PART FOUR- OPTICS AND LATTICE DESIGN

Lattice design

Synchrotrons and high energy particle colliders -> circular machines -> so we need plenty of dipoles in the lattice to bend the particles around the ring -> this creates the design orbit of the machine Then 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.

A storage ring

Injection

Extraction

Collimation

R.F.

Dipoles

Focusing elements

Dipoles

Example – the LHC

The LHC

existing collimators

AFP220

150 m 190 m TCL5

TCL4

TCL6 ~225 m

possible new collimator

ER The FODO cell

Recall that two quadrupoles 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 contain nothing, a bend, some diagnostics, an RF cavity or even an experiment!

The dynamics in a FODO cell

To understand 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 strength Recall

And we multiply these matrices in sequence

First element!

The FODO cell

Let’s be concrete and take some real numbers K = +/- 0.54102 m-2

lq = 0.5 m L = 2.5 m This gives, if we do the maths

This is the one period map of the FODO cell, and so has the form

There is lots of information here! And, really, we are just doing by hand what an optics code would do…

Properties of our FODO cell

Is 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 this FODO cell will give stable dynamics in this plane. What is the phase advance per cell? Recall

And 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 use

And find that beta=11.611 m and alpha=0 (what does the ellipse look like?) What does MAD compute? Try it yourself! (www.cern.ch/mad)

A thin lens FOD0 cell

We can also make our life easier and compute the matrix for our FODO cell using the thin lens matrices. Again, starting from the middle of QF we have

Doing the mathematics we end up with the matrix in terms of L and f

This contains lots of information. Straight away we can ask for what parameters is the FODO cell going to gives stable motion. This means

We can also write the cell phase advance in terms of the parameters:

Transfer line stability

Our stability equation from the previous slide seems slightly odd at first

It seems to say motion is stable when focusing is weak! So strong quadrupoles are not necessarily better!

Beta in a FODO cell

Now can compute the lattice functions in the cell. Note that beta in the focusing and defocusing quadrupoles are maximised there, and this maximum depends solely on the cell length and phase advance

Question – how would you compute the the evolution of the 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 terms of length and phase?

Using

We get

In the D quad

Exercise

We 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 f1 and a defocusing quadrupole with focal length –f2

The LHC cell

LHC cell optics

Beta rise in a drift

So we build our ring out of dipoles 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,

Beta rise in a drift

If alpha is zero, the we have

And so the beta function around the symmetry point increases rapidly

This is very bad for accelerator designers! What happens can be understood in terms of the ellipse. 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 of Liouville theorem, 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 Dr Liouville.

Mini beta insertions

This 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. Around 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…aperture

Lattice design

1) 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 linear lattice is constructed based on the building blocks. Linear lattice should fulfill accelerator physics criteria and provide global quantities such as circumference, emittance, betatron tunes, 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 (DBA), Triple Bend Achromat (TBA), Quadruple Bend Achromat (QBA), or Multi-Bend Achromat (MBA or nBA) types. 6) Combined-function or separated-function magnets are selected. 7) Maximum magnetic field strengths are constrained. (room- temperature 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 solution of the off-momentum particle, we need to put sextupole magnets 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 sextupole fields 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 correction, etc., need to be developed. (dipole correctors, skew quadrupoles, beam position monitors, etc)

Designing the lattice

Accelerator codes simply assume a piecewise-continuous representation of the accelerator structure. but because of edge focusing the number of matrices is not the same as the number of elements.

But reality is not the same as your code

PART FIVE- ERROR AND MISALIGNMENTS

Real life and field errors

Recall that we started with an arbitrary field and made an expansion

Where the first term is the dipole steering field, the second is the quadrupole term, the next is the sextupole term and so on. To realise these 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 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 supply 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 errors.

A dipole error (from a quad offset in this case)

Closed orbit distortion

The design orbit defined by all of the dipoles in the ring is known as the closed design orbit. This is the perfect situation If there is a small dipole kick – the orbit will distort, and this distortion will 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 orbit

Closed orbit distortion

The new closed orbit must obey the periodicity of the ring Imagine we have a dipole kick error at some location s0

A short analysis gives the closed orbit distortion to be

Resonance

Note 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 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 minimising magnet errors and staying away from dangerous values of the tune. Here we should avoid integer tune values. But we’ll soon see there are may other resonances which occur at other tune values. Generally resonances occur

Quadrupole errors

Imagine we had an extra quadruple in our ring, or a quadrupole field error. This would perturb the beam, and cause 1) A change in the tune of the machine 2) 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 particle

If we represent the rest of the machine by the one turn map, then the effect on the global dynamics of the machine can be calculated from the matrix product

The shift of the tune

This gives, doing the matrix multiplication

This is the perturbed one-turn map, and all symbols with 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 like

Equating the traces of these two matrices gives

Which relates the unpertubed and perturbed tune. If q is small, then the perturbed tune is close to the unperturbed tune.

The tune shift perturbation

Let’s assume the tune shift is small, and write

If we then expand the cosine function using a standard identity

We can write, because the tune shift is small,

So we arrive at

And so, cancelling sine and cosine terms from both sides, we arrive at a very important formula

And the tune shift is

Tune shifts

So 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 focusing This means a pure quadrupole field error would shift the tune one way in one plane and the other way in the other plane However, we can also get tune shifts from space-charge, beam-beam effects and electron clouds, which can cause same-sign tune shift in both planes The 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.

A distribution of quadrupole errors

If we have a distribution of quadrupole errors around the ring, the approximate tune shift can be calculated from

This 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 large beta functions in the arcs of the LHC. This means the proton beams will be sensitive to field errors.

Beta beat

What about the change in beta function due to our quadrupole error q at s0? Skipping the derivation (which is short and standard), we obtain

Note the beta perturbation is a function of s, so is a ‘beta wave’ around the ring The distortion oscillates at twice the betatron frequency, which is 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 half-integer. This is resonance

Beta beat at the LHC

PART SIX- DISPERSION, CHROMATICITY, EMITTANCE

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 on-momentum particles However, in general, a particle’s momentum will be p + (something small)

How do we introduce this into our formalism. Recall we had

Which we can write as

And expanding the right hand side and keeping small terms.

See relativity course

The inhomogeneous equation of motion

Now we write the momenta in terms of the deviation ‘delta’

Expand the vertical magnetic field and binomially approximate the momentum term (linear!)

If we plug these results back into the equation of motion we get the following

Now we expand all of the brackets, keeping only terms linear in x and delta, and using

we obtain the inhomogeneous equation of motion, like before but with a right-hand side term not containing x or its derivative. This is the inhomogeneous term and leads to dispersion:

Homogeneous and inhomogeneous Hill’s equations

Now we see a little more structure to Hill’s equations. It now looks like

Which 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 off-momentum 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 motion of a particle is given by the sum of two terms : the betatron motion term and an off-momentum dispersion term

We can think of xi(s) as a closed orbit term, around which xh(s) oscillates. Let’s define a special orbit, D(s), which is followed by a particle with delta=1

Dispersion

So our newly defined dispersion function D(s) it’s actually an orbit, and the one a particle with dp/p=1 would have as it’s an orbit, it is focused by the lattice the motion of the particle is the sum of our old x(s) and the dispersion

One 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?

Dispersion

Dispersion

Central design orbit = closed for p=p0 Closed orbit for p < p0

Closed orbit for p > p0

Lattice property

Particle’s momentum error

How 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 solution of the resulting inhomogeneous equation

We 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 equation above immediately gives

And so our general solution for D(s) is the sum

The matrix equation for D(s)

How do we determine A and B? We use the initial conditions at s=0

Inserting these into our general solution yields

Hence we can write the dispersion function as

Which we can write as a matrix equation

The dispersion

Note the upper-left 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 quadrupole 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 dispersion

We can write the general motion for x and x’ as a matrix equation

Dispersion in a short sector dipole and a quadrupole

For a short sector dipole with bending angle theta small compared to 1

We can find a simplified matrix for its entrance to its exit

This is useful for quick calculations and corresponds to a thin lens kick for an off-momentum particle. A quadrupole has no driving term for the dispersion and the 3x3 map is given by

Dispersion in a FODO cell

Consider 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 arrangement

Looking at only the x motion we find the one-cell map

Which evaluates to

Here L is the length of each dipole, theta is the bend angle and f is the quadrupole focal length. The upper 2x2 was obtained before, and now we have info on the dispersion.

Dispersion in a FODO cell

The dispersion in the middle of QF must satisfy the closed orbit condition

And if we solve the resulting equation, noting that (from our previous FODO analysis)

We get, for the dispersion in QF

We can get the dispersion elsewhere by transforming this vector using our 3x3 maps. For example in the middle of QD we get

LHC cell optics

What does a code calculate for the dispersion?

D(s) is created (or driven) by dipoles, focused by quadrupoles and will grow in a drift if the angular dispersion D’ is non-zero

Dispersion around ATLAS in the LHC

Beta functions and dispersion in the LHC arcs at injection

Beta functions

Beam sizes

Dispersion

Lattice building blocks

We’ve seen how to combine alternating gradient quadrupoles to make a focusing 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 quadrupoles that can give stable motion For example a FODO cell to transport a beam

We can talk about dispersion-free lattices, which are important in many applications. These allow bending of the beam without generating dispersion. Examples are Chasman-green, triple-bend achromat

We also can build dispersion suppressor, which match the periodic dispersion

in the arc (perhaps made of FODO cells) into a dispersion-free straight. We can also displace the beam transversely without generating dispersion

using a sequence of only bends. Sometimes called a geometrical achromat.

Let’s look at achromats in more detail….

The double bend achromat (DBA)

If the dispersion function is non-zero 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 cancelling dispersion” In principle, dispersion can be suppressed by one focusing quadrupole and one bending magnet. With one focusing quad in the middle between two dipoles, one can get the achromat condition, which means no additional dispersion is driven by the structure. Due to mirror symmetry of the lattice w.r.t. to the middle quad D’ at quad center should be zero (see upcoming plot). This is so called double bend achromat (DBA) structure. We generally need quads outside DBA section to match the betatron functions, tunes, etc. Similarly, one can design triple bend achromat (TBA), quadruple bend achromat (QBA), and multi-bend achromat (MBA or nBA) structure.

DBA

Consider a simple DBA cell with a single quadrupole in the middle. We use the thin lens approximation and write down the dispersion matching condition i.e. We want some dispersion Dc in the middle of the quadrupole and feed into the system zero dispersion

Here f is quadrupole focal length, theta and L are the bend parameters and L is the distance between the quadrupole and bend centers. In essence we match to the Dc’=0 condition at the middle of the quadrupole. Or the quad turns over the sign of the dispersion generated by the bend. The required focal length and resulting Dc is hence

Note 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.

DBA structure with a single quadrupole (sometimes called Chasman-Green)

A DBA structure with a quadrupole triplet (vertical)

The long straight section of the LHeC collider (optical work done by CI)

The 4 magnet chicane

Momentum compaction

So, a momentum offset changes the horizontal orbit of a particle through dispersion. Ideally, a machine with only horizontal bends does not generate any vertical dispersion However, dispersion does generate a longitudinal effect, as the total circumference of an off momentum particle’s trip around the machine will a different to the reference particle What is this circumference, or path length, error? Consider this situation:

Momentum compaction

The path length deviation is given by

The change in circumference of the machine is given by an integral over the whole ring

For the case where the closed orbit distortion is given by a momentum error

We define the linear momentum compaction factor

So we then have an integral around the ring to compute the momentum compaction factor

so

Typical lattices and momentum compaction

The momentum compaction factor is an important lattice design parameter Typically it is given by

A large value means the path length varies a lot for off-momentum 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 <D> > 0, so the particles tend to orbit outer side of the ring. In this case the momentum compaction factor is > 0 (and the orbit gets longer). An isochronous lattice is designed to counter this natural tendency, normally to some order in the momentum deviation.

Chromaticity

Consider some particles of slightly different energy passing through a FODO cell

Higher 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.

Chromaticity

If 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 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 have

Expanding out the brackets and dropping terms of higher order, we are left with our equation of motion, this time with a chromatic term

Where we defined (as normal)

Chromaticity

We already know how to compute the effect of a quadrupole field error. Recall the tune shift from a quadrupole error k(s) in our lattice

We can think of this chromatic terms as a quadrupole field error of strength

A similar analysis in the vertical plane would have a found a chromatic perturbation of

Which means we can write down the tune-shift arising from the chromatic perturbation term,

An expression which is linear in the momentum deviation.

Chromaticity

It’s conventional to define the tune change per unit delta as the chromaticity

Where we call the chromaticity ‘natural’ as it arises from the quadrupoles. Any lattice with quadrupoles naturally generates this chromaticity. Similarly in the vertical plane

Note as the horizontal beta function is biggest in horizontally focusing quadrupoles (and vice versa) means the natural chromaticity is normally negative in both planes. The linear chromaticity is sometimes written as Q

For a FODO cell we can show that

Chromaticity in the LHC

Why is chromaticity bad?

Chromaticity is naturally generated by any focusing lattice. So when we have non-zero 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 tune-space. This is a plot of tune at HERA, showing uncorrected chromaticity and not a clean tune signal

And if we correct it

So how do we do this? Chromaticity basically comes about when a particle which is slightly off-momentum 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 sextupole!

Sextupoles

A sextupole field has field components given by

Note 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 sextupole couples the beam planes! An off-momentum particle passing through the sextupole has displacement

Where we define the sexupole strength by

And so the fields seen by the particle are found by substitution

There are many terms here, some helpful and some harmful. The helpful ones for us are

The horizontal dispersion function has made each sextupole into 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.

Sextupoles

So the dispersion effectively makes the sextupole into a quadrupole with a momentum dependent focusing gradient

This 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 of the terms are good and fix our chromaticity, but some are bad and introduce non-linearities and coupling into our accelerator ring.

It is difficult to represent sextupole in our linear formalism, and often the best way to understand the impact of sextupole fields is to track particles with matrices, and stopping to be more careful every time a sextupole is encountered. This leads to the study of a machine’s dynamic aperture, or what amplitude of particle can survive or many turns.

Lattice design (continued) 1) 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 linear lattice is constructed based on the building blocks. Linear lattice should fulfill accelerator physics criteria and provide global quantities such as circumference, emittance, betatron tunes, 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 (DBA), Triple Bend Achromat (TBA), Quadruple Bend Achromat (QBA), or Multi-Bend Achromat (MBA or nBA) types. 6) Combined-function or separated-function magnets are selected. 7) Maximum magnetic field strengths are constrained. (room- temperature 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 solution of the off-momentum particle, we need to put sextupole magnets 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 sextupole fields 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 correction, etc., need to be developed. (dipole correctors, skew quadrupoles, beam position monitors, etc)

Beams of particles and emittance So far we’ve defined the emittance of a particle, which was our constant in the Courant-Snyder analysis. The motion of an individual particle is completely specified by its emittance and initial phase. Different particles have different emittances and initial phases but they all have the same Courant-Snyder functions (how true is this?!) For example, the particle with x=x’=0 will have zero emittance and 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 characterise a beam of particles, each with their own emittance.

Beams of particles and emittance

In general, we have more than one particle in our beam, each with its own value for the emittance (or Courant-Synder invariant).

We choose one of the particle’s emittance to represent the emittance of the entire beam. For example, characterise the beam by the emittance of the particle for which 95% of the beam particles are within the ellipse of this particle. Another useful definition, when dealing with complicated distributions, is the RMS emittance, which we find by averaging over the beam distribution

An aside : Louville’s theorem

Liouville’s theorem : the density of points representing particles in 6-D (x, p) phase space is conserved if any forces conservative and differentiable Radiation 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 Hamiltonian system Note: acceleration keeps (x,p) phase space constant, but reduces (x, x’) phase space , so there is no violation of Liouville theorem Transfer maps derived from a Hamiltonian have a mathematical property called symplecticity, which is deeply linked to Louville’s theorem. But this is beyond this course… But sympleciticty in 2D phase space is equivalent to det(M)=1

The end

What 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 coordinate system for future analysis.

ds

rho theta

Beam moments

For a complex and non-linear beam distribution, we often work with the moments of the beam distribution

Optimise the system

So 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 optimise:

Let’s find the value of l which minimises the maximum beta, for a given beta0

So we pick beta_0=l to minimise the maximum beta growth.

Measuring the chromaticity

Tune

Tune is the number of oscillations made per turn What is measurable and relevant is the non-integer part of the tune .28 .31

Courant-Snyder parameter evolution in a drift

In a drift space of length L we have

And so

The lattice functions evolve A particle evolves

Is a drift stable?

Exercise

Consider 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 quadrupole and what expression tells you how the lattice functions evolve when you know the transfer matrix?) Contrast the behaviour 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?

A quadruple bend achromat structure