Difference between scattering amplitude and scattering lenght

Question

I'm studying neutron scattering theory and I noticed that one usually writes the scattered wave as a spherical wave: $$\psi \sim \frac{-b} {r}e^{ikr}$$ where $b$ is known as scattering lenght. From usual scattering theory (for instance Sakurai's chapter) one shows that the scattered wave can be written as (suppose $\vec{k} \parallel \hat{z}$): $$ \psi \sim e^{ikz}+f(\theta, \phi)\frac{e^{ikr}}{r}. $$ So my doubts:

1) Why (in the first eq.) do we forget about the transmitted plane wave? Is it just not important now?.

2) What is the relationship between the scattering amplitude $f$ and $b$? For instance in Sakurai's book a scattering lenght "$a$" is basically defined as a certain limit (page 403) and then it turns out that $\sigma_{tot}= 4 \pi a^2$ which I see is also a property of $b$ ! So I would say they're the same.. but in the book $a$ reduces to $f$ only in the case of an $S$ wave scattering with $l=0$.. While from the first and the second equation I'd say $b=-f(\theta, \phi)$ no matter what $l$ is! So what is their conection??. Thanks!!

Answer 1

With regard to your first question, the transmitted plane wave doesn't undergo any scattering from the potential. This is made explicit by the representation of the scattered wavefunction as a sum of an incident planewave and an outgoing spherical wave. As such, the transmitted wave doesn't have anything to tell us about the scattering event. All of that information is contained in the outgoing spherical wave.

As to the second question, your physical intuition is correct. By replacing $f(\theta, \phi) = -b$ we've explicitly taken the low energy limit and assumed that only the s-wave channel is important for this process. Imagine that you have a scattering potential with rotational symmetry, then expanding the scattering amplitude in terms of partial waves $$f(\theta) = \sum_{l=0}^\infty(2l+1)f_l P_l(\cos{\theta}),$$ you can see that if $f(\theta) = -b$ then we must have $f_0 = -b$ and $f_l = 0, \ l>0$ (recall that the Legendre polynomials are orthogonal, so they must vanish here term by term and not merely as a sum). If we wanted to include higher partial waves then there would have to be some non-spherically symmetric terms in the scattering amplitude.

In general, one has to specify a scattering length for each partial wave separately. If we let the s-wave scattering length be $a_s$ and neglect all higher partial waves, but do not take the 0 energy limit, then $$f(k) = \frac{1}{k\cot{\delta_s(k)}-ik} \approx \frac{1}{-\frac{1}{a_s}-ik}.$$ The full expansion for $$k \cot{\delta_s(k)} = -\frac{1}{a_s} + \frac{1}{2} r_s k^2 + \mathcal{O}(k^4)$$ is known as the Effective Range Expansion. You can see that for $k \to 0$ you recover the expressions you quoted for the scattering amplitude and total cross-section .