Phase space Limiting Particles - Revision history

Vanwdani: /* 4.1.2 Phase space Limiting Particles */

2017-07-26T16:47:43Z

4.1.2 Phase space Limiting Particles

Vanwdani: /* 4.1.2 Phase space Limiting Particles */

2017-07-26T16:31:09Z

4.1.2 Phase space Limiting Particles

Vanwdani at 14:59, 30 May 2017

2017-05-30T14:59:26Z

Vanwdani at 14:47, 30 May 2017

2017-05-30T14:47:18Z

Vanwdani at 14:46, 30 May 2017

2017-05-30T14:46:24Z

Vanwdani at 14:32, 30 May 2017

2017-05-30T14:32:25Z

Vanwdani at 14:24, 30 May 2017

2017-05-30T14:24:41Z

Vanwdani: Created page with "Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z compone…"

2017-05-30T14:10:24Z

Created page with "Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z compone…"

New page

Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z component of the momentum must decrease by the relation:

<center><math>p^2=p_x^2+p_y^2+p_z^2</math></center>

In the Center of Mass frame, this becomes:

<center><math>p_x^{*2}+p_y^{*2} = p^{*2}-p_z^{*2}</math></center>

Since the momentum in the CM frame is a constant, this implies that pz must decrease. We can use the variable rapidity:

<center><math>y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)</math></center>

where

<center><math> P^+ \equiv E+p_z</math></center>

<center><math> P^- \equiv E-p_z</math></center>

this implies that as
<center><math>p_z \rightarrow 0 \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow 1 \Rrightarrow \ln 1 \rightarrow 0 \Rrightarrow y=0</math></center>

For forward travel in the light cone:
<center><math>p_z \rightarrow E \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow \infin \Rrightarrow \ln \infin \rightarrow \infin \Rrightarrow y \rightarrow \infin </math></center>

This corresponds to the scattered electron proven [[DV_Calculations_of_4-momentum_components|earlier]].

For backward travel in the light cone:
<center><math>p_z \rightarrow -E \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow 0 \Rrightarrow \ln 0 \rightarrow -\infin \Rrightarrow y \rightarrow -\infin </math></center>

Similarly, this corresponds to the Moller electron.

For a particle that transforms from the Lab frame to the CM frame where the particle is not within the light cone:

<center><math>p_x^2+p_y^2=52.589054^2+9.272868^2=53.400MeV > 53.015 MeV (E) \therefore p_z \rightarrow imaginary</math></center>

These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the number of particles that will be detected.

← Older revision		Revision as of 16:31, 26 July 2017
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	Since the momentum in the CM frame is a constant, this implies that pz must decrease. We can use the variable rapidity:		Since the momentum in the CM frame is a constant, this implies that pz must decrease. We can use the variable rapidity:
		+
		+
		+	<center><math>y \equiv \arctan{\frac{\vec p}{E}}</math></center>
		+
		+
		+
		+	<center><math>y \equiv \frac {1}{2} \ln \left(\frac{E+\vec p}{E-\vec p}\right)</math></center>
		+
		+

	<center><math>y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)</math></center>		<center><math>y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)</math></center>

@@ Line 61: / Line 61: @@
 <center><math>\textbf{\underline{Navigation}}</math>
-[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
+[[Center_of_Mass_for_Stationary_Target|<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>]]
+[[Determining_Momentum_Components_After_Collision_in_CM_Frame|<math>\vartriangleright </math>]]
 </center>

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

← Older revision		Revision as of 14:46, 30 May 2017
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		+	=4.1.2 Phase space Limiting Particles=


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	These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the number of particles that will be detected.		These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the number of particles that will be detected.
		+
		+
		+
		+	----

@@ 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 22: / Line 22: @@
 Since the momentum in the CM frame is a constant, this implies that pz must decrease.  We can use the variable rapidity:
@@ Line 34: / Line 27: @@
 <center><math>y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)</math></center>
 this implies that as
@@ Line 60: / Line 48: @@
 <center><math>p_x^2+p_y^2=52.589054^2+9.272868^2=53.400MeV > 53.015 MeV (E) \therefore p_z \rightarrow imaginary</math></center>
-These particles are outside the light cone and are more timelike, thus not visible in normal space.  This will reduce the number of particles that will be detected.
+These particles are outside the light cone and are more timelike, thus not visible in normal space.  This will reduce the range in theta that Moller electrons will be detected.

← Older revision		Revision as of 14:24, 30 May 2017
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		+	<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>
		+
		+
		+
		+
		+
	Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z component of the momentum must decrease by the relation:		Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z component of the momentum must decrease by the relation:

Line 39:		Line 51:

	These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the number of particles that will be detected.		These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the number of particles that will be detected.
		+
		+
		+
		+	<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>