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: $\pi a b = 3.14 \times 3.791 \times 1.189 = 14.154~mm*rad = 14154~mm*mrad = 0.014154~m*rad$

$\epsilon_{x,y} = 14154~mm*mrad$

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

$\beta_x = \frac{\sigma_x^2}{\epsilon_x} = \frac{0.00341^2}{0.014154} = 0.00082154~m$

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

$\sigma_y = 3.43 \pm 0.01 ~mm$

$\beta_y = \frac{\sigma_y^2}{\epsilon_y} = \frac{0.00343^2}{0.014154} = 0.00083121~ m$

Transverse Beam Divergence

X

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

$\gamma_x = \frac{\sigma_{x'}^2}{\epsilon_x} = \frac{0.732^2}{0.014154} = 37.857$

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

Y

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

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

$\gamma_y = \frac{\sigma_{y'}^2}{\epsilon_y} = \frac{0.732^2}{0.014154} = 37.857$

Energy and Momentum Distributions

En

P

Py vs Px

Conclusion

Positrons coming out of W-target:

$\epsilon_{x,y} = 14154~mm*mrad$

$\sigma_x = 3.41 \pm 0.01 ~mm$

$\sigma_y = 3.43 \pm 0.01 ~mm$

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

$\beta_x = \frac{\sigma_x^2}{\epsilon_x} = \frac{0.00341^2}{0.014154} = 0.00082154~m$

$\beta_y = \frac{\sigma_y^2}{\epsilon_y} = \frac{0.00343^2}{0.014154} = 0.00083121~m$