Born Approximation and the Form Factor

Derivation

Born Approximation and the Form Factor

D-274 Home PU-304 Threads waves · fields · chance Depends on time-dependent-perturbation-theory, greens-function-helmholtz

Statement

For a localized potential \(V(\mathbf r)\), the stationary scattering state with incident wavevector \(\mathbf k\) satisfies the Lippmann–Schwinger equation built on the outgoing Helmholtz Green's function. Replacing the exact state inside the interaction region by the incident plane wave (the first Born approximation) gives the scattering amplitude as the three-dimensional Fourier transform of the potential, \(f_{\mathrm B}(\mathbf q)=-\frac{m}{2\pi\hbar^{2}}\int e^{i\mathbf q\cdot\mathbf r}\,V(\mathbf r)\,d^{3}r\), evaluated at the momentum transfer \(\mathbf q=\mathbf k-\mathbf k'\) with \(|\mathbf q|=2k\sin(\theta/2)\). When \(V\) is generated by an extended charge (or matter) distribution, the amplitude factorizes into a point amplitude times a dimensionless form factor \(F(\mathbf q)=\int\rho(\mathbf r)\,e^{i\mathbf q\cdot\mathbf r}\,d^{3}r\), so that \(\frac{d\sigma}{d\Omega}=\left(\frac{d\sigma}{d\Omega}\right)_{\!\text{point}}\lvert F(\mathbf q)\rvert^{2}\).

Why it matters

The Born approximation is the workhorse of scattering theory: it converts a hard integral equation into a single Fourier transform, so that measuring an angular distribution is, to leading order, measuring the Fourier components of the target. This is the theoretical basis of electron and X-ray diffraction, neutron scattering, and — most famously — the elastic electron scattering experiments (Hofstadter) that first mapped the finite size and radial charge profile of nuclei and the proton.

The form factor cleanly separates the kinematics of a point Coulomb (or nuclear) interaction from the structure of the target. Its small-\(q\) slope delivers the mean-square radius model-independently, and its diffraction minima encode the sharpness of the charge boundary. Almost every statement of the form "the proton radius is \(0.84\ \mathrm{fm}\)" is a statement about \(F(q)\) near \(q=0\).

Assumptions

Localized (short-range) potential.If \(V\) does not fall faster than \(1/r\), the asymptotic separation into "plane wave + outgoing spherical wave" fails and \(f\) is not well defined; the bare Coulomb tail must be handled by screening or by the Coulomb-modified partial-wave phases. Elastic scattering, single channel.Energy is conserved so \(|\mathbf k'|=|\mathbf k|=k\); dropping this couples the equation to inelastic channels and the simple Fourier relation no longer holds. Weak coupling: the incident wave is barely distorted inside \(V\).The Born series is truncated at first order, i.e. \(\psi\to e^{i\mathbf k\cdot\mathbf r}\) under the integral. If the potential is strong or supports quasi-bound states, higher Born terms (multiple scattering) are not negligible and the amplitude is no longer a single Fourier transform. Static, spin-independent, local potential.Velocity dependence, exchange, or nonlocality add extra \(\mathbf q\)- and energy-structure; the factorization into a single form factor generally breaks (e.g. one then needs both electric and magnetic form factors \(G_E,G_M\)).

Derivation

\[ \left(\nabla^{2}+k^{2}\right)\psi(\mathbf r)=U(\mathbf r)\,\psi(\mathbf r),\qquad U(\mathbf r)\equiv\frac{2m}{\hbar^{2}}V(\mathbf r),\quad k^{2}=\frac{2mE}{\hbar^{2}} \]

Rewrite the time-independent Schrödinger equation \(-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V\psi=E\psi\) as an inhomogeneous Helmholtz equation with source \(U\psi\). A

\[ \psi(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}+\int G_{0}^{+}(\mathbf r-\mathbf r')\,U(\mathbf r')\,\psi(\mathbf r')\,d^{3}r',\qquad G_{0}^{+}(\mathbf R)=-\frac{e^{ikR}}{4\pi R} \]

Invert the Helmholtz operator using the outgoing Green's function (prior result, greens-function-helmholtz) and add the homogeneous solution \(e^{i\mathbf k\cdot\mathbf r}\) fixed by the incident-beam boundary condition. This is the Lippmann–Schwinger equation. B

\[ R=\lvert\mathbf r-\mathbf r'\rvert=r\sqrt{1-\tfrac{2\,\hat{\mathbf r}\cdot\mathbf r'}{r}+\tfrac{r'^{2}}{r^{2}}}\;\xrightarrow{r\gg r'}\;r-\hat{\mathbf r}\cdot\mathbf r'+O\!\left(\tfrac{r'^{2}}{r}\right) \]

Take the detector far from a target of finite extent, \(r\gg r'\), and expand \(R\). Keep the phase to first order but set \(R\to r\) in the slowly varying \(1/R\) prefactor. B

\[ \frac{e^{ikR}}{R}\;\longrightarrow\;\frac{e^{ikr}}{r}\,e^{-ik\,\hat{\mathbf r}\cdot\mathbf r'}=\frac{e^{ikr}}{r}\,e^{-i\mathbf k'\cdot\mathbf r'},\qquad \mathbf k'\equiv k\,\hat{\mathbf r} \]

Substitute the expansion. The scattered wavevector \(\mathbf k'\) points from the target to the detector and has magnitude \(k\) (elastic). This isolates a common outgoing factor \(e^{ikr}/r\). A

\[ \psi(\mathbf r)\;\xrightarrow{r\to\infty}\;e^{i\mathbf k\cdot\mathbf r}+f(\theta,\phi)\,\frac{e^{ikr}}{r},\qquad f=-\frac{1}{4\pi}\int e^{-i\mathbf k'\cdot\mathbf r'}\,U(\mathbf r')\,\psi(\mathbf r')\,d^{3}r' \]

Read off the scattering amplitude by matching to the standard asymptotic form. This expression for \(f\) is still exact — no approximation yet — but implicit, since \(\psi\) appears on the right. B

\[ \psi(\mathbf r')\approx e^{i\mathbf k\cdot\mathbf r'}\quad\Longrightarrow\quad f_{\mathrm B}=-\frac{1}{4\pi}\int e^{-i\mathbf k'\cdot\mathbf r'}\,U(\mathbf r')\,e^{i\mathbf k\cdot\mathbf r'}\,d^{3}r' \]

First Born approximation: replace the full state under the integral by the undistorted incident plane wave. Legal when the correction from one scattering event is small, i.e. the leading term of the Born series dominates. C

\[ f_{\mathrm B}(\mathbf q)=-\frac{m}{2\pi\hbar^{2}}\int e^{i\mathbf q\cdot\mathbf r}\,V(\mathbf r)\,d^{3}r,\qquad \mathbf q\equiv\mathbf k-\mathbf k' \]

Combine the two plane-wave phases, \(e^{i(\mathbf k-\mathbf k')\cdot\mathbf r'}=e^{i\mathbf q\cdot\mathbf r'}\), restore \(U=2mV/\hbar^{2}\), and rename \(\mathbf r'\to\mathbf r\). The amplitude is the Fourier transform of \(V\) at the momentum transfer \(\mathbf q\). A

\[ q=\lvert\mathbf k-\mathbf k'\rvert=\sqrt{k^{2}+k^{2}-2k^{2}\cos\theta}=2k\sin\!\frac{\theta}{2} \]

Use \(|\mathbf k|=|\mathbf k'|=k\) and the law of cosines with scattering angle \(\theta\) between \(\mathbf k\) and \(\mathbf k'\). For a central potential \(f_{\mathrm B}\) depends only on \(q\). A

\[ V(\mathbf r)=g\!\int\frac{\rho(\mathbf r')}{\lvert\mathbf r-\mathbf r'\rvert}\,d^{3}r'\ \Longrightarrow\ f_{\mathrm B}(\mathbf q)=\underbrace{\left(-\frac{m}{2\pi\hbar^{2}}\,\frac{4\pi g}{q^{2}}\right)}_{f_{\text{point}}(\mathbf q)}\underbrace{\int\rho(\mathbf r)\,e^{i\mathbf q\cdot\mathbf r}\,d^{3}r}_{F(\mathbf q)} \]

When \(V\) is the Coulomb potential of a normalized distribution \(\rho\) (\(\int\rho\,d^{3}r=1\)), the Fourier transform of a convolution factorizes: FT\((1/r)=4\pi/q^{2}\) times FT\((\rho)=F(\mathbf q)\). The structure separates from the point kinematics. C

\[ \frac{d\sigma}{d\Omega}=\lvert f_{\mathrm B}\rvert^{2}=\left(\frac{d\sigma}{d\Omega}\right)_{\!\text{point}}\lvert F(\mathbf q)\rvert^{2},\qquad F(0)=\int\rho\,d^{3}r=1 \]

Square the amplitude. The point cross section (Rutherford, if \(f_{\text{point}}\propto1/q^{2}\)) is modulated by \(|F|^{2}\); normalization forces \(F\to1\) as \(q\to0\), recovering the point result at forward angles. A

Result

\[ f_{\mathrm B}(\mathbf q)=-\frac{m}{2\pi\hbar^{2}}\int e^{i\mathbf q\cdot\mathbf r}\,V(\mathbf r)\,d^{3}r,\qquad \frac{d\sigma}{d\Omega}=\left(\frac{d\sigma}{d\Omega}\right)_{\!\text{point}}\!\lvert F(\mathbf q)\rvert^{2},\quad F(\mathbf q)=\int\rho(\mathbf r)\,e^{i\mathbf q\cdot\mathbf r}\,d^{3}r \]

Reading. Scattering at momentum transfer \(\mathbf q\) reads off a single Fourier component of the potential. A pointlike target scatters with a smooth Rutherford-like amplitude; any spatial extent multiplies it by the form factor \(F(\mathbf q)\), which starts at unity and falls off over \(q\sim1/(\text{size})\). Diffraction minima in \(|F|^{2}\) are the interference of waves scattered from different parts of the distribution. The small-\(q\) slope, \(F\simeq1-\tfrac16 q^{2}\langle r^{2}\rangle\), gives the mean-square radius without any shape assumption.

Units check. \([\,m\,V\,L^{3}/\hbar^{2}\,]=\dfrac{\mathrm{kg}\cdot\mathrm{J}\cdot\mathrm{m}^{3}}{(\mathrm{J\,s})^{2}}=\dfrac{\mathrm{kg\,m^{3}}}{\mathrm{J\,s^{2}}}=\dfrac{\mathrm{kg\,m^{3}}}{(\mathrm{kg\,m^{2}s^{-2}})\,\mathrm{s^{2}}}=\mathrm{m}\), so \(f\) has the dimension of length and \(|f|^{2}\) of area, as a differential cross section must. \(F\) is dimensionless (\(\rho\) has units \(\mathrm{m^{-3}}\), \(d^{3}r\) has \(\mathrm{m^{3}}\)).

Limiting cases

Point target \(\rho\to\delta^{3}(\mathbf r)\): \(F(\mathbf q)\to1\) for all \(q\), and \(d\sigma/d\Omega\) reduces to the bare point cross section.
Forward scattering \(q\to0\) (\(\theta\to0\)): \(F\to1-\tfrac16 q^{2}\langle r^{2}\rangle\); the intercept fixes charge normalization, the slope fixes \(\langle r^{2}\rangle\).
Screened Coulomb / Yukawa \(V=\beta e^{-\mu r}/r\): \(f_{\mathrm B}=-\dfrac{2m\beta}{\hbar^{2}(q^{2}+\mu^{2})}\); letting \(\mu\to0\) reproduces the exact Rutherford amplitude.
Low energy \(k\to0\): \(f_{\mathrm B}\to-\dfrac{m}{2\pi\hbar^{2}}\int V\,d^{3}r=-a_{s}\), a constant (\(s\)-wave scattering length), isotropic.
High-\(q\) (large angle): \(F\) probes the short-distance / boundary structure; a sharp edge gives oscillatory diffraction zeros, a soft edge gives a smooth exponential/Gaussian falloff.

Breaks when

Strong potential. When the distortion of the incident wave inside \(V\) is not small (roughly \(\frac{m}{\hbar^{2}}|V_0|a^{2}\gtrsim1\) at low energy, or near a resonance/bound state at threshold), higher Born terms — multiple scattering — dominate and \(f\) is no longer a single Fourier transform.
Long-range / unscreened Coulomb. The \(1/r\) tail makes the Born integral \(\int e^{i\mathbf q\cdot\mathbf r}/r\,d^{3}r\) only conditionally convergent; the true wavefunction carries a logarithmic phase \(\propto\ln r\), so the naive amplitude picks up an (unobservable) divergent Coulomb phase and partial-wave sums fail to truncate.
Inelastic / relativistic regimes. If channels open (target excitation, particle production) or the projectile is relativistic with spin, elastic \(|\mathbf k'|=|\mathbf k|\) and the single scalar \(F\) break down — one needs coupled channels or the pair \(G_E,G_M\) with the recoil (Mott) kinematics.
Slow, heavy projectiles (large \(\eta\)). When the Sommerfeld parameter \(\eta=Z_1Z_2 e^2/(4\pi\varepsilon_0\hbar v)\gtrsim1\), the semiclassical/strong-coupling regime, first-order Born badly misestimates the magnitude even where its \(q\)-dependence looks right.

Failure modes

Sign/direction of \(\mathbf q\). Writing \(\mathbf q=\mathbf k'-\mathbf k\) instead of \(\mathbf k-\mathbf k'\): harmless for \(|F|^{2}\) of a real even \(\rho\), but flips the phase and corrupts any interference or complex-\(F\) calculation.
Forgetting elastic constraint. Using \(q=k-k'\) as a scalar difference of magnitudes (which is zero!) rather than \(q=2k\sin(\theta/2)\).
Dropping the \(4\pi\) or the \(m/2\pi\hbar^2\) prefactor when comparing to Rutherford; a common error is a stray factor of 2 from \(U=2mV/\hbar^2\).
Confusing \(F\) with \(|F|^{2}\). The amplitude carries \(F(\mathbf q)\); the cross section carries \(|F(\mathbf q)|^{2}\). Reporting a "radius from the cross-section slope" without the square root is off by \(\sqrt2\).
Non-normalized \(\rho\). If \(\int\rho\,d^{3}r\neq1\), \(F(0)\neq1\) and the extracted radius absorbs a spurious normalization.
Using Born for the Coulomb magnitude at low energy. Born happens to reproduce the exact Rutherford formula — a famous accident — but students over-generalize and trust Born magnitudes for strong short-range potentials where it is wrong.

Discussion

The central lesson is a duality between real space and momentum space: an experiment that varies the scattering angle sweeps the momentum transfer \(q=2k\sin(\theta/2)\), and each angle samples one Fourier component of the target's potential (or charge density). Wide-angle data at large \(q\) resolve fine spatial detail; the diffraction pattern of \(|F(q)|^{2}\) is the momentum-space "image" whose inverse transform is the density. This is exactly the wave-optics of a diffraction grating, transplanted to matter waves — the reason the same mathematics governs X-ray crystallography, electron microscopy, and nuclear form-factor measurements.

The factorization of Step 9 is why the Born approximation is so powerful pedagogically and experimentally: it cleanly quarantines what we assume (the point interaction, e.g. Coulomb) from what we measure (the structure \(F\)). The point cross section is known analytically; dividing the data by it isolates \(|F(q)|^{2}\), and the small-\(q\) expansion \(F=1-\tfrac16 q^2\langle r^2\rangle+\tfrac1{120}q^4\langle r^4\rangle-\dots\) extracts radial moments in a nearly model-independent way. Hofstadter's Nobel-winning nuclear radii, and the modern proton-radius determinations, are readings of this slope.

A subtler point is that the Born approximation is the first term of a systematic series generated by iterating the Lippmann–Schwinger equation, \(f=f^{(1)}+f^{(2)}+\dots\), where \(f^{(2)}\) involves two insertions of \(U\) connected by \(G_0^+\) — a genuine two-step (intermediate off-shell) scattering. The optical theorem, \(\sigma_{\text{tot}}=\frac{4\pi}{k}\,\mathrm{Im}\,f(0)\), exposes the limits of truncation: the first Born amplitude for a real \(V\) is real at \(\theta=0\), so \(\mathrm{Im}\,f^{(1)}(0)=0\) and unitarity is only satisfied at the next order. This is why Born conserves probability only approximately and why total cross sections need at least \(f^{(2)}\).

Common misconceptions. (i) "Born requires high energy." It requires weak coupling; high energy helps because the wave spends less time being distorted, but a genuinely weak potential is well described at any energy. (ii) "Born reproduces Rutherford, so it is exact for Coulomb." It reproduces the Rutherford formula for the cross section by a special cancellation; the underlying wavefunction and phase are not exact, and interference terms (e.g. identical-particle Mott scattering) require the true Coulomb phases. (iii) "\(F\) is the charge density." \(F\) is its Fourier transform; you recover \(\rho(r)\) only after an inverse transform over the full measured \(q\)-range, which experiments never fully cover — hence radii (moments) are robust but detailed profiles are model-dependent.

Worked examples

Example 1 — Born amplitude and cross section for a screened Coulomb (Yukawa) potential. An electron scatters from the screened potential of a hydrogen-like target, \(V(r)=-\dfrac{Ze^{2}}{4\pi\varepsilon_0}\dfrac{e^{-\mu r}}{r}\), with \(Z=1\) and screening length \(a=1/\mu=a_0=0.529\ \text{Å}\). Beam energy \(E=1\ \mathrm{keV}\); find \(d\sigma/d\Omega\) at \(\theta=10^{\circ}\).

\[ f_{\mathrm B}(q)=-\frac{m}{2\pi\hbar^{2}}\int e^{i\mathbf q\cdot\mathbf r}\left(-\frac{Ze^{2}}{4\pi\varepsilon_0}\right)\frac{e^{-\mu r}}{r}\,d^{3}r=\frac{m}{2\pi\hbar^{2}}\,\frac{Ze^{2}}{4\pi\varepsilon_0}\cdot\frac{4\pi}{q^{2}+\mu^{2}} \]

Use the standard Yukawa transform \(\int e^{i\mathbf q\cdot\mathbf r}\frac{e^{-\mu r}}{r}d^{3}r=\frac{4\pi}{q^{2}+\mu^{2}}\). A

\[ f_{\mathrm B}(q)=\frac{2m}{\hbar^{2}}\,\frac{Ze^{2}}{4\pi\varepsilon_0}\,\frac{1}{q^{2}+\mu^{2}} \]

Collect constants; \(4\pi\) cancels one factor. Symbolic form before numbers. A

\[ k=\frac{\sqrt{2mE}}{\hbar}=\frac{\sqrt{2(9.11\times10^{-31})(1.602\times10^{-16})}}{1.055\times10^{-34}}=1.62\times10^{11}\ \mathrm{m^{-1}}=16.2\ \text{Å}^{-1} \]

Convert \(E=1\,\mathrm{keV}=1.602\times10^{-16}\,\mathrm J\) to the incident wavenumber. A

\[ q=2k\sin\frac{\theta}{2}=2(16.2)\sin 5^{\circ}=2.82\ \text{Å}^{-1},\qquad \mu=\frac1{a_0}=1.89\ \text{Å}^{-1} \]

\(q^{2}+\mu^{2}=7.95+3.57=11.5\ \text{Å}^{-2}=1.15\times10^{21}\ \mathrm{m^{-2}}\). A

\[ f_{\mathrm B}=\underbrace{\frac{2m}{\hbar^{2}}}_{1.64\times10^{38}\,\mathrm{J^{-1}m^{-2}}}\underbrace{\frac{e^{2}}{4\pi\varepsilon_0}}_{2.31\times10^{-28}\,\mathrm{J\,m}}\frac{1}{1.15\times10^{21}\,\mathrm{m^{-2}}}=3.3\times10^{-11}\ \mathrm m=0.33\ \text{Å} \]

Multiply, with \(Z=1\). Prefactor \(=1.64\times10^{38}\times2.31\times10^{-28}=3.78\times10^{10}\ \mathrm{m^{-1}}\); divide by \(1.15\times10^{21}\). B

\[ \frac{d\sigma}{d\Omega}=\lvert f_{\mathrm B}\rvert^{2}=(0.33\ \text{Å})^{2}\approx1.1\times10^{-1}\ \text{Å}^{2}=1.1\times10^{-21}\ \mathrm{m^{2}}\ (\approx1.1\times10^{7}\ \mathrm{barn}) \]

Reading. Small-angle electron scattering is strongly forward-peaked; screening tames the \(1/q^{4}\) Rutherford divergence, replacing \(q^2\) by \(q^2+\mu^2\) so \(d\sigma/d\Omega\) stays finite as \(\theta\to0\). Units check. \(f\) came out in metres; its square is an area.

Example 2 — Form factor of a uniformly charged sphere; nuclear size from a diffraction minimum. Model a gold nucleus (\(A=197\)) as a uniform charge ball of radius \(R\). Derive \(F(q)\), its rms radius, and the first diffraction minimum, then find the electron scattering angle at \(E_e=250\ \mathrm{MeV}\).

\[ \rho(r)=\frac{3}{4\pi R^{3}}\ (r\le R),\qquad F(q)=\int\rho\,e^{i\mathbf q\cdot\mathbf r}\,d^{3}r=\frac{3}{4\pi R^{3}}\int_0^{R}\!\!\frac{4\pi r^{2}\sin qr}{qr}\,dr \]

Spherical symmetry lets the angular integral give \(\int e^{i\mathbf q\cdot\mathbf r}d\Omega=4\pi\,\frac{\sin qr}{qr}\). B

\[ F(q)=\frac{3}{R^{3}q}\int_0^{R} r\sin qr\,dr=\frac{3}{(qR)^{3}}\big[\sin qR-qR\cos qR\big] \]

Evaluate \(\int_0^R r\sin qr\,dr=\frac{\sin qR-qR\cos qR}{q^{2}}\). Check \(F(0)=1\) by small-\(x\) expansion. A

\[ \langle r^{2}\rangle=\int r^{2}\rho\,d^{3}r=\frac{3}{4\pi R^{3}}\!\int_0^{R}\!4\pi r^{4}dr=\frac{3}{5}R^{2}\ \Rightarrow\ \sqrt{\langle r^{2}\rangle}=\sqrt{0.6}\,R \]

Second moment of a uniform ball; consistent with \(F\simeq1-\tfrac16 q^2\langle r^2\rangle=1-\tfrac1{10}q^2R^2\). A

\[ R=1.2\,A^{1/3}\ \mathrm{fm}=1.2\,(197)^{1/3}=6.98\ \mathrm{fm},\qquad \sqrt{\langle r^{2}\rangle}=0.775\times6.98=5.41\ \mathrm{fm} \]

Insert numbers. \(A^{1/3}=5.81\). A

\[ F(q)=0\ \Rightarrow\ \tan qR=qR\ \Rightarrow\ (qR)_{\min}=4.493\ \Rightarrow\ q_{\min}=\frac{4.493}{6.98\ \mathrm{fm}}=0.644\ \mathrm{fm^{-1}} \]

First nonzero root of \(\sin x-x\cos x\). This is the first diffraction dip in \(|F|^2\). B

\[ k\approx\frac{E_e}{\hbar c}=\frac{250\ \mathrm{MeV}}{197.3\ \mathrm{MeV\,fm}}=1.267\ \mathrm{fm^{-1}},\qquad \sin\frac{\theta}{2}=\frac{q_{\min}}{2k}=\frac{0.644}{2.534}=0.254 \]

Ultrarelativistic electrons: \(k\simeq E/\hbar c\). B

\[ F(q)=\frac{3\big[\sin qR-qR\cos qR\big]}{(qR)^{3}},\qquad \theta_{\min}=2\arcsin(0.254)\approx29.4^{\circ} \]

Reading. The first minimum of the elastic cross section at \(\approx29^{\circ}\) directly measures \(R\); its angular position scales as \(1/R\), so bigger nuclei diffract at smaller angles. Units check. \(qR\) is dimensionless (\(\mathrm{fm^{-1}}\times\mathrm{fm}\)); \(F\) is dimensionless; \(\theta\) in degrees.

Problems

Starting from elastic kinematics (\(|\mathbf k'|=|\mathbf k|=k\)), derive \(q=2k\sin(\theta/2)\) and find the maximum possible momentum transfer.
Solution
\(q^2=|\mathbf k-\mathbf k'|^2=2k^2(1-\cos\theta)=4k^2\sin^2(\theta/2)\Rightarrow q=2k\sin(\theta/2)\). It is maximal at backscattering \(\theta=180^\circ\): \(q_{\max}=2k\), corresponding to reversing the momentum, \(\hbar q_{\max}=2\hbar k=2p\).
Find the first Born amplitude for a delta-shell potential \(V(\mathbf r)=\gamma\,\delta(r-a)\) (with \(\gamma\) of units \(\mathrm{J\,m}\)). At what \(q\) does \(f\) first vanish?
Solution
\(f_{\mathrm B}=-\frac{m}{2\pi\hbar^2}\gamma\int\delta(r-a)e^{i\mathbf q\cdot\mathbf r}d^3r\). The angular integral over the shell gives \(a^2\cdot4\pi\frac{\sin qa}{qa}=\frac{4\pi a\sin qa}{q}\), so \(f_{\mathrm B}=-\frac{2m\gamma a}{\hbar^2}\,\frac{\sin qa}{qa}\). It vanishes when \(qa=n\pi\) (\(n=1,2,\dots\)), first at \(q=\pi/a\).
For the Yukawa potential \(V=\beta e^{-\mu r}/r\), obtain \(f_{\mathrm B}(q)\) and its low-energy limit; interpret the latter as a scattering length.
Solution
\(f_{\mathrm B}=-\frac{m}{2\pi\hbar^2}\beta\frac{4\pi}{q^2+\mu^2}=-\frac{2m\beta}{\hbar^2(q^2+\mu^2)}\). As \(k\to0\), \(q\to0\): \(f_{\mathrm B}(0)=-\frac{2m\beta}{\hbar^2\mu^2}\equiv-a_s\), a constant, isotropic amplitude. The Born scattering length is \(a_s=\frac{2m\beta}{\hbar^2\mu^2}\); positive \(\beta\) (repulsion) gives \(a_s>0\).
An exponential charge density \(\rho(r)=\frac{1}{8\pi a^{3}}e^{-r/a}\) (normalized). Show \(F(q)=(1+q^2a^2)^{-2}\) (the "dipole" form factor) and find \(\langle r^2\rangle\).
Solution
Normalization: \(\int\rho\,d^3r=\frac{1}{2a^3}\int_0^\infty r^2e^{-r/a}dr=\frac{1}{2a^3}(2a^3)=1\). Fourier transform: \(\int e^{i\mathbf q\cdot\mathbf r}e^{-r/a}d^3r=\frac{8\pi a^3}{(1+q^2a^2)^2}\), so \(F(q)=\frac{1}{8\pi a^3}\cdot\frac{8\pi a^3}{(1+q^2a^2)^2}=(1+q^2a^2)^{-2}\). Mean square radius: \(\langle r^2\rangle=\frac{1}{2a^3}\int_0^\infty r^4e^{-r/a}dr=\frac{1}{2a^3}(24a^5)=12a^2\), so \(\sqrt{\langle r^2\rangle}=\sqrt{12}\,a\). Check small-\(q\): \(F\approx1-2q^2a^2=1-\tfrac16 q^2(12a^2)\). ✓
An elastic-scattering experiment finds \(F(q_1)=0.80\) at \(q_1=0.50\ \mathrm{fm^{-1}}\), in the small-\(q\) regime. Estimate the rms charge radius. State one reason the estimate is only approximate.
Solution
Small-\(q\): \(F\simeq1-\tfrac16 q^2\langle r^2\rangle\Rightarrow\langle r^2\rangle=\frac{6(1-F)}{q^2}=\frac{6(0.20)}{(0.50)^2}=\frac{1.2}{0.25}=4.8\ \mathrm{fm^2}\). Thus \(\sqrt{\langle r^2\rangle}\approx2.19\ \mathrm{fm}\). Approximate because the expansion truncates the \(q^4\langle r^4\rangle\) term (non-negligible unless \(q\sqrt{\langle r^2\rangle}\ll1\); here \(q\sqrt{\langle r^2\rangle}\approx1.1\), so higher moments and Coulomb-distortion corrections matter and the single point should really be a fit of the slope as \(q\to0\)).