Phi Dependent Components - Revision history

Vanwdani at 15:00, 30 May 2017

2017-05-30T15:00:29Z

Vanwdani at 14:54, 30 May 2017

2017-05-30T14:54:54Z

Vanwdani at 14:37, 30 May 2017

2017-05-30T14:37:49Z

Vanwdani at 14:33, 30 May 2017

2017-05-30T14:33:08Z

Vanwdani at 14:25, 30 May 2017

2017-05-30T14:25:56Z

Vanwdani: Created page with "Since only the z direction is considered to be the relativistic direction of motion, this implies that the x and y components are not effected by a Lorentz transformation and rem…"

2017-05-30T14:11:52Z

Created page with "Since only the z direction is considered to be the relativistic direction of motion, this implies that the x and y components are not effected by a Lorentz transformation and rem…"

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Since only the z direction is considered to be the relativistic direction of motion, this implies that the x and y components are not effected by a Lorentz transformation and remain the same in the CM and Lab frame. Holding the angle Phi constant at an initial value of 10 degrees, allows us to find the x and y components.

<center>[[File:xy_lab.png | 400 px]]</center>
<center>'''Figure 4: Definition of Moller electron variables in the Lab Frame in the x-y plane.'''</center>

<center>Similarly, <math>\phi '_2=\arccos \left( \frac{p^'_{2(x) Lab}}{p^'_{2(xy)}} \right)</math></center>

<center>where <math>p_{2(xy)}^'=\sqrt{(p_{2(x)}^')^2+(p^'_{2(y)})^2}</math></center>

<center><math>(p^'_{2(xy)})^2=(p^'_{2(x)})^2+(p^'_{2(y)})^2</math></center>

<center>and using <math>p^2=p_{(x)}^2+p_{(y)}^2+p_{(z)}^2</math></center>

<center>this gives <math>(p^'_{2})^2=(p^'_{2(xy)})^2+(p^'_{2(z)})^2</math></center>

<center><math>\Longrightarrow (p'_{2})^2-(p'_{2(z)})^2 = (p'_{2(xy)})^2</math></center>

<center><math>\Longrightarrow p_{2(xy)}^'=\sqrt{(p^'_{2})^2-(p^'_{2(z)})^2}</math></center>

<center>which gives<math>\phi '_2 = \arccos \left( \frac{p_{2(x)}'}{\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}}}\right)</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>\Longrightarrow p_{2(x)}'=\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}} \cos(\phi)</math>
|}

<center>Similarly, using <math>p_{2}^2=p_{2(x)}^2+p_{2(y)}^2+p_{2(z)}^2</math></center>

<center><math>\Longrightarrow p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}=p_{2(y)}^{'\ 2}</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>p_{2(y)}'=\sqrt{p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}}</math>
|}

Checking on the sign from the cosine results for <math>\phi '_2</math>

We have the limiting range that <math>\phi</math> must fall within:
{| class="wikitable" align="center"
| style="background: grey" | <math>-\pi \le \phi '_2 \le \pi\ Radians</math>
|}

<center>[[File:xy_plane.png | 400px]]</center>

Examining the signs of the components which make up the angle <math>\phi</math> in the 4 quadrants which make up the xy plane:

{| class="wikitable" align="center"
| style="background: red" | <math>For\ 0 \ge \phi '_2 \ge \frac{-\pi}{2}\ Radians</math>
|-
| <center>px=POSITIVE</center>
|-
| <center>py=NEGATIVE</center>
|}

{| class="wikitable" align="center"
| style="background: red" | <math>For\ 0 \le \phi '_2 \le \frac{\pi}{2}\ Radians</math>
|-
| <center>px=POSITIVE</center>
|-
| <center>py=POSITIVE</center>
|}

{| class="wikitable" align="center"
| style="background: red" | <math>For\ \frac{-\pi}{2} \ge \phi '_2 \ge -\pi\ Radians</math>
|-
| <center>px=NEGATIVE</center>
|-
| <center>py=NEGATIVE</center>
|}

{| class="wikitable" align="center"
| style="background: red" | <math>For\ \frac{\pi}{2} \le \phi '_2 \le \pi\ Radians</math>
|-
|<center>px=NEGATIVE</center>
|-
| <center>py=POSITIVE</center>
|}

@@ Line 106: / Line 106: @@
 <center><math>\textbf{\underline{Navigation}}</math>
-[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
+[[Theta_Dependent_Components|<math>\vartriangleleft </math>]]
-[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<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>]]
+[[Electron_Center_of_Mass_Frame|<math>\vartriangleright </math>]]
 </center>

@@ Line 10: / Line 10: @@
+=4.1.3.2 Phi Dependent Components=
 Since only the z direction is considered to be the relativistic direction of motion, this implies that the x and y components are not effected by a Lorentz transformation and remain the same in the CM and Lab frame.  Holding the angle Phi constant at an initial value of 10 degrees, allows us to find the x and y components.
@@ Line 98: / Line 98: @@
 | <center>p<sub>y</sub>=POSITIVE</center>
 |}

@@ Line 1: / Line 1: @@
 <center><math>\textbf{\underline{Navigation}}</math>
-[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
+[[Theta_Dependent_Components|<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>]]
+[[Electron_Center_of_Mass_Frame|<math>\vartriangleright </math>]]
 </center>

@@ Line 2: / Line 2: @@
 [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
-[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<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>]]

@@ Line 1: / Line 1: @@
 Since only the z direction is considered to be the relativistic direction of motion, this implies that the x and y components are not effected by a Lorentz transformation and remain the same in the CM and Lab frame.  Holding the angle Phi constant at an initial value of 10 degrees, allows us to find the x and y components.
@@ Line 85: / Line 98: @@
 | <center>p<sub>y</sub>=POSITIVE</center>
 |}