Describing Energy Deposit in CsI Crystals

To determine the energy resolution oe E =E of CsI crystals the width of energy deposit distributions in single crystals or clusters of crystals has to be calculated. Such distributions have typically a tail towards lower energies whereas peak shape and hig

BABAR

Note

# 294

May 7, 1996

Describing Energy Deposit in CsI CrystalsR. Seitz Institut fur Kern- und Teilchenphysik Technische Universitat Dresden

To determine the energy resolution E=E of CsI crystals the width of energy deposit distributions in single crystals or clusters of crystals has to be calculated. Such distributions have typically a tail towards lower energies whereas peak shape and high energy side are approximately gaussian. This note discusses analytical functions describing the energy deposit in CsI and compares their performance to t calorimeter beamtest data.

Abstract

contact: rse@slac.stanford.edu

To determine the energy resolution oe E =E of CsI crystals the width of energy deposit distributions in single crystals or clusters of crystals has to be calculated. Such distributions have typically a tail towards lower energies whereas peak shape and hig

1 IntroductionDistributions of deposited energy in CsI crystals show a non-Gaussian tail on the lower energy side. The tail in these curves results from uctuations in the shower development for di erent events. This leads for example to a di erent amount of side-leakage, front or rear face leakage of shower particles out of the crystal, resulting in a tail towards lower energies. The resolution in this case is de ned as: E= FWHM=2:36: E E In this de nition FWHM is Full Width at Half Maximum of the curve and E the energy value at the maximum of the distribution. These values can be determined using a t to the distributions, ideally using an analytical function which allows to calculate the desired quantities or obtaining them directly as t results.

2 Analytical descriptions2.1 Landau Density multiplied with a GaussianBABAR-note 231 2] contains a t to calculate the energy resolution, which is the product of the Landau density distribution: Z ( )= 21 exp( s ln s ds); and a Gaussian. The functional form used for tting the energy deposit curve is:#" mL? x exp? 1 E? mG: f (E )= n (1) 22

n is a normalisation factor, mL the mean and L the width of the Landau distribution. mG and G are mean and width of the additional Gaussian. Disadvantage of this function is the complicated form involving the calculation of the integral and the di culty to access FWHM. To determine the energy resolution E=E, FWHM would have to be calculated numerically. 1

L

G

To determine the energy resolution oe E =E of CsI crystals the width of energy deposit distributions in single crystals or clusters of crystals has to be calculated. Such distributions have typically a tail towards lower energies whereas peak shape and hig

2.2 Gaussian with power law tailThis function is used in BABAR-note 183 6]. The energy deposit is described by a gaussian t, together with a power law for the tail on the left side at lower energies. Connected are the two components of the function at a point which is times the of the Gaussian away from the mean. The mean m, the factor and the Gaussian are t parameters. In total ve parameters describe the function: ! (x? m) ) x> m?: f (x)= C exp? 2 !p p= x< m?: f (x)= C m?? x+ p= exp(? 2 ): (2)2 2 2

FWHM is calculated as sum of the Gaussian (FWHMright) and the power law tail (FWHMleft) FWHM: FWHMright= FWHMleft=

p

If> 2 ln 2 the FWHM of the t is Gaussian.

p

1 ((2 exp(?2 )) p? 1) p+2

2 ln 2

!

:

2.3 Logarithmic Normal DistributionIn statistics, the Logarithmic Normal Distribution 3] describes events whose logarithm is Gaussian distributed.

The peak region of the function is approximately Gaussian with a tail to one side (see g.1(c)). This shape was used to t energy deposits in a beamtest for the CsI calorimeter of the Crystal Barrel experiment in 1988 5]: 2 !3 n exp 4? ln(m? E )? 5: f (E )= m? E (3)2

Analytical expressions for Emax and FWHM=Emax exist:

Emax= m? e2

(

1? 2 2 );

To determine the energy resolution oe E =E of CsI crystals the width of energy deposit distributions in single crystals or clusters of crystals has to be calculated. Such distributions have typically a tail towards lower energies whereas peak shape and hig

and

FWHM= 2 sinh( 2 ); Emax me 22?? 1 p which is connected to E=E by the factor 2 ln 4= 2:36. Another parametrisation of the same curve developed at Novosibirsk 1] has the following form:( )

p

2 0 6 1 B ln 1+ f (E )= n exp 6? 2 B 6@ 4

(E? )

sinh(

p p ln 4)ln 4

3 1 7 C C+ 7: (4) 7 A 52 2

n is the normalisation, the mean value, a parameter for the tail. is directly FWHM/2.36. Although this parametrisation seems to be more complicated, it has the advantage of delivering directly E and Emax. Equations 3 and 4 can be reduced to a common functional form: 2A?2A ! (C? x) B2? (C? x)? C?x=B2; f (x)= N exp B (5) which is calculated explicitly in appendix A. For practical purposes the parametrisations 3 and 4 are more useful then equation 5 since their convergence is better and the parameters of function 4 are the quantities of interest FWHM/2.36 and Emax. The BELLE Experiment uses for the analysis of their beamtest data 4] the same function, as parametrised in eq. 3.2 1 ln( ) 2

3 ComparisonTo compare the four described functions, raw electron data of 215 MeV beam energy, taken in October 1995 during the BABAR CsI beamtest at PSI, are used as t example.

3

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