Born approximation

The Born approximation replaces the unknown total field inside a scattering integral by the known incident field, thereby retaining only single scattering. In nonrelativistic quantum mechanics, first-order Lippmann-Schwinger iteration gives

For a weak constant spherical potential of radius $r_0$, the scattering amplitude is proportional to $[\sin x-x\cos x]/x^3$, where $x=q{r}_0$. Its forward limit is nonzero. Positive zeros satisfy $\tan x=x$, with first root $x_1\simeq 4.49341$, so an elastic incident wavenumber needs $k\ge x_1/(2r_0)$ to access a zero. The asymptotic roots are $x_n=\pi (n+1/2)-1/[\pi (n+1/2)]+O(n^{-3})$; increasing $k$ alone does not remove the low-order root correction.

For $V(r)=A{e}^{-\kappa r}/r$, with real $A$ and $\kappa >0$, the first Born approximation gives the scattering amplitudeIndeed, orienting the polar axis along the momentum transfer gives the angular integral $4\pi \sin (qr)/(qr)$, and the remaining integral is ${\int }_0^{\infty }{e}^{-\kappa r}\sin (qr)dr=q/(q^2+{\kappa }^2)$. The differential scattering cross-section is $|f_B(q){|}^2$.

The limit $\kappa \to 0^{+}$ at fixed nonzero $q$ produces the Rutherford differential cross-section for a Coulomb potential. The unscreened interaction has long range, so this limiting calculation by itself does not establish the validity of a naive Born approximation to the full Rutherford scattering state. The exact Rutherford scattering amplitude has an additional phase, while its modulus agrees with the Rutherford differential cross-section.

For a real shell potential $V(r)=V_0\delta (r-R)$, define $g=2m{V}_{0}R/{\hslash }^2$ and $\mathrm{sinc}u=\sin u/u$, with value one at zero. Angular integration of its three-dimensional Fourier transform gives $4\pi V_0{R}^2\mathrm{sinc}(qR)$, where the momentum transfer has wave-number magnitude $q=2k\sin (\theta /2)$. Substitution in the first Born approximation proves the displayed scattering amplitude. It is real. Its squared modulus is a nonzero order-$V_0^2$ differential scattering cross-section, whose optical theorem counterpart requires the order-$V_0^2$ imaginary part of the forward amplitude from the next Born term. Thus squaring the first-order amplitude does not supply a consistently unitary second-order amplitude.