# Difference between scattering amplitude and scattering lenght

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!!

• I'm thinking maybe because in this phenomena the energy of the neutron is always small so basically $l=0$ is the only relevant case?
– user78618
Aug 14 '15 at 15:38

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 .