DV RunGroupC Moller

Simulating the Moller scattering background for EG1

Step 1

Determine the Moller background using an LH2 target to check the physics in GEANT4

Incident electron energy varies from 1-11 GeV.

LH2 target is a cylinder with a 1.5 cm diameter and 1 cm thickness.

  (Following dimensions listed on page 8 of File:PHY02-33.pdf)

Numbers Moller electrons per incident electron.

  While 2nd and 3rd generations are created, only 2 2nd generation daughter particles are created for 1E6 incident particles.  All knock on electrons are not counted.

Momentum and angular distributions. 11GeV

In the collision of two particles of mass m_1 and m_2, the total energy in the center of mass frame can be written

[math]E_{cm}=sqrt[2]{(E_1+E_2)^2-(p_1+p_2)^2}[/math]

E_cm=[m₁²+m₂²+2E₁E₂(1-β₁β₂cosθ)]^1/2

where θ is the angle between the particles. In the frame where one particle (m₂) is at rest

E_cm=(m₁²+m₂²+2E_{1 lab}m₂)^1/2

The velocity of the center of mass in the lab frame is

β_cm=p_lab/(E_{1 lab}+m₂),

where p_lab≡p_{1 lab} and

γ_cm(E_{1 lab}+m₂)/E_cm

This gives the momenta of the particles in the center of mass to have equal magnitude, but opposite directions

p_cm=p_labm₂/E_cm

For an incoming electron with momentum of 11GeV, we should find the momentum in the center of mass to be around 50 MeV which is not confirmed in the the plots.

Sample output from GEANT4 simulation:

 KE_i          Px_i         Py_i         Pz_i         x_i          y_i          z_i          KE_f          Px_f          Py_f          Pz_f          x_f          y_f          z_f           KE_m           Px_m         Py_m          Pz_m          x_m          y_m          z_m

11000 0 0 11000.5 0 0 -510 10999.1 0.433025 -0.858867 10999.6 0 0 -509.276 0.905324 -0.433025 0.858867 0.905366 0 0 -509.276

Comparing experimental vs. theoretical for Møller differential cross section 11GeV

Using the equation from Halzen and Martin (p121) to approximate Moller scattering

[math]\frac{d\sigma}{d\Omega}=\frac{m^2 \alpha^2}{16 p^2}\left(\frac{1}{\sin^4 \frac{\theta}{2}}+\frac{1}{\cos^4 \frac{\theta}{2}}-\frac{1}{\sin^2 \frac{\theta}{2}\cos^2 \frac{\theta}{2}}\right)[/math]

where [math]\alpha = \frac{e^2}{4\pi}[/math]

and [math]e^2 = 1.43299764 MeV - fm[/math]

Converting the number of electrons to nano-barns,

[math]L=\frac{i_{scattered}}{\sigma} \approx i_{scattered}\times \rho_{target}\times l_{target}[/math]

where \rho_{target} is the density of the target material, l_{target} is the length of the target, and i_{scattered} is the

[math]L=\frac{70.85 kg}{1 m^3}\times \frac{1 mole}{2.02 g} \times \frac{1000g}{1 kg} \times \frac{6\times10^{23} atoms}{1 mole} \times \frac{1cm}{100 cm} \times \frac{1 m}{ } \times \frac{10^{-23} m^2}{barn} =2.10\times 10^{-2} barns[/math]

[math]\frac{1}{L \times 10^7}=4.75\times 10^{-6} barns[/math]

Step 2

Replace the LH2 target with an NH3 target and compare with LH2 target.

Step 3

Determine impact of Solenoid magnet on Moller events

Papers used

A polarized target for the CLAS detectorFile:PHY02-33.pdf

An investigation of the spin structure of the proton in deep inelastic scattering of polarized muons on polarized protons File:1819.pdf

EG12