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.
Slide 3
A storage ring Injection Extraction Collimation R.F. Dipoles
Focusing elements Dipoles
Slide 4
Example the LHC
Slide 5
The LHC existing collimators AFP220 150 m 190 m TCL5 TCL4 TCL6
~225 m possible new collimator
Slide 6
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!
Slide 7
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!
Slide 8
The FODO cell Lets be concrete and take some real numbers K =
+/- 0.54102 m -2 l q = 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
Slide 9
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)
Slide 10
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:
Slide 11
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!
Slide 12
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
Slide 13
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 f 1 and a
defocusing quadrupole with focal length f 2
Slide 14
The LHC cell
Slide 15
LHC cell optics
Slide 16
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 whilewhat 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,
Slide 17
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.
Slide 18
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 quadrupolesaperture
Slide 19
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)
Slide 20
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.
Slide 21
But reality is not the same as your code
Slide 22
PART FIVE- ERROR AND MISALIGNMENTS
Slide 23
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.
Slide 24
A dipole error (from a quad offset in this case)
Slide 25
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
Slide 26
Closed orbit distortion The new closed orbit must obey the
periodicity of the ring Imagine we have a dipole kick error at some
location s 0 A short analysis gives the closed orbit distortion to
be
Slide 27
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 its 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 well soon see there are may other resonances which
occur at other tune values. Generally resonances occur
Slide 28
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). Lets 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
Slide 29
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.
Slide 30
The tune shift perturbation Lets 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
Slide 31
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.
Slide 32
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.
Slide 33
Beta beat What about the change in beta function due to our
quadrupole error q at s 0 ? 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 its 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
Slide 34
Beta beat at the LHC
Slide 35
PART SIX- DISPERSION, CHROMATICITY, EMITTANCE
Slide 36
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 particles 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
Slide 37
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:
Slide 38
Homogeneous and inhomogeneous Hills equations Now we see a
little more structure to Hills 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 x i (s) as a closed orbit term, around which x h (s)
oscillates. Lets define a special orbit, D(s), which is followed by
a particle with delta=1
Slide 39
Dispersion So our newly defined dispersion function D(s) its
actually an orbit, and the one a particle with dp/p=1 would have as
its 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?
Slide 40
Dispersion
Slide 41
Central design orbit = closed for p=p 0 Closed orbit for p <
p 0 Closed orbit for p > p 0 Lattice property Particles momentum
error
Slide 42
How do we calculate D(s)? We need to find a solution to the
inhomogeneous Hills 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
Slide 43
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
Slide 44
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
Slide 45
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
Slide 46
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. Lets 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.
Slide 47
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
Slide 48
LHC cell optics
Slide 49
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
Slide 50
Dispersion around ATLAS in the LHC
Slide 51
Beta functions and dispersion in the LHC arcs at injection Beta
functions Beam sizes Dispersion
Slide 52
Lattice building blocks Weve 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. Lets look at
achromats in more detail.
Slide 53
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.
Slide 54
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 D c 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 D c =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 D c is hence Note the dispersion at the quad becomes
higher for longer distances and bigger bend angles. This analysis
shows its possible but in practice we need extra quads for matching
and maybe a reduction of the required quad strength by splitting
the central quad.
Slide 55
DBA structure with a single quadrupole (sometimes called
Chasman-Green)
Slide 56
A DBA structure with a quadrupole triplet (vertical)
Slide 57
The long straight section of the LHeC collider (optical work
done by CI)
Slide 58
The 4 magnet chicane
Slide 59
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 particles trip around
the machine will a different to the reference particle What is this
circumference, or path length, error? Consider this situation:
Slide 60
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
Slide 61
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 > 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.
Slide 62
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.
Slide 63
Chromaticity If the machine tunes depend on the momentum
deviation, we can write linearly Where weve 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 weve 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)
Slide 64
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.
Slide 65
Chromaticity Its 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
Slide 66
Chromaticity in the LHC
Slide 67
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
Slide 68
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!
Slide 69
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.
Slide 70
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 its not all perfect.
Remember we ignored plenty of terms in the fields of the
sextupolessome 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 machines dynamic aperture, or what
amplitude of particle can survive or many turns.
Slide 71
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)
Slide 72
Beams of particles and emittance So far weve 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.
Slide 73
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 particles
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
Slide 74
An aside : Louvilles theorem Liouvilles 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 Louvilles theorem. But
this is beyond this course But sympleciticty in 2D phase space is
equivalent to det(M)=1
Slide 75
The end
Slide 76
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
Slide 77
Beam moments For a complex and non-linear beam distribution, we
often work with the moments of the beam distribution
Slide 78
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: Lets find the
value of l which minimises the maximum beta, for a given beta 0 So
we pick beta_0=l to minimise the maximum beta growth.
Slide 79
Measuring the chromaticity
Slide 80
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
Slide 81
Courant-Snyder parameter evolution in a drift In a drift space
of length L we have And so The lattice functions evolveA particle
evolves Is a drift stable?
Slide 82
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?