1. Lecture 3 - Magnets and Transverse Dynamics I
ACCELERATOR PHYSICS
Lecture 3
Longitudinal Motion
TT 2012
E. J. N. Wilson

2. Physics of the electron v. protons
When we collide Protons we collide complex assemblies of three quarks
Only two quarks interact
Their available energy is on average only 10% of the total centre of mass energy
We do not know which quarks they are
Hence in some ways 100 GeV LEP is as good as 1000 GeV TEVATRON
HENCE THE RETURN TO LEP AFTER SPS
AND THEN LHC AFTER LEP

3. Newton & Einstein
Almost all modern accelerators accelerate particles to speeds very close to that of light.
In the classical Newton regime the velocity of the particle increases with the square root of the kinetic energy.
As v approaches c it is as if the velocity of the particle "saturates"
One can pour more and more energy into the particle, giving it a shorter De Broglie wavelength so that it probes deeper into the sub-atomic world
Velocity increases very slowly and asymptotically to that of light

4. Center of mass v. Fixed target

5. Luminosity
Imagine a blue particle colliding with a beam of cross section area - A
Probability of collision is
For N particles in both beams
Suppose they meet f times per second at the revolution frequency
Event rate
Make big
Make small
LUMINOSITY

6. Phase stability
PHS.AD5

7. Bucket and pendulum q q
The “bucket” of synchrotron motion is just that of the rigid pendulum
Linear motion at small amplitude
Metastable fixed point at the top
Continuous rotation outside

8. Analogy with gravity What keeps particles in the machine
There is a solution to Hills Equation
It is closed and symmetric
It is closer to the axis at vertically Defocusing Quadrupoles
Deflection is larger in F than D and cancels the force of gravity elsewhere.
We could call the shape the “suspension” function.
Fig. cas 1.4C

9. Dispersion Low momentum particle is bent more
It should spiral inwards but:
There is a displaced (inwards) closed orbit
Closer to axis in the D’s
Extra (outward) force balances extra bends
D(s) is the “dispersion function”
Fig. cas C

10. Dispersed beam cross sections
These are real cross-section of beam
The central and extreme momenta are shown
There is of course a continuum between
The vacuum chamber width must accommodate the full spread
Half height and half width are:

11. Physics of the electron v. protons
When we collide Protons we collide complex assemblies of three quarks
Only two quarks interact
Their available energy is on average only 10% of the total centre of mass energy
We do not know which quarks they are
Hence in some ways 100 GeV LEP is as good as 1000 GeV TEVATRON
HENCE THE RETURN TO LEP AFTER SPS
AND THEN LHC AFTER LEP

12. Transition - does an accelerated particle catch up - it has further to go
Is a function of two, momentum
dependent, terms b and R.
and
Using partial differentials to define
a slip factor:
This changes from negative to positive
and is zero at ‘ transition’ when:
GAMMA TRANSITION

13. Synchrotron motion Recall Elliptical trajectory for small amplitude
Note that frequency is rate of change of phase
From definition of the slip factor h
Substitute and differentiate again
But the extra acceleration is
THUS

14. Synchrotron motion (continued)
This is a biased rigid pendulum
For small amplitudes
Synchrotron frequency
Synchrotron “tune”

15. A chain of buckets