Emittance Calculation
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Transcript of Emittance Calculation
Emittance Calculation
Chris Rogers,Imperial College/RAL
Septemebr 2004
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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
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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
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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
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Class Diagram
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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
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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!
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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.
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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
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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)