Chapter 22 Exercises: Scattering Theory

Section A: Scattering Setup and Cross Sections (Problems 1--6)

Problem 1 (Basic)

A beam of neutrons (mass $m_n = 1.675 \times 10^{-27}\;\text{kg}$) with kinetic energy $E = 1\;\text{eV}$ is incident on a target.

(a) Calculate the de Broglie wavelength $\lambda = 2\pi/k$ and the wavenumber $k$.

(b) If the scattering amplitude is $f(\theta) = -a$ (a constant, independent of angle --- called "s-wave scattering"), what is the differential cross section?

(c) What is the total cross section? Express your answer in barns ($1\;\text{barn} = 10^{-24}\;\text{cm}^2$) for $a = 5\;\text{fm}$.

Problem 2 (Basic)

Starting from the probability current $\mathbf{j} = (\hbar/2mi)(\psi^*\nabla\psi - \psi\nabla\psi^*)$:

(a) Compute $\mathbf{j}$ for a plane wave $\psi = Ae^{i\mathbf{k}\cdot\mathbf{r}}$.

(b) Compute the radial component of $\mathbf{j}$ for a spherical outgoing wave $\psi = Af(\theta)e^{ikr}/r$ at large $r$.

(c) Show that the differential cross section $d\sigma/d\Omega = |f(\theta)|^2$ follows from the ratio of scattered flux to incident flux.

Problem 3 (Basic)

The scattering amplitude for a certain potential is $f(\theta) = (a + b\cos\theta)e^{i\delta}$ where $a, b$ are real constants and $\delta$ is a phase.

(a) What is the differential cross section $d\sigma/d\Omega$?

(b) Compute the total cross section by integrating over solid angle. Show that the cross terms between $a$ and $b\cos\theta$ vanish.

(c) What physical significance does the interference between the constant and $\cos\theta$ terms have in terms of forward-backward asymmetry?

Problem 4 (Intermediate)

The scattering length $a_s$ is defined as $a_s = -\lim_{k\to 0} f(k, \theta=0)$ (more precisely, $f \to -a_s$ as $k \to 0$ for s-wave scattering).

(a) Show that $a_s = -\tan\delta_0/k$ in the limit $k \to 0$.

(b) For a hard sphere of radius $R$, show that $a_s = R$.

(c) The scattering length for neutron-proton scattering in the spin-triplet state is $a_t = 5.42\;\text{fm}$, and in the spin-singlet state is $a_s = -23.7\;\text{fm}$. What are the corresponding low-energy cross sections $\sigma = 4\pi a^2$? Why is the singlet cross section so much larger, despite the interaction being weaker?

Problem 5 (Intermediate)

A detector at angle $\theta = 30°$ subtends a solid angle $\Delta\Omega = 10^{-3}\;\text{sr}$. The incident beam has flux $\Phi = 10^{8}\;\text{particles/s/cm}^2$ and the target has areal density $n_t = 10^{20}\;\text{cm}^{-2}$.

(a) If $d\sigma/d\Omega = 50\;\text{mb/sr}$ at $\theta = 30°$, how many particles per second does the detector register?

(b) If the experiment runs for 1 hour, what is the total count? What is the statistical uncertainty (assuming Poisson statistics)?

(c) What minimum run time is needed to measure $d\sigma/d\Omega$ to 1% precision?

Problem 6 (Intermediate)

Show that for a central potential, the scattering amplitude is independent of the azimuthal angle $\phi$. Hint: Consider the symmetry of the problem when the beam is along $\hat{z}$ and the potential is spherically symmetric. What is the conserved quantum number?

Section B: Born Approximation (Problems 7--13)

Problem 7 (Basic)

Compute the Born approximation scattering amplitude for the Gaussian potential $V(r) = V_0 e^{-r^2/a^2}$.

(a) Evaluate the Fourier transform $\tilde{V}(\mathbf{q}) = \int V(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}\,d^3\mathbf{r}$. Hint: The Fourier transform of a Gaussian is a Gaussian.

(b) Write the differential cross section in terms of $q = 2k\sin(\theta/2)$.

(c) At what angle $\theta_{\min}$ is the cross section $1/e^2$ of its forward value? How does $\theta_{\min}$ depend on $ka$?

Problem 8 (Basic)

For the Yukawa potential $V(r) = V_0 e^{-\mu r}/r$:

(a) Verify that $\tilde{V}(\mathbf{q}) = 4\pi V_0/(q^2 + \mu^2)$.

(b) Compute the total cross section in the Born approximation by integrating $d\sigma/d\Omega$ over all angles. Hint: Use the substitution $u = q^2 = 4k^2\sin^2(\theta/2)$.

(c) Show that in the limit $\mu \to 0$, the total cross section diverges. Explain why this is physically expected for the Coulomb potential.

Problem 9 (Intermediate)

The exponential potential $V(r) = V_0 e^{-r/a}$ models the nuclear force at intermediate distances.

(a) Compute the Born scattering amplitude. Hint: The Fourier transform is $\tilde{V}(\mathbf{q}) = 8\pi V_0 a^3/(1 + q^2 a^2)^2$.

(b) Show that the angular dependence at high energy ($ka \gg 1$) produces a diffraction pattern with the first minimum at $\theta \sim 1/(ka)$.

(c) Compare the Born cross section at $\theta = 0$ and $\theta = 90°$ for $ka = 5$. How forward-peaked is the scattering?

Problem 10 (Intermediate)

Born validity criterion. For a square well potential $V(r) = -V_0$ for $r < a$, $V(r) = 0$ for $r > a$:

(a) Write down the low-energy Born validity criterion $2mV_0 a^2/\hbar^2 \ll 1$ and interpret it physically.

(b) Show that this is equivalent to requiring that there are no bound states in the potential. Hint: The condition for the first bound state in a spherical well is $V_0 a^2 \approx \pi^2\hbar^2/(8m)$.

(c) Write down the high-energy criterion $2mV_0 a/(\hbar^2 k) \ll 1$. For a neutron scattering off a nucleus with $V_0 = 40\;\text{MeV}$ and $a = 4\;\text{fm}$, at what kinetic energy does the Born approximation become reliable?

Problem 11 (Intermediate)

Second Born approximation. For the Yukawa potential, the second Born correction to the scattering amplitude is:

$$f^{(2)}(\theta) = -\frac{m}{2\pi\hbar^2}\int\frac{\tilde{V}(\mathbf{k}' - \mathbf{k}'')}{k^2 - k''^2 + i\epsilon}\tilde{V}(\mathbf{k}'' - \mathbf{k})\frac{d^3\mathbf{k}''}{(2\pi)^3}$$

(a) Argue on dimensional grounds that $f^{(2)}/f^{(1)} \sim mV_0/(k\hbar^2)$ (up to logarithmic corrections).

(b) For the Yukawa potential with $V_0 = 1\;\text{a.u.}$, $\mu = 1\;\text{a.u.}$, estimate the energy above which the second Born correction is less than 10% of the first.

(c) Explain why the second Born amplitude is complex even for a real potential. What does the imaginary part signify physically?

Problem 12 (Advanced)

Born approximation for the screened Coulomb potential (Thomas-Fermi model).

In metals, the Coulomb potential of an impurity is screened: $V(r) = (Ze^2/4\pi\epsilon_0)(e^{-r/\lambda_{\text{TF}}}/r)$ where $\lambda_{\text{TF}}$ is the Thomas-Fermi screening length.

(a) Write the Born differential cross section in terms of the momentum transfer $q$.

(b) Show that the total cross section is finite (unlike bare Coulomb) and compute it.

(c) The transport cross section (relevant for electrical resistivity) is $\sigma_{\text{tr}} = \int(1 - \cos\theta)(d\sigma/d\Omega)\,d\Omega$. Compute $\sigma_{\text{tr}}$ for the screened Coulomb potential. Why does the $(1 - \cos\theta)$ factor suppress forward scattering?

Problem 13 (Advanced)

Born approximation in momentum space. Show that the Born approximation can be written in Dirac notation as:

$$\langle\mathbf{k}'|T^{(1)}|\mathbf{k}\rangle = \langle\mathbf{k}'|\hat{V}|\mathbf{k}\rangle$$

where $|\mathbf{k}\rangle$ are plane wave states normalized to $\langle\mathbf{k}|\mathbf{k}'\rangle = (2\pi)^3\delta^3(\mathbf{k} - \mathbf{k}')$.

(a) Derive the position-space Born amplitude $f_{\text{Born}}(\theta)$ from this expression by inserting a complete set of position states.

(b) Show that the second Born approximation corresponds to $\langle\mathbf{k}'|T^{(2)}|\mathbf{k}\rangle = \langle\mathbf{k}'|\hat{V}\hat{G}_0^{(+)}\hat{V}|\mathbf{k}\rangle$.

(c) Interpret each factor in the second Born term as a physical process: scattering, propagation, scattering.

Section C: Partial Wave Analysis and Phase Shifts (Problems 14--22)

Problem 14 (Basic)

The Rayleigh expansion of a plane wave is $e^{ikz} = \sum_{l=0}^{\infty}(2l+1)i^l j_l(kr)P_l(\cos\theta)$.

(a) Verify this for $l = 0$ by showing that $j_0(kr) = \sin(kr)/(kr)$ and $\int_{-1}^{1}e^{ikr\cos\theta}d(\cos\theta) = 2j_0(kr)$.

(b) At what radius $r$ does the $l = 5$ partial wave have its first maximum in $|j_5(kr)|$? Express your answer in terms of $1/k$.

(c) Argue physically why partial waves with $l \gg ka$ (where $a$ is the range of the potential) do not contribute significantly to scattering.

Problem 15 (Basic)

For a hard sphere of radius $a$, the $s$-wave ($l = 0$) phase shift is $\delta_0 = -ka$.

(a) Compute the $s$-wave cross section $\sigma_0 = (4\pi/k^2)\sin^2(ka)$.

(b) In the low-energy limit $ka \ll 1$, show that $\sigma_0 \approx 4\pi a^2$.

(c) What happens at $ka = \pi$? What about $ka = n\pi$ for integer $n$? This is called a Ramsauer-Townsend effect.

Problem 16 (Intermediate)

Show that the partial wave expansion of the total cross section is:

$$\sigma_{\text{tot}} = \frac{4\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)\sin^2\delta_l$$

by substituting $f(\theta) = (1/k)\sum_l(2l+1)e^{i\delta_l}\sin\delta_l P_l(\cos\theta)$ into $\sigma = \int|f|^2 d\Omega$ and using the orthogonality relation $\int_{-1}^{1}P_l P_{l'}\,d(\cos\theta) = 2\delta_{ll'}/(2l+1)$.

Problem 17 (Intermediate)

Levinson's theorem states that $\delta_l(k=0) = n_l\pi$, where $n_l$ is the number of bound states with angular momentum $l$.

(a) For a square well with $l = 0$, verify Levinson's theorem numerically by computing $\delta_0(k \to 0)$ for a well that supports exactly one bound state, and for a well that supports exactly two.

(b) Explain why this theorem makes physical sense: what does the potential do to the wavefunction at zero energy if there is a bound state just below threshold?

(c) The deuteron is the only bound state of the neutron-proton system (in the $^3S_1$ channel). What is $\delta_0(k=0)$ for neutron-proton scattering in this channel?

Problem 18 (Intermediate)

For a finite square well $V(r) = -V_0$ for $r < a$, $V = 0$ for $r > a$, the $s$-wave ($l = 0$) interior solution is $u(r) = A\sin(Kr)$ where $K = \sqrt{k^2 + 2mV_0/\hbar^2}$.

(a) Match the interior and exterior solutions at $r = a$ to derive $\tan\delta_0 = (k\tan(Ka) - K\tan(ka))/(K + k\tan(ka)\tan(Ka))$.

(b) Simplify for $ka \ll 1$ (low energy) to obtain $\tan\delta_0 \approx ka[\tan(Ka)/(Ka) - 1]$.

(c) Show that $\delta_0$ passes through $\pi/2$ when $Ka \approx (n + 1/2)\pi$ for integer $n$, i.e., when the well nearly supports a new bound state. These are the s-wave resonances.

Problem 19 (Intermediate)

The effective range expansion for $s$-wave scattering is:

$$k\cot\delta_0 = -\frac{1}{a_s} + \frac{1}{2}r_{\text{eff}}k^2 + \cdots$$

where $a_s$ is the scattering length and $r_{\text{eff}}$ is the effective range.

(a) Show that $\sigma \approx 4\pi a_s^2/(1 + k^2 a_s^2)$ if only the first term is retained.

(b) For a hard sphere of radius $R$, show that $a_s = R$ and $r_{\text{eff}} = 2R/3$.

(c) What is the condition for the effective range expansion to converge? Why does it fail for the Coulomb potential?

Problem 20 (Intermediate)

Classical limit of partial wave analysis. For a hard sphere at high energy ($ka \gg 1$):

(a) Argue that partial waves with $l \leq ka$ are "classical" (their classical impact parameter $b = l/k$ is less than $a$) and scatter strongly, while $l > ka$ miss the sphere.

(b) Using $\delta_l \approx ka\sqrt{1 - l^2/(ka)^2} - l\cos^{-1}(l/(ka))$ for $l < ka$ (the semiclassical approximation), and $\delta_l \approx 0$ for $l > ka$, show that $\sigma_{\text{tot}} \approx 2\pi a^2$ in this limit.

(c) The factor of 2 (compared to the classical geometric cross section $\pi a^2$) is sometimes called the "extinction paradox." Explain why the extra $\pi a^2$ comes from diffraction around the sphere's edge.

Problem 21 (Advanced)

Resonance in a centrifugal barrier. Consider a square well potential $V = -V_0$ for $r < a$, $V = 0$ for $r > a$, with $l = 1$.

(a) The effective potential is $V_{\text{eff}}(r) = -V_0 + \hbar^2 l(l+1)/(2mr^2)$ for $r < a$ and $V_{\text{eff}}(r) = \hbar^2 l(l+1)/(2mr^2)$ for $r > a$. Sketch $V_{\text{eff}}$ and identify the potential barrier.

(b) Find the $l = 1$ phase shift by solving the radial equation exactly (the interior solution involves $j_1$ and the exterior involves $j_1$ and $n_1$) and matching at $r = a$.

(c) Show that a $p$-wave resonance occurs when the energy of the incident particle matches a quasi-bound state behind the centrifugal barrier. For $V_0 = 50\;\text{MeV}$, $a = 2\;\text{fm}$, $m = m_n$, find the resonance energy numerically.

Problem 22 (Advanced)

Prove the optical theorem directly from the Lippmann-Schwinger equation without using the partial wave expansion.

Hint: Start from $\operatorname{Im}[f(0)] = \operatorname{Im}\langle\mathbf{k}|T|\mathbf{k}\rangle$ (with appropriate normalization) and use the relation $T = V + VG_0^{(+)}T$ together with $\operatorname{Im}[G_0^{(+)}(E)] = -\pi\delta(E - H_0)$ to obtain:

$$\operatorname{Im}\langle\mathbf{k}|T|\mathbf{k}\rangle = -\pi\int|\langle\mathbf{k}'|T|\mathbf{k}\rangle|^2\delta(E_k - E_{k'})\,d^3\mathbf{k}'/(2\pi)^3$$

Show that the right side equals $-k\sigma_{\text{tot}}/(4\pi)$.

Section D: Coulomb Scattering and Applications (Problems 23--26)

Problem 23 (Basic)

The Rutherford cross section is $d\sigma/d\Omega = (Z_1 Z_2 e^2/(4E))^2/\sin^4(\theta/2)$ (Gaussian units).

(a) Show that this diverges as $\theta \to 0$. What is the physical origin of this divergence?

(b) Compute $d\sigma/d\Omega$ for 5 MeV alpha particles ($Z_1 = 2$) scattering off gold ($Z_2 = 79$) at $\theta = 90°$, $\theta = 10°$, and $\theta = 170°$. Express your answers in $\text{fm}^2/\text{sr}$.

(c) At what angle is the Rutherford cross section equal to $1\;\text{barn/sr}$ for this system?

Problem 24 (Intermediate)

The distance of closest approach. For Rutherford scattering, the classical distance of closest approach for a head-on collision ($\theta = 180°$) is $d = Z_1 Z_2 e^2/(2E)$.

(a) Compute $d$ for 7.7 MeV alpha particles on gold. Compare with the nuclear radius $R \approx 1.2 A^{1/3}\;\text{fm}$ for gold ($A = 197$).

(b) Geiger and Marsden found that the Rutherford formula broke down for aluminum at energies above about 6 MeV. Use this to estimate the nuclear radius of aluminum ($A = 27$).

(c) Why does the classical and quantum Coulomb cross sections agree exactly? Is this a general property of all potentials? If not, what is special about $1/r$?

Problem 25 (Advanced)

Coulomb phase shifts. The exact Coulomb phase shifts are:

$$\sigma_l = \arg\Gamma(l + 1 + i\eta)$$

where $\eta = Z_1 Z_2 e^2 m/(2\hbar^2 k)$ (Gaussian units) is the Sommerfeld parameter.

(a) For $\eta = 1$ (intermediate regime), compute $\sigma_l$ for $l = 0, 1, 2, 3$ using the relation $\Gamma(z+1) = z\Gamma(z)$ and $\sigma_0 = \arg\Gamma(1 + i)$.

(b) Show that the Coulomb total cross section (summed over partial waves) is indeed infinite, consistent with the divergence of the Rutherford integral.

(c) When a nuclear potential is added to the Coulomb potential, the phase shifts become $\delta_l = \sigma_l + \delta_l^N$ where $\delta_l^N$ is the nuclear phase shift. The nuclear scattering amplitude is then $f_N(\theta) = (1/2ik)\sum_l(2l+1)(e^{2i\delta_l^N} - 1)e^{2i\sigma_l}P_l$. Explain why the Coulomb phases appear as overall phases multiplying each partial wave.

Problem 26 (Advanced)

Mott scattering. For identical particles (e.g., $\alpha$-$\alpha$ scattering), the scattering amplitude must be symmetrized:

$$f_{\text{Mott}}(\theta) = f(\theta) + f(\pi - \theta)$$

(a) Write the Mott differential cross section and show that it contains an interference term $2\operatorname{Re}[f(\theta)f^*(\pi - \theta)]$.

(b) Using the Rutherford amplitude, compute the Mott cross section. Show that it exhibits oscillations as a function of $\theta$ that are absent in classical Rutherford scattering.

(c) What happens at $\theta = 90°$? Why is this a particularly interesting angle for studying nuclear effects?

Section E: Advanced Topics and Synthesis (Problems 27--30)

Problem 27 (Intermediate)

S-matrix poles and bound states. For a square well potential with $l = 0$, the S-matrix is:

$$S_0(k) = e^{-2ika}\frac{K\cot(Ka) + ik}{K\cot(Ka) - ik}$$

where $K = \sqrt{k^2 + 2mV_0/\hbar^2}$.

(a) Show that the bound state energies correspond to $S_0(k)$ having a pole at $k = i\kappa$ ($\kappa > 0$), which requires $K\cot(Ka) = -\kappa$ where $K = \sqrt{\kappa^2 - k^2} = \sqrt{2m(V_0 - |E|)/\hbar^2}$ at the bound state.

(b) Show that this is exactly the bound state condition obtained from matching wavefunctions at $r = a$.

(c) What is the residue of $S_0(k)$ at a bound-state pole? Show it is purely imaginary.

Problem 28 (Advanced)

The optical theorem for inelastic scattering. When absorption is present, $|S_l| = \eta_l < 1$.

(a) Show that the elastic cross section is $\sigma_{\text{el}} = (\pi/k^2)\sum_l(2l+1)|1 - \eta_l e^{2i\delta_l}|^2$.

(b) Show that the reaction (absorption) cross section is $\sigma_{\text{reac}} = (\pi/k^2)\sum_l(2l+1)(1 - \eta_l^2)$.

(c) For maximum absorption ($\eta_l = 0$) of a single partial wave $l$, show that $\sigma_{\text{el}} = \sigma_{\text{reac}} = \pi(2l+1)/k^2$ --- the elastic and reaction cross sections are equal. This is the "black disk" limit.

(d) In the black disk limit with all $l \leq L$ absorbed ($\eta_l = 0$ for $l \leq L$), show that $\sigma_{\text{tot}} = 2\pi(L+1)^2/k^2$ and that the elastic and reaction cross sections are each half of $\sigma_{\text{tot}}$.

Problem 29 (Advanced)

Born approximation vs. partial waves: a quantitative comparison. Consider the Yukawa potential $V(r) = V_0 e^{-\mu r}/r$.

(a) Compute the Born cross section at $\theta = 90°$ for $V_0 = 3$, $\mu = 1$, and $k = 1, 2, 5, 10$ (all in atomic units).

(b) Compute the exact (partial wave) cross section at $\theta = 90°$ for the same parameters. At what energy do Born and exact results agree to within 10%?

(c) Plot the ratio $d\sigma_{\text{Born}}/d\sigma_{\text{exact}}$ as a function of $k$ at $\theta = 90°$. Verify that the Born approximation improves at high energy.

Problem 30 (Advanced)

Synthesis problem: analyzing a scattering experiment. The differential cross section for neutrons scattering off a certain nucleus has been measured at $E = 2\;\text{MeV}$:

(a) Fit this data using $f(\theta) = A_0 + A_1\cos\theta + A_2 P_2(\cos\theta)$ (retaining only $l = 0, 1, 2$ partial waves). Extract the coefficients $A_l$.

(b) From the partial wave coefficients, extract the phase shifts $\delta_0$, $\delta_1$, and $\delta_2$.

(c) Compute the total cross section both by integrating the fit and by the partial wave sum, and verify they agree.

(d) Check the optical theorem: does $\sigma_{\text{tot}} = (4\pi/k)\operatorname{Im}[f(0)]$?

(e) Based on your phase shifts, is there evidence for a resonance in any partial wave? Justify your conclusion.