Difference between revisions of "Relativistic Differential Cross-section"

From New IAC Wiki
Jump to navigation Jump to search
 
(83 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
<center><math>\underline{\textbf{Navigation}}</math></center>
 +
 +
<center>
 +
[[Lorentz_Transformation_to_Lab_Frame|<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>]]
 +
</center>
 +
 
=Relativistic Differential Cross-section=
 
=Relativistic Differential Cross-section=
  
Line 6: Line 14:
  
  
<center><math>dQ=(2\pi)^4\delta^4(p_1 + p_2 - p_1^' - p_2^')\frac{d^3p_1^'}{()^3 2E_1^'}\frac{d^3p_2^'}{()^3 2E_2^'}</math></center>
+
<center><math>dQ=(2\pi)^4\delta^4 \left(\vec p_{1} +\vec p_{2} - \vec p_{1}^{'} -\vec p_{2}^{'} \right)\frac{d^3 \vec p_{1}^{'}}{(2\pi)^3 2E_{1}^{'}}\frac{d^3 \vec p_{2}^{'}}{(2\pi)^3 2E_{2}^{'}}</math></center>
  
  
Line 12: Line 20:
  
  
<center><math>F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|</math></center>
 
  
  
The invariant form of F is
 
  
 +
<center><math>F_{cms}=4 \vec p_{1}^{*}\sqrt {s}</math></center>
  
<center><math>F=4\sqrt{(\vec {p}_1 \cdot \vec {p}_2)^2-(m_1m_2)^2}</math></center>
 
  
  
<center><math>F_{cms}=4p_i\sqrt {s}</math></center>
+
<center><math>d\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>
  
  
<center><math>d^3p_1^'=p^{'3}_1 d_p^'  d\Omega</math></center>
 
  
 +
<center><math>d^3 \vec p_{1}^{'}=\vec p^{'3}_{1} d \vec p^{'}  d\Omega</math></center>
  
<center><math>(E_1^')^2=(p_1^')^2+(m_1)^2</math></center>
+
 +
<center><math>(E_{1}^{'})^2=(\vec p_{1}^{'})^2+(m_{1})^{2}</math></center>
  
  
<center><math>E_1^' d E_1^'= p_1^' d p_1^'</math></center>
+
<center><math>E_{1}^{'} d E_{1}^{'}= \vec p_{1}^{'} d \vec p_{1}^{'}</math></center>
  
  
  
<center><math>dQ=\frac{1}{(4\pi)^2}\delta (E_1+E_2-E_1^'-E_2^')\frac{p_1^'dE_1^'}{E_2^'}d\Omega</math><\center>
+
<center><math>dQ=\frac{1}{(4\pi)^2}\delta (E_{1}+E_{2}-E_{1}^{'}-E_{2}^{'})\frac{\vec p_{1}^{'}dE_{1}^{'}}{E_{2}^{'}}d\Omega</math><\center>
  
  
<center><math>W_i \equiv E_1+E_2 \qquad \qquad W_f \equiv E_1^'+E_2^'</math></center>
+
<center><math>W_{i} \equiv E_{1}+E_{2} \qquad \qquad W_f \equiv E_{1}^{'}+E_{2}^{'}</math></center>
  
  
  
<center><math>dW_f=dE_1^'+dE_2^'=\frac{p_1^' dp_1^'}{E_1^'}+\frac{p_2^' dp_2^'}{E_2^'}</math></center>
+
<center><math>dW_f=dE_{1}^{'}+dE_{2}^{'}=\frac{\vec p_{1}^{'} d \vec p_{1}^{'}}{E_{1}^{'}}+\frac{p_{2}^{'} dp_{2}^{'}}{E_{2}^{'}}</math></center>
  
  
 
In the center of mass frame  
 
In the center of mass frame  
<center><math>p_1^'=p_2^'=p_f^' \rightarrow p_1^' dp_1^'=p_2^' dp_2^'=p_f^' dp_f^'</math></center>
+
<center><math>|\vec p_{1}^{'}|=|\vec p_{2}^{'}|=|\vec p_{f}^{'}| \rightarrow |\vec p_{1}^{'} d \vec p_{1}^{'}|=|\vec p_{2}^{'} d \vec p_{2}^{'}|=|\vec p_{f}^{'} d \vec p_{f}^{'}|</math></center>
 +
 
 +
 
 +
 
 +
<center><math>dW_{f}=\frac{W_{f}}{E_{2}^{'}}dE_{1}^{'}</math></center>
 +
 
  
  
 +
<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_{f} dW_{f}}{W_{f}}d\Omega</math></center>
  
<center><math>dW_f=\frac{W_f}{E_2^'}dE_1^'</math></center>
 
  
 +
<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_{f}}{\sqrt {s}}d\Omega</math></center>
  
  
<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{p_f dW_f}{W_f}d\Omega</math></center>
+
<center><math>\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_{f}}{\mathbf p_{i}}|\mathcal {M}|^2</math></center>
  
  
<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\frac{p_f}{\sqrt s}d\Omega</math></center>
+
----
  
 +
<center><math>\underline{\textbf{Navigation}}</math></center>
  
<center><math>\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_f}{\mathbf p_i}|\mathcal {M}|^2</math></center>
+
<center>
 +
[[Lorentz_Transformation_to_Lab_Frame|<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>]]
 +
</center>

Latest revision as of 20:51, 29 December 2018

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

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

Relativistic Differential Cross-section

[math]d\sigma=\frac{1}{F}|\mathcal{M}|^2 dQ[/math]

dQ is the invariant Lorentz phase space factor


[math]dQ=(2\pi)^4\delta^4 \left(\vec p_{1} +\vec p_{2} - \vec p_{1}^{'} -\vec p_{2}^{'} \right)\frac{d^3 \vec p_{1}^{'}}{(2\pi)^3 2E_{1}^{'}}\frac{d^3 \vec p_{2}^{'}}{(2\pi)^3 2E_{2}^{'}}[/math]


and F is the flux of incoming particles



[math]F_{cms}=4 \vec p_{1}^{*}\sqrt {s}[/math]


[math]d\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ[/math]


[math]d^3 \vec p_{1}^{'}=\vec p^{'3}_{1} d \vec p^{'} d\Omega[/math]


[math](E_{1}^{'})^2=(\vec p_{1}^{'})^2+(m_{1})^{2}[/math]


[math]E_{1}^{'} d E_{1}^{'}= \vec p_{1}^{'} d \vec p_{1}^{'}[/math]


[math]dQ=\frac{1}{(4\pi)^2}\delta (E_{1}+E_{2}-E_{1}^{'}-E_{2}^{'})\frac{\vec p_{1}^{'}dE_{1}^{'}}{E_{2}^{'}}d\Omega[/math]<\center>


[math]W_{i} \equiv E_{1}+E_{2} \qquad \qquad W_f \equiv E_{1}^{'}+E_{2}^{'}[/math]


[math]dW_f=dE_{1}^{'}+dE_{2}^{'}=\frac{\vec p_{1}^{'} d \vec p_{1}^{'}}{E_{1}^{'}}+\frac{p_{2}^{'} dp_{2}^{'}}{E_{2}^{'}}[/math]


In the center of mass frame

[math]|\vec p_{1}^{'}|=|\vec p_{2}^{'}|=|\vec p_{f}^{'}| \rightarrow |\vec p_{1}^{'} d \vec p_{1}^{'}|=|\vec p_{2}^{'} d \vec p_{2}^{'}|=|\vec p_{f}^{'} d \vec p_{f}^{'}|[/math]


[math]dW_{f}=\frac{W_{f}}{E_{2}^{'}}dE_{1}^{'}[/math]


[math]dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_{f} dW_{f}}{W_{f}}d\Omega[/math]


[math]dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_{f}}{\sqrt {s}}d\Omega[/math]


[math]\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_{f}}{\mathbf p_{i}}|\mathcal {M}|^2[/math]



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

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