Download - Emittance Calculation
![Page 1: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/1.jpg)
Emittance Calculation
Chris Rogers,Imperial College/RAL
Septemebr 2004
1
![Page 2: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/2.jpg)
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
![Page 3: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/3.jpg)
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
![Page 4: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/4.jpg)
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
![Page 5: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/5.jpg)
Class Diagram
5
![Page 6: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/6.jpg)
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
![Page 7: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/7.jpg)
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
![Page 8: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/8.jpg)
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.
![Page 9: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/9.jpg)
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
![Page 10: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/10.jpg)
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…
![Page 11: Emittance Calculation](https://reader036.vdocument.in/reader036/viewer/2022072016/56813174550346895d97ed01/html5/thumbnails/11.jpg)
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)