What is the formula for the differential cross section and macro cross section?

[queenstudy]

hello , can anyone give the formula of the differential cross section and the macro cross section because in each web i see a different notation thank you

 

[Einj]

There are very different formulas for the differential cross section that depend on what is more convenient for your particular case. However I found often usefull the formula for the differential cross section of a process "two particle in many particles":

$$A(p_1)+B(p_2)\rightarrow \sum_{i=1}^n{C_i(k_i)}$$

the differential cross section in this case can be written as:

$$d\sigma=\frac{(2\pi)^4\delta\left(p_1+p_2-\sum_i{k_i}\right)}{4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}}{\left|M(p_1,p_2;k_i)\right|}^2\prod_{j=1}^n{\frac{d^3k_j}{(2\pi)^32\omega_j(k_j)}}$$

where M is the matrix element of the process.
The total cross section is obviousli the integral of the differential cross section. However you usually define it as:

$$dN_r=dN_f \cdot \sigma \cdot n_b\cdot d$$

where $N_r$ is the number of particles produced by the reaction, $N_f$ is the number of particles in the beam, $n_b$ is the density of targets and d is the thickness of the target.

You can find some usefull formulas and a very good treatment of this subject here: http://www.staff.science.uu.nl/~wit00103/ftip/Ch03.pdf

 

[queenstudy]

thanks for the help but the first 2 formulas are new to me but the third is what i am looking for but there is another syntax for it i still don't remember it but thanks any way