Calculating the differential cross-sections for the different materials, and placing them as well as the theoretical differential cross-section into a plot:

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Figure 8d: The Moller scattering angle theta for 4E7 incident 11 GeV electrons in the Center of Mass frame of reference for 3 different target materials.
Density of target material
Material Density (g/cm3)
C 2.26
NH3 0.86
LH2 0.07

Comparing this to the theoretical differential cross section

As shown earlier , we find that the differential cross section scale is [math]\frac{d\sigma}{d\Omega}\approx 16.2\times 10^{-2}mb=16.2\mu b[/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 ρtarget is the density of the target material, ltarget is the length of the target, and iscattered is the number of incident particles scattered.

For LH2:

[math]\rho_{target}\times l_{target}=\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{1m^3}{(100 cm)^3} \times \frac{1 cm}{ } \times \frac{10^{-24} cm^{2}}{barn} =2.10\times 10^{-2} barns[/math]

[math]\frac{1}{\rho_{target}\times l_{target} \times 4\times 10^7}=1.19\times 10^{-6} barns[/math]

For Carbon:

[math]\rho_{target}\times l_{target}=\frac{2.26 g}{1 cm^3}\times \frac{1 mole}{12.0107 g} \times \times \frac{6\times10^{23} atoms}{1 mole} \times \frac{1 cm}{ } \times \frac{10^{-24} cm^{2}}{barn} =1.13\times 10^{-1} barns[/math]

[math]\frac{1}{\rho_{target}\times l_{target} \times 4\times 10^7}=2.21\times 10^{-7} barns[/math]

For Ammonia:

[math]\rho_{target}\times l_{target}=\frac{.8 g}{1 cm^3}\times \frac{1 mole}{17 g} \times \frac{6\times10^{23} atoms}{1 mole} \times \frac{1 cm}{ } \times \frac{10^{-24} cm^2}{barn} =2.82\times 10^{-2} barns[/math]

[math]\frac{1}{\rho_{target}\times l_{target} \times 4\times 10^7}=8.87\times 10^{-7} barns[/math]

Combing plots in Root:

new TBrowser();
TH1F *LH2=new TH1F("LH2","LH2",360,90,180);

TH1F *C12=new TH1F("C12","C12",360,90,180);

TH1F *NH3=new TH1F("NH3","NH3",360,90,180);

Figure 8c: The Moller electron differential cross-section for 4E7 incident 11 GeV electrons in the Center of Mass frame of reference.
Molar Mass of Target Material
Material Molar Mass (g/mole)
NH3 17
C 12
LH2 2

Number of Electrons in Target Material
Material Number of Electrons
NH3 10
C 6
LH2 2

[math]\Longrightarrow[/math]The molar mass is proportional to the number of electrons in the target material. The theoretical Moller differential cross-section is an expression for a single scattering electron. NH3, as a molecule, is composed of 10 electrons that all have an equal momentum and probability of scattering with the incident electron in the CM frame.

Comparing this with a plot of the Moller scattering angle theta,

Moller Electron Momentum in Lab Frame
Figure 8b: The Moller electron momentum distribution for 4E7 incident 11 GeV electrons in the Lab frame of reference.
Density of target material
Material Density (g/cm3)
C 2.26
NH3 0.86
LH2 0.07

[math]\Longrightarrow[/math]In the lab frame, the incident electron, and the transfer of its momentum, depends on the stationary target material density. The more dense material requires more energy to break free an electron from the target material. This higher energy results in a smaller maximum scattering angle theta for the Moller electron in the Lab frame.

Adjusting the theoretical differential cross-section for 10 times less,

TH1F *Combo=new TH1F("TheoryExperiment","Theoretical and Experimental Differential Cross-Section CM Frame",360,90,180);
TH1F *Combo2=new TH1F("TheoryExperiment2","Theoretical and Experimental Differential Cross-Section CM Frame",360,90,180);
Adjusted MollerXSect NH3.png