Particle by Particle Emittance Measurement to High Precision

1. Particle by Particle Emittance Measurement to High Precision Chris Rogers Imperial College/RAL 17th March 2005

2. Overview • Detail the resolution of the MICE tracker in measuring phase space variables • See this creates a bias in the emittance measurement • Propose a correction routine • Performance plots by Malcolm Ellis

3. Bunch Emittance • Define the bunch emittance in some 2N dimensional space by • Where V is the covariance matrix with elements • Here U is the position vector in some space • The MICE baseline is to take U to be the phase space vector e.g. (t,E,x,px,y,py) • This is analogous to the volume of a hyper-ellipsoid in 2N dimensions • 2Nth root gives it area-like dimensions • Use of mechanical phase space makes it self-normalising • Emittance resolution is determined by resolution in ui

4. Resolution of Tracker - x, y • Transverse position resolution • Difference between x and y caused by threefold rotational symmetry in tracker design • At equilibrium emittance ~ smallest emittance we wish to measure rms ~ 2.5%

5. Resolution of Tracker - px, py • Resolution in transverse momenta • RMS ~ 10%

6. Emittance Bias - approx • We want to relate the resolution in our phase space variables to emittance • First approximation -> addition in quadrature • Hence we can recover the true emittance from the measured emittance • Except for statistical fluctuations from the error pdf and the true pdf (Central Limit Theorem) • In order to be sure of factoring this offset out we require: • Says2error<~15% of s2meas to give a bias ins2meas<1% • Our transverse phase space variables have achieved this

7. Emittance Bias - detail • In the detail it turns out the error pdf is not independent of the true pdf • Originates from Kalman? From dead fibres? • Recall the covariance matrix has elements • Writing uimeas=uitrue+duierror and substituting into the equation for covariance we find

8. Emittance Bias - transverse downstream • We get an offset in emittance in our Monte Carlo • Different in upstream and downstream trackers • Can be negative or positive • Different for different emittances upstream

9. Emittance Bias - detail • We can write this offset in matrix form… • Where R has elements • This is the term that comes from a correlation between the true pdf and the errors • Then the emittance can be calculated from |Vtrue|

10. Corrected Emittance upstream downstream • By using the formula for the bias and taking the resolutions from our monte carlo we can correct the offset

11. Resolution of Tracker - E • Energy resolution is just below 14% • Again, taken at equilibrium emittance • We do not have good figures for our TOF resolution yet • Subsequent slides will show emittance as a function of TOF resolution

12. Emittance Bias - Longitudinal • This is the uncorrected emittance bias as a function of TOF resolution • In FSII the bunch has • s(E)~ 25 MeV • s(t) ~ 500 ps • We measure the resolution with this bunch in mind

13. Corrected Emittance • We apply the same correction technique • This is the corrected emittance resolution as a function of TOF resolution

14. Summary • We have seen the following tracker resolutions: • We understand how to relate these resolutions to emittance

16. Cartoon MICE Stage III Tracker Coil -> 4T constant Bz