The Born Approximation

Derivation

The Born Approximation

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Statement

For a particle of mass \(m\) scattering off a static potential \(V(\mathbf{r})\), the first Born approximation replaces the exact scattering state in the integral for the amplitude by the incident plane wave, yielding the first-order scattering amplitude \(f^{(1)}(\mathbf{k}',\mathbf{k}) = -\dfrac{m}{2\pi\hbar^{2}}\displaystyle\int e^{i\mathbf{q}\cdot\mathbf{r}'}\,V(\mathbf{r}')\,d^{3}r'\) — the Fourier transform of the potential evaluated at momentum transfer \(\mathbf{q}=\mathbf{k}-\mathbf{k}'\) — and hence the differential cross section \(\dfrac{d\sigma}{d\Omega}=\bigl|f^{(1)}\bigr|^{2}\).

Why it matters

The Born approximation is the workhorse of scattering theory. It turns the impossible-looking problem of solving the full Schrödinger equation with an outgoing-wave boundary condition into a single quadrature: take the Fourier transform of the potential. Because it is linear in \(V\), it makes the potential directly readable from experiment — the measured angular distribution is (to first order) the modulus-squared Fourier transform of the interaction, which is exactly why elastic scattering of electrons, neutrons, and X-rays reconstructs charge densities, nuclear shapes, and crystal structure.

It is also the conceptual bridge between waves, matter, and chance in this unit: a matter wave diffracts off a potential (waves), the transferred momentum \(\hbar\mathbf{q}\) is a mechanical bookkeeping of the collision (matter), and the squared amplitude is a probability per unit solid angle (chance). Every higher-order scattering result — the Born series, distorted-wave methods, form factors — is measured against this baseline.

Assumptions

The potential is static and localized (falls off faster than \(1/r\) so the scattering amplitude and cross section are well defined).If \(V\) is long-ranged (pure Coulomb), the asymptotic plane wave and the \(e^{ikr}/r\) outgoing form are both wrong — the Coulomb phase distorts the wave logarithmically at all radii and the naive Fourier integral diverges.

Scattering is elastic and energy-conserving, so \(|\mathbf{k}'|=|\mathbf{k}|=k\).If inelastic channels open, a single scalar \(V(\mathbf r)\) and one energy shell no longer describe the dynamics; one needs a coupled-channel or optical potential.

The scattered wave is a small perturbation of the incident wave everywhere the potential acts.If dropped, replacing \(\psi\) by \(e^{i\mathbf k\cdot\mathbf r}\) inside the integral is not justified and the first term of the Born series is not a good approximation — resonances and bound-state effects are missed.

The outgoing-wave Green's function \(G_0^{(+)}\) with the \(+i\varepsilon\) prescription is used, selecting purely outgoing scattered spherical waves.If the \(-i\varepsilon\) (incoming) or standing-wave Green's function is used instead, the boundary condition is wrong and \(f\) acquires an unphysical sign/phase, breaking the optical theorem.

Derivation

\[ \psi^{(+)}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}+\int G_0^{(+)}(\mathbf r,\mathbf r')\,U(\mathbf r')\,\psi^{(+)}(\mathbf r')\,d^{3}r' \]

The Lippmann–Schwinger equation in position space (assumed prior result), with reduced potential \(U=\tfrac{2m}{\hbar^{2}}V\) and \(G_0^{(+)}=-\dfrac{1}{4\pi}\dfrac{e^{ik|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}\) the outgoing Green's function of \((\nabla^{2}+k^{2})\). A

\[ \psi^{(+)}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}-\frac{m}{2\pi\hbar^{2}}\int \frac{e^{ik|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}\,V(\mathbf r')\,\psi^{(+)}(\mathbf r')\,d^{3}r' \]

Insert \(U=\tfrac{2m}{\hbar^{2}}V\) and the explicit Green's function; the constants combine as \(\tfrac{2m}{\hbar^{2}}\cdot\tfrac{1}{4\pi}=\tfrac{m}{2\pi\hbar^{2}}\). Pure substitution. A

\[ |\mathbf r-\mathbf r'| \xrightarrow{r\to\infty} r-\hat{\mathbf r}\cdot\mathbf r'+O\!\left(\frac{r'^{2}}{r}\right),\qquad \frac{e^{ik|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}\longrightarrow \frac{e^{ikr}}{r}\,e^{-i\mathbf k'\cdot\mathbf r'} \]

Far-field expansion for a detector at \(\mathbf r=r\hat{\mathbf r}\) with the source region bounded. Define the outgoing wavevector \(\mathbf k'\equiv k\hat{\mathbf r}\); the phase is kept to first order while the slowly varying \(1/|\mathbf r-\mathbf r'|\to 1/r\). Valid because \(V\) is localized. B

\[ \psi^{(+)}(\mathbf r)\xrightarrow{r\to\infty} e^{i\mathbf k\cdot\mathbf r}+f(\mathbf k',\mathbf k)\,\frac{e^{ikr}}{r},\qquad f(\mathbf k',\mathbf k)=-\frac{m}{2\pi\hbar^{2}}\int e^{-i\mathbf k'\cdot\mathbf r'}\,V(\mathbf r')\,\psi^{(+)}(\mathbf r')\,d^{3}r' \]

Compare with the standard asymptotic form \(\psi\sim e^{i\mathbf k\cdot\mathbf r}+f\,e^{ikr}/r\) that defines the scattering amplitude \(f\). This is still exact — no approximation yet. B

\[ \psi^{(+)}(\mathbf r')\;\approx\;e^{i\mathbf k\cdot\mathbf r'} \]

The first Born approximation: inside the integral, replace the exact state by the incident plane wave. Legitimate when the scattered correction is small on the support of \(V\) (weak or short-ranged potential / high energy). This is the one and only approximation. B

\[ f^{(1)}(\mathbf k',\mathbf k)=-\frac{m}{2\pi\hbar^{2}}\int e^{-i\mathbf k'\cdot\mathbf r'}\,V(\mathbf r')\,e^{i\mathbf k\cdot\mathbf r'}\,d^{3}r' =-\frac{m}{2\pi\hbar^{2}}\int e^{i(\mathbf k-\mathbf k')\cdot\mathbf r'}\,V(\mathbf r')\,d^{3}r' \]

Substitute step 5 into step 4 and combine the two exponentials. Define the momentum transfer \(\mathbf q\equiv\mathbf k-\mathbf k'\); the amplitude is now the Fourier transform of \(V\). A

\[ f^{(1)}(\theta)=-\frac{2m}{\hbar^{2}q}\int_{0}^{\infty} r'\,V(r')\,\sin(qr')\,dr',\qquad q=2k\sin\!\frac{\theta}{2} \]

For a central potential \(V(\mathbf r')=V(r')\), do the angular integral \(\int e^{i\mathbf q\cdot\mathbf r'}d\Omega'=4\pi\,\dfrac{\sin(qr')}{qr'}\); elasticity gives \(q^{2}=|\mathbf k-\mathbf k'|^{2}=2k^{2}(1-\cos\theta)=4k^{2}\sin^{2}(\theta/2)\). B

\[ \frac{d\sigma}{d\Omega}=\bigl|f^{(1)}(\mathbf k',\mathbf k)\bigr|^{2} =\left(\frac{m}{2\pi\hbar^{2}}\right)^{2}\left|\int e^{i\mathbf q\cdot\mathbf r'}V(\mathbf r')\,d^{3}r'\right|^{2} \]

The differential cross section is the modulus-squared amplitude (probability current of the outgoing spherical wave per unit solid angle, divided by the incident flux). This identification is standard from the asymptotic form in step 4. A

Result

\[ f^{(1)}(\mathbf q)=-\frac{m}{2\pi\hbar^{2}}\int e^{i\mathbf q\cdot\mathbf r'}\,V(\mathbf r')\,d^{3}r',\qquad \frac{d\sigma}{d\Omega}=\bigl|f^{(1)}(\mathbf q)\bigr|^{2},\qquad \mathbf q=\mathbf k-\mathbf k',\; q=2k\sin\frac{\theta}{2} \]

Reading. The scattering amplitude at scattering angle \(\theta\) is minus a constant times the spatial Fourier transform of the potential, sampled at the momentum transfer \(\hbar\mathbf q\). Large-angle scattering probes short-wavelength (small-scale) features of \(V\); forward scattering (\(q\to 0\)) sees only the volume integral \(\int V\,d^{3}r\). The cross section, being the square, loses the phase — so scattering measures the modulus of the Fourier transform, and reconstructing \(V\) requires a phase assumption or additional data.

Units check. \(\left[\dfrac{m}{\hbar^{2}}\right]=\dfrac{\mathrm{kg}}{\mathrm{J^{2}s^{2}}}\), \([V]=\mathrm{J}\), \([d^{3}r]=\mathrm{m^{3}}\). Product: \(\dfrac{\mathrm{kg}}{\mathrm{J^{2}s^{2}}}\cdot\mathrm{J}\cdot\mathrm{m^{3}}=\dfrac{\mathrm{kg\,m^{3}}}{\mathrm{J\,s^{2}}}\). With \(\mathrm{J}=\mathrm{kg\,m^{2}/s^{2}}\), \(\mathrm{J\,s^{2}}=\mathrm{kg\,m^{2}}\), leaving \(\mathrm{m}\). So \(f\) has units of length and \(d\sigma/d\Omega=|f|^{2}\) has units \(\mathrm{m^{2}\,sr^{-1}}\) — an area per solid angle, as required.

Limiting cases

Forward scattering, \(q\to 0\): \(f^{(1)}\to-\dfrac{m}{2\pi\hbar^{2}}\int V\,d^{3}r\), a constant set by the "strength" \(\int V\,d^{3}r\); the cross section is finite and isotropic near \(\theta=0\) for any short-ranged \(V\).
Low energy, \(ka\ll 1\): \(e^{i\mathbf q\cdot\mathbf r'}\approx 1\) over the range \(a\) of the potential, so \(f^{(1)}\) becomes angle-independent (pure s-wave) with \(f^{(1)}\to-\dfrac{m}{2\pi\hbar^{2}}\int V\,d^{3}r\), the Born scattering length.
Yukawa \(\to\) Coulomb (\(\mu\to 0\)): the screened potential's \(1/(q^{2}+\mu^{2})\) amplitude reduces to \(1/q^{2}\), and \(d\sigma/d\Omega\) reproduces the Rutherford formula \(\propto \sin^{-4}(\theta/2)\) exactly.
High energy, \(ka\gg1\): \(q\) can be large, so the transform is sampled far out; the cross section falls off rapidly with angle, concentrating scattering into a forward cone of width \(\sim 1/(ka)\).

Breaks when

Strong or resonant potentials. When \(\tfrac{2m|V_0|a^{2}}{\hbar^{2}}\gtrsim 1\) at low energy, replacing \(\psi\) by the incident plane wave is invalid; near a bound state or resonance the true amplitude has poles the first Born term cannot produce, and the series does not converge.
Long-range (Coulomb) potentials taken literally. The pure \(1/r\) tail makes \(\int e^{i\mathbf q\cdot\mathbf r}V\,d^{3}r\) formally divergent and the asymptotic plane-wave/outgoing-wave forms invalid; one must screen (\(\mu\neq0\)) and take a limit, and the exact wavefunction is logarithmically distorted at all radii.
Near threshold with an s-wave bound state at zero energy. The scattering length diverges, \(d\sigma/d\Omega\) is huge and non-perturbative, and no finite-order Born term captures it.
Optical theorem test fails. The first Born amplitude is real for a real central potential, so \(\operatorname{Im}f^{(1)}(0)=0\) and it predicts zero total cross section via the optical theorem — the leading nonzero imaginary part only appears at second order, signalling the approximation's incompleteness.

Failure modes

Forgetting the momentum transfer is a vector difference. Writing \(q=k-k'=0\) for elastic scattering; in fact \(|\mathbf k|=|\mathbf k'|\) but \(\mathbf q=\mathbf k-\mathbf k'\neq 0\), with magnitude \(q=2k\sin(\theta/2)\).
Using the wrong sign of \(\mathbf q\) in the exponent. Since \(d\sigma/d\Omega=|f|^{2}\) and \(V\) is real, \(e^{+i\mathbf q\cdot\mathbf r}\) vs \(e^{-i\mathbf q\cdot\mathbf r}\) does not change the cross section — but it matters for interference/form-factor phases; students often carry an inconsistent convention into multi-scatterer problems.
Dropping the reduced-mass/factor bookkeeping. Using bare mass instead of reduced mass in a two-body collision, or losing the \(2m/\hbar^{2}\) in converting \(V\) to \(U\), giving a cross section off by a numerical factor.
Treating \(d\sigma/d\Omega\) as \(|f|\) rather than \(|f|^{2}\). The amplitude is length; the cross section is its square.
Applying it to Coulomb without screening and being surprised by a divergent integral, or claiming the Rutherford result "just falls out" without the \(\mu\to0\) limit.
Believing \(f^{(1)}\) real means no total cross section physically. It is an artifact of first order; the flux is not actually conserved to first order because unitarity is only restored order by order.

Discussion

The single most important structural fact is that the amplitude is linear in the potential and equal to its Fourier transform. This is why the language of "form factors" dominates scattering physics: if the target is a distribution of identical scatterers, \(V(\mathbf r)=\sum_j v(\mathbf r-\mathbf r_j)\), the amplitude factorizes into a single-scatterer transform \(\tilde v(\mathbf q)\) times a structure factor \(\sum_j e^{i\mathbf q\cdot\mathbf r_j}\). Bragg peaks in crystallography, nuclear charge form factors from electron scattering, and the atomic scattering factor all live inside this one equation.

The momentum transfer \(q=2k\sin(\theta/2)\) is the resolution knob. Because the transform is sampled at \(q\), scattering "sees" spatial structure on the scale \(1/q\). To resolve a feature of size \(d\) one needs \(q\gtrsim 1/d\), i.e. either large angle or short de Broglie wavelength — this is the scattering-theory statement of the diffraction limit and the reason high-energy beams are needed to probe small structures.

The reality of \(f^{(1)}\) for a real potential exposes a subtlety about unitarity. The optical theorem \(\sigma_{\text{tot}}=\tfrac{4\pi}{k}\operatorname{Im}f(0)\) demands an imaginary forward amplitude, yet \(f^{(1)}\) is real, so \(\sigma_{\text{tot}}^{(1)}=0\) even though \(\int|f^{(1)}|^{2}d\Omega\neq0\). The resolution is that the optical theorem couples orders: \(\operatorname{Im}f\) at order \(n\) is fixed by \(|f|^{2}\) at order \(n-1\). The leading total cross section computed two ways — integrating \(|f^{(1)}|^{2}\), or taking \(\operatorname{Im}f^{(2)}(0)\) — must agree, and their agreement is a standard consistency check on the Born series. This is the perturbative face of unitarity.

Common misconceptions. (i) The Born approximation is not a low-energy approximation — it is a weak-scattering approximation, and it actually improves at high energy where the potential has little time to act. (ii) It does not require a spherically symmetric potential; the central-potential radial form (step 7) is a convenience, not a limitation — the general result is the full 3-D Fourier transform. (iii) The cross section does not measure the potential, only the modulus of its transform; the lost phase is the origin of the "phase problem" in structure determination.

Worked examples

Screened Coulomb (Yukawa) potential \( V(r)=\dfrac{\beta\,e^{-\mu r}}{r}\).

Set up the radial Born integral of step 7. A

\[ f^{(1)}(\theta)=-\frac{2m}{\hbar^{2}q}\int_{0}^{\infty} r\cdot\frac{\beta e^{-\mu r}}{r}\sin(qr)\,dr =-\frac{2m\beta}{\hbar^{2}q}\int_{0}^{\infty} e^{-\mu r}\sin(qr)\,dr \]

The \(r\) cancels; use \(\int_0^\infty e^{-\mu r}\sin(qr)\,dr=\dfrac{q}{\mu^{2}+q^{2}}\). A

\[ f^{(1)}(\theta)=-\frac{2m\beta}{\hbar^{2}}\,\frac{1}{\mu^{2}+q^{2}},\qquad \frac{d\sigma}{d\Omega}=\left(\frac{2m\beta}{\hbar^{2}}\right)^{2}\frac{1}{(\mu^{2}+q^{2})^{2}} \]

Symbolic result. Now put numbers for an electron: \(Z=1\) so \(\beta=\dfrac{e^{2}}{4\pi\varepsilon_0}=1.44~\mathrm{eV\,nm}\); screening length \(1/\mu=0.05~\mathrm{nm}\Rightarrow\mu=20~\mathrm{nm^{-1}}\); electron energy \(E=100~\mathrm{eV}\Rightarrow k=5.12~\mathrm{\AA^{-1}}=51.2~\mathrm{nm^{-1}}\). B

\[ \theta=90^\circ:\quad q=2k\sin45^\circ=72.4~\mathrm{nm^{-1}},\quad \frac{2m\beta}{\hbar^{2}}=\frac{1.44~\mathrm{eV\,nm}}{0.0381~\mathrm{eV\,nm^{2}}}=37.8~\mathrm{nm^{-1}} \]

Using \(\hbar^{2}/2m=0.0381~\mathrm{eV\,nm^{2}}\) for the electron; \(\mu^{2}+q^{2}=400+5242=5642~\mathrm{nm^{-2}}\). B

\[ f^{(1)}=-\frac{37.8}{5642}~\mathrm{nm}=-6.7\times10^{-3}~\mathrm{nm}=-0.067~\mathrm{\AA},\qquad \frac{d\sigma}{d\Omega}=4.5\times10^{-3}~\mathrm{\AA^{2}\,sr^{-1}} \]

Reading. A sub-ångström amplitude gives a cross section of a few thousandths of an ångström-squared per steradian at \(90^\circ\) — a realistic elastic electron–atom value. Letting \(\mu\to0\) turns \((\mu^2+q^2)^{-2}\) into \(q^{-4}\), i.e. Rutherford, \(d\sigma/d\Omega=(\beta/4E)^2\sin^{-4}(\theta/2)\).

Attractive square well \( V(r)=-V_0\) for \(r<a\), \(0\) otherwise.

Insert into the radial Born integral. A

\[ f^{(1)}(\theta)=\frac{2mV_0}{\hbar^{2}q}\int_{0}^{a} r\sin(qr)\,dr =\frac{2mV_0}{\hbar^{2}q^{3}}\bigl[\sin(qa)-qa\cos(qa)\bigr] \]

Using \(\int_0^a r\sin(qr)\,dr=\dfrac{\sin(qa)}{q^{2}}-\dfrac{a\cos(qa)}{q}\); the sign flips positive because \(V=-V_0\). B

\[ \frac{d\sigma}{d\Omega}=\left(\frac{2mV_0}{\hbar^{2}q^{3}}\right)^{2}\bigl[\sin(qa)-qa\cos(qa)\bigr]^{2} \]

Symbolic result. Numbers for a neutron: \(V_0=10~\mathrm{MeV}\), \(a=2~\mathrm{fm}\), \(E=50~\mathrm{MeV}\), with \(\hbar^{2}/2m_n=20.7~\mathrm{MeV\,fm^{2}}\Rightarrow k=\sqrt{50/20.7}=1.55~\mathrm{fm^{-1}}\). Validity: \(\dfrac{mV_0a}{\hbar^{2}k}=0.31<1\), so Born is applicable. B

\[ \theta=30^\circ:\; q=2k\sin15^\circ=0.805~\mathrm{fm^{-1}},\; qa=1.61,\; \sin(qa)-qa\cos(qa)=0.999+0.063=1.06 \]

\(\dfrac{2mV_0}{\hbar^{2}}=\dfrac{10}{20.7}\cdot 2 =0.483~\mathrm{fm^{-2}}\) (i.e. \(V_0/(\hbar^2/2m)\)), and \(q^{3}=0.522~\mathrm{fm^{-3}}\). C

\[ f^{(1)}=\frac{0.483\times1.06}{0.522}~\mathrm{fm}=0.98~\mathrm{fm},\qquad \frac{d\sigma}{d\Omega}=0.97~\mathrm{fm^{2}\,sr^{-1}}=9.7~\mathrm{mb\,sr^{-1}} \]

Reading. About 10 millibarn per steradian at \(30^\circ\), a typical fast-neutron elastic value. In the low-energy limit \(qa\to0\) the bracket \(\to (qa)^3/3\), so \(f^{(1)}\to \dfrac{2mV_0a^{3}}{3\hbar^{2}}\), independent of angle — pure isotropic s-wave scattering, as expected when the wavelength dwarfs the well.

Problems

A \(200~\mathrm{eV}\) electron scatters to \(\theta=60^\circ\). Compute the momentum transfer \(q\) in \(\mathrm{nm^{-1}}\).
Solution
\(k=0.512\sqrt{E[\mathrm{eV}]}~\mathrm{\AA^{-1}}=0.512\sqrt{200}=7.24~\mathrm{\AA^{-1}}=72.4~\mathrm{nm^{-1}}\). Then \(q=2k\sin(\theta/2)=2(72.4)\sin30^\circ=2(72.4)(0.5)=72.4~\mathrm{nm^{-1}}\). (Coincidentally \(q=k\) because \(2\sin30^\circ=1\).)
Derive the total cross section for the Yukawa potential in first Born approximation by integrating \(d\sigma/d\Omega\) over solid angle.
Solution
\(\dfrac{d\sigma}{d\Omega}=\left(\dfrac{2m\beta}{\hbar^{2}}\right)^{2}\dfrac{1}{(\mu^{2}+q^{2})^{2}}\) with \(q^{2}=2k^{2}(1-\cos\theta)\). Let \(u=\cos\theta\), \(d\Omega=2\pi\,du\), \(u\in[-1,1]\), and \(A=\mu^2+2k^2\): \(\displaystyle\int_{-1}^1\dfrac{du}{(A-2k^2u)^2}=\dfrac{1}{2k^2}\left[\dfrac{1}{A-2k^2u}\right]_{-1}^{1}=\dfrac{1}{2k^2}\left(\dfrac{1}{\mu^2}-\dfrac{1}{\mu^2+4k^2}\right)=\dfrac{2}{\mu^2(\mu^2+4k^2)}\). Hence \(\sigma_{\text{tot}}=2\pi\left(\dfrac{2m\beta}{\hbar^2}\right)^2\dfrac{2}{\mu^2(\mu^2+4k^2)}=\left(\dfrac{2m\beta}{\hbar^2}\right)^2\dfrac{4\pi}{\mu^2(\mu^2+4k^2)}\). As \(\mu\to0\) this diverges — the Coulomb total cross section is infinite, as expected for an infinite-range force.
A three-dimensional delta potential \(V(\mathbf r)=g\,\delta^{3}(\mathbf r)\). Find \(f^{(1)}\) and \(d\sigma/d\Omega\), and comment on the angular dependence.
Solution
\(f^{(1)}=-\dfrac{m}{2\pi\hbar^2}\int e^{i\mathbf q\cdot\mathbf r}g\,\delta^3(\mathbf r)\,d^3r=-\dfrac{mg}{2\pi\hbar^2}\), independent of \(q\) and hence of \(\theta\). So \(\dfrac{d\sigma}{d\Omega}=\left(\dfrac{mg}{2\pi\hbar^2}\right)^2\), perfectly isotropic — a point (zero-range) scatterer produces pure s-wave, angle-independent scattering, the extreme of the low-energy limit.
Gaussian potential \(V(r)=V_0\,e^{-r^{2}/2a^{2}}\). Compute \(f^{(1)}(\mathbf q)\).
Solution
Use the 3-D Gaussian Fourier transform \(\int e^{i\mathbf q\cdot\mathbf r}e^{-r^2/2a^2}d^3r=(2\pi a^2)^{3/2}e^{-q^2a^2/2}\). Therefore \(f^{(1)}(\mathbf q)=-\dfrac{m}{2\pi\hbar^2}V_0(2\pi a^2)^{3/2}e^{-q^2a^2/2}=-\dfrac{mV_0 a^3}{\hbar^2}\sqrt{2\pi}\,e^{-q^2a^2/2}\), and \(\dfrac{d\sigma}{d\Omega}=\dfrac{2\pi m^2V_0^2a^6}{\hbar^4}e^{-q^2a^2}\). The Gaussian shape maps to a Gaussian in \(q\): a smooth potential gives a smooth, rapidly-forward-peaked cross section with angular width set by \(qa\sim1\).
For the square well of Worked Example 2 (\(V_0=10~\mathrm{MeV}\), \(a=2~\mathrm{fm}\), neutron), evaluate \(d\sigma/d\Omega\) in the forward direction \(\theta\to0\), and state whether the first Born approximation is trustworthy at \(E=50~\mathrm{MeV}\).
Solution
As \(q\to0\), \(\sin(qa)-qa\cos(qa)\to (qa)^3/3\), so \(f^{(1)}\to\dfrac{2mV_0}{\hbar^2 q^3}\cdot\dfrac{(qa)^3}{3}=\dfrac{2mV_0 a^3}{3\hbar^2}\). Numerically \(\dfrac{2mV_0}{\hbar^2}=0.483~\mathrm{fm^{-2}}\), \(a^3=8~\mathrm{fm^3}\): \(f^{(1)}(0)=\dfrac{0.483\times8}{3}=1.29~\mathrm{fm}\), so \(\dfrac{d\sigma}{d\Omega}\big|_0=1.66~\mathrm{fm^2\,sr^{-1}}=16.6~\mathrm{mb\,sr^{-1}}\). Validity at high energy needs \(\dfrac{mV_0a}{\hbar^2 k}\ll1\); here it is \(0.31\), comfortably below 1, so the first Born approximation is reasonable (though not exact) at this energy. At low energy the relevant parameter \(\dfrac{2mV_0a^2}{\hbar^2}=0.483\times4=1.9\gtrsim1\), so Born would be unreliable near threshold.