Input to the energy calibration procedure is a set of reconstructed
values (x,cos(theta)), where x is the reduced positron energy E/Emax.

To do the calibration theta range is split into a number of bins; for
each such bin x spectrum is produced. Spectrum edge x_max is
determined from a fit for each of the theta bins. Then obtained values
of x_max (and their errors) is used in a linear fit:

       x_max(t) = (1+beta) - alpha*t

where  t == 1/|cos(theta)|. This linear dependence is rigorously valid
for the planar detector geometry. The fit gives values for

    (1+beta), the energy scale 
    alpha,    an energy loss parameter

To imitate reconstructed (x,theta) values a program based on TWIST
GEANT 1.4 was run. Initial kinematics was provided by the standard
DOSP card: a stopping distribution of polarized muons in the target.
Energy loss and multiple scattering were on. Also, the STRA=1 option
to simulate energy loss in thin materials was used. Previous postings
discuss use of this option, see

http://stoney.phys.ualberta.ca/~e614/Projects/E614-S1/00006/
http://stoney.phys.ualberta.ca/~e614///Projects/E614MAIL/00470/index.html

A complete copy of one of e614.ffcards file used is attached.  Other
files had different RNDM and TRIG. All output files were merged to
provide a statistics equivalent to 1.2*10^{7} decays in the whole
Michel spectrum.

Initial angle and energy of positrons from muon decay (x_true,
theta_true) and their energy at position abs(z)==4.16cm x1 were
recorded. 

Two variants of the analysis were made: one using as input

    (x1 smeared with sigma=0.26MeV Gaussian,  cos(theta_true) )

another

    (x_true smeared with sigma=0.26MeV Gaussian,  cos(theta_true) )

To fit x_max both analyses use the Heaviside step function, imitating
the sharp Michel spectrum edge, smeared with a Gaussian, to imitate
energy resolution.

The first analysis gives beta=0 within it's errors as expected (since
MC energy has not been scaled). But the calibration procedure applied
to the second data set gives beta = -0.0016 +- 6.8e-5, an answer that
is wrong by more than 20 sigma.

Some of suggested explanations: 

    - a more precise than Heaviside function Michel spectrum
      description around the edge is required;

    - an account of a non symmetrical (x_rec - x_true) smearing
      shape is required;

    - angular smearing may be important.

Further investigation is in progress...

Andrei

================================================================
NOTE: there was a question about horizontal scale of the slide

http://stoney.phys.ualberta.ca/~e614/Projects/E614-S1/00006/eloss_thin1.ps

The scale is GeV. Positron emitted with theta ~ 40 degrees loses
0.14MeV before leaving the central volume.
================================================================

C Input card file for E614
C updated Feb. 21/01 DRG
CLIST
C
C
C ============ RNDM
C   Input random numbers (see GEANT manual)
C
C RNDM 1585406068 68420761
C
RNDM 1375734345   346349937
C
C ============ RUNG
C   Input run number (see GEANT manual)
C
RUNG 20  1
C
C ============ TIME
C   Time factors (see GEANT manual)
C
TIME 2=10. 3=-1.00
C
C ============ TRIG
C   Number of events to run (see GEANT manual)
C
TRIG 2000000
C
C   Maximum number of steps and maximum path length
C
CC MAXS 30000 300.
MAXS 30000 3000.
C
C ============= DEBU
C   Debug card (see GEANT manual):
C   1st entry: first event # to debug
C   2nd entry: last event # to debug
C   3rd entry: debug every nth event in between
C
C DEBU 1 100 1
CC DEBU -1 50 1
DEBU 0 0 0
C
C ============= SWIT
C   Control switch definitions (see GEANT manual re SWIT 1 and 2)
C   ISWIT(1) = 2 the content of the temporary stack for secondaries in
C                 the common /GCKING/ is printed;
C   ISWIT(2) = 2 Prints out end of step information when DEBUG is active
C
C   ISWIT(10) is used to turn on the output of the hits
C      = 0 no digitization performed (for fast execution)
C      = 1 writes simulated TDC data to disk file
C      = 2 writes ASCII data to disk file
C
CC (my debug): SWIT 2 2 0 0 0 0 0 0 0 3
SWIT 0 0 0 0 0 0 0 0 0 0
CCC SWIT 0 0 0 0 0 0 0 0 0 3
C
PRIN 'MATE' 'TMED'
C PRIN 'PART' 'MATE' 'TMED' 'VOLU' 'SETS'
C
C ============= MONO
C   Mono-energetic particle:
C   1st entry: part. type
C   2nd, 3rd:  total momentum(MeV/c), theta (degrees) (phi random)
C   4th-6th: px,py,pz only if entry 2 is <= 0.
C   7th-9th: starting point of track (x,y,z)
C   10th,11th: sigma x, sigma y of track starting point
C   12th: halfwidth of flat starting point dist. in z 
C
C MONO 2 0.0 0.0 6.53 2.38 39.39 0.0 0.0 0.0 0.0 0.0 0.0
C MONO 2 0.0 0.0 11.85 6.84 37.59 0.0 0.0 0.0 0.0 0.0 0.0
C MONO 2 40.0 30.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
MONO -2 28.0 12.0 0.0 0.0 0.0 0.0 0.0 -52.0 0.5 0.5 0.00375
C
C ============= COSM
C   Cosmic Rays:
C   1st entry : any positive integer to use cosmic generator
C                 (0 or -ve to be ignored)
C   2nd-3rd   : minimum and maximum momenta allowed (respectively, MeV)
C                for the cosmic ray
C   4th       : 'window' height above detector (y) in cm
C   5th-6th   : the z coordinates of the ends of the 'window'
C                 (assumed to be rectangular)
C                  [enter the more negative coordinate first]
C   7th-8th   : the x coordinates of the ends of the window
C
C   Assumes independant 1/E**2 and cos**theta distributions
C 
COSM -1 1000.0 100000.0 130.8 -146.5 146.5 -132.0 132.0
C
C ============= BEAM
C   Beam parameters:
C   1st entry: part. type
C   2nd-4th: initial momentum(3)(MeV/c)
C   5th: mom. spread(sigma in %)
C   6th,7th: sigma thetax,sigma thetay (radians)
C   8th-10th: origin(3) 
C   11th,12th: sigma x, sigma y 
C   13th: distance to project rays (- upstream, + downstream) 
BEAM -65 0.0 0.0 29.7  0.5 0.006 0.006  0.0 0.0 0.0  0.6 0.6 -200.
C BEAM 65 0.0 0.0 65.0  0.5 0.006 0.006  0.0 0.0 0.0  0.6 0.6 -200.
C
C ========== SORC: ISORC PSORC(14) ==========
C
C     For iswit(4).eq.0 and iswit(5).eq.0
C     -----------------------------------
C     ISORC     = GEANT Particle type
C     PSORC( 1) = Particle gun origin x [cm]
C     PSORC( 2) = Particle gun origin y [cm]
C     PSORC( 3) = Particle gun origin z [cm]
C     PSORC( 4) = Particle gun aim - rotation around x-axis (deg)
C     PSORC( 5) = Particle gun aim - rotation around y-axis (deg)
C     PSORC( 6) = Particle gun aim - rotation around z-axis (deg)
C     PSORC( 7) = sigma of x position (cm)   
C     PSORC( 8) = sigma of y position (cm)   
C     PSORC( 9) = horizontal emittance [mr] 
C     PSORC(10) = vertical   emittance [mr] 
C     PSORC(11) = Particle momentum [MeV/c]
C     PSORC(12) = 1 +- deltaP/P             
C     PSORC(13) = Beam focus position in z
C     PSORC(14) = Surface muons no = 0, yes > 0
C 
SORC -65 0. 0. -300. 0. 0. 0. 0.0 1.0 15.0 10.0 27.0 1.01 -142.5 1.
C
C ============= RAYS
C   Read-in beam rays:
C   1st entry: part. type
C   2nd: central beam momentum(MeV/c)
C   3rd-5th: coordinates of focus 
C   6th,7th: overall direction of beam (theta,phi) (degrees)
C                [SO FAR ONLY THETA=0, PHI=0 ALLOWED]
C   8th: distance to project rays (- upstream, + downstream) 
C
RAYS -65 29.7 0.0 0.0 -150.0  0.0 0.0  -50.0
C
C ============= DOSP
C   Distribution and orientation of initial stopped particles
C   1st entry: part. type
C   2nd-4th: centroid of stopping distribution
C   5th,6th: sigma x, sigma y of stopping dist.
C   7th: half-width of flat stopping dist. in z 
C   8th,9th: upper, lower limits of initial spin polar angle (degrees) 
C            (ignored for all particles except mu+ with spin)
C
DOSP 65 0.0 0.0 0.0 0.45 0.45 0.00375 180.0 179.2
C
C ============= AUTO
C   AUTO 1 (default) allows automatic calulation of tracking parameters
C                            (except EPSIL)
C   AUTO 0 turns off automatic calculation unless parameters are < 0
C
AUTO 0
C
C ============= STPL
C   Step length limiting parameters
C   1st entry: > 0 enable parameter change, < 0 - disable (use defaults)
C   1st entry should be one of: (if want different values for
C     9 = High Purity Aluminium, 21 = Mylar, 17 = Gold, 6 = Graphite
C     10 = Iron, 22 Aluminized Mylar, others to come in future.
C   2nd-6th: tmaxfd, stemax, deemax, epsil, stmin 
C   7th-10th: stemax, deemax, epsil and stmin for the target material
C   11th-13th: 3 spares
C
STPL 21 -1.0 -.5 -.25 .0001 -.1 -0.1 -.0001 .0001 -.0001 0. 0. 0.
C
C ============= BFLD
C   Magnetic field
C   IFIELD: 3 -> uniform field, 1 -> non-uniform field, 0 -> no field
C   FIELDM: maximum field in kGauss (see GEANT manual, routine GSTMED)
C
BFLD 3 22.0
C
C ========== HSTA: LHSTA
C GEANT histograms
C HSTA 'TIME' provides histogram of CPU time per event
C
HSTA 'TIME'
C
C STAT 'VOLU'
C
C ============= CUTS
C   Low energy cutoffs - no tracking below these values (GeV)
C 
CUTS 0.0005 0.00002 0.0005 0.0005 0.00001
C
C ============= PIBR
C   Pion decay branch
C   1 : pi+ -> mu+ nu   mu has no spin
C   2 : pi+ -> e+ nu
C   3 : pi+ -> mu+ n    mu has spin
C
PIBR 1
C
C ============= MCHL
C   Set Michel parameters 
C   First entry not used (can be any integer)
C   2nd through 5th entries: rho, delta, xi, eta 
C
MCHL 1 0.75 0.75 1.00 0.00
C
C ============= ULIM
C   Set Michel spectrum generation limits
C   xmin, xmax            -- dimensionless positron energy
C   cthetamin, cthetamax  -- cos(theta) of the positron
C			   IMPORTANT: theta is relative to muon spin
C				NOT to the detector
C
C The last entry is currently ignored
C   up_and_down (integer) -- if nonzero, angular limits will be made 
C 				symmetric (NOT IMPLEMENTED)
C
C ULIM 0. 1. -1. 1. 0
ULIM  0.85 1.  -1. 1.  0
C ============= DPOL
C   Set muon depolarization parameters
C   1st flag to turn on depolarization > 0 = on, 0 or < 0 = off
C   2nd polarization mean life(sec)
C
DPOL 0 0.0000022
C
C
C ========== GABS: IGAS ABS_LENGTH GAS_MIXTURE ==========
C
C     Identity of Gas     == igas  = 20 -> He/CO2
C                                  = 21 -> He/Ar
C                                  = 22 -> He/Xe
C     Absorber gas length == abs_length [cm] (full length)
C     Gas mixture ratio   == gas_mixture (fraction of heavier gas)
C
GABS 20 15.0 0.5
C
C ========== PABR: IPLASTIC  PLAS_LNGTH PLAS_POSITN ==========
C
C     ipalstic   == 0  -> no plastic degrader
C                == 31 -> Mylar
C                == 26 -> Scintillator
C     plas_lngth  == [cm] (full length)
C     plas_radius == [cm]
C     plas_positn == position of plastic degrader [cm]
C
PABR 31 0.004 8.0 -78.1
C
C ========== ROOM: P_ATM TEMP_C ==========
C
C     Atmospheric pressure == P_atm [Torr]
C     Room temperature     == temp_C [deg C]
C
ROOM 760.0 20.0
C
C ========== HEBG: ==========
C       Helium bag between end of beampipe and detector volume
C	 ihebag > 0 means insert HEBG.
C               < 1 means insert HEBG full of air.
C        ibag_gas - type of gas in bag
C                    13 magnetic air
C                    16 magnetic helium
C         
HEBG 1 13
C
C ========== TECM: ==========
C       Time Expansion Chamber properties inputs via ffcard TECM
C	 itecm - 0 means do not insert a TEC.
C        P_atm - atmospheric pressure
C        temp_C - temperature in deg. C
C
TECM 0 20.0 20.0
C
C ========== YOKE: ==========
C	 iyoke > 0 means insert YOKE etc.
C
YOKE 0
C
C =============  Physics flags See GEANT Manual ==========
C
C   DCAY =0 decay not allowed
C   DCAY =1 allow decay, track secondaries
C   DCAY =2 allow decay, don't generate secondaries
C
C decay in flight
DCAY 1
C Positron annihilation: 1(D) generate photons
ANNI 0
C compton scattering: 1(D) = with generation of e-; 
COMP 0
C Pair production: 1(D): stop photon generate e+e-
PAIR 0
C Bremsstrahlung: 1(D) bremsstrahlung with generation of gamma
BREM 1
C Photoelectric effect: 1(D)=stop photon, generate electron
PHOT 0
C muon-nuclear interactions
MUNU 0
C continuous energy loss: 
C	1 = with delta-rays; 
C	2(D) = Landau fluctuations, no delta
LOSS 2
C AG: energy loss in thin materials (0/1)
STRA 1
C multiple scattering: 1=Moli`ere 
MULS 1
HADR 0
ERAN 0.000001 1.0 160
STOP

Description: Examples of the spectrum edge fits. , Filename: x1-gausstep1-typical.ps

Description: x_true based energy calibration , Filename: xtrue-gausstep1-ec.ps

Description: x1 based energy calibration , Filename: x1-gausstep1-ec.ps

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