Classical Limit formula for differential cross section for Hard Sphere

[aim1732]

I am looking for the derivation to an approximation formula for the differential cross section for hard sphere scattering in the limit of high energy. The paper that mentioned this had referred to Methods of Theoretical Physics, PM Morse and H. Feshbach page 1484 but I have no access to the text.Can somebody please help me out?The actual formula has like three special functions in a summation.

 

[Bill_K]

[Taking my copy of Morse and Feshbach from beneath its helium-filled bell jar...]

Page 1484 is devoted to showing that the total cross section approximates 2πa². Then on p 1485 it says, "Later in this chapter we shall show that when ka is very large, the scattered intensity becomes S = a²/4r² + (a²/4r²)cot²(θ/2) J₁²(ka sin θ). The first term of this is the reflected wave, with uniform distribution in all directions... The second term corresponds to the shadow-forming wave..."

Is this getting close?

 

[aim1732]

Thank you but you know I already have the result.I do not know how to get there however.Yes it is exactly what I am looking for but I think I would require a bit more.

 

[DrDu]

You may also have a look at Mie's classic paper:

http://diogenes.iwt.uni-bremen.de/vt/laser/papers/SAND78-6018-Mie-1908-translation.pdf

 

[Bill_K]

Thank you but you know I already have the result.I do not know how to get there however.Yes it is exactly what I am looking for but I think I would require a bit more.

Please try to make it clear from the outset what it is you have and what it is you want. You said it involved "three special functions in a summation." There is nothing on p 1484 like that. The formula I quoted is derived much later in the chapter, using Green's functions and steepest descent, but I see no point in pursuing it only to be told you have it already.

 

[DrDu]

[Taking my copy of Morse and Feshbach from beneath its helium-filled bell jar...]

I got mine when the library of the Helmholtz Zentrum "modernized" their library and everyone could take books no longer wanted for free before the rest got binned. It is quite sad what has become of a formerly reknowned organization.

 

[vanhees71]

What? They through out Morse/Feshbach? I'm shocked. Also GSI has no library anymore. Fortunately my colleagues there saved the theory books for their hand library in the theory department.

 

[aim1732]

Please try to make it clear from the outset what it is you have and what it is you want. You said it involved "three special functions in a summation." There is nothing on p 1484 like that. The formula I quoted is derived much later in the chapter, using Green's functions and steepest descent, but I see no point in pursuing it only to be told you have it already.

Well I am extremely sorry if you got me wrong.I am actually reading this paper that scoops this result out of Morse Feshbach and I had no means to see how the result was arrived at.I said it involves summation of three special functions because that is how I thought it should be done.If I had seen or read the book before there would have been little point for this thread but I have no idea how the result is arrived at in the text.All I have is this reference in the paper.

I am sorry if you feel offended or something.I am just a greenhorn undergrad doing a project that involves this kind of grad level stuff.I am quite overwhelmed by the mathematics so please excuse me for this.