A quadruple bend achromatstructure - PDF document

A quadruple bend achromatstructureThe e
A quadruple bend achromatstructureThe endAn aside : Louville’stheoremLiouville’stheorem : the density of points representing particles in 6D (x, p) phase space is conserved if any forces conservative and differentiable adiation and dissipation do not satisfy this requirement, but magnetic forces and (Newtonian) gravitational forces do. There must be no or very slow time dependence in the Ham

iltonian system Note: acceleration keeps
iltonian system Note: acceleration keeps (x,p) phase space constant, but reduces (x, x’) phase space , so there is no violation of Liouvilletheorem Transfer maps derived from a Hamiltonian have a mathematical property called symplecticity, which is deeply linked to Louville’stheorem. But this is beyond this course…But symplecitictyin 2D phase space is equivalent to det(M)=1 Beams of par

ticles and emittanceIn general, we have
ticles and emittanceIn general, we have more than one particle in our beam, each with its own value for the emittance (or CourantSynderinvariant). We choose one of the particle’s emittance to represent the emittance of the entire beam.For example, characterisethe beam by the emittance of the particle for which 95% of the beam particles are within the ellipse of this particle.Another useful definit

ion, when dealing with complicated distr
ion, when dealing with complicated distributions, is the RMS emittance, which we find by averaging over the beam distributionBeams of particles and emittanceSo far we’ve defined the emittanceof a particle, which was our constant in the CourantSnyder analysis. The motion of an individual particle is completely specified by its emittanceand initial phase. Different particles have different emittance

sand initial phases but they all have th
sand initial phases but they all have the same CourantSnyder functions (how true is this?!)For example, the particle with x=x’=0 will have zero emittanceand always stay at x=x’=0. This is the ideal particle. But, we always have more than one particle in our beam and so need to understand how to characterisea beam of particles, each with their own emittanceLattice design (continued)1) For a mo

dern accelerator, lattice design work us
dern accelerator, lattice design work usually takes some years to finalize the design parameters. It is an iterative process, involving users, funding, accelerator physics, accelerator subsystems, civil engineering, etc. 2) It starts from major parameters such as energy, size, etc. 3) Then linear lattice is constructed based on the building blocks. Linear lattice should fulfill accelerator physics crite

ria and provide global quantities such a
ria and provide global quantities such as circumference, emittance, betatrontunes, magnet strengths, and some other machine parameters. 4) Design codes such as MAD, OPA, BETA, Tracy, Elegant, AT, BeamOptics,.... are used for the matching of lattice functions and parameters calculations. 5) Usually, a design with periodic cells is needed in a circular machine. The cell can be FODO, Double Bend Achromat(D

BA), Triple Bend Achromat(TBA), Quadrupl
BA), Triple Bend Achromat(TBA), Quadruple Bend Achromat(QBA), or MultiBend Achromat(MBA or nBA) types. 6) Combinedfunction or separatedfunction magnets are selected. 7) Maximum magnetic field strengths are constrained. (roomtemperature or superconducting magnets, bore radius or chamber profile, etc.) 8) Matching or insertion sections are matched to get desired machine functions. 9) To get stable solutio

n of the offmomentum particle, we need t
n of the offmomentum particle, we need to put sextupolemagnets and RF cavities in the lattice beam line. Such nonlinear elements induce nonlinear beam dynamics and the dynamic acceptances in the transverse and longitudinal planes need to be carefully studied in order to get sufficient acceptances. (for long beam current lifetime and high injection efficiency) 10) For the modern high performance machines

, strong sextupolefields to correct high
, strong sextupolefields to correct high chromaticity will have large impact on the nonlinear beam dynamics and it is the most challenging and laborious work at this stage. 11) In the real machine, there are always imperfections in the accelerator elements. So, one need to consider engineering/alignment limitation or errors, vibrations, etc. Correction schemes such as orbit correction, coupling correcti

on, etc., need to be developed. (dipole
on, etc., need to be developed. (dipole correctors, skew quadrupoles, beam position monitors, etcSextupolesSo the dispersion effectively makes the sextupoleinto a quadrupole with a momentum dependent focusing gradientThis means we can compensate the chromaticity in the ring, and undo the tune shift, by adjusting the sextupoles. But it’s not all perfect. Remember we ignored plenty of terms in the

fields of the sextupoles……some
fields of the sextupoles……some of the terms are good and fix our chromaticity, but some are bad and introduce nonlinearitiesand coupling into our accelerator ring. It is difficult to represent sextupolein our linear formalism, and often the best way to understand the impact of sextupolefields is to track particles with matrices, and stopping to be more careful every time a sextupoleis encount

ered. This leads to the study of a machi
ered. This leads to the study of a machine’s dynamic aperture, or what amplitude of particle can survive or many turns.Sextupolessextupolefield has field components given byNote the field is quadratic in x and y, and also (for the first time) we see products of x and y in our equations. A sextupolecouples the beam planes!An offmomentum particle passing through the sextupolehas displacementWhere

we define the sexupolestrength byAnd so
we define the sexupolestrength byAnd so the fields seen by the particle are found by substitutionThere are many terms here, some helpful and some harmful. The helpful ones for us areThe horizontal dispersion function has made each sextupoleinto a quadrupole with an effective gradient SD.delta. We can use these to cancel the natural chromaticity in the lattice and cancel the chromatic tune shift.And if

we correct itSo how do we do this?Chrom
we correct itSo how do we do this?Chromaticity basically comes about when a particle which is slightly offmomentum sees a different quadrupole field than it should and this particle is focused differently to the others.So in essence we need a correcting device which has some kind of momentum dependent focusing. A sextupoleWhy is chromaticity bad?Chromaticity is naturally generated by any focusing latt

ice. So when we have nonzero k we have
ice. So when we have nonzero k we have chromaticity. And it tends to be negative in both planes.It tells us how much the tune shifts for a unity shift in the momentum deviation.So given the beam has an energy spread, it tells us the spread of the tune of the beam.So tune is a blob in tunespace.This is a plot of tune at HERA, showing uncorrected chromaticity and not a clean tune signalChromaticity in t

he LHCChromaticityIt’s conventiona
he LHCChromaticityIt’s conventional to define the tune change per unit delta as the chromaticityWhere we call the chromaticity ‘natural’ as it arises from the quadrupoles. Any lattice with quadrupoles naturally generates this chromaticity. Similarly in the vertical planeNote as the horizontal beta function is biggest in horizontally focusing quadrupoles (and vice versa) means the natura

l chromaticity is normally negative in b
l chromaticity is normally negative in both planes.The linear chromaticity is sometimes written as QFor a FODO cell we can show thatChromaticityWe already know how to compute the effect of a quadrupole field error.Recall the tune shift from a quadrupole error k(s) in our latticeWe can think of this chromatic terms as a quadrupole field error of strengthA similar analysis in the vertical plane would h

ave a found a chromatic perturbation ofW
ave a found a chromatic perturbation ofWhich means we can write down the tuneshift arising from the chromatic perturbation term, An expression which is linear in the momentum deviation. ChromaticityIf the machine tunes depend on the momentum deviation, we can write linearly Where we’ve defined the linear chromaticity. To analyse this we return to the equations of motion, but this time keeping all

terms linear in x and delta. Recall we
terms linear in x and delta. Recall we had We proceed in the same way as we’ve done before, but when we expand the various terms we keep the term (x.delta) we previously dropped. So we haveExpanding out the brackets and dropping terms of higher order, we are left with our equation of motion, this time with a chromatic termWhere we defined (as normal)ChromaticityConsider some particles of slightly

different energy passing through a FODO
different energy passing through a FODO cellHigher momentum particles have a higher rigidity, so experience weaker effects when passing through magnetic fields. This means focusing is momentum dependent and so the machine tune will depend on momentum deviation.Typical lattices and momentum compactionThe momentum compaction factor is an importantlattice design parameterTypically it is given by A larg

e value means the path length varies a l
e value means the path length varies a lot for offmomentum particles. This means the particles tend to spread out and the bunch length becomes long.Similarly, a small value means a shorter bunch length. (Perhaps it really should be called momentum dilution factor!!!)Typically Dက က 0, so the particles tend to orbit outer side of the ring. In this case the momentum compaction factor isက 0 (and the

orbit gets longer).An isochronous lattic
orbit gets longer).An isochronous lattice is designed to counter this natural tendency, normally to some order in the momentum deviation. Momentum compactionThe path length deviation is given by The change in circumference of the machine is given by an integral over the whole ringFor the case where the closed orbit distortion is given by a momentum errorWe define the linearmomentum compaction factorS

o we then have an integral around the ri
o we then have an integral around the ring to compute the momentum compaction factorso Momentum compactionSo, a momentum offset changes the horizontal orbit of a particle through dispersion.Ideally, a machine with only horizontal bends does not generate any vertical dispersionHowever, dispersion does generate a longitudinal effect, as the total circumference of an off momentum particle’s trip aro

und the machine will a different to the
und the machine will a different to the reference particleWhat is this circumference, or path length, error?Consider this situation:The 4 magnet chicaneThe long straight section of the LHeCcollider (optical work done by CI)A DBA structure with a quadrupoletriplet (vertical)DBA structure with a single quadrupole(sometimes called ChasmanGreen)DBAConsider a simple DBA cell with a single quadrupolein

the middle. We use the thin lens approxi
the middle. We use the thin lens approximation and write down the dispersion matching condition i.e. We want some dispersion Din the middle of the quadrupoleand feed into the system zero dispersionHere f is quadrupolefocal length, theta and L are the bend parameters and L is the distance between the quadrupoleand bend centers. In essence we match to the D’=0 condition at the middle of the quadrupol

e. Or the quad turns over the sign of th
e. Or the quad turns over the sign of the dispersion generated by the bend. The required focal length and resulting Dis henceNote the dispersion at the quad becomes higher for longer distances and bigger bend angles.This analysis shows it’s possible but in practice we need extra quads for matching and maybe a reduction of the required quad strength by splitting the central quad.The double bend ach

romat(DBA) If the dispersion function is
romat(DBA) If the dispersion function is nonzero the orbit of particles depends on particle momenta. An achromatic system means we design it such that beam positions do not depend on momenta at each end. What this means is an arrangement of magnets, including bends, which does not generate any dispersion through the structure. A single bend is not achromatic as zero dispersion going in would not lead to

zero dispersion going out. “self c
zero dispersion going out. “self cancelling dispersion”In principle, dispersion can be suppressed by one focusing quadrupoleand one bending magnet. With one focusing quad in the middle between two dipoles, one can get the achromatcondition, which means no additional dispersion is driven by the structure.Due to mirror symmetry of the lattice w.r.t. to the middle quad’ at quad center shoul

d be zero (see upcoming plot). This is s
d be zero (see upcoming plot). This is so called double bend achromat(DBA) structure. We generally need quads outside DBA section to match the betatronfunctions, tunes, etc. Similarly, one can design triple bend achromat(TBA), quadruple bend achromat(QBA), and multibend achromat(MBA or nBA) structure. Lattice building blocksWe’ve seen how to combine alternating gradient quadrupolesto make a focusi

ng structure in both planes. This is cal
ng structure in both planes. This is called the FODO cell and is an example of a basic optical building block we use to construct lattices.There are many possible configurations of dipoles and quadrupolesthat can give stable motion For example a FODO cell to transport a beam We can talk about dispersionfree lattices, which are important in many applications. These allow bending of the beam witho

ut generating dispersion. Examples are C
ut generating dispersion. Examples are Chasmangreen, triplebend achromatWe also can build dispersion suppressor, which match the periodic dispersion in the arc (perhaps made of FODO cells) into a dispersionfree straight. We can also displace the beam transversely without generating dispersion using a sequence of only bends. Sometimes called a geometrical achromatLet’s look at achromatsin more detai

l….Beta functions and dispersion i
l….Beta functions and dispersion in the LHC arcs at injectionBeta functionsBeam sizesDispersionExerciseConsider a thin lens quadrupole with focal length f.Work out the change in the lattice functions through this quadrupole..(Hint: what is the transfer matrix of the quadrupoleand what expression tells you how the lattice functions evolve when you know the transfer matrix?)Contrast the behaviou

r of a focusing (f � 0) and a def
r of a focusing (f � 0) and a defocusing (f 0) quadrupole on the change in the beta function through the lens. (Hint: look at how alpha changes).As a result, where would you expect to find the maximum horizontal beta function in a beam transport channel composed of alternating focusing and defocusing quadrupoles?Courantnyder parameter evolution in a driftIn a drift space of length L we haveAnd

soThe lattice functions evolveA partic
soThe lattice functions evolveA particle evolvesIs a drift stable?TuneTune is the number of oscillations made per turnWhat is measurable and relevant is the noninteger part of the tune.28 .31Measuring the chromaticityOptimisethe systemSo if we squeeze beta at the symmetry point, we cannot avoid the resulting rapid rise in the function in the surrounding drift. However, we can optimiseLet’s f

ind the value of l which minimisesthe ma
ind the value of l which minimisesthe maximum beta, for a given betaSo we pick beta_0=l to minimisethe maximum beta growth.Beam momentsFor a complex and nonlinear beam distribution, we often work with the moments of the beam distributionWhat about the bending?Our first task to figure out the geometry, and define the curved reference orbit using a layout of dipole magnets. This forms the fundamental

footprint of the machine and defines our
footprint of the machine and defines our coordinate system for future analysis. dsrhothetaDispersion around ATLAS in the LHCDispersion in a short sector dipole and a quadrupoleFor a short sector dipole with bending angle theta small compared to 1We can find a simplified matrix for its entrance to its exitThis is useful for quick calculations and corresponds to a thin lens kick for an offmomentum p

article. quadrupolehas no driving term f
article. quadrupolehas no driving term for the dispersion and the 3x3 map is given byThe dispersionNote the upperleft 2 by 2 matrix is just the transfer matrix for a dipole we have already derived. This means the dispersion function obeys the matrix equations we have already derived. In a dipole there is also dispersion produced, or driven.This means the dispersion function in a quadrupole obeys the qu

adrupole transfer matrix. So the dispers
adrupole transfer matrix. So the dispersion function is focused in a quadrupole. However, there is no extra dispersion driven in a quadrupole. (so M13 and M23 are zero in the matrix on the last slide)Finally, as the motion is given as the sum of the betatron motion and the dispersionWe can write the general motion for x and x’ as a matrix equationThe matrix equation for D(s)How do we determine A

and B? We use the initial conditions at
and B? We use the initial conditions at s=0Inserting these into our general solution yieldsHence we can write the dispersion function asWhich we can write as a matrix equationHow do we calculate D(s)?We need to find a solution to the inhomogeneous Hill’s equation and add it to the general solution of the homogeneous equation.Imagine we have no gradients and delta=1. This means that D(s) is a sol

ution of the resulting inhomogeneous equ
ution of the resulting inhomogeneous equationWe have already solved the homogeneous equation, and found our matrix solutions.This means we need to find a particular solution of the inhomogeneous equation and add this solution to the solution of the homogeneous equation. Since the RHS is a constant, then a valid choice of a particular solution is a constant Inserting this into the inhomogeneous equatio

n above immediately givesAnd so our gene
n above immediately givesAnd so our general solution for D(s) is the sumDispersionCentral design orbit = closed for p=pClosed orbit for p pClosed orbit for p � Lattice propertyParticlemomentum errorDispersionDispersionSo our newly defined dispersion function D(s)it’s actually an orbit, and the one a particle with /p=1 would haveas it’s an orbit, it is focused by the latticethe mo

tion of the particle is the sum of our o
tion of the particle is the sum of our old x(s) and the dispersionOne way of viewing this equation is thinking of the dispersive term as an closed orbit around the machine, and the particle oscillates around this dispersive orbit through the usual betatron oscillations. This like a dipole error closed orbit distortion.What are typical values? Homogeneous and inhomogeneous Hill’s equationsNow we s

ee a little more structure to Hill’
ee a little more structure to Hill’s equations. It now looks likeWhich looks like the homogeneous version apart from a term linear in delta (and not x)The extra term on the RHS will drive the x motion of an offmomentum particle, which we shall call horizontal dispersion, or simply dispersion. Note there is no dispersion driving term in the vertical plane. The general solution for the horizontal mo

tion of a particle is given by the sum o
tion of a particle is given by the sum of two terms : the betatron motion term and an offmomentum dispersion termWe can think of x(s) as a closed orbit term, around which (s) oscillates.Let’s define a special orbit, D(s), which is followed by a particle with delta=1The inhomogeneous equation of motionNow we write the momenta in terms of the deviation ‘delta’xpand the vertical magnetic f

ield and binomially approximate the mome
ield and binomially approximate the momentum term (linear!) If we plug these results back into the equation of motion we get the followingNow we expand all of the brackets, keeping only terms linear in x and delta, and usinge obtain the inhomogeneous equation of motion, like before but with a righthand side term not containing x or its derivative. This is the inhomogeneous term and leads to dispersion:

What happens for a particle momentum not
What happens for a particle momentum not quite at the design value?So far we have considered beam motion when the particles have the design momentum p.We refer to these particles as onmomentum particlesHowever, in general, a particle’s momentum will be p + (something small)How do we introduce this into our formalism. Recall we hadWhich we can write asAnd expanding the right hand side and keeping

small terms. See relativity coursePART
small terms. See relativity coursePART SIXDISPERSION, CHROMATICITY, EMITTANCEBeta beat at the LHCBeta beatWhat about the change in beta function due to our quadrupole error q at sSkipping the derivation (which is short and standard), we obtainNote the beta perturbation is a function of s, so is a ‘beta wave’ around the ringThe distortion oscillates at twice the betatron frequency, which is

why it’s called a beta beat.Note a
why it’s called a beta beat.Note also the strength of the distortion is proportional to quad error and also the beta function at the position of the quadrupole error.Finally, we have a sin(2 pi nu) term in the denominator. This means the expression will get very large whenever the tune approaches a halfinteger. This is resonanceA distribution of quadrupole errorsIf we have a distribution of quadr

upole errors around the ring, the approx
upole errors around the ring, the approximate tune shift can be calculated fromThis can also be used to measure the beta functions. We vary a single quadrupole in the ring, and measure the tune, as the response is proportional to the beta function. In general the beta function tells you how sensitive the beam is to perturbations.For example, for LHC luminosity upgrades we may have to live with very larg

e beta functions in the arcs of the LHC.
e beta functions in the arcs of the LHC. This means the proton beams will be sensitive to field errors.Tune shiftsSo we have Note the following important features:the perturbed tune increases if �q 0, which corresponds to a focusing quadrupole i.e. focus more means more oscillations. So we get a positive tune shift for increased particle focusingThis means a pure quadrupole field error would sh

ift the tune one way in one plane and th
ift the tune one way in one plane and the other way in the other planeHowever, we can also get tune shifts from spacecharge, beambeam effectsand electron clouds, which can cause samesign tune shift in both planesThe effect of the quadrupole error in proportional to the local beta function. This is a common feature that the beta function magnifies local field errors.The tune shift perturbationLet’

;s assume the tune shift is small, and w
;s assume the tune shift is small, and writeIf we then expand the cosine function using a standard identityWe can write, because the tune shift is small,So we arrive atAnd so, cancelling sine and cosine terms from both sides, we arrive at a very important formulaAnd the tune shift isThe shift of the tuneThis gives, doing the matrix multiplicationThis is the perturbed oneturn map, and all symbols with

a ‘0’ subscript represent the
a ‘0’ subscript represent the values of the unperturbed machine.If we denote the tune and lattice functions of the perturbed machine by a subscript ‘p’, then the one turn map looks likeEquating the traces of these two matrices givesWhich relates the unpertubedand perturbed tune. If q is small, then the perturbed tune is close to the unperturbed tune. Quadrupole errorsImagine we ha

d an extra quadruple in our ring, or a q
d an extra quadruple in our ring, or a quadrupole field error. This would perturb the beam, and cause1) A change in the tune of the machine2) A change in the beta function of the machine (beta beat).Let’s calculate it. Imagine our quadrupole error had integrated strength KL=+q.This means it has a matrix which kicks the x’ of the particleIf we represent the rest of the machine by the one turn

map, then the effect on the global dynam
map, then the effect on the global dynamics of the machine can be calculated from the matrix productResonanceNote that our expression for the closed orbit distortion has an overall factor of This means that every time the tune becomes an integer, the argument of the sine becomes a multiple of pi, and so this factor diverges.This means the closed orbit distortion gets very large. This is an example of

resonance.Imagine the tune was 2pi in a
resonance.Imagine the tune was 2pi in a machine. Then the particle would encounter a dipole error at just one point in the machine every turn, and at the same point in it’s betatron oscillation. This means the effect of the dipole error adds up turn after turn after turn.We avoid this by minimisingmagnet errors and staying away from dangerous values of the tune. Here we should avoid integer tune va

lues.But we’ll soon see there are m
lues.But we’ll soon see there are may other resonances which occur at other tune values. Generally resonances occurClosed orbit distortionThe new closed orbit must obey the periodicity of the ringImagine we have a dipole kick error at some location sA short analysis gives the closed orbit distortion to beWhat does a code calculate for the dispersion?D(s) is created (or driven) by dipoles, focus

ed by quadrupoles and will grow in a dri
ed by quadrupoles and will grow in a drift if the angular dispersion D’ is nonzeroLHC cell opticsDispersion in a FODO cellThe dispersion in the middle of QF must satisfy the closed orbit conditionAnd if we solve the resulting equation, noting that (from our previous FODO analysis)We get, for the dispersion in QFWe can get the dispersion elsewhere by transforming this vector using our 3x3 maps. F

or example in the middle of QD we getDi
or example in the middle of QD we getDispersion in a FODO cellConsider a FODO cell with thin lens quadrupoles. Now we know dispersion is driven by dipoles we can calculate the dispersion function in the same way we computed the beta functions in a FODO cell.Let’s do it at the middle of the F quad, so we have a magnetic arrangementLooking at only the x motion we find the onecell mapWhich evaluates

to Here L is the length of each dipole,
to Here L is the length of each dipole, theta is the bend angle and f is the quadrupolefocal length. The upper 2x2 was obtained before, and now we have info on the dispersion. Closed orbit distortionThe design orbit defined by all of the dipoles in the ring is known as the closed design orbit.This is the perfect situationIf there is a small dipole kick the orbit will distort, and this distortion will

run around the entire ring. So a small k
run around the entire ring. So a small kick at some location will be seen anywhere in the ring!This closed orbit distortion defines a position dependent orbit offset around the ring. In effect the particles no longer oscillate around the design orbit but around a new closed orbitDesigning the latticeAccelerator codes simply assume a piecewisecontinuous representation of the accelerator structure.but b

ecause of edge focusing the number of ma
ecause of edge focusing the number of matrices is notthe same as the number of elements.Lattice design1) For a modern accelerator, lattice design work usually takes some years to finalize the design parameters. It is an iterative process, involving users, funding, accelerator physics, accelerator subsystems, civil engineering, etc. 2) It starts from major parameters such as energy, size, etc. 3) Then l

inear lattice is constructed based on th
inear lattice is constructed based on the building blocks. Linear lattice should fulfill accelerator physics criteria and provide global quantities such as circumference, emittancebetatrontunes, magnet strengths, and some other machine parameters. 4) Design codes such as MAD, OPA, BETA, Tracy, Elegant, AT, BeamOptics,.... are used for the matching of lattice functions and parameters calculations. 5) Usu

ally, a design with periodic cells is ne
ally, a design with periodic cells is needed in a circular machine. The cell can be FODO, Double Bend Achromat(DBA), Triple Bend Achromat(TBA), Quadruple Bend Achromat(QBA), or MultiBend Achromat(MBA or nBA) types. 6) Combinedfunction or separatedfunction magnets are selected. 7) Maximum magnetic field strengths are constrained. (roomtemperature or superconducting magnets, bore radius or chamber profile

, etc.) 8) Matching or insertion section
, etc.) 8) Matching or insertion sections are matched to get desired machine functions. 9) To get stable solution of the offmomentum particle, we need to put sextupolemagnets and RF cavities in the lattice beam line. Such nonlinear elements induce nonlinear beam dynamics and the dynamic acceptances in the transverse and longitudinal planes need to be carefully studied in order to get sufficientceptances

. (for long beam current lifetime and hi
. (for long beam current lifetime and high injection efficiency) 10) For the modern high performance machines, strong sextupolefields to correct high chromaticity will have large impact on the nonlinear beam dynamics and it is the most challenging and laborious work at this stage. 11) In the real machine, there are always imperfections in the accelerator elements. So, one need to consider engineering/al

ignment limitation or errors, vibrations
ignment limitation or errors, vibrations, etc. Correction schemes such as orbit correction, coupling correction, etc., need to be developed. (dipole correctors, skew quadrupoles, beam position monitors, etc Mini beta insertionsThis is the region around ATLAS in the LHC. Here we have a waist at ATLAS IP, and sitting at +/22 m are strong quadrupoles (in a fact a triplet) to make the beam waist. A

round these we have matching quadrupoles
round these we have matching quadrupoles to match the beta function back into the periodic solution in the LHC arc FODO cells. Problem : Large beta in the triplet quadrupoles…apertureBeta rise in a driftIf alpha is zero, the we have And so the beta function around the symmetry point increases rapidlyThis is very bad for accelerator designers!What happens can be understood in terms of the ellipse

. The area of the ellipse is constant, s
. The area of the ellipse is constant, so squeezing beta means we increase gamma � the beam rapidly diverges after it leaves the symmetry point. This is an example ofLiouvilletheorem, which states the the area occupied by a beam in phase space is constant as it moves through the accelerator.We saw our ellipse area was constant, which is LiouvilleBeta rise in a driftSo we build our ring out of di

poles and FODO cells.What about an exper
poles and FODO cells.What about an experiment or a region free of magnets for diagnostics?We need to stop focusing for a while………what will happen?Remember we derived the expression for the evolution of the beta function in a drift?What happens to our beta function in a drift, around a symmetry point. Here, the beta function becomes a minimum, and so alpha is zero. For example,LHC cell o

pticsThe LHC cellExerciseWe looked at
pticsThe LHC cellExerciseWe looked at a FODO cell where the focal length of the focusing and defocusing quadrupoles was the same. Imagine they were different.What would this mean for the phase advance in the x and the z plane?Assuming thin lens optics, compute the phase advance in each plane for a focusing quadrupole with focal length fand a defocusing quadrupole with focal length Beta in a FODO cell

Now can compute the lattice functions i
Now can compute the lattice functions in the cell. Note that beta in the focusing and defocusing quadrupoles are maximisedthere, and this maximum depends solely on the cell length and phase advanceQuestion how would you compute the the evolution ofhe beta function in a FODO cell if you know the beta in the focusing quadrupole?How about expressions for the maximum and minimum beta in a FODO cell in te

rms of length and phase?UsingWe getIn t
rms of length and phase?UsingWe getIn the D quadTransfer line stabilityOur stability equation from the previous slide seems slightly odd at firstIt seems to say motion is stable when focusing is weak!So strong quadrupolesare not necessarily better!A thin lens FOD0 cellWe can also make our life easier and compute the matrix for our FODO cell using the thin lens matrices. Again, starting from the mid

dle of QF we haveDoing the mathematics w
dle of QF we haveDoing the mathematics we end up with the matrix in terms of L and fThis contains lots of information. Straight away we can ask for what parameters is the FODO cell going to gives stable motion. This meansWe can also write the cell phase advance in terms of the parameters:Properties of our FODO cellIs the FODO cell stable? For this we need the (mod) trace of the one turn map to be less

than or equal to 2. Here it is 1.415. So
than or equal to 2. Here it is 1.415. So this FODO cell will give stable dynamics in this plane.What is the phase advance per cell? RecallAnd so the phase advance per cell is 45 degrees. This is a “45 degree cell”.What are the lattice functions at the point of the one turn map? For us, this is in the middle of the focusing quadrupole. Well, we useAnd find that beta=11.611 m and alpha=0 (what

does the ellipse look like?)What does MA
does the ellipse look like?)What does MAD compute? Try it yourself! (www.cern.ch/mad)The FODO cellLet’s be concrete and take some real numbersK = +/0.54102 m= 0.5 m= 2.5 Thisgives, wedo themathsThis is the one period map of the FODO cell, and so has the formThere is lots of information here!And, really, we are just doing by hand what an optics code would do…The dynamics in a FODO cellTo un

derstand the beam dynamics in a FODO cel
derstand the beam dynamics in a FODO cell we need to compute the one period map.To do this we simply multiple the matrices of the components of the cell together, conventionally starting in the middle of one of the quadrupoles, which means we start and end with a quadrupole matrix of half strengthRecall And we multiply these matrices in sequenceFirst element!ER The FODO cellRecall that two quadrupole

s of opposite polarity could provide foc
s of opposite polarity could provide focusing in both planes at the same time. This is the fundamental building block of the FODO lattice.The basic building block of this periodic structure is the FODO cell, consisting of a horizontally focusing quadrupole (F), a space (O), a defocusing quadrupole (D) and a space (O).We can repeat the FODO cell to make a FODO channel. Note the drift space (O) can conta

in nothing, a bend, some diagnostics, an
in nothing, a bend, some diagnostics, an RF cavity or even an experiment!The LHCexisting collimatorsAFP220150 m190 m~225 mpossible new collimatorExample the LHCA storage ringInjectionExtractionCollimationR.F.DipolesFocusing elementsDipolesLattice designSynchrotrons and high energy particle colliders� circular machines� so we need plenty of dipoles in the lattice to bend the part

icles around the ring� this creat
icles around the ring� this creates the design orbit of the machineThen once the design orbit is sorted out, we need to design the magnetic lattice, and position the quadrupoles and higher order magnets. This is lattice design.PART FOUROPTICS AND LATTICE DESIGNA dipole error (from a quad offset in this case)Real life and field errorsRecall that we started with an arbitrary field and made an ex

pansionWhere the first term is the dipo
pansionWhere the first term is the dipole steering field, the second is the quadrupole term, the next is the sextupoleterm and so on.To realisethese fields we build the magnets and specify some field quality. These magnets will never be perfect, and most synchrotrons specify a field quality of one part in 10,000.Therefore these magnets will have mostly the field component you want, but will have small

contributions of higher order field com
contributions of higher order field components. What do these do to the beam?We also need to place magnets in the correct place. A quadrupole can be misaligned, so generating an additional dipole field in the beam. (can you show this?)Or a power upply to a dipole or a quadrupole may be slightly away from design.The bottom line is our lattice is never as we designed it and we need to deal with field er