Flux of Incoming Particles

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The number of particles in a beam passing through a unit area per unit time is

[math]\frac{#\ of\ beam\ particles}{time\times Volume}=[/math]
[math]\frac{|\vec v_1|2E_1}{V}[/math]


The number of stationary target particles per unit volume is


[math]\frac{#\ of\ target\ particles}{Volume}[/math]


[math]\frac{2E_2}{V}[/math]


Initial flux=[math]|\vec v_1|\frac{2E_1}{V} \frac{2E_2}{V}[/math]


[math]F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|=4|E_1E_2\vec v_{21}|[/math]


where [math]v_{21}[/math] is the relative velocity between the particles in the frame where particle 1 is at rest


[math]\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)= E_{1}E_{2}[/math]


Using the relativistic definition of energy

[math]E^2 \equiv p^2+m^2=m^2[/math]


[math]\rightarrow \mathbf P_1 \cdot \mathbf P_2 =mE_{2}[/math]


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


[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]

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


[math] v_{21}=\frac{|\vec p_{21}|}{E_{21}}[/math]


[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]


The invariant form of F is

[math]F=4\sqrt{(\mathbf P_1 \cdot \mathbf P_2)^2-m^4}[/math]


[math]\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec 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]


[math]\mathbf P_1 \cdot \mathbf P_2 = E_1^2+\vec p_1\,^{*2}[/math]


[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]


[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]



[math]F_{cms}=4 \sqrt {4m^2\vec p_1\,^{*2}+4\vec p_1 \,^{*4}}[/math]


[math]F_{cms}=4 \sqrt {\vec p_1\,^{*2}4(m^2+\vec p_1 \,^{*2})}[/math]


As shown earlier

[math]s_{CM}=4(m^2+\vec p_1 \,^{*2})[/math]



[math]F_{cms}=4 \sqrt {\vec p_1^*\,^2s}[/math]