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 distributions.

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

[math]E_{cm}=[m\lt sub\gt 1\lt /sub\gt \lt sup\gt 2\lt /sup\gt +m\lt sub\gt 2\lt /sub\gt \lt sup\gt 2\lt /sup\gt +2E\lt sub\gt 1\lt /sub\gt E\lt sub\gt 2\lt /sub\gt (1-β\lt sub\gt 1\lt /sub\gt β\lt sub\gt 2\lt /sub\gt cosθ)]\lt sup\gt 1/2\lt /sup\gt \lt /center\gt where θ is the angle between the particles. In the frame where one particle ''(m\lt sub\gt 2\lt /sub\gt )'' is at rest ''\lt center\gt E\lt sub\gt cm\lt /sub\gt =(m\lt sub\gt 1\lt /sub\gt \lt sup\gt 2\lt /sup\gt +m\lt sub\gt 2\lt /sub\gt \lt sup\gt 2\lt /sup\gt +2E\lt sub\gt 1 lab\lt /sub\gt m\lt sub\gt 2\lt /sub\gt )\lt sup\gt 1/2\lt /sup\gt \lt /center\gt '' The velocity of the center of mass in the lab frame is ''\lt center\gt β\lt sub\gt cm\lt /sub\gt =p\lt sub\gt lab\lt /sub\gt /(E\lt sub\gt 1 lab\lt /sub\gt +m\lt sub\gt 2\lt /sub\gt ),\lt /center\gt '' where ''p\lt sub\gt lab\lt /sub\gt ≡p\lt sub\gt 1 lab\lt /sub\gt '' and ''\lt center\gt γ\lt sub\gt cm\lt /sub\gt (E\lt sub\gt 1 lab\lt /sub\gt +m\lt sub\gt 2\lt /sub\gt )/E\lt sub\gt cm\lt /sub\gt \lt /center\gt '' This gives the momenta of the particles in the center of mass to have equal magnitude, but opposite directions ''\lt center\gt p\lt sub\gt cm\lt /sub\gt =p\lt sub\gt lab\lt /sub\gt m\lt sub\gt 2\lt /sub\gt /E\lt sub\gt cm\lt /sub\gt \lt /center\gt '' For an incoming electron with momentum of 11GeV, we should find the momentum in the center of mass to be around '''53 MeV''' which is confirmed in the the plots. Sample output from GEANT4 simulation: {| border=1 |- ! KE\lt sub\gt i\lt /sub\gt !! Px\lt sub\gt i\lt /sub\gt !! Py\lt sub\gt i\lt /sub\gt !! Pz\lt sub\gt i\lt /sub\gt !! x\lt sub\gt i\lt /sub\gt !! y\lt sub\gt i\lt /sub\gt !! z\lt sub\gt i\lt /sub\gt !! KE\lt sub\gt f\lt /sub\gt !! Px\lt sub\gt f\lt /sub\gt !! Py\lt sub\gt f\lt /sub\gt !! Pz\lt sub\gt f\lt /sub\gt !! x\lt sub\gt f\lt /sub\gt !! y\lt sub\gt f\lt /sub\gt !! z\lt sub\gt f\lt /sub\gt !! KE\lt sub\gt m\lt /sub\gt !! Px\lt sub\gt m\lt /sub\gt !! Py\lt sub\gt m\lt /sub\gt !! Pz\lt sub\gt m\lt /sub\gt !! x\lt sub\gt m\lt /sub\gt !! y\lt sub\gt m\lt /sub\gt !! z\lt sub\gt m\lt /sub\gt |- | 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 |} Running a GEANT simulation just to put this set into a .dat file, then using the .dat file through the Moller_OG.C file in ROOT, we find in the Lab Frame: Fnl4Mom.P 10999.599651 Fnl4Mom.E 10999.610609 Mol4Mom.P 1.320928 Mol4Mom.E 1.416324 Fnl4Mom.Theta(degrees) 0.005013 Fnl4Mom.Theta(radians) 0.000087 Mol4Mom.Theta(degrees) 46.756514 Mol4Mom.Theta(radians) 0.815641 Fnl4Mom.P 52.455386 Fnl4Mom.E 54.704969 Mol4Mom.P 52.455386 Mol4Mom.E 52.457875 Fnl4Mom.Theta(degrees) 1.051202 Fnl4Mom.Theta(radians) 0.018338 Mol4Mom.Theta(degrees) 179.040096 Mol4Mom.Theta(radians) 3.123255 [[File:FnlMomCM.png]][[File:MolMomCM.png]] [[File:FnlTheta.png]][[File:MolTheta.png]] [[File:FnlThetaCM.png]][[File:MolThetaCM.png]] '''[[Comparing experimental vs. theoretical for Møller differential cross section]]''' 11GeV Using the equation from Halzen and Martin (p121) to approximate Moller scattering (in the Center of Mass Frame) \lt center\gt \lt math\gt \frac{d\sigma}{d\Omega}=\frac{m^2 \alpha^2}{16 p^4}\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{1}{137}[/math]

Plugging in the values expected for a scattering electron:

[math]m^2 = \frac{(.000511 GeV)^2}{c^4}=2.6\times 10^{-7} GeV^2/c^4[/math]

[math]\alpha^2=\frac{1}{137^2}=7.3\times 10^{-3}[/math]

[math]p^4=\frac{(.053 GeV)^4}{c^4}=7.9\times 10^{-6} GeV^4/c^4[/math]

Using unit analysis on the term outside the parantheses, we find that the differential cross section for an electron at this momentum should be around

[math]\frac{m^2 \alpha^2}{16 p^4}=\frac{1.5\times 10^{-5}}{ GeV^{2}}[/math]

Using the conversion of

[math]\frac{1}{1GeV^{2}}=.3892 mb [/math]

We find that the differential cross section is [math]5.8\times 10^{-6} mb=5.8 nb[/math]

Converting the number of electrons to barns,

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

where ρ_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 4\times 10^7}=1.19\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