1. Lecture 3 Transverse Optics II
Professor Emmanuel Tsesmelis
Directorate Office, CERN
Department of Physics, University of Oxford
ACAS School for Accelerator Physics 2012
Australian Synchrotron, Melbourne
November/December 2012

2. Contents – Lecture 3 Phase Space Liouville’s Theorem
Emittance & Acceptance
Normalised Phase Space
Matrix Formalism
Lattices
Tune Calculations
Phase Space & Betatron Tune
Introduction to Accelerators

3. Phase Space Ellipse (1) So now have an expression for x and x’ and
If plot x’ versus x as  goes from 0 to 2, get an ellipse, which is called the phase space ellipse.
 = 3/2 x’ x
 = 0 = 2
Introduction to Accelerators

4. Phase Space Ellipse (2)
As particle moves around the machine, the shape of the ellipse will change as  changes under the influence of the quadrupoles.
However, the area of the ellipse () does not change x’ x x’ x
Area = · r1· r2
 is called the transverse emittance and is determined by the initial beam conditions.
The units are meter·radians, but in practice we use more often mm·mrad.
Introduction to Accelerators

5. Liouville’s Theorem (1)
Conservation law of phase space.
Consider beam of particles as cloud of points within closed contour in transverse space.
Contour is typically an ellipse.

6. Liouville’s Theorem (2)
FODO Cell
According to Liouville emittance area conserved as beam circulates in synchrotron or passes down transport line irrespective of magnetic focusing or bending operation.
narrow waist
& c) diverging beam
d) broad max. at F lens centre

7. Liouville’s Theorem (3)
Equation of ellipse
Related to Twiss parameters
Called Courant & Snyder invariant)
Strictly speaking:
Particle density in phase space is constant if particles move in external B-field or in general field in which forces do not depend upon velocity.
Parameters of phase-space ellipse
containing emittance at point in lattice
between quadrupoles.

8. Exceptions to Liouville’s Theorem
Liouville’s Theorem not applicable when:
Space-charge forces within beam play a role
Particles emit synchrotron light (velocity-dependent effect).
Liouville’s Theorem can thus be reliably applied to proton beams (do not normally emit synchrotron light) and to electrons travelling for few turns in synchrotron.

9. Normalized Phase Space
by’ y
Circle of radius
By multiplying the y-axis by β the phase space is normalized and the ellipse turns into a circle.
Introduction to Accelerators

10. Normalized Emittance & Hamiltonian Mechanics
Normalized emittance is:
As beam is accelerated in a synchrotron, normalized emittance is conserved and physical emittance within RHS falls inversely with momentum.
Close to light velocity, implies that it is inversely proportional to energy:
Beam dimensions shrink as 1/(p01/2).
Adiabatic damping

11. Chains of Accelerators
Adiabatic damping implies that proton accelerators need full aperture at injection.
Economical to split single large ring into chain of accelerators.
Smaller-radius rings have large apertures
Higher-energy rings with large radius have smaller apertures.
CERN Accelerator Complex

12. Beam Distribution in Real Space (1)x’ x x’ x
The projection of the ellipse on the x-axis gives the physical transverse beam size.
Therefore, the variation of (s) around the machine will describe how the transverse beam size will vary.
Introduction to Accelerators

13. Beam Distribution in Real Space (2)
Particles in beam distributed in population which appears Gaussian when projected in vertical or horizontal plane.
Protons – emittance boundary chosen to include 90% of Gaussian beam at 2
Electrons – safer to put boundary at 6 to 10 (redistribution of position due to quantum emission and damping).
Difference (factor 4) between emittance for protons and electrons:

14. Emittance & Acceptance
To be rigorous we should define the emittance slightly differently.
Observe all the particles at a single position on one turn and measure both their position and angle.
This will give a large number of points in our phase space plot, each point representing a particle with its co-ordinates x, x’.
beam x’ x
emittance
acceptance
The emittance is the area of the ellipse, which contains all, or a defined percentage, of the particles.
The acceptance is the maximum area of the ellipse, which the emittance can attain without losing particles.
Introduction to Accelerators

15. Acceptance
Defined as the size of the hole which is the vacuum chamber transformed into phase space:
Particle location that grazes an obstacle (e.g. vacuum chamber) defines acceptance.

16. Matrix Formalism
Represent the particles transverse position and angle by a column matrix.
As the particle moves around the machine the values for x and x’ will vary under influence of the dipoles, quadrupoles and drift spaces.
These modifications due to the different types of magnets can be expressed by a Transport Matrix M
If x1 and x1’ are known at some point s1 then can calculate particle’s position and angle after the next magnet at position s2 using:
Introduction to Accelerators

17. How to Apply the Formalism
If want to know how a particle behaves in the accelerator as it moves around using the matrix formalism, need to:
Split machine into separate element as dipoles, focusing and defocusing quadrupoles, and drift spaces.
Find the matrices for all of these components.
Multiply them all together.
Calculate what happens to an individual particle as it makes one or more turns around the machine.
Introduction to Accelerators

18. Matrix for a Drift Space
A drift space contains no magnetic field.
A drift space has length L.
x2 = x1 + L.x1’ L
x1’ x1
x1’ small }
Introduction to Accelerators

19. Matrix for a Quadrupole (1)
deflection
A quadrupole of length L.
Remember By  x and the deflection due to the magnetic field is:
Provided L is small }
Introduction to Accelerators

20. Matrix for a Quadrupole (2)
Found that:
Define the focal length of the quadrupole as f=
Introduction to Accelerators

21. What Next?
For our purposes, treat dipoles as simple drift spaces as they bend all the particles by the same amount.
Have Transport Matrices corresponding to drift spaces and quadrupoles.
These matrices describe the real discrete focusing of quadrupoles.
Now must combine these matrices with solution to Hill’s equation, since they describe the same motion……
Introduction to Accelerators

22. Matrices & Hill’s Equation (1)
Multiply the matrices of drift spaces and quadrupoles together to form a transport matrix that describes a larger section of accelerator.
These matrices will move particle from one point (x(s1),x’(s1)) on phase space plot to another (x(s2),x’(s2)), as shown in the matrix equation below.
The elements of this matrix are fixed by the elements through which the particles pass from point s1 to point s2.
However, can also express (x,x’) as solutions of Hill’s equation.
and
Introduction to Accelerators

23. Matrices & Hill’s Equation (2)
Assume that transport matrix describes a complete turn around the machine.
Therefore: (s2) = (s1)
Let  be the change in betatron phase over one complete turn.
Then get for x(s2):
Introduction to Accelerators

24. Matrices & Hill’s Equation (4)
So, for the position x at s2 obtain…
Equating the ‘sine’ terms gives:
Which leads to:
Equating the ‘cosine’ terms gives:
Which leads to:
Can be repeated for c and d.
Introduction to Accelerators

25. Matrices & Twiss Parameters
Previously defined:
These are called TWISS Parameters
Also  is the total betatron phase advance over one complete turn:
Tune: Number of betatron oscillations per turn
Transport matrix becomes:
Introduction to Accelerators

26. Lattice Parameters
This matrix describes one complete turn around machine and will vary depending on the starting point (s).
If start at any point and multiply all of the matrices representing each element all around the machine, can calculate α, β, γ and μ for that specific point, which will then give β(s) and Q.
If repeat this many times for many different initial positions (s), can calculate the Lattice Parameters for all points around the machine.
Introduction to Accelerators

27. Lattice Calculations and Codes
Obviously  (or Q) is not dependent on the initial position ‘s’, but can calculate the change in betatron phase, d, from one element to the next.
Computer codes like “MAD” or “Transport” vary lengths, positions and strengths of the individual elements to obtain the desired beam dimensions or envelope ‘β(s)’ and the desired ‘Q’.
Often a machine is made of many individual and identical sections (FODO cells). In that case only calculate a single cell and not the whole machine, as the functions β(s) and dμ will repeat themselves for each identical section.
The insertion section has to be calculated separately.
Introduction to Accelerators

28. The (s) and Q Relation , where μ = Δ over a complete turn
But we also found:
This leads to:
Over one complete turn
Increasing the focusing strength decreases the size of the beam envelope (β) and increases Q and vice versa.
Introduction to Accelerators

29. Tune Corrections (1)
What happens if change the focusing strength slightly?
The Twiss matrix for ‘FODO’ cell is given by:
Add a small QF quadrupole, with strength dK and length ds.
This will modify the ‘FODO’ lattice, and add a horizontal focusing term:
The new Twiss matrix representing the modified lattice is:
Introduction to Accelerators

30. Change in phase advance
Tune Corrections (2)
This gives
This extra quadrupole will modify the phase advance  for the FODO cell.
1 =  + d
New phase advance
Change in phase advance
If d is small then can ignore changes in β
So the new Twiss matrix is just:
Introduction to Accelerators

31. Tune Corrections (3)
These two matrices represent the same FODO cell, therefore:
Which equals:
Combining and comparing the first and the fourth terms of these two matrices gives:
Introduction to Accelerators

32. Tune Corrections (4) Remember 1 =  + d and dμ is small , but:
dQ = dμ/2π
In the horizontal plane this is a QF
If follow the same reasoning for both transverse planes for both QF and QD quadrupoles QF QD
Introduction to Accelerators

33. Tune Corrections (5)
Let dkF = dk for QF and dkD = dk for QD
bhF, bvF = b at QF and bhD, bvD = b at QD
Then:
This matrix relates the change in the tune to the change in strength of the quadrupoles.
Can invert this matrix to calculate change in quadrupole field needed for a given change in tune.
Introduction to Accelerators

34. Phase Space & Betatron Tune
If fold out a trajectory of a particle that makes one turn in machine with a tune of Q = 3.333, get: 2p y
by’ y
This is the same as going times around on the circle in phase space.
The net result is times around on the circle.
q is the fractional part of Q
So, q =
2πq
Introduction to Accelerators