Emittance Calculation

Chris Rogers,Imperial College/RAL

Septemebr 2004

1

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

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

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

Class Diagram

5

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

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

Solution? - 4D Hamiltonian We can introduce a four dimensional Hamiltonian

H=(Pu – Au)2/m Pu is the canonical momentum 4-vector Au is the 4 potential Equations of motion are now linear in terms of the

independent variable t given byt = i/gi

Weird huh? Actually, this is in Goldstein Classical Mechanics. He points out that the “normal” Hamiltonian is not covariant, and not particularly relativistic.

9

Evolution in drift The evolution in drift is now given by

xu(t) = xu(0) + Pu /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

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…

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)