Difference between revisions of "Phase space Limiting Particles"

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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 \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>y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)</math></center>
  
<center><math> P^- \equiv E-p_z</math></center>
 
  
 
this implies that as  
 
this implies that as  
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<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>
 
<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.
  
  
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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>]]
+
[[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>
 
</center>

Latest revision as of 16:47, 26 July 2017

[math]\textbf{\underline{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]


4.1.2 Phase space Limiting Particles

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:

[math]p^2=p_x^2+p_y^2+p_z^2[/math]

In the Center of Mass frame, this becomes:

[math]p_x^{*2}+p_y^{*2} = p^{*2}-p_z^{*2}[/math]


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


[math]y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)[/math]


this implies that as

[math]p_z \rightarrow 0 \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow 1 \Rrightarrow \ln 1 \rightarrow 0 \Rrightarrow y=0[/math]


For forward travel in the light cone:

[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]

This corresponds to the scattered electron proven earlier.

For backward travel in the light cone:

[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]


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:

[math]p_x^2+p_y^2=52.589054^2+9.272868^2=53.400MeV \gt 53.015 MeV (E) \therefore p_z \rightarrow imaginary[/math]

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.




[math]\textbf{\underline{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]