Center of Mass for Stationary Target - Revision history

Vanwdani: /* 4.1.1 Center of Mass for Stationary Target */

2017-05-30T18:36:08Z

4.1.1 Center of Mass for Stationary Target

Vanwdani: /* 4.1.1 Center of Mass for Stationary Target */

2017-05-30T18:35:37Z

4.1.1 Center of Mass for Stationary Target

Vanwdani: /* 4.1.1 Center of Mass for Stationary Target */

2017-05-30T18:32:43Z

4.1.1 Center of Mass for Stationary Target

Vanwdani: /* 4.1.1 Center of Mass for Stationary Target */

2017-05-30T18:32:04Z

4.1.1 Center of Mass for Stationary Target

Vanwdani: /* 4.1.1 Center of Mass for Stationary Target */

2017-05-30T18:31:36Z

4.1.1 Center of Mass for Stationary Target

Vanwdani at 14:59, 30 May 2017

2017-05-30T14:59:05Z

Vanwdani at 14:45, 30 May 2017

2017-05-30T14:45:44Z

Vanwdani at 14:44, 30 May 2017

2017-05-30T14:44:37Z

Vanwdani at 14:32, 30 May 2017

2017-05-30T14:32:05Z

Vanwdani at 14:24, 30 May 2017

2017-05-30T14:24:15Z

@@ Line 138: / Line 138: @@
 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>

@@ Line 128: / Line 128: @@
 <center><gallery widths=500px heights=400px>
-File:Init_e_Mom_CM.png|'''Figure 5.1.3:''' Initial momentum for a lab frame stationary particle as seen in the center of mass frame.
+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 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>
@@ Line 136: / Line 136: @@
 <center><gallery widths=500px heights=400px>
-File:Init_e_Theta_CM.png|'''Figure 5.1.5:''' An Isotropic CM frame distribution of bin hits in the DC for superlayer 1, layer 1
+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:''' An Isotropic lab frame distribution of bin hits in the DC for superlayer 1, layer 1.
+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.

@@ Line 136: / Line 136: @@
 <center><gallery widths=500px heights=400px>
-File:File:Init_e_Theta_CM.png|'''Figure 5.1.5:''' An Isotropic CM frame distribution of bin hits in the DC for superlayer 1, layer 1
+File:Init_e_Theta_CM.png|'''Figure 5.1.5:''' An Isotropic CM frame distribution of bin hits in the DC for superlayer 1, layer 1
 File:Init_Mol_Theta_CM.png|'''Figure 5.1.6:''' An Isotropic lab frame distribution of bin hits in the DC for superlayer 1, layer 1.
 </gallery></center>

@@ Line 128: / Line 128: @@
 <center><gallery widths=500px heights=400px>
-File:File:Init_e_Mom_CM.png|'''Figure 5.1.3:''' Initial momentum for a lab frame stationary particle as seen in the center of mass frame.
+File:Init_e_Mom_CM.png|'''Figure 5.1.3:''' Initial momentum for a lab frame stationary particle as seen in the center of mass frame.
 File:Init_Mol_Mom_CM.png|'''Figure 5.1.4:''' Initial momentum for lab frame electron incident at 11GeV as seen in the center of mass frame.
 </gallery></center>

← Older revision		Revision as of 18:31, 30 May 2017
Line 12:		Line 12:

	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( \begin{matrix}E^_{1}+E^_{2}\\ 0 \\ 0 \\ 0\end{matrix} \right)=\left(\begin{matrix}\gamma^* & 0 & 0 & -\beta^* \gamma^\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta^ \gamma^* & 0 & 0 & \gamma^* \end{matrix} \right) . \left( \begin{matrix}E_{1}+E_{2}\\ 0 \\ 0 \\ p_{1(z)}+p_{2(z)}\end{matrix} \right)</math></center>
		+
		+
		+	For the case of a stationary electron, this simplifies to:
		+
		+	<center><math>\left( \begin{matrix} E^* \\ p^_{x} \\ p^_{y} \\ p^_{z}\end{matrix} \right)=\left(\begin{matrix}\gamma^ & 0 & 0 & -\beta^* \gamma^\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta^\gamma^* & 0 & 0 & \gamma^* \end{matrix} \right) . \left( \begin{matrix}E_{1}+m\\ 0 \\ 0 \\ p_{1(z)}+0\end{matrix} \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>
		+
		+
		+	<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( \begin{matrix} E^_{1}+E^_{2} \\ p^_{1(x)}+p^_{2(x)} \\ p^_{1(y)}+p^_{2(y)} \\ p^_{1(z)}+p^_{2(z)}\end{matrix} \right)=\left(\begin{matrix}\gamma^* & 0 & 0 & -\beta^* \gamma^\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta^ \gamma^* & 0 & 0 & \gamma^* \end{matrix} \right) . \left( \begin{matrix}E_{1}+m\\ 0 \\ 0 \\ p_{1(z)}+0\end{matrix} \right)</math></center>


Line 17:		Line 74:


		+	<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( \begin{matrix} E^_{1}\\ p^_{1(x)} \\ p^_{1(y)} \\ p^_{1(z)}\end{matrix} \right)=\left(\begin{matrix}\gamma^* & 0 & 0 & -\beta^* \gamma^\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta^ \gamma^* & 0 & 0 & \gamma^* \end{matrix} \right) . \left( \begin{matrix}E_{1}\\ 0 \\ 0 \\ p_{1(z)}\end{matrix} \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.
		+
		+
		+
		+
		+	<center><gallery widths=500px heights=400px>
		+	File:File:Init_e_Mom_CM.png\|'''Figure 5.1.3:''' Initial momentum for a lab frame stationary particle as seen in the center of mass frame.
		+	File:Init_Mol_Mom_CM.png\|'''Figure 5.1.4:''' Initial momentum for lab frame electron incident at 11GeV 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><gallery widths=500px heights=400px>
		+	File:File:Init_e_Theta_CM.png\|'''Figure 5.1.5:''' An Isotropic CM frame distribution of bin hits in the DC for superlayer 1, layer 1
		+	File:Init_Mol_Theta_CM.png\|'''Figure 5.1.6:''' An Isotropic lab frame distribution of bin hits in the DC for superlayer 1, layer 1.
		+	</gallery></center>

−	~~<center>[[File:Init_e_Theta_CM.png\|600 px]][[File:Init_Mol_Theta_CM.png\|600 px]]</center>~~

@@ Line 34: / Line 34: @@
 <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>]]
-[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance|<math>\vartriangleright </math>]]
+[[Phase_space_Limiting_Particles|<math>\vartriangleright </math>]]
 </center>

@@ Line 9: / Line 9: @@
+=4.1.1  Center of Mass for Stationary Target=
 For an [[DV_Calculations_of_4-momentum_components#Center_of_Mass_Frame|incoming electron of 11GeV striking a stationary electron]] we would expect:
@@ Line 24: / Line 24: @@
 <center>[[File:Init_e_Theta_CM.png|600 px]][[File:Init_Mol_Theta_CM.png|600 px]]</center>

← Older revision		Revision as of 14:24, 30 May 2017
Line 1:		Line 1:
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\triangle </math>]]
		+	[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance\|<math>\vartriangleright </math>]]
		+
		+	</center>
		+
		+
		+
		+
		+
	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:

Line 12:		Line 24:

	<center>[[File:Init_e_Theta_CM.png\|600 px]][[File:Init_Mol_Theta_CM.png\|600 px]]</center>		<center>[[File:Init_e_Theta_CM.png\|600 px]][[File:Init_Mol_Theta_CM.png\|600 px]]</center>
		+
		+
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\triangle </math>]]
		+	[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance\|<math>\vartriangleright </math>]]
		+
		+	</center>