1. Emittance Calculation Chris Rogers, Imperial College/RAL Septemebr 2004 1

2. Two Strands G4MICE Analysis Code Calc 2/4/6D Emittance Apply statistical weights, cuts, etc Theory Phase Space/Geometric Emittance aren’t good for high emittance beams Looking at new ways to calculate emittance 2

3. Analysis Code Aims For October Collaboration Meeting: Plot emittance down the MICE Beamline Trace space, phase space, canonical momenta Enable tracker analysis Apply statistical weights to events 3

4. Progress Analysis Code can now Calculate emittance Apply statistical weights Weight events such that they look Gaussian Cut events that don’t make it to the downstream tracker, fall outside a certain pos/mom range Still can’t do canonical coordinates 4

5. Class Diagram 5

6. Some Results Phase Space Emittances in constant Bz - Top left: 2D trans emittance. Top right: 2D long emittane. Bottom left: 4D trans emittance. Bottom right: 6D emittance 6

7. Theory Emittance is not defined with highly dispersive beams in mind Geometric emittance - calc’d using p/pz Normalisation fails for non-symmetric highly dispersive beams Phase Space Emittance - calc’d using p Non-linear equations of motion => emittance increases in drift/solenoid Looks like heating even though in drift space! 7

8. 8 Solution? - 4D Hamiltonian We can introduce a four dimensional Hamiltonian H=(P u – A u ) 2 /m P u is the canonical momentum 4-vector A u is the 4 potential Equations of motion are now linear in terms of the independent variable t given by t =  i /g i Weird huh? Actually, this is in Goldstein Classical Mechanics. He points out that the “normal” Hamiltonian is not covariant, and not particularly relativistic.

9. 9 Evolution in drift The evolution in drift is now given by x u (t) = x u (0) + P u  /m This is linear so emittance is a constant But proper time is not a physical observable Need to do simulation work Need to approach multiple scattering with caution Stochastic process

10. 10 Evolution in Fields The Lorentz forces are Lorentz invariant so particle motion is still linear inside linear B-fields. That is dP/d  = q(dx/d  x B) We can show this using more rigorous methods This means that all of our old conditions for linear motion are still obeyed in the 4-space However, in a time-varying field it is less clear how to deal with motion of a particle. An RF cavity is sinusoidal in time - but what does it look like in proper time t? I don’t know…

11. Summary Analysis code coming along Can apply statistical weights Theory proving interesting Need to look at RF, solenoids Other avenues? Absolute density, etc Other aspects (e.g. Holzer method)