Optimization of Positron Capturing

• Thickness of the Tungsten target: 1.25 mm
• Beam size at Tungsten target: Gaussian, $\sigma_{x,y} = 3~mm$

Optimization with ELEGANT Simulation on the Region from Gun Exit to Tungsten Target

Emittance: $\epsilon_{x,y} = 13~\mu m$ (Not sure yet, need to be calculated)

Average Peak $Q = 14~pC$

Beam Energy: $10~MeV$

Energy Spread: $\sigma_{\delta}=4.23%$

Alpha Function: $\alpha =$

Beta Function: $\beta =$

RMS longitudinal Bunch Length: $\sigma_{z} = 10.6~ps$

GEANT4 Simulation for 2 mm W-Target

Thickness of the Tungsten target: 2 mm

Electron Beam:

Beam size at Tungsten target: Gaussian, $\sigma_{x,y} = 3~mm$

Beam divergence at target: 0

Beam Energy: $10~MeV$

Energy Spread: $\sigma_{\delta}=4.23%$

A volume called "detector" is placed at the of the W-target to get the information on the only of the outgoing positrons from the W-target. Followings are from "detector".

Phase Spaces of e+ Created

Phase space for positrons coming out W-target. Include 100 % of the particles

X phase space

Y phase space

Phase Spaces ellipse of 39% particles

Total positrons created are 75483, and 39 % of them will be 29438. Core 39% area of the phase space ellipse is RMS emittance, $\epsilon_{rms}$.

Long and short axis of these two ellipse are: a = 3.791 mm; b = 1.189 rad;

Area of this ellipse would be: $\pi a b = 3.14 \times 3.791 \times 1.189 = 14.03~mm*rad = 1403~mm*mrad = 0.01403~m*rad$

$\epsilon_{x,y} = 14030~mm*mrad$

X phase space

Y phase space

Transverse Beam Size

X

 FCN=1008.69 FROM MIGRAD    STATUS=CONVERGED      62 CALLS          63 TOTAL
                    EDM=5.73734e-06    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     8.62885e+02   4.11205e+00   4.92422e-02  -8.37091e-04
  2  Mean        -1.58596e-04   1.25728e-02   1.95074e-04  -1.22250e-03
  3  Sigma        3.41790e+00   1.03579e-02   1.17273e-05  -8.88590e-02

$\sigma_x = 3.41 \pm 0.01 ~mm$

$\sigma_x = \sqrt{\epsilon_x \beta_x}$

So

$\beta_x = \frac{\sigma_x^2}{\epsilon_x} = \frac{0.00341^2}{0.01403} = 0.00082154~m$

Y

 FCN=1016.5 FROM MIGRAD    STATUS=CONVERGED      62 CALLS          63 TOTAL
                    EDM=4.46233e-06    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     8.58714e+02   4.10556e+00   4.91913e-02  -7.80242e-04
  2  Mean         6.57793e-03   1.26416e-02   1.96791e-04   5.52501e-03
  3  Sigma        3.43480e+00   1.04943e-02   1.18410e-05  -3.65830e-01

$\sigma_y = 3.43 \pm 0.01 ~mm$

$\beta_y = \frac{\sigma_y^2}{\epsilon_y} = \frac{0.00343^2}{0.01403} = 0.00083121~ m$

Transverse Beam Divergence

XP

FCN=1205.9 FROM MIGRAD    STATUS=CONVERGED      60 CALLS          61 TOTAL
                    EDM=9.41429e-08    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     2.08367e+03   9.54533e+00   1.29688e-01   1.45076e-05
  2  Mean         1.65233e-03   2.93970e-03   4.98630e-05  -1.42668e-01
  3  Sigma        7.32175e-01   2.40413e-03   1.53830e-05   8.15997e-02

$\sigma_{x'} = 0.732 \pm 0.002 ~rad$

$\sigma_{x'} = \sqrt{\epsilon_x \gamma_x}$

So

$\gamma_x = \frac{\sigma_{x'}^2}{\epsilon_x} = \frac{0.732^2}{0.01403} = 37.857$

$\gamma_x = \frac{1 + \alpha_x^2}{\beta_x}$

So

$\alpha_x = \sqrt{\gamma_x \beta_x-1} = \sqrt{37.857 \times 0.00082154 -1} = \sqrt{0.0311 -1}$

YP

FCN=1278 FROM MIGRAD    STATUS=CONVERGED      60 CALLS          61 TOTAL
                    EDM=2.39023e-07    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     2.08210e+03   9.52221e+00   1.33471e-01   4.78861e-05
  2  Mean         3.97056e-03   2.94127e-03   5.13379e-05   1.97481e-01
  3  Sigma        7.31984e-01   2.38977e-03   1.57624e-05   2.99074e-01

$\sigma_{y'} = 0.732 \pm 0.002 ~rad$

$\gamma_y = \frac{\sigma_{y'}^2}{\epsilon_y} = \frac{0.732^2}{0.01403} = 37.857$

Energy and Momentum Distributions

En

P

Py vs Px

Conclusion

Positrons coming out of W-target:

$\epsilon_{x,y} = 14030~mm*mrad$

$\sigma_x = 3.41 \pm 0.01 ~mm$

$\sigma_y = 3.43 \pm 0.01 ~mm$

$\sigma_{x',y'} = 0.732 \pm 0.002 ~rad$

$\beta_x = \frac{\sigma_x^2}{\epsilon_x} = \frac{0.00341^2}{0.01403} = 0.00082154~m$

$\beta_y = \frac{\sigma_y^2}{\epsilon_y} = \frac{0.00343^2}{0.01403} = 0.00083121~m$

Target Placed 20 cm from Quadrupole 4

x

In mm unit

FCN=314.927 FROM MIGRAD    STATUS=CONVERGED      67 CALLS          68 TOTAL
                    EDM=7.23115e-08    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     6.26496e+01   1.31186e+00   9.22753e-03  -2.86419e-04
  2  Mean        -1.55703e+00   2.98157e-01   2.40433e-03   7.26565e-04
  3  Sigma        1.34518e+01   2.49312e-01   4.42013e-05  -3.10903e-02

y

In mm unit

FCN=203.003 FROM MIGRAD    STATUS=CONVERGED      68 CALLS          69 TOTAL
                    EDM=3.25951e-09    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     6.07799e+01   1.28695e+00   7.08046e-03   1.59797e-05
  2  Mean         6.24506e-01   3.19737e-01   2.20644e-03  -2.42181e-04
  3  Sigma        1.47606e+01   3.17675e-01   4.40665e-05   5.56943e-03

xp

In rad unit

FCN=265.085 FROM MIGRAD    STATUS=CONVERGED      61 CALLS          62 TOTAL
                    EDM=3.01383e-09    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     8.80056e+01   1.73095e+00   1.18113e-02  -2.55131e-05
  2  Mean         6.16198e-04   1.10788e-03   8.81846e-06  -5.90190e-03
  3  Sigma        6.18004e-02   7.08426e-04   2.61954e-05  -2.34453e-02

yp

In rad unit

FCN=329.265 FROM MIGRAD    STATUS=CONVERGED      62 CALLS          63 TOTAL
                    EDM=1.51428e-08    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     9.27538e+01   1.80938e+00   1.39977e-02  -3.41258e-05
  2  Mean        -1.83324e-03   1.06761e-03   9.45216e-06   1.51733e-01
  3  Sigma        6.01380e-02   6.22891e-04   2.62088e-05  -1.60567e-02

En

In MeV unit

FCN=431.695 FROM MIGRAD    STATUS=CONVERGED      79 CALLS          80 TOTAL
                    EDM=3.43935e-09    STRATEGY= 1      ERROR MATRIX ACCURATE 
 EXT PARAMETER                                   STEP         FIRST   
 NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
  1  Constant     9.06730e+01   2.25831e+00   1.58988e-02  -4.86385e-05
  2  Mean         2.73284e+00   3.48882e-02   3.44084e-04   2.48074e-04
  3  Sigma        1.69962e+00   4.06010e-02   5.82766e-05  -6.72485e-03

Pn

In MeV/c unit

GEANT output

Root files and data files at

aztec.iac.isu.edu:/home/setisadi/pos_Tar_Opt/ana/dif_tar_pos

SteppingVerbose

 if( fTrack->GetVolume()->GetName() =="Detector4"  &&  fTrack->GetDefinition()->GetPDGEncoding()==-11 )//e+
     { 
         float Px, Py, Pz, Px2, Py2, Pz2;
         Px = fTrack->GetMomentum().x();
         Py = fTrack->GetMomentum().y();
         Pz = fTrack->GetMomentum().z();
         Px2 = pow(Px, 2);
         Py2 = pow(Py, 2);
         Pz2 = pow(Pz, 2);

float xp, yp, P2, En, DelEn;

         xp = atan(Px/Pz);
         yp = atan(Py/Pz);

P2 = Px2 + Py2 + Pz2; En = sqrt ( P2 + 0.511*0.511 ); DelEn = 10 - En;

         if(Px2>0 || Py2>0 || Pz2>0 ){
         outfile_detector4

         <<fTrack->GetKineticEnergy() << " "
         << fTrack->GetPosition().x()<< " "
         << fTrack->GetPosition().y()<< " "
         << fTrack->GetPosition().z()<< " "
         << fTrack->GetMomentum().x() << " "
         << fTrack->GetMomentum().y() << " "
         << fTrack->GetMomentum().z() << " "

<< xp << " " << yp << " " << En << " " << DelEn << " "

         << G4endl;

}

So, the data file contains columns of

KE x y z Px Py Pz xp yp En DelEn

detector

Detector right after the W target.

detector.dat detector_T.root

detector1

Detector 25 mm from W target.

detector1.dat detector1_25mm.root

detector8

Detector 200 mm from W target.

detector8.dat detector8_200mm.root

Ratio of the positrons on detectors

NT = 215111

N1 = 103506

N8 = 3787

NT:N1=2.08

NT:N8=56.8

N1:N8=27.3

Conclusions

Positrons coming out of W-target:

$\epsilon_{x,y} = 14030~mm*mrad$

$\sigma_x = 3.41 \pm 0.01 ~mm$

$\sigma_y = 3.43 \pm 0.01 ~mm$

$\sigma_{x',y'} = 0.732 \pm 0.002 ~rad$

$\beta_x = \frac{\sigma_x^2}{\epsilon_x} = \frac{0.00341^2}{0.01403} = 0.00082154~m$

$\beta_y = \frac{\sigma_y^2}{\epsilon_y} = \frac{0.00343^2}{0.01403} = 0.00083121~m$

$\gamma_x = \frac{1 + \alpha_x^2}{\beta_x}$

$\gamma_y = \frac{\sigma_{y'}^2}{\epsilon_y} = \frac{0.732^2}{0.01403} = 37.857$

Imaginary $\alpha_x$ function?

$\alpha_x = \sqrt{\gamma_x \beta_x-1} = \sqrt{37.857 \times 0.00082154 -1} = \sqrt{0.0311 -1}$