What is physical explanation of a scattering length? Could someone please explain the physical meaning of a scattering length? I can not find a satisfying answer.

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*From what I understand, if the scattering length is positive, the bigger it is, the stronger repulsion is for particles?


*Or is it like an imaginary sphere around particle where the sphere's radius is equal to scattering length and effectively this imaginary sphere behaves like a hard particle which can collide with other imaginary spheres?
For convenience I have drawn a little picture of what I described above:
 A: Your second point is a nice physical picture in thinking about it. But it doesn't get to the repulsive/attractive interactions for positive/negative scattering lengths.
A helpful way to understand that is by considering square well potentials of length $L$ (ranges from $r=0$ to $r=L$) and strength $V_0$. The scattering length in this case corresponds to the position where the wavefunction turns and is 'pushed' out of the potential well.
If $V_0$ is positive and infinite, scattering length is $+L$, since it corresponds to a hard wall and the wavefunction is pushed out, right at $+L$. For non-infinite but positive potentials, the wavefunction is 'pushed' out at a length smaller than $+L$. This corresponds to a scattering length $a$, where $0<a<L$. A weaker potential allows you to probe more regions inside the potential, the limiting case being zero scattering length when $V_0=0$.
For $V_0<0$, an attractive potential, the wavefunction is 'pushed' out at negative values of $r$ (negative scattering length), which basically corresponds to the wavefunction being 'pulled' in all the way.
