Optimization of Positron Capturing

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

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

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

Average Peak [math] Q = 14~pC [/math]

Beam Energy: [math] 10~MeV [/math]

Energy Spread: [math] \sigma_{\delta}=4.23% [/math]

Alpha Function: [math] \alpha = [/math]

Beta Function: [math] \beta = [/math]

RMS longitudinal Bunch Length: [math] \sigma_{z} = 10.6~ps[/math]

GEANT4 Simulation for 2 mm W-Target

Thickness of the Tungsten target: 2 mm

Electron Beam:

Beam size at Tungsten target: Gaussian, [math]\sigma_{x,y} = 3~mm[/math]

Beam divergence at target: 0

Beam Energy: [math] 10~MeV [/math]

Energy Spread: [math] \sigma_{\delta}=4.23% [/math]

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, [math]\epsilon_{rms}[/math].

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

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

[math]\epsilon_{x,y} = 14030~mm*mrad[/math]

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

[math]\sigma_x = 3.41 \pm 0.01 ~mm[/math]

[math]\sigma_x = \sqrt{\epsilon_x \beta_x}[/math]

So

[math]\beta_x = \frac{\sigma_x^2}{\epsilon_x} = \frac{0.00341^2}{0.01403} = 0.00082154~m[/math]

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

[math]\sigma_y = 3.43 \pm 0.01 ~mm[/math]

[math]\beta_y = \frac{\sigma_y^2}{\epsilon_y} = \frac{0.00343^2}{0.01403} = 0.00083121~ m[/math]

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

[math]\sigma_{x'} = 0.732 \pm 0.002 ~rad[/math]

[math]\sigma_{x'} = \sqrt{\epsilon_x \gamma_x} [/math]

So

[math]\gamma_x = \frac{\sigma_{x'}^2}{\epsilon_x} = \frac{0.732^2}{0.01403} = 37.857 [/math]

[math]\gamma_x = \frac{1 + \alpha_x^2}{\beta_x} [/math]

So

[math] \alpha_x = \sqrt{\gamma_x \beta_x-1} = \sqrt{37.857 \times 0.00082154 -1} = \sqrt{0.0311 -1} [/math]

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

[math]\sigma_{y'} = 0.732 \pm 0.002 ~rad[/math]

[math]\gamma_y = \frac{\sigma_{y'}^2}{\epsilon_y} = \frac{0.732^2}{0.01403} = 37.857 [/math]

Energy and Momentum Distributions

En

P

Py vs Px

Conclusion

Positrons coming out of W-target:

[math]\epsilon_{x,y} = 14030~mm*mrad[/math]

[math]\sigma_x = 3.41 \pm 0.01 ~mm[/math]

[math]\sigma_y = 3.43 \pm 0.01 ~mm[/math]

[math]\sigma_{x',y'} = 0.732 \pm 0.002 ~rad[/math]

[math]\beta_x = \frac{\sigma_x^2}{\epsilon_x} = \frac{0.00341^2}{0.01403} = 0.00082154~m[/math]

[math]\beta_y = \frac{\sigma_y^2}{\epsilon_y} = \frac{0.00343^2}{0.01403} = 0.00083121~m[/math]

Target Placed 20 cm from Quadrupole 4

x

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

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