Bessel Functions as Solutions to Scattering Integrals?

[Tidewater]

Hello All.

I'm currently in a crash course on X-ray Diffraction and Scattering Theory, and I've reached a point where I have to learn about Bessel Functions, and how they can be used as solutions to integrals of certain functions which have no solution. Or at least, that's as much as I understand. So far I've been able to learn about what Bessel Functions are, how they are derived/defined, etc., what simple functions (like the sinc function) are equal to some of them. But what I haven't found is something like a Fourier Transform Pair Table, only for Bessel Functions and their related unsolvable integrals.

The example I was given is

j₀(√(x²+y²+z²)) = 1/(4∏)∫∫sinθe^(i(xsinθcos\varphi+ysinθsin\varphi+zcosθ))dθd\varphi

where the integration is from \varphi=0→2∏ and θ=0→∏

I'm still not entirely sure on the geometric representation of this expression, though it seems ellipsoid (at least the expression within the exponential does).

The idea which was expressed to me was that, through memorization, one should be able to recognize which integrals of this nature (∫e^ikr) are unsolvable, and which Bessel Functions provide the 'solutions'. To me, this says there must be a Fourier-transform-pair-esque table somewhere. Am I wrong in my thinking?

Thanks for any help you can give!

 

[Tidewater]

Update: I still have yet to find any such table. Any help is appreciated on this matter.

 

[SteamKing]

It's not clear what you are trying to find.

The Bessel functions of various kinds take one argument and produce one functional value. They are not transforms in the sense of a Fourier transform. Bessel functions themselves can be shown to have their own Fourier transforms and Laplace transforms. These functions are also orthogonal, however, and if one was sufficiently masochistic, other functions could be represented in terms of a summation of different Bessel functions.

Bessel functions are derived from the solution to certain ODEs and are generally evaluated using infinite series or by evaluating certain definite integrals.

This article http://en.wikipedia.org/wiki/Bessel_function will hit the highlights of these functions.

 

[Tidewater]

Thanks SteamKing, I understand most of that. I recognize that Bessel functions aren't transforms, like Fourier transforms. But what I'm trying to find is a table of sorts, of relations between Bessel functions and integrals of the form ∫e^(ikr) or ∫sinθe^(ikr), where r is a position vector in 3D space, and k is a 3D scattering function.

I'm under task to learn how to recognize when a Bessel function is to be used as a 'solution' to such an integral, but I can't find any relation between Bessel functions and ∫e^(ikr) integrals anywhere.

 

[Tidewater]

Does anyone have any idea what I'm talking about, or am I just spouting nonsense?

 

[SteamKing]

Well, you are not making much sense, but that might be because you haven't specified what your problem is.

You say integrals of the form e^(ikr) or sinθ e^(ikr) are 'unsolvable', but in what sense do you mean this? Are you trying to evaluate these integrals? Is 'r' some function other than a radius?

The example formula in your OP looks like your are trying to evaluate a double integral.
Are you trying to verify the numerical equivalence between J0(r) and the definite integral from the OP?

This integral might not be 'solvable', to use your term, because the indefinite integral cannot be expressed in terms of elementary functions, but that is no impediment to obtaining a numerical evaluation.

The CDF of a normal distribution is not 'solvable' in terms of elementary functions, but numerical values of the CDF are quite well known and used constantly.

 

[Tidewater]

Well okay, to clarify, the way this was explained to me (in second-language English spoken by my Japanese supervisor) is that in some cases of diffraction scattering from an aspherical sample (such as a metallofullerene), the definite spherical integration over θ and φ which defines the intensity (I think) is 'unsolvable' (his wording), and a Bessel function is used as (what I assume to be) an approximation.

The formula in the OP is an example in a Japanese textbook (so I'm afraid I don't completely know the context) of a case where an 'unsolvable' (again, his wording) intensity integration can be equated to the spherical Bessel function j0.

In terms of the definite integral being numerically evaluated, my understanding is that because the variable r (or in the case of this function, the variables x, y, and z) is present in the integrand, and is not being integrated over, a numerical evaluation is out of the question.

And as to specifying what my problem is, I need to understand the thought process or mechanism behind the relation between Bessel functions and these sort of integrals, and I have yet to find any mention of such a relation anywhere aside from in papers concerning scattering from fullerenes and carbon nanotubes in which this technique is utilized, though such a reference doesn't exactly explain why such a relation exists.

 

[SteamKing]

In terms of the definite integral being numerically evaluated, my understanding is that because the variable r (or in the case of this function, the variables x, y, and z) is present in the integrand, and is not being integrated over, a numerical evaluation is out of the question.

If 'r' is not the variable of integration, then it can be treated as a constant. That is Calc 101. Your integral in the OP can be evaluated numerically, since the variables of integration are 'theta' and 'phi'. Granted, iterated integrals are more involved when numerically evaluated, but the principle remains valid.

I think your colleague is trying to explain that by evaluating the Bessel function instead, you can avoid having to do a complicated numerical evaluation of a double integral. IMO, what you are lacking is an understanding on how to manipulate the integral definition of J0 to show that it is equivalent to the integral in the OP. I don't think that such an understanding will come easily unless you can find the time to study Bessel functions in more detail. Entire books have been written showing the derivation of these functions and their use. I don't know your math background, but in order to study these works, you should have studied calculus up to and including complex analysis.

There are several recent threads by PF user 'yungman' where he tries to verify some Bessel function J0 derivations.