chamber. The pumping holes are shielded by the central chamber walls. Pictures from [ ] and [ ]

Figure . :

 . Figure . : Horizontal (top plot) and vertical (bottom plot) transverse dipolar impedance as a function of frequency for the the HE-LHC injection energy options. The models are compared to the LHC and the HL-LHC models. Solid lines represent the real part and dashed lines the imaginary part.

Figure . :

 . Figure . : Horizontal (top plot) and vertical (bottom plot) transverse dipolar impedance as a function of frequency for the HE-LHC top energy. The model is compared to the LHC and the HL-LHC models. Solid lines represent the real part and dashed lines the imaginary part.

  .

  . The TMCI intensity threshold is found when the growth rate becomes non-zero. At the injection energies of GeV and . TeV, the thresholds are found at × p.p.b. and . × p.p.b. Compared to the nominal bunch intensity of . × p.p.b., the two injection energy scenarios have therefore large stability margins.

Figure . :

 . Figure . : Instability growth rate in the vertical plane as a function of bunch intensity. The top plot is for the GeV injection option, the central plot for the . TeV injection option and the bottom plot for the . TeV top energy. Three di ferent transverse damper settings are represented: damper deactivated and damping times of and turns. The mode coupling instability appears for the cases without the transverse damper, at × p.p.b. and . × p.p.b. for the two injection energies considered. At top energy it appears at × p.p.b., an intensity smaller than the nominal bunch intensity.

Figure

  Figure A. : Collimator impedance before weighting by β functions.

  Figure A. : Example of two collimators with identical β functions.

  Figure A. : Example of two collimators with di ferent β functions. This case mimics an association of two LHC collimators, the TCSG.D L and the TCSG.B L , which are respectively vertical and horizontal collimators.

Figure

  Figure A. : Example of two collimators with di ferent β functions.

Figure

  Figure C. : Measurements results of the beam TCP collimators induced tune-shifts.

  Figure D. shows the slicing of a bunch made of × macroparticles using slices. The bunch has a Gaussian longitudinal distribution as can be seen in the upper plot with the number of macroparticles per slice. The slices at the edges of the longitudinal space have a low number of macroparticles. This is con rmed by the bottom plot of Fig. D. : the slices with index to and to have no macroparticle.

Figure

  Figure D. : Particle distribution with the UniformBinSlicer in PyHEADTAIL. Top plot shows the number of macroparticles in each slice, as well as the longitudinal limits of the slices. Bottom plot shows the longitudinal position of each macroparticle as a function of the slice index.

Figure

  Figure D. : Most unstable mode growth-rate versus bunch intensity versus intensity for the LHC impedance model. The UniformBinSlicer is used, changing the number of bins. No improvement to the convergence is made even for bin numbers larger than .

Table . :

 . Magnetic parameters for protons beams of various accelerators in operation or projected at CERN. Unless noted, the parameters correspond to the maximum energy reached by the beam in the accelerator.

	Accelerator	Mag. eld / T L acc / m η dipole / % (Bρ) / T m p / GeV/c
	LHC	.
	LHC injection	.
	SPS	.
	PS	.
	PSB	.	.
	HL-LHC	.
	FCC-hh
	Magnetic eld reached in the main dipoles during Run II (

Table . :

 . Beam chamber parameters used to obtain the resistive wall wake and impedance functions.

	Parameter	Value
	Geometry	Cylindrical
	Radius b	mm
	Length L	km
	Material	Copper at	K
	Conductivity σ c	. MS m -
	Conductivity obtained from the copper re-
	sistivity at	K ρ c = 17 nΩ m [ ].

Table . :

 . Impedance and beam parameters for the two-particle model simulations.

	Parameter	Value
	Impedance model	Transverse broad-band resonator
	Machine
	Circumference / m	.
	Transverse tune Q x0 Momentum compaction factor α c Synchrotron tune Q s	. . × -× -
	Beam
	Number of bunches
	Number of macroparticles
	σ bunch length τ b / ns	.
	Bunch intensity / p.p.b.	and ×
	Chromaticity Q

  . ) Equation . is called Sacherer's integral equation [ ]. It includes as well a damping term treated as an impedance. It is an eigensystem since the radial function R m (τ ) of azimuthal mode m is itself a function of all the radial functions.This type of distribution leads to an expansion over Laguerre polynomials of Sacherer's integral. It becomes in the end an eigenvalue problem, which is then solved numerically. The radial functions g 0 (τ ) and R m (τ ) are decomposed over Laguerre polynomials[ ]

				. Treatment of coherent beam instabilities
	E
	Sacherer's integral equation . can be solved for di ferent shapes of the longitudinal phase space
	distribution [ ]. The code DELPHI assumes that the unperturbed longitudinal distribution is Gaussian
	with a RMS bunch length σ
	g 0 (r) =	1 2πσ	exp -	r 2 2σ 2 .	( . )

Table . :

 . Machine and beam parameters for DELPHI and PyHEADTAIL simulations for the head-tail instability study.

	Parameter		Value
	Impedance
	Impedance model		LHC Resistive wall at-top
	Machine
	Circumference / m			.
	Transverse tunes Q x,y		. / .
	Momentum compaction factor α c	. × -
	RF voltage / MV
	Harmonic number
	Synchrotron tune Q s		.	× -
	Beam
	Number of bunches
	σ bunch length / ns		.
	Bunch intensity /	p.p.b.	and
	Chromaticity Q		-to
	Impedance and wake model as described in [ , ] and available at [ ].

Table . :

 . DELPHI and PyHEADTAIL speci c parameters for the head-tail instability study.

	Parameter	Value
	PyHEADTAIL
	Software version	. .
	Number of slices for
	the longitudinal distribution
	Longitudinal cut / σ z	±
	Number of macroparticles
	Number of turns	×
	DELPHI
	Plane simulated	Horizontal
	Convergence criterion	× -

DELPHI convergence criterion crit checks at a given iteration i the imaginary part of the mode imaginary part. If |

Table . :

 . Machine and beam parameters for DELPHI and PyHEADTAIL simulations.

	Parameter	Value
	Impedance
		LHC	at-top
	Impedance model	Broad-band resonator
	Beam
	Number of bunches
	σ bunch length τ b / ns	.
	Bunch intensity / ×	p.p.b.	to
	Chromaticity Q
	LHC impedance and wake model as described in [ ] and available at [ ].

Table . :

 . DELPHI and PyHEADTAIL speci c parameters for the TMCI simulations.

	Parameter	Value
	PyHEADTAIL
	Software version	. .
	Number of slices for the longitudinal
	distribution
	Longitudinal cut / σ z	±
	Number of macroparticles
	Number of turns	×
	DELPHI
	Plane simulated	Horizontal
	Convergence criterion	× -

.

  Simulations of the Transverse Mode Coupling in the LHC

		600
	I oct at top energy / A	100 200 300 400 500		Operational Predicted
		0
	Figure . : Octupoles current used in operation (in blue) versus predicted from the impedance model (in red)
	during Run II. The maximum current of	A which can be reached in the magnets is highlighted as
	a dashed line. Courtesy X.Bu fat [ ]
	. S	T	M	C	LHC

As seen in part . , the Transverse Mode Coupling Instability is a fast instability which arises at high bunch intensity and at a chromaticity corrected to zero. The LHC currently operates with a high chromaticity of Q ∼ 15 to mitigate electron-cloud induced instabilities [ ]. Moreover instabilities in this region of small chromaticities were observed during the LHC Run I ( -) [ ].

Table . :

 . Machine and beam parameters for TMCI simulations with DELPHI.

	Parameter		Value
	Machine
	Impedance model		LHC	at-top
	Circumference / m			.
	Transverse tunes Q x,y Momentum compaction factor α c	.	. / . × -
	RF voltage / MV
	Harmonic number
	Synchrotron tune Q s		.	× -
	Beam
	Number of bunches
	σ bunch length / ns		.
	Bunch intensity / ×	p.p.b.	to
	Chromaticity Q		to
	LHC impedance and wake model as described in [ ] and
	available at [ ].

Table . :

 . Primary and secondary collimators gaps settings used during the tune shift versus intensity measurement.

	Collimator setting / σ coll
	TCP	TCSG	Con guration name Remark
			( / )	Relaxed setting, HL-LHC mock-up
		.	( / . )	nominal setting
	.		( / )	Tight settings

Table . :

 . TMCI intensity threshold inferred from the measurements presented in Table.and from DELPHI simulations. The third column corresponds to DELPHI results. The fourth and fth columns are the thresholds estimated from the mode tune shift, without and with the correction for the quadrupolar tune shift e fect. The last column shows the measurement results.

				Threshold / 10 11 p.p.b.
	Conf.	DELPHI	DELPHI tune shift DELPHI tune shift corrected Measured
	B H ( / . )	.	.	.	.
	( / . )	.	.	.	.
	B V ( / . )	.	.	.	.
	( / . )	.	.	.	.
	B H ( / . )	.	.	.	.
	( / . )	.	.	.	.
	( . / . )	.	.	.	.
	B V ( / . )	.	.	.	.
	( / . )	.	.	.	.
	( . / . )	.	.	.	.

Table . :

 . Measurement to simulation ratios of the tune shifts versus intensity in the LHC, for both beams and planes. The measurements for the di ferent collimators con gurations are averaged and compared to simulations results which account for the quadrupolar tune shift.

		Horizontal Vertical
	Beam	.	.
	Beam	.	.

Table . :

 . List of single collimators measured during Run II.

	Type	Collimator name Location Collimation plane
	Primary	D L	IR	Vertical
		C L	IR	Horizontal
		B L	IR	Skew
		L	IR	Horizontal
	Secondary A L	IR	Skew
		D L	IR	Vertical
		B L	IR	Horizontal
		B R	IR	Skew
		D R	IR	Skew
		E R	IR	Skew
		R	IR	Horizontal
	Tertiary	TCTPV	IR	Vertical
		TCTPH	IR	Horizontal

Table . :

 . Table . reports those ratios for the di ferent years [ ]. List of ratios between measured and predicted tune shifts values for both beams and planes. Example of a beam vertical position signal. The y-axis is in logarithmic scale. With this scale the t should be realized on an linearly growing part. However for this instability signal the part to t is less clear as the instability seems to develop at di ferent rates between turn and and turn and .

	. Assessment of the LHC impedance and stability limits with beam based measurements
	) / arb. units	10 2 10 3 10 4	Instability signal
	log( y	10 1
		10 0
		0	10000	20000	30000	40000	50000	60000
					Turns
	Figure . :
	Year Measurement					B H B V B H B V
		Tune shift vs. IR secondary collimators gaps	.	.	.	.
		Tune shift vs. bunch intensity			.	.	.	.
		Tune-shift vs. bunch intensity			.	.	.	.
		Full machine tune shift at at-top		.	-	-	-
		Growth-rate vs. negative chromaticity		.	.	.	.
		Tune shifts measurements with all IR secondary collimators, detailed in [ ].
		Quadrupolar impedance e fect not taken into account.
		Quadrupolar impedance e fect taken into account.
		Measurement from Beam Transfer Function (BTF) detailed in [ ].

Table . :

 . Resistivity of the materials selected for the LHC collimation upgrade, compared to the current CFC.

	Material	Resistivity / nΩ m
	CFC
	MoGr
	TiN ,
	Mo
	Further data on the electric properties of materials cur-
	rently used in the LHC can be found in [ ].

Table . :

 . Scenarios considered for the study of TMCI in HL-LHC. The table shows the materials used for the di ferent collimators families and the number of collimators upgraded. , represented in orange, is close to the LS scenarios of the HL-HC collimation upgrade, represented in green and red, for the MHz to GHz range.

	Scenario name	TCP	TCSG
	LHC		CFC	CFC
	LHC	, TCSGs at 14σ coll	CFC	CFC
	HL-LHC LS , uncoated TCSGs	MoGr, CFC	MoGr, CFC
	HL-LHC LS , coated TCSGs	MoGr, CFC	Mo+MoGr, CFC
	HL-LHC nal, uncoated TCSGs	MoGr, CFC	MoGr
	HL-LHC nal, coated TCSGs	MoGr, CFC	Mo+MoGr
	at 14σ

coll

Table . :

 . Results of DELPHI simulations for the LHC and HL-LHC impedance reduction scenarios. Both the tune shift versus intensity and the TMCI threshold are reported.

	The Transverse Mode Coupling Instability in future machines
			10 3
		Ωm -1 x ) / M (Z dip	10 1 10 2	LHC LHC HL-LHC LS , uncoated TCSGs , TCSGs at σ coll HL-LHC LS , coated TCSGs
				HL-LHC nal, uncoated TCSGs
				HL-LHC nal, coated TCSGs
			10 0
			10 3
		(Z dip x ) / M Ωm -1	10 1 10 2
			10 0
			10 3	10 4	10 5	10 6	10 7	10 8	10 9	10 10
					Frequency / Hz
	Scenario				Tune shift / 10 11 p.p.b. × Q s	-1	TMCI
	LHC					-.			.
	LHC	, TCSGs at 14σ coll		-.			.
	HL-LHC LS , uncoated TCSGs		-.			.
	HL-LHC LS , coated TCSGs		-.			.
	HL-LHC nal, uncoated TCSGs		-.			.
	HL-LHC nal, coated TCSGs		-.			.

Table . :

 . Measured and simulated tune shifts as a function of beam intensity for the LHC nominal collimator settings and the HL-LHC mock-up scenario. The third column reports DELPHI simulations results. The fourth column is the correction factor to account for the quadrupolar impedance e fect and the fth column is the simulation results with the correction factor applied.

		Tune shift / 10 12 p.p.b. × Q s	-1
	Conf.	Sim. Correction / % Sim. w/ quad. Measured
	B H ( / ) -.

Table . :

 . TMCI intensity threshold inferred from the measurements presented in Table.and from DELPHI simulations. The third column corresponds to DELPHI results. The fourth and fth columns are the threshold estimated from the mode tune shift, without and with the correction for the quadrupolar tune shift e fect. The last column shows the measurement results. The nominal LHC case and the HL-LHC-like case are presented.

	0.2
	0.0
	-0.2
	-0.4
	-0.6
	-0.8
	-1.0
	-1.2
	0.0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0
				Bunch intensity / 10 11 p.p.b
		Conf.	DELPHI	DELPHI tune shift DELPHI tune shift corrected Measured
	B H ( / )	.	.			.		.
		( / . )	.	.			.		.
	B V ( / )	.	.			.		.
		( / . )	.	.			.		.
	B H ( / )	.	.			.		.
		( / . )	.	.			.		.
	B V ( / )	.	.			.		.
		( / . )	.	.			.		.

Threshold / 10 11 p.p.b.

  Using the previous relations, the impedances can all be expressed as a function of structure 1 dipolar impedance component

					A. Impedance of the single vertical collimator
	The total impedance for horizontal and vertical planes is thus
	Z dip+quad x,tot	=	1 β x	(3β x,2 )Z dip x,1 ,	(A. )
	Z dip+quad y,tot
								2 ,	(A. )
	Z quad x,tot =	β x,1 β x		Z quad x,1 +		β x,2 β x	Z quad x,2 ,	(A. )
	Z dip y,tot =	β y,1 β y		Z dip y,1 +		β y,2 β y	Z dip y,2 ,	(A. )
	Z quad y,tot =	β y,1 β y		Z quad y,1 +		β y,2 β y	Z quad y,2 .	(A. )
	Z dip x,tot =			β x,1 β x	Z dip x,1 +		β x,2 β x	Z dip y,1 ,	(A. )
	Z quad x,tot = -		β x,1 β x	Z dip x,1 +	1 2	β x,2 β x	Z dip y,1 ,	(A. )
	Z dip y,tot =			β y,1 β y	Z dip y,1 +		β y,2 β y	Z dip x,1 ,	(A. )
	Z quad y,tot =	1 2		β y,1 β y	Z dip y,1 -		β y,2 β y	Z dip x,1 .	(A. )
	Moreover we have Z dip x,1 = Z quad y,1 and Z dip y,1 = 2Z quad y,1 , Z dip y,1 = 2Z dip x,1 and the previous equations
	become
	Z dip x,tot =	1 β x	(β x,1 + 2β x,2 )Z dip x,1 ,	(A. )
	Z quad x,tot =	1 β x	(-β x,1 + β x,2 )Z dip x,1 ,	(A. )
	Z dip y,tot =	1 β x	(2β y,1 + β y,2 )Z dip x,1 ,	(A. )
	Z quad y,tot =	1 β x	(β y,1 -β y,2 )Z dip x,1 .	(A. )

√β 0

During the rst attempt (top plot), a tune shift induced by the collimator gap change is visible but the measurement precision is reduced by tune drifts. During the second attempt the collimator gap was opened and closed in quick successions while kicking the beam with the ADT (bottom plot). A tune drift was still present but could be tted and the tune data corrected.

A. Conclusion

A. C

We saw that the impedance of a pair of at structures with di ferent orientations can lead to tune shifts compensation or adjonction depending on the optics functions used in each structure. For the LHC and its numerous collimators, this means that the quadrupolar part of the impedance might partially compensate the e fects of the dipolar impedance on the tune shifts. This depends of course as well on the collimators gaps used. Collimator name TCP7 5σ/ TCSG7 6.5σ TCP7 5σ/ TCSG7 6σ TCP7 4.5σ/ TCSG7 5.5σ TCP7 4σ/ TCSG7 5σ Figure B. : Physical gaps of the LHC IR collimators in mm for the di ferent scenarios investigated. The nominal LHC con guration in is displayed in blue, whereas the other colors show tighter collimators settings. Only the primary and secondary collimators of IR settings were modi ed and are thus showed in the gure. In the tightest setting presented here the collimator gaps are reduced by ≈ %.

B. C TMCI

B. Collimator gaps and beam parameters for the HL-LHC TMCI simulations The normalised emittance used to compute the collimators gaps are di ferent between LHC and HL-LHC/HE-LHC. In LHC ε LHC n = 3.5 µm whereas in both HL-LHC and HL-LHC ε n = 2.5 µm.

D. C

The convergence study made with the two beam slicers available in PyHEADTAIL showed a great di ference when applied to the LHC impedance model. With the UniformBinSlicer, results diverge from DELPHI simulations whereas the UniformChargeSlicer allowed to obtain converges results using at least . We saw that when the number of slices with the UniformBinSlicer is increased, bins sampling the edges of the distribution can be empty. This could introduce numerical artefacts leading to the discrepancy found in part . . . The issue could be solved by using the UniformChargeSlicer in PyHEADTAIL. With this slicer, all bins have the same number of macroparticles. With enough slices, results were found to be close to DELPHI predictions.