Partial-Wave Analysis and Phase Shifts

Derivation

Partial-Wave Analysis and Phase Shifts

D-362 Home PU-401 Threads waves · matter · symmetry Depends on The Lippmann-Schwinger Equation, spherical-harmonics

Statement

For a particle of energy \(E=\hbar^2 k^2/2m\) scattering off a spherically symmetric, finite-range potential \(V(r)\), the elastic scattering amplitude admits the exact partial-wave expansion \(f(\theta)=\frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell+1)\,e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta)\), where each real phase shift \(\delta_\ell(k)\) is read off from the asymptotic form \(u_\ell(r)\sim \sin\!\left(kr-\tfrac{\ell\pi}{2}+\delta_\ell\right)\) of the \(\ell\)-th radial solution outside the range of the potential. Equivalently \(f_\ell=\dfrac{e^{2i\delta_\ell}-1}{2ik}\).

Why it matters

Partial-wave analysis converts a three-dimensional scattering problem into a countable set of one-dimensional radial problems, one per angular momentum \(\ell\). The entire influence of the potential on elastic scattering is compressed into a real number \(\delta_\ell(k)\) for each \(\ell\), and at low energy only the first one or two channels contribute, so a single number (the \(s\)-wave phase shift, and hence the scattering length) can control an entire cold-collision physics.

Because \(\delta_\ell\) is real for a Hermitian potential, unitarity (conservation of probability) is manifest: each channel simply reshuffles flux without absorbing it. This makes phase shifts the natural language for resonances, bound-state thresholds (Levinson's theorem), and effective-range theory, and the framework generalises directly to nuclear and particle scattering.

Assumptions

The potential is central, \(V=V(r)\).If \(V\) depends on angle, \(\ell\) and \(m\) are no longer good quantum numbers, the channels couple, and a single Legendre series in \(P_\ell(\cos\theta)\) no longer diagonalises the problem.

The potential has finite range: \(V(r)\to 0\) faster than \(1/r\) as \(r\to\infty\).For a Coulomb tail \(V\sim 1/r\) the phase never settles to a constant; one gets logarithmic phase distortions and must use Coulomb-modified partial waves instead.

Scattering is elastic and the interaction is Hermitian.If channels open (inelasticity) or \(V\) is complex (absorption), \(\delta_\ell\) acquires an imaginary part; one replaces \(e^{2i\delta_\ell}\) by \(S_\ell=\eta_\ell e^{2i\delta_\ell}\) with inelasticity \(0\le\eta_\ell\le1\).

The energy is fixed and positive, and the incident state is a stationary plane wave.If dropped, the time-independent expansion is invalid; wave-packet dynamics and threshold/virtual-state behaviour require analytic continuation of \(S_\ell(k)\) into complex \(k\).

The radial solution is regular at the origin, \(u_\ell(0)=0\).Admitting the irregular solution \(\sim r^{-\ell}\) would violate normalisability and mis-define \(\delta_\ell\); regularity is what fixes the phase uniquely.

Derivation

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

Time-independent Schrödinger equation for a stationary scattering state at energy \(E>0\); the boundary condition (below) selects the outgoing solution. A

\[ \psi(\mathbf r)\;\xrightarrow[r\to\infty]{}\; e^{ikz} + f(\theta)\,\frac{e^{ikr}}{r} \]

Scattering boundary condition: incident plane wave along \(z\) plus an outgoing spherical wave whose angular modulation defines the amplitude \(f(\theta)\). Azimuthal symmetry (central \(V\), \(\hat z\) incidence) kills \(\phi\)-dependence. A

\[ \psi(r,\theta)=\sum_{\ell=0}^{\infty} R_\ell(r)\,P_\ell(\cos\theta),\qquad R_\ell(r)=\frac{u_\ell(r)}{r} \]

Separation of variables. With no \(\phi\)-dependence only \(m=0\) harmonics survive, and \(Y_\ell^0\propto P_\ell(\cos\theta)\) (assumes spherical-harmonics). Writing \(R_\ell=u_\ell/r\) will linearise the radial operator. A

3′

\[ -\frac{\hbar^2}{2m}\frac{d^2u_\ell}{dr^2}+\left[V(r)+\frac{\hbar^2}{2m}\frac{\ell(\ell+1)}{r^2}\right]u_\ell=E\,u_\ell \]

Substituting \(R_\ell=u_\ell/r\) into \(\nabla^2\) uses \(\nabla^2(u/r)=(u''/r)\) for the radial part, turning the 3D equation into a 1D radial equation with an added centrifugal barrier. B

\[ r>R_0:\quad \frac{d^2u_\ell}{dr^2}+\left[k^2-\frac{\ell(\ell+1)}{r^2}\right]u_\ell=0 \]

Outside the range \(R_0\) of the potential (\(V=0\)) the radial equation is the free spherical Bessel equation; its general solution is a combination of \(j_\ell(kr)\) and \(n_\ell(kr)\). The irregular \(n_\ell\) is now allowed because the origin is excluded from this region. B

\[ R_\ell(r)\;\xrightarrow[r>R_0]{}\; B_\ell\big[\cos\delta_\ell\, j_\ell(kr)-\sin\delta_\ell\, n_\ell(kr)\big] \]

Most general regular-at-origin-matched exterior solution, parametrised so that a single real angle \(\delta_\ell\) measures the admixture of \(n_\ell\). With \(V=0\) exactly, \(\delta_\ell=0\); the potential's only trace is this mixing angle. B

\[ j_\ell(kr)\xrightarrow{kr\gg\ell}\frac{\sin\!\left(kr-\tfrac{\ell\pi}{2}\right)}{kr},\qquad n_\ell(kr)\xrightarrow{kr\gg\ell}-\frac{\cos\!\left(kr-\tfrac{\ell\pi}{2}\right)}{kr} \]

Standard large-argument asymptotics of spherical Bessel functions. Substituting these into Step 5 and using \(\cos\delta_\ell\sin x+\sin\delta_\ell\cos x=\sin(x+\delta_\ell)\) collapses the two terms into one shifted sine. B

\[ R_\ell(r)\;\xrightarrow[r\to\infty]{}\;\frac{B_\ell}{kr}\,\sin\!\left(kr-\frac{\ell\pi}{2}+\delta_\ell\right) \]

Definition of the phase shift: relative to the free radial wave \(\sin(kr-\ell\pi/2)/kr\), the interacting wave is shifted in phase by exactly \(\delta_\ell\). This is the quantity to extract by integrating the interior equation out to \(R_0\) and matching logarithmic derivatives. A

\[ e^{ikz}=e^{ikr\cos\theta}=\sum_{\ell=0}^{\infty}(2\ell+1)\,i^{\ell}\,j_\ell(kr)\,P_\ell(\cos\theta) \]

Rayleigh (plane-wave) expansion: the incident free wave decomposed into partial waves. This lets us compare the incident and full solutions channel by channel. B

\[ e^{ikz}\;\xrightarrow[r\to\infty]{}\;\sum_{\ell}(2\ell+1)\,i^{\ell}\,\frac{\sin\!\left(kr-\tfrac{\ell\pi}{2}\right)}{kr}\,P_\ell(\cos\theta) =\sum_\ell\frac{2\ell+1}{2ik}\,i^\ell\,\frac{e^{i(kr-\ell\pi/2)}-e^{-i(kr-\ell\pi/2)}}{r}P_\ell \]

Insert the asymptotic \(j_\ell\) and write the sine as incoming (\(e^{-ikr}\)) and outgoing (\(e^{+ikr}\)) spherical waves. The incoming component is fixed by the source and cannot be altered by scattering. B

\[ f(\theta)=\sum_{\ell=0}^{\infty}(2\ell+1)\,f_\ell\,P_\ell(\cos\theta) \]

The outgoing spherical wave \(f(\theta)e^{ikr}/r\) is likewise expanded in \(P_\ell\); completeness/orthogonality of Legendre polynomials makes the expansion unique, defining the partial-wave amplitudes \(f_\ell\). A

\[ \underbrace{\frac{2\ell+1}{2ik}i^\ell(-1)\,i^{-\ell}}_{\text{incoming }e^{-ikr}}\ \text{must match} \implies \text{full }\psi:\ \frac{i^\ell}{2ik}\left[(1+2ikf_\ell)e^{ikr}-e^{-ikr}(-1)^{?}\right] \]

Assemble the full asymptotic \(\psi=e^{ikz}+f e^{ikr}/r\) per channel and demand that its interior origin be regular. The coefficient of the incoming wave \(e^{-ikr}\) is unchanged by \(V\); only the outgoing coefficient can pick up a phase — this is unitarity. C

\[ 1+2ik\,f_\ell = e^{2i\delta_\ell}\;\equiv\;S_\ell \]

Comparing the outgoing coefficient of the full solution (Step 7 recast into \(e^{\pm ikr}\) form) with that of the free wave (Step 9): conservation of flux forces the outgoing amplitude to differ from the incoming one only by a unit-modulus phase \(S_\ell=e^{2i\delta_\ell}\). Solving for \(f_\ell\) follows. C

\[ f_\ell=\frac{S_\ell-1}{2ik}=\frac{e^{2i\delta_\ell}-1}{2ik}=\frac{e^{i\delta_\ell}\sin\delta_\ell}{k} \]

Algebra: \(e^{2i\delta}-1=e^{i\delta}(e^{i\delta}-e^{-i\delta})=2i\,e^{i\delta}\sin\delta\). Each partial amplitude is fixed entirely by its real phase shift. A

\[ f(\theta)=\frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell+1)\,e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta) \]

Substitute Step 13 into Step 10. This is the target result: the full elastic amplitude as a Legendre series with one real phase shift per channel. A

Result

\[ f(\theta)=\frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell+1)\,e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta),\qquad f_\ell=\frac{e^{2i\delta_\ell}-1}{2ik} \]

Reading. Every effect of a central, finite-range potential on elastic scattering is stored in the set of real phase shifts \(\{\delta_\ell(k)\}\). The \(\ell\)-th outgoing partial wave is the free wave advanced (or retarded) in phase by \(\delta_\ell\); an attractive potential pulls the wave inward, \(\delta_\ell>0\), a repulsive one pushes it out, \(\delta_\ell<0\). Integrating cross-terms gives the total elastic cross section \(\sigma=\frac{4\pi}{k^2}\sum_\ell(2\ell+1)\sin^2\delta_\ell\), and the forward direction obeys the optical theorem \(\sigma=\frac{4\pi}{k}\,\mathrm{Im}\,f(0)\).

Units check. \([1/k]=\text{m}\); \(e^{i\delta_\ell}\sin\delta_\ell\) and \((2\ell+1)P_\ell\) are dimensionless, so \([f]=\text{m}\), and \([|f|^2]=[d\sigma/d\Omega]=\text{m}^2/\text{sr}\), as required. In \(\sigma=\frac{4\pi}{k^2}\sum(2\ell+1)\sin^2\delta_\ell\), \([1/k^2]=\text{m}^2\), giving an area.

Limiting cases

Low energy / \(s\)-wave dominance: the centrifugal barrier suppresses \(\delta_\ell\sim (kR_0)^{2\ell+1}\), so as \(k\to0\) only \(\ell=0\) survives; scattering is isotropic and \(f\to -a\), defining the scattering length \(a=-\lim_{k\to0}\delta_0/k\).
Weak potential (Born limit): \(\sin\delta_\ell\approx\delta_\ell\), \(e^{i\delta_\ell}\approx1\), and \(\delta_\ell\approx-\frac{2m}{\hbar^2 k}\int_0^\infty V(r)\,[j_\ell(kr)]^2 r^2\,dr\), linear in \(V\).
Resonance: when \(\delta_\ell\) sweeps through \(\pi/2\), \(\sin^2\delta_\ell=1\) saturates the unitarity bound and the partial cross section hits its maximum \(\frac{4\pi}{k^2}(2\ell+1)\) — a Breit–Wigner peak.
Hard sphere, \(s\)-wave: \(u_0(R_0)=0\) forces \(\delta_0=-kR_0\), giving \(\sigma\to4\pi R_0^2\) as \(k\to0\) (four times the geometric cross section).
High energy / large \(\ell\): partial waves up to \(\ell_{\max}\approx kR_0\) contribute (semiclassical impact parameter \(b=\ell/k\)); the sum becomes effectively continuous and reproduces classical/eikonal scattering.

Breaks when

Long-range (Coulomb) potentials, \(V\sim1/r\): the asymptotic wave never reduces to a pure shifted sine — it carries a divergent logarithmic phase \(\sim\eta\ln(2kr)\) — so a constant \(\delta_\ell\) cannot be defined and the plain expansion fails; one must use Coulomb wavefunctions and add a Coulomb phase.
Inelastic or absorptive channels: if flux is lost (complex \(V\), or open reaction channels), \(|S_\ell|<1\) and \(\delta_\ell\) becomes complex; the real-phase-shift statement is invalid and must be replaced by \(S_\ell=\eta_\ell e^{2i\delta_\ell}\) with a separate reaction cross section.
Non-central potentials or spin–orbit coupling: \(\ell\) (or \(m\)) is no longer conserved, channels mix, and a single Legendre series in \(P_\ell(\cos\theta)\) does not diagonalise the \(S\)-matrix.
Slow convergence at high energy: when \(kR_0\gg1\), thousands of \(\ell\) contribute and the partial-wave sum, though formally exact, is computationally useless — eikonal/WKB methods are used instead.

Failure modes

Dropping the \((2\ell+1)\) weight: writing \(f=\sum f_\ell P_\ell\) with \(f_\ell=e^{i\delta_\ell}\sin\delta_\ell/k\) — the degeneracy factor from the \(Y_\ell^0\) normalisation is then double-counted or lost, corrupting \(\sigma\).
Sign of the phase convention: taking \(u_\ell\sim\sin(kr-\ell\pi/2-\delta_\ell)\) flips every \(\delta_\ell\) and makes attractive potentials look repulsive.
Using \(\sigma=\frac{4\pi}{k^2}\sum(2\ell+1)\delta_\ell^2\) beyond the Born regime: replacing \(\sin^2\delta_\ell\) by \(\delta_\ell^2\) violates the unitarity bound near a resonance.
Forgetting the interference (cross) terms: computing \(d\sigma/d\Omega\) by summing \(|f_\ell|^2\) separately instead of \(|\sum(2\ell+1)f_\ell P_\ell|^2\) discards the angular interference that produces the actual differential pattern.
Retaining the irregular solution at \(r=0\): including \(n_\ell\) inside the potential region gives a non-normalisable, unphysical \(u_\ell(0)\neq0\) and a meaningless phase.
Confusing scattering length sign: quoting \(a=+\delta_0/k\) rather than \(a=-\lim\delta_0/k\), reversing attractive vs. repulsive low-energy behaviour and the sign of any mean-field interaction.

Discussion

The deep content of the derivation is unitarity. Because probability is conserved, the interacting solution can differ from the free one only by rotating the phase of the outgoing wave while leaving the incoming amplitude untouched. That single fact forces \(|S_\ell|=|e^{2i\delta_\ell}|=1\) and reduces the entire scattering operator, in each angular-momentum sector, to one real number. The \(S\)-matrix \(S_\ell=e^{2i\delta_\ell}\) is the prototype of the general \(S\)-matrix of quantum field theory, and its analytic structure in complex \(k\) encodes the full spectrum: poles on the positive imaginary axis are bound states, poles just below the real axis are resonances.

Phase shifts also connect scattering to bound states through Levinson's theorem, \(\delta_\ell(0)-\delta_\ell(\infty)=n_\ell\pi\), where \(n_\ell\) is the number of bound states of angular momentum \(\ell\). Thus a purely asymptotic measurement — how much the wave is phase-shifted at large \(r\) — counts the discrete states buried inside the potential. Near threshold the \(s\)-wave phase behaves as \(k\cot\delta_0=-1/a+\tfrac12 r_e k^2+\dots\), the effective-range expansion, which lets two numbers (scattering length \(a\), effective range \(r_e\)) parametrise all low-energy collisions independent of the potential's fine details — the basis of universality in ultracold atoms and low-energy nuclear physics.

Physically, \(\delta_\ell>0\) means the attractive potential pulls the radial wave crest inward: the wave "arrives sooner" than the free wave, a positive spatial advance. The differential cross section is then an interference pattern among the \(P_\ell(\cos\theta)\), so anisotropy in the angular distribution is a direct fingerprint of how many partial waves are active — pure \(s\)-wave scattering is isotropic, while a sharp forward or backward peak signals many contributing \(\ell\).

At the rigorous level, the expansion is an exact rewriting of the Lippmann–Schwinger equation \(|\psi^+\rangle=|\phi\rangle+G_0^+ V|\psi^+\rangle\) in the angular-momentum basis, where the free resolvent \(G_0^+\) is diagonal in \(\ell\). The on-shell \(T\)-matrix element then satisfies \(f_\ell=-\frac{2m}{\hbar^2}\,\frac{\pi}{k}\,\langle k\ell|T|k\ell\rangle\), and unitarity of \(T\) (the optical theorem) is equivalent to the statement that \(\delta_\ell\) is real. This is also where analyticity enters: \(S_\ell(k)\) continued to the second Riemann sheet exposes resonance poles at \(k=k_R-i\Gamma/(2v)\), giving the Breit–Wigner form \(\delta_\ell\approx\arctan\frac{\Gamma/2}{E_R-E}\).

Common misconceptions. A phase shift is not an energy or a probability — it is a dimensionless angle measured only in the asymptotic region; its whole value comes from the region where the potential acts, even though it is read off where the potential is zero. And \(\delta_\ell\) is defined only modulo \(\pi\) as a phase, but the physical, continuous branch (fixed by \(\delta_\ell(\infty)=0\) and continuity in \(k\)) is what carries the bound-state count via Levinson's theorem.

Worked examples

\(s\)-wave phase shift and cross section for an attractive square well

Well: \(V=-V_0\) for \(r<R_0\), zero outside; parameters \(V_0=20\ \text{MeV}\), \(R_0=2.0\ \text{fm}\), incident neutron lab energy \(E=1.0\ \text{MeV}\), reduced mass \(m\approx m_n\) (\(\hbar^2/2m=20.7\ \text{MeV·fm}^2\)). Low energy \(\Rightarrow\) keep only \(\ell=0\). A

\[ k=\sqrt{\frac{2mE}{\hbar^2}}=\sqrt{\frac{E}{20.7}}=\sqrt{\frac{1.0}{20.7}}\ \text{fm}^{-1}=0.220\ \text{fm}^{-1} \]

Outside wavenumber from \(E=\hbar^2k^2/2m\). A

\[ K=\sqrt{\frac{2m(E+V_0)}{\hbar^2}}=\sqrt{\frac{21.0}{20.7}}\ \text{fm}^{-1}=1.007\ \text{fm}^{-1} \]

Interior wavenumber; the well raises the local kinetic energy to \(E+V_0\). A

\[ \tan(kR_0+\delta_0)=\frac{k}{K}\tan(KR_0) \]

Matching logarithmic derivatives of \(u_0=\sin(Kr)\) (interior, regular) and \(u_0\propto\sin(kr+\delta_0)\) (exterior) at \(r=R_0\). B

\[ KR_0=2.014\ \text{rad},\quad \tan(KR_0)=-2.14,\quad \frac{k}{K}\tan(KR_0)=-0.468 \]

Numerics: \(\tan(kR_0+\delta_0)=-0.468\Rightarrow kR_0+\delta_0=-0.437\) (or \(+2.705\) rad choosing the physical continuous branch). B

\[ kR_0=0.440,\quad \delta_0=2.705-0.440=2.265\ \text{rad}\ \ (\equiv -0.877\ \text{mod }\pi),\quad \sin^2\delta_0=0.588 \]

Subtract the free phase \(kR_0\). Use \(\sin^2\delta_0\) (branch-independent) for the cross section. B

\[ \sigma_0=\frac{4\pi}{k^2}\sin^2\delta_0=\frac{4\pi}{(0.220)^2}(0.588)\ \text{fm}^2=1.53\times10^{2}\ \text{fm}^2\approx1.5\ \text{barn} \]

Reading. A sizeable \(s\)-wave phase shift produces a cross section of order a barn, typical of low-energy neutron scattering; \(\sin^2\delta_0\) near \(0.6\) shows this well is close to (but not at) a resonance.

Units check. \([1/k^2]=\text{fm}^2\); \(1\ \text{barn}=100\ \text{fm}^2\), so \(153\ \text{fm}^2=1.53\) barn.

Hard-sphere scattering length and low-energy cross section

Impenetrable sphere of radius \(R_0=1.0\ \text{fm}\): \(u_\ell(R_0)=0\). Take the \(k\to0\) \(s\)-wave limit. A

\[ u_0(r)\propto\sin(kr+\delta_0),\qquad u_0(R_0)=0\ \Rightarrow\ \sin(kR_0+\delta_0)=0 \]

Exterior free solution (\(V=0\) for \(r>R_0\)) with the wall boundary condition at \(r=R_0\). A

\[ kR_0+\delta_0=0\ \Rightarrow\ \delta_0=-kR_0 \]

Choosing the branch continuous with \(\delta_0\to0\) as \(V\to0\) is impossible here (the wall is always present), so take the smallest-magnitude root; negative \(\delta_0\) signals repulsion. B

\[ a\equiv-\lim_{k\to0}\frac{\delta_0}{k}=-\frac{-kR_0}{k}=R_0=1.0\ \text{fm} \]

Definition of scattering length; for a hard sphere it equals the radius exactly. A

\[ \sigma_0=\frac{4\pi}{k^2}\sin^2\delta_0\xrightarrow{k\to0}\frac{4\pi}{k^2}(kR_0)^2=4\pi R_0^2 \]

Since \(\sin\delta_0\approx\delta_0=-kR_0\) at small \(k\), the \(1/k^2\) cancels. B

\[ \sigma_0=4\pi R_0^2=4\pi(1.0\ \text{fm})^2=12.6\ \text{fm}^2 \]

Reading. The zero-energy hard-sphere cross section is four times the geometric area \(\pi R_0^2\) — a purely quantum (diffractive) enhancement — and the scattering length equals the physical radius.

Units check. \([R_0^2]=\text{fm}^2\); \(4\pi(1.0)^2=12.57\ \text{fm}^2\), an area.

Problems

(A) A partial-wave analysis at \(k=0.50\ \text{fm}^{-1}\) gives \(\delta_0=30^\circ\) and \(\delta_1=10^\circ\), with all higher shifts negligible. Compute the total elastic cross section.
Solution
\(\sigma=\frac{4\pi}{k^2}\big[(1)\sin^2 30^\circ+(3)\sin^2 10^\circ\big]\). \(\sin^2 30^\circ=0.250\), \(\sin^2 10^\circ=0.0302\Rightarrow 3(0.0302)=0.0905\). Sum \(=0.3405\). \(\frac{4\pi}{(0.50)^2}=\frac{12.566}{0.25}=50.27\ \text{fm}^2\). \(\sigma=50.27\times0.3405=17.1\ \text{fm}^2\).
(A) Show from the result that the maximum possible contribution of the \(\ell=2\) channel to the cross section at \(k=1.0\ \text{fm}^{-1}\) is \(\frac{4\pi}{k^2}(2\ell+1)\), and give its numeric value.
Solution
The partial cross section is \(\sigma_\ell=\frac{4\pi}{k^2}(2\ell+1)\sin^2\delta_\ell\), maximal when \(\sin^2\delta_\ell=1\) (\(\delta_\ell=\pi/2\), the unitarity limit / resonance). For \(\ell=2\), \(2\ell+1=5\): \(\sigma_2^{\max}=\frac{4\pi}{(1.0)^2}\times5=20\pi=62.8\ \text{fm}^2\).
(B) Using the optical theorem, verify the total cross section of Problem 1 by computing \(\frac{4\pi}{k}\,\mathrm{Im}\,f(0)\). (Recall \(P_\ell(1)=1\).)
Solution
\(\mathrm{Im}\,f(0)=\frac1k\sum(2\ell+1)\sin^2\delta_\ell\) since \(\mathrm{Im}(e^{i\delta}\sin\delta)=\sin^2\delta\) and \(P_\ell(1)=1\). So \(\frac{4\pi}{k}\mathrm{Im}f(0)=\frac{4\pi}{k^2}\sum(2\ell+1)\sin^2\delta_\ell\), identical to \(\sigma\). Numerically \(\mathrm{Im}f(0)=\frac{1}{0.50}(0.250+0.0905)=0.681\ \text{fm}\); \(\frac{4\pi}{0.50}(0.681)=25.13\times0.681=17.1\ \text{fm}^2\). Matches.
(B) For a weak attractive potential \(V(r)=-V_0 e^{-r/R_0}\) with \(V_0=5\ \text{MeV}\), \(R_0=1.5\ \text{fm}\), estimate the Born \(s\)-wave phase shift at \(k=0.30\ \text{fm}^{-1}\). Use \(\hbar^2/2m=20.7\ \text{MeV·fm}^2\) and the small-\(kr\) approximation \(j_0(kr)\approx1\) over the potential range.
Solution
Born \(s\)-wave: \(\delta_0\approx-\frac{2m}{\hbar^2 k}\int_0^\infty V(r)[j_0(kr)]^2 r^2 dr\). With \(j_0\approx1\): \(\int_0^\infty(-V_0 e^{-r/R_0})r^2 dr=-V_0(2R_0^3)=-5(2)(1.5)^3=-33.75\ \text{MeV·fm}^3\). Prefactor \(\frac{2m}{\hbar^2 k}=\frac{1}{20.7\times0.30}=0.1610\ \text{MeV}^{-1}\text{fm}^{-1}\). \(\delta_0\approx-0.1610\times(-33.75)=+5.43\). This exceeds \(1\ \text{rad}\), so the "weak/Born" assumption is in fact violated — flagging that \(V_0R_0^2/(\hbar^2/2m)=5(2.25)/20.7=0.54\) is not \(\ll1\) once accumulated; the sign (attractive \(\Rightarrow\delta_0>0\)) is the robust conclusion. (Accept \(\delta_0\gtrsim0.5\ \text{rad}\), attractive.)
(C) A neutron–nucleus \(\ell=1\) phase shift rises sharply through \(90^\circ\) as the energy passes \(E_R=0.35\ \text{MeV}\), fitting \(\delta_1=\arctan\!\frac{\Gamma/2}{E_R-E}\) with width \(\Gamma=0.10\ \text{MeV}\). Find the \(p\)-wave cross section at resonance (\(E=E_R\)) given \(k=0.13\ \text{fm}^{-1}\) there, and at \(E=E_R+\Gamma\).
Solution
At \(E=E_R\): denominator \(E_R-E=0\Rightarrow\delta_1=\pi/2\), \(\sin^2\delta_1=1\). \(\sigma_1=\frac{4\pi}{k^2}(2\cdot1+1)(1)=\frac{4\pi}{(0.13)^2}(3)=\frac{12.566}{0.0169}(3)=743.6\times3=2.23\times10^{3}\ \text{fm}^2\approx22\ \text{barn}\). At \(E=E_R+\Gamma\): \(\frac{\Gamma/2}{E_R-E}=\frac{0.05}{-0.10}=-0.5\), \(\delta_1=\arctan(-0.5)=-26.57^\circ\) (i.e. the resonant branch \(180-26.57=153.4^\circ\)); either way \(\sin^2\delta_1=\sin^2(26.57^\circ)=0.200\). \(\sigma_1=743.6\times3\times0.200=446\ \text{fm}^2\approx4.5\ \text{barn}\). The cross section falls to one-fifth its peak a width above resonance, the Breit–Wigner Lorentzian shape.