Difference between revisions of "Center of Mass for Stationary Target"

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For an [[DV_Calculations_of_4-momentum_components#Center_of_Mass_Frame|incoming electron of 11GeV striking a stationary electron]] we would expect:
 
For an [[DV_Calculations_of_4-momentum_components#Center_of_Mass_Frame|incoming electron of 11GeV striking a stationary electron]] we would expect:
 +
 +
 +
Inspecting the Lorentz transformation to the Center of Mass frame:
 +
 +
 +
<center><math>\left( E1+E2000
\right)=\left(γ00βγ01000010βγ00γ
\right) . \left( E1+E200p1(z)+p2(z)
\right)</math></center>
 +
 +
 +
For the case of a stationary electron, this simplifies to:
 +
 +
<center><math>\left( Epxpypz
\right)=\left(γ00βγ01000010βγ00γ
\right) . \left( E1+m00p1(z)+0
\right)</math></center>
 +
 +
 +
which gives,
 +
 +
 +
<center><math>\Longrightarrow\begin{cases}
 +
E^*=\gamma^* (E_{1}+m)-\beta^* \gamma^* p_{1(z)} \\
 +
p^*_{z}=-\beta^* \gamma^*(E_{1}+m)+\gamma^* p_{1(z)}
 +
\end{cases}</math></center>
 +
 +
 +
Solving for <math>\beta^*</math>, with <math>p^*_{z}=0</math>
 +
 +
<center><math>\Longrightarrow \beta^* \gamma^*(E_{1}+m)=\gamma^* p_{1(z)}</math></center>
 +
 +
 +
{| class="wikitable" align="center"
 +
| style="background: gray"      | <math>\Longrightarrow \beta^*=\frac{p_{1}}{(E_{1}+m)}</math>
 +
|}
 +
 +
 +
 +
Similarly, solving for <math>\gamma^*</math> by substituting in <math>\beta^*</math>
 +
 +
 +
<center><math>E^*=\gamma^* (E_{1}+m)-\frac{p_{1}}{(E_{1}+m)} \gamma^* p_{1(z)}</math></center>
 +
 +
 +
<center><math>E^*=\gamma^* \frac{(E_{1}+m)^2}{(E_{1}+m)}-\gamma^*\frac{(p_{1(z)})^2}{(E_{1}+m)}</math></center>
 +
 +
 +
Using the fact that <math>E^*=[(E_{1}+E_{2})^2-(\vec p_{1}+\vec p_{2})^2]^{1/2}</math>
 +
 +
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<center><math>E^*=\gamma^* \frac{E^*\ ^2}{(E_{1}+m)}</math></center>
 +
 +
{| class="wikitable" align="center"
 +
| style="background: gray"      | <math>\Longrightarrow \gamma^*=\frac{(E_1+m)} {E^*}</math>
 +
|}
 +
 +
 +
 +
 +
Using the relation
 +
 +
<center><math>\left( E1+E2p1(x)+p2(x)p1(y)+p2(y)p1(z)+p2(z)
\right)=\left(γ00βγ01000010βγ00γ
\right) . \left( E1+m00p1(z)+0
\right)</math></center>
  
  
 
<center><math>\Longrightarrow \left( E2p2(x)p2(y)p2(z)
\right)=\left(γ00βγ01000010βγ00γ
\right) . \left( m000
\right)</math></center>
 
<center><math>\Longrightarrow \left( E2p2(x)p2(y)p2(z)
\right)=\left(γ00βγ01000010βγ00γ
\right) . \left( m000
\right)</math></center>
 +
 +
 +
<center><math>\Longrightarrow\begin{cases}
 +
E^*_{2}=\gamma^* (m) \\
 +
p^*_{2(z)}=-\beta^* \gamma^* (m)
 +
\end{cases}</math></center>
 +
 +
 +
<center><math>\Longrightarrow\begin{cases}
 +
E^*_{2}=\frac{(E_{1}+m)}{E^*} (m) \\
 +
p^*_{2(z)}=-\frac{p_{1}}{(E_{1}+m)} \frac{(E_{1}+m)}{E^*} (m_{2})
 +
\end{cases}</math></center>
 +
 +
 +
<center><math>\Longrightarrow\begin{cases}
 +
E^*_{2}=\frac{(11000 MeV+.511 MeV)}{106.031 MeV} (.511 MeV) \approx 53.015 MeV\\
 +
p^*_{2(z)}=-\frac{11000 MeV}{106.031 MeV} (.511 MeV) \approx -53.013 MeV
 +
\end{cases}</math></center>
 +
 +
 +
<center><math>\Longrightarrow \left( E1p1(x)p1(y)p1(z)
\right)=\left(γ00βγ01000010βγ00γ
\right) . \left( E100p1(z)
\right)</math></center>
 +
 +
 +
<center><math>\Longrightarrow\begin{cases}
 +
E^*_{1}=\gamma^* (E_{1})-\beta^* \gamma^* p_{1(z)} \\
 +
p^*_{1(z)}=-\beta^* \gamma^*(E_{1})+\gamma^* p_{1(z)}
 +
\end{cases}</math></center>
 +
 +
 +
 +
<center><math>\Longrightarrow\begin{cases}
 +
E^*_{1}=\frac{(E_{1}+m)}{E^*} (E_{1})-\frac{p_{1(z)}}{(E_{1}+m)} \frac{(E_{1}+m)}{E^*} p_{1(z)} \\
 +
p^*_{1(z)}=-\frac{p_{1}}{(E_{1}+m)} \frac{(E_{1}+m)}{E^*}(E_{1})+\frac{(E_{1}+m)}{E^*} p_{1(z)}
 +
\end{cases}</math></center>
 +
 +
 +
 +
<center><math>\Longrightarrow\begin{cases}
 +
E^*_{1}=\frac{(11000 MeV+.511 MeV)}{106.031 MeV} (11000 MeV)-\frac{11000 MeV}{106.031 MeV} 11000 MeV \approx 53.015 MeV\\
 +
p^*_{1(z)}=-\frac{11000 MeV}{106.031 MeV}(11000 MeV)+\frac{(11000 MeV+.511 MeV)}{106.031 MeV} 11000 MeV \approx 53.013 MeV
 +
\end{cases}</math></center>
 +
 +
 +
 +
<center><math>p^*_{1} =\sqrt {(p^*_{1(x)})^2+(p^*_{1(y)})^2+(p^*_{1(z)})^2} \Longrightarrow p^*_{1}=p^*_{1(z)}</math></center>
 +
 +
 +
<center><math>p^*_{2} =\sqrt {(p^*_{2(x)})^2+(p^*_{2(y)})^2+(p^*_{2(z)})^2} \Longrightarrow p^*_{2}=p^*_{2(z)}</math></center>
 +
 +
 +
This gives the momenta of the particles in the center of mass to have equal magnitude, but opposite directions.
 +
  
  
  
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<center><gallery widths=500px heights=400px>
 +
File:Init_e_Mom_CM.png|'''Figure 5.1.3:''' Initial momentum for lab frame electron incident at 11GeV as seen in the center of mass frame.
 +
File:Init_Mol_Mom_CM.png|'''Figure 5.1.4:''' Initial momentum for a lab frame stationary particle as seen in the center of mass frame.
 +
</gallery></center>
  
<center>[[File:Init_e_Mom_CM.png|600 px]][[File:Init_Mol_Mom_CM.png|600 px]]</center>
 
  
  
  
<center>[[File:Init_e_Theta_CM.png|600 px]][[File:Init_Mol_Theta_CM.png|600 px]]</center>
+
<center><gallery widths=500px heights=400px>
 +
File:Init_e_Theta_CM.png|'''Figure 5.1.5:''' Initial incoming angle theta for a lab frame electron incident at 11GeV as seen in the center of mass frame.
 +
File:Init_Mol_Theta_CM.png|'''Figure 5.1.6:''' Initial incoming angle theta for a lab frame stationary electron as seen in the center of mass frame.
 +
</gallery></center>
  
  
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<center><math>\textbf{\underline{Navigation}}</math>
 
<center><math>\textbf{\underline{Navigation}}</math>
  
[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
+
[[Uniform_distribution_in_Energy_and_Theta_LUND_files|<math>\vartriangleleft </math>]]
 
[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]]
 
[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]]
[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance|<math>\vartriangleright </math>]]
+
[[Phase_space_Limiting_Particles|<math>\vartriangleright </math>]]
  
 
</center>
 
</center>

Latest revision as of 18:36, 30 May 2017

\underline{Navigation}


4.1.1 Center of Mass for Stationary Target

For an incoming electron of 11GeV striking a stationary electron we would expect:


Inspecting the Lorentz transformation to the Center of Mass frame:


(E1+E2000)=(γ00βγ01000010βγ00γ).(E1+E200p1(z)+p2(z))


For the case of a stationary electron, this simplifies to:

(Epxpypz)=(γ00βγ01000010βγ00γ).(E1+m00p1(z)+0)


which gives,


{E=γ(E1+m)βγp1(z)pz=βγ(E1+m)+γp1(z)


Solving for β, with pz=0

βγ(E1+m)=γp1(z)


β=p1(E1+m)


Similarly, solving for γ by substituting in β


E=γ(E1+m)p1(E1+m)γp1(z)


E=γ(E1+m)2(E1+m)γ(p1(z))2(E1+m)


Using the fact that E=[(E1+E2)2(p1+p2)2]1/2


E=γE 2(E1+m)
γ=(E1+m)E



Using the relation

(E1+E2p1(x)+p2(x)p1(y)+p2(y)p1(z)+p2(z))=(γ00βγ01000010βγ00γ).(E1+m00p1(z)+0)


(E2p2(x)p2(y)p2(z))=(γ00βγ01000010βγ00γ).(m000)


{E2=γ(m)p2(z)=βγ(m)


{E2=(E1+m)E(m)p2(z)=p1(E1+m)(E1+m)E(m2)


{E2=(11000MeV+.511MeV)106.031MeV(.511MeV)53.015MeVp2(z)=11000MeV106.031MeV(.511MeV)53.013MeV


(E1p1(x)p1(y)p1(z))=(γ00βγ01000010βγ00γ).(E100p1(z))


{E1=γ(E1)βγp1(z)p1(z)=βγ(E1)+γp1(z)


{E1=(E1+m)E(E1)p1(z)(E1+m)(E1+m)Ep1(z)p1(z)=p1(E1+m)(E1+m)E(E1)+(E1+m)Ep1(z)


{E1=(11000MeV+.511MeV)106.031MeV(11000MeV)11000MeV106.031MeV11000MeV53.015MeVp1(z)=11000MeV106.031MeV(11000MeV)+(11000MeV+.511MeV)106.031MeV11000MeV53.013MeV


p1=(p1(x))2+(p1(y))2+(p1(z))2p1=p1(z)


p2=(p2(x))2+(p2(y))2+(p2(z))2p2=p2(z)


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









\underline{Navigation}