Derivation of the differential cross section

[Gavroy]

hi

i am currently looking for a derivation of the differential cross section, that is not an abuse of mathematics, cause all derivation i found use differentials that are treated like fractions and so on?

greetings

Gavroy

 

[xts]

I can't really get what's your problem...

If you need an intuitive example of differential cross section, try the one for scattering particles at a given angle. Differential cross section $d\sigma$ is an area on which you have to aim incoming particle to get it scattered at the angle between $\theta$ and $\theta+d\theta$

 

[timthereaper]

When you say "differential cross section", are you talking about the particle physics concept?

 

[chrisbaird]

Do you want a derivation of the differential cross section variable itself? If so, there isn't really a derivation, per se. The differential cross section is just defined to mean something useful: the amount of scattering into a infinitesimal solid angle (in units of an effective cross-sectional scattering area). If you mean a derivation of how the differential cross section depends on the system variables, that will depend on the specifics of the system. Here is an example from http://faculty.uml.edu/cbaird/95.658%282011%29/Lecture8.pdf" .

 

[Gavroy]

I can't really get what's your problem...

If you need an intuitive example of differential cross section, try the one for scattering particles at a given angle. Differential cross section $d\sigma$ is an area on which you have to aim incoming particle to get it scattered at the angle between $\theta$ and $\theta+d\theta$

this is exactly what i found in many physics books, but i am looking for a derivation that does not use differentials and infinitesimal small variables. i am looking for a derivation, that does not use as I would call them "vague mathematical concepts".

and to clearify what i want to get:
$\frac{d \sigma}{d \Omega}= \frac{b}{sin (\theta)} \frac{db}{d\theta}$

 

[xts]

i am looking for a derivation that does not use differentials and infinitesimal small variables. i am looking for a derivation, that does not use as I would call them "vague mathematical concepts".

I am afraid you are looking for impossible, as those infinitesimal small steps are a very foundation of calculus...
Like looking for algebra without vague concept of multiplication.

 

[chrisbaird]

this is exactly what i found in many physics books, but i am looking for a derivation that does not use differentials and infinitesimal small variables. i am looking for a derivation, that does not use as I would call them "vague mathematical concepts".

and to clearify what i want to get:
$\frac{d \sigma}{d \Omega}= \frac{b}{sin (\theta)} \frac{db}{d\theta}$

You want a derivation of the differential cross section that does not use differentials? That's like asking for a description of the sun that never uses the word "sun". If you don't trust differentials, than treat everything as macroscopic angles and do the derivation, then at the end take the limit of infinitesimal segments to recover calculus.

 

[Gavroy]

it is not that i do not trust calculus

for example the derivative in the equation is well-defined

$\frac{d}{d \Omega} : C^1 (\mathbb{R}) \rightarrow C(\mathbb{R})$

cause everybody here knows what a derivative is

but what do you mean, when you are referring to an infinitesimal small number?

 

[xts]

Try to explain how do you define the operator:

$\frac{d}{d \Omega}$

without using the term of "infinitesimal" changes. I believe you are able to provide mathematically correct definition. Then take the phrases you used in your explanation and you will have the definition of 'infinitesimality' fitting to your taste.