Relativistic Differential Cross-section - Revision history

Vanwdani at 20:51, 29 December 2018

2018-12-29T20:51:32Z

Vanwdani: /* Relativistic Differential Cross-section */

2018-12-29T20:50:40Z

Relativistic Differential Cross-section

Vanwdani: /* Relativistic Differential Cross-section */

2018-12-29T20:50:16Z

Relativistic Differential Cross-section

Vanwdani: /* Relativistic Differential Cross-section */

2018-12-29T20:49:44Z

Relativistic Differential Cross-section

Vanwdani at 20:45, 29 December 2018

2018-12-29T20:45:42Z

Vanwdani at 20:43, 29 December 2018

2018-12-29T20:43:14Z

Vanwdani: /* Relativistic Differential Cross-section */

2017-07-04T16:59:59Z

Relativistic Differential Cross-section

Vanwdani: /* Relativistic Differential Cross-section */

2017-07-04T16:49:22Z

Relativistic Differential Cross-section

Vanwdani: /* Relativistic Differential Cross-section */

2017-07-04T16:46:09Z

Relativistic Differential Cross-section

Vanwdani: /* Relativistic Differential Cross-section */

2017-07-04T16:44:00Z

Relativistic Differential Cross-section

← Older revision		Revision as of 20:51, 29 December 2018
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		+	<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=

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	<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>\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>\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>

← Older revision		Revision as of 20:50, 29 December 2018
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−	<center><math>(E_{1}^')^2=(\vec p_{1}^{'})^2+(m_{1})^{2}</math></center>	+	<center><math>(E_{1}^{'})^2=(\vec p_{1}^{'})^2+(m_{1})^{2}</math></center>

← Older revision		Revision as of 20:50, 29 December 2018
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−	<center><math>(~~E_1~~^')^2=(\vec p_{1}^{'})^2+(m_{1})^{2}</math></center>	+	<center><math>(E_{1}^')^2=(\vec p_{1}^{'})^2+(m_{1})^{2}</math></center>

@@ Line 36: / Line 36: @@
-<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{\vec p_1^' d \vec p_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
-<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>|\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>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>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>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}\frac{\vec p_{f}}{\sqrt {s}}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>\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_{f}}{\mathbf p_{i}}|\mathcal {M}|^2</math></center>

@@ Line 15: / Line 15: @@
-<center><math>F_{cms}=4 \vec p_1^*\sqrt {s}</math></center>
+<center><math>F_{cms}=4 \vec p_{1}^{*}\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\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>
-<center><math>d^3 \vec p_1^'=\vec p^{'3}_1 d \vec 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=(\vec 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^'= \vec p_1^' d \vec 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{\vec 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>

← Older revision		Revision as of 20:43, 29 December 2018
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−	<center><math>dQ=(2\pi)^4\delta^4(\vec ~~p_1~~ +\vec ~~p_2~~ - \vec ~~p_1~~^' -\vec ~~p_2~~^')\frac{d^3 \vec ~~p_1~~^'}{(2\pi)^3 ~~2E_1~~^'}\frac{d^3 \vec ~~p_2~~^'}{(2\pi)^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>

← Older revision		Revision as of 16:59, 4 July 2017
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−	~~<center><math>F=2E_1 2E_2\|\vec {v}_1-\vec {v}_2\|=4\|E_1E_2\vec v_{21}\|</math></center>~~
−
−
−	~~where <math>v_{21}</math> is the relative velocity between the particles in the frame where particle 1 is at rest~~
−
−
−	~~<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)= E_{1}E_{2}</math></center>~~
−
−
−	~~Using the relativistic definition of energy~~
−
−	~~<center><math>E^2 \equiv p^2+m^2=m^2</math></center>~~
−
−
−
−	~~<center><math>\rightarrow \mathbf P_1 \cdot \mathbf P_2 =mE_{2}</math></center>~~
−
−
−	~~Letting <math>E_{21}\equiv E_2</math> be the energy of particle 2 wiith respect to particle 1, the relativistic energy equation can be rewritten such that~~
−
−
−	~~<center><math>\|p_{21}^2\| =E_{21}^2-m^2=\frac{(\mathbf P_1 \cdot \mathbf P_2)^2}{m^2}-m^2=\frac{(\mathbf P_1 \cdot \mathbf P_2)^2-m^4}{m^2}</math></center>~~
−
−	~~where similarly <math>p_{21}</math> is defined as the momentum of particle 2 with respect to particle 1.~~
−
−
−
−
−	~~The relative velocity can be expressed as~~
−
−
−	~~<center><math> v_{21}=\frac{\|\vec p_{21}\|}{E_{21}}</math></center>~~
−
−
−
−	~~<center><math>F=2E_1 2E_2\|\vec {v}_1-\vec {v}_2\|=4\|mE_{21}\vec v_{12}\|=4\|mE_{21}\frac{\|\vec p_{21}\|}{E_{21}}\|=4m\|\vec p_{21}\|</math></center>~~
−
−
−
−	~~The invariant form of F is~~
−
−	~~<center><math>F=4\sqrt{(\mathbf P_1 \cdot \mathbf P_2)^2-m^4}</math></center>~~
−
−
−	~~<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)</math></center>~~
−
−
−
−	~~where in the center of mass frame <math>E_1=E_2</math> and <math> \vec p_1^=-\vec p_2^</math>~~
−
−
−	~~<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_1^2+\vec p_1\,^{*2}</math></center>~~
−
−
−	~~<center><math>F=4\sqrt{(E_1^2+\vec p_1\,^{2})^2-m^4}=4\sqrt{(\vec p_1 \,^{2}+m^2+\vec p_1\,^{*2})^2-m^4}</math></center>~~
−
−
−
−	~~<center><math>F=4\sqrt{(2\vec p_1\,^{2}+m^2)^2-m^4}=4\sqrt{4\vec p_1\,^{4}+m^4+4\vec p_1\,^{*2}m-m^4}</math></center>~~
−
−
−
−
−	~~<center><math>F_{cms}=4 \sqrt {4m^2\vec p_1\,^{2}+4\vec p_1 \,^{4}}</math></center>~~
−
−
−
−	~~<center><math>F_{cms}=4 \sqrt {\vec p_1\,^{2}4(m^2+\vec p_1 \,^{2})}</math></center>~~
−
−
−
−	~~As shown earlier~~
−	~~<center><math>s_{CM}=4(m^2+\vec p_1 \,^{*2})</math></center>~~
−
−
−
−
−	~~<center><math>F_{cms}=4 \sqrt {\vec p_1^*\,^2s}</math></center>~~

@@ Line 63: / Line 63: @@
-<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_1^2+\vec p_1\ ^{*2}</math></center>
+<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_1^2+\vec p_1\,^{*2}</math></center>
-<center><math>F=4\sqrt{(E_1^2+\vec p_1\ ^{*2})^2-m^4}=4\sqrt{(\vec p_1 \ ^{*2}+m^2+\vec p_1\ ^{*2})^2-m^4}</math></center>
+<center><math>F=4\sqrt{(E_1^2+\vec p_1\,^{*2})^2-m^4}=4\sqrt{(\vec p_1 \,^{*2}+m^2+\vec p_1\,^{*2})^2-m^4}</math></center>
-<center><math>F=4\sqrt{(2\vec p_1\ ^{*2}+m^2)^2-m^4}=4\sqrt{4\vec p_1\ ^{*4}+m^4+4\vec p_1\ ^{*2}m-m^4}</math></center>
+<center><math>F=4\sqrt{(2\vec p_1\,^{*2}+m^2)^2-m^4}=4\sqrt{4\vec p_1\,^{*4}+m^4+4\vec p_1\,^{*2}m-m^4}</math></center>
-<center><math>F_{cms}=4 \sqrt {4m^2\vec p_i\ ^{*2}+4\vec p_1 \ ^{*4}}</math></center>
+<center><math>F_{cms}=4 \sqrt {4m^2\vec p_1\,^{*2}+4\vec p_1 \,^{*4}}</math></center>
-<center><math>F_{cms}=4 \sqrt {\vec p_i\ ^{*2}4(m^2+\vec p_1 \ ^{*2})}</math></center>
+<center><math>F_{cms}=4 \sqrt {\vec p_1\,^{*2}4(m^2+\vec p_1 \,^{*2})}</math></center>
 As shown earlier
-<center><math>s_{CM}=4(m^2+\vec p_1 \ ^{*2})</math></center>
+<center><math>s_{CM}=4(m^2+\vec p_1 \,^{*2})</math></center>
-<center><math>F_{cms}=4 \sqrt {\vec p_1^*\ ^2s}</math></center>
+<center><math>F_{cms}=4 \sqrt {\vec p_1^*\,^2s}</math></center>
@@ Line 97: / Line 97: @@
-<center><math>d\sigma=\frac{1}{4 \vec p_1\ ^*\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>
+<center><math>d\sigma=\frac{1}{4 \vec p_1\,^*\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>

@@ Line 60: / Line 60: @@
-where in the center of mass frame <math>E_1=E_2</math> and <math> p_1^*=-p_2^*</math>
+where in the center of mass frame <math>E_1=E_2</math> and <math> \vec p_1^*=-\vec p_2^*</math>
@@ Line 66: / Line 66: @@
-<center><math>F=4\sqrt{(E_1^2+\vec p_1\ ^{*2})^2-m^4}=4\sqrt{(p_1 \ ^{*2}+m^2+p_1\ ^{*2})^2-m^4}</math></center>
+<center><math>F=4\sqrt{(E_1^2+\vec p_1\ ^{*2})^2-m^4}=4\sqrt{(\vec p_1 \ ^{*2}+m^2+\vec p_1\ ^{*2})^2-m^4}</math></center>
-<center><math>F=4\sqrt{(2p_1\ ^{*2}+m^2)^2-m^4}=4\sqrt{4p_1\ ^{*4}+m^4+4p_1\ ^{*2}m-m^4}</math></center>
+<center><math>F=4\sqrt{(2\vec p_1\ ^{*2}+m^2)^2-m^4}=4\sqrt{4\vec p_1\ ^{*4}+m^4+4\vec p_1\ ^{*2}m-m^4}</math></center>