This page provides a comprehensive theoretical treatment of how to calculate the cross-section of a nuclear reaction using the Partial Wave Method in quantum mechanics. It begins by establishing the classical concept of the impact parameter and its quantisation into discrete angular momentum values (ℓ). The discussion then transitions to a quantum mechanical treatment, representing the incident particle as a plane wave and expanding it into a series of partial waves. The key focus is on how the presence of a nuclear potential modifies the outgoing spherical wave, leading to phase shifts (δℓ​) and attenuation factors (ηℓ​). By analysing the asymptotic behaviour of the total wavefunction, the page derives the expressions for the scattering amplitude (f(θ)), the differential elastic scattering cross-section (dσ/dΩ), and the total cross-section in terms of these quantum parameters. This method is fundamental to understanding particle-target interactions in nuclear and particle physics.

Consider a projectile \(x\) with momentum \(p_x\) and zero intrinsic spin incident on a nucleus that is assumed to be a perfectly absorbing sphere of radius \(R\).
The transverse distance between the trajectory of the projectile and the nuclear centre is called the \emph{impact parameter}, denoted by \(b\). For nuclear forces of short range, the path of the incoming particle remains essentially straight when \(b>R\). Only when \(b<R\) can the projectile interact strongly with the target.

The orbital angular momentum of the system is
\begin{equation}

L = p_x \times b
\end{equation}

Quantum mechanics restricts this to discrete values,
\begin{equation}

L = \ell \hbar, \qquad \ell = 0,1,2,\dots
\end{equation}

Thus, whenever
\begin{equation}

\ell \frac{\hbar}{p_x} \leq b < (\ell+1)\frac{\hbar}{p_x},
\end{equation}

the corresponding orbital quantum number is \(\ell\).

This permits us to picture the nuclear target as being surrounded by concentric annular regions (“partial waves”), each associated with a given \(\ell\) [Fig.}.
Since \(\hbar/p_x\) is essentially the de Broglie wavelength \(\lambda\), these circular zones have radii \(\ell\lambda, (\ell+1)\lambda, \dots\).

The geometrical cross section associated with the \(\ell\)-th zone is

\begin{equation}
\sigma_\ell = \pi\left[(\ell+1)^2 – \ell^2\right]\lambda^2
= (2\ell+1)\pi \lambda^2 .
\tag{1}
\end{equation}

Clearly, the number of particles removed from the beam cannot exceed the number incident, hence the total reaction cross section is bounded above by

\begin{equation}
\sigma_{\text{reac}} \leq \sum_{\ell}(2\ell+1)\pi \lambda^2 .
\tag{2}
\end{equation}

This limitation is not relevant for purely elastic scattering.

For complete absorption by the target, the interaction occurs only if \(b<R\).
This defines a maximum value of the angular momentum quantum number:

\begin{equation}
b_{\max} = \frac{\ell_{\max}\hbar}{p_x} = R .
\end{equation}

Thus
\begin{equation}
\ell_{\max} = \frac{R}{\hbar/p_x} = \frac{R}{\lambda} .
\tag{3}
\end{equation}

Accordingly, the total absorption cross section can be written as

\begin{equation}
\sigma_{\text{tot}} = \sum_{\ell=0}^{\ell_{\max}} (2\ell+1)\pi \lambda^2 .
\end{equation}

The summation evaluates to

\begin{equation}
\sigma_{\text{tot}} = \pi \lambda^2 \sum_{\ell=0}^{\ell_{\max}} (2\ell+1)
= \pi \lambda^2 \bigg(\frac{R}{\lambda}+1\bigg)^2 .
\tag{4}
\end{equation}

Therefore,
\begin{equation}
\sigma_{\text{tot}} \;\leq\; \pi (R+\lambda)^2 .
\end{equation}

For large \(\ell_{\max}\) (small \(\lambda\), i.e., high incident energy), this expression approaches the geometrical area \(\pi R^2\).
For very low energies, where \(\lambda\) is large compared to \(R\), the absorption cross section is governed primarily by the de Broglie wavelength.

Quantum Mechanical Treatment of Partial Waves

Let us now analyse the problem quantum mechanically.
Consider a collimated beam of monoenergetic particles moving along the \(z\) axis.
The incident plane wave can be represented as
\begin{equation}
\psi_{\text{inc}} = e^{ikz} = e^{ikr\cos\theta} ,\end{equation}
where \(k\) is the wave number of the incoming particle.

This plane wave can be expanded into a series of spherical Bessel functions \(j_\ell(kr)\) multiplied by Legendre polynomials \(P_\ell(\cos\theta)\) as
\begin{equation}
e^{ikr\cos\theta} = \sum_{\ell=0}^{\infty} (2\ell+1)\, i^\ell \, j_\ell(kr)\, P_\ell(\cos\theta).
\tag{5}
\end{equation}

At large distances \(r\to\infty\), the asymptotic form of the spherical Bessel function is
\begin{equation}

j_\ell(kr) \;\xrightarrow{r\to\infty}\; \frac{1}{kr}\,\sin\big(kr – \tfrac{\ell\pi}{2}\big).
\end{equation}

Hence, the asymptotic form of the incident wave is
\begin{equation}
\psi_{\text{inc}} \;\approx\; \frac{1}{kr}\sum_{\ell=0}^{\infty} (2\ell+1)\, i^\ell
\sin!\left(kr – \tfrac{\ell\pi}{2}\right) P_\ell(\cos\theta).
\tag{6}
\end{equation}

When a nucleus is present, the outgoing spherical wave is modified: both its amplitude and phase are altered due to scattering.
Thus, in place of the sine term above, we substitute a more general expression containing a phase shift \(\delta_\ell\) and an attenuation factor \(\eta_\ell\) (with \(0\leq \eta_\ell \leq 1\)).
Accordingly, the total wavefunction becomes

\begin{equation}
\psi(r,\theta) = \frac{1}{2ikr}\sum_{\ell=0}^{\infty} (2\ell+1)P_\ell(\cos\theta)
\Big[ \eta_\ell e^{i(kr – \ell\pi/2)} – e^{-i(kr – \ell\pi/2)} \Big].
\tag{7}
\end{equation}

Here \(\eta_\ell\) is a complex coefficient.
If \(|\eta_\ell|=1\), only the phase is changed (elastic scattering).
If \(|\eta_\ell| < 1\), some absorption has occurred (compound nucleus formation or reaction).

At large \(r\), the scattered part of the wave has the form:
\begin{equation}
\psi_{\text{sc}} = f(\theta)\,\frac{e^{ikr}}{r},
\tag{8}
\end{equation}
where \(f(\theta)\) is the scattering amplitude.

Thus, the total wavefunction can be expressed as
\begin{equation}
\psi(r,\theta) = \psi_{\text{inc}} + \psi_{\text{sc}}
= e^{ikz} + f(\theta)\,\frac{e^{ikr}}{r}.
\end{equation}

Comparing this with the partial-wave expansion (Eq. 7) gives an explicit expression for the scattering amplitude:
\begin{equation}
f(\theta) = \frac{1}{2ik}\sum_{\ell=0}^{\infty} (2\ell+1) P_\ell(\cos\theta)\,(\eta_\ell – 1).
\tag{9}
\end{equation}

Scattering Cross Section

The incident plane wave \(\psi_{\text{inc}} = e^{ikz}\) is normalised such that there is only one particle per unit volume.
Hence, the incident flux \(j_{\text{inc}}\), which is the number of particles crossing unit area per unit time, is
\begin{equation}
j_{\text{inc}} = v,
\end{equation}
where \(v\) is the velocity of the incident particles.

The probability of scattering into a solid angle \(d\Omega\) is expressed in terms of the \(\textbf{differential scattering cross section}\) as
\begin{equation}
w(\theta) = \frac{d\sigma}{d\Omega} d\Omega,
\end{equation}
where \({\sigma}{\Omega}\) is the elastic scattering cross section per unit solid angle in the direction \(\theta\).

If \(\theta\) is the number of scattered particles per unit volume per unit solid angle, then
\begin{equation}
I(\theta) = |f(\theta)|^2 \, d\Omega,
\end{equation}
so that the scattered flux becomes
\begin{equation}
j_{\text{sc}} = v |f(\theta)|^2 d\Omega.
\end{equation}

Comparing the scattered flux with the classical definition, we identify
\begin{equation}
\frac{d\sigma}{d\Omega} = |f(\theta)|^2.
\end{equation}

Scattering Amplitude in Partial Waves

We can expand \(f(\theta)\) as a series in Legendre polynomials:
\begin{equation}
f(\theta) = \sum_{\ell=0}^\infty f_\ell(\theta),
\end{equation}
where \(f_\ell(\theta)\) represents the scattering amplitude for the \(\ell\)-th partial wave.

From earlier results,
\begin{equation}
f(\theta) = \frac{1}{2ik} \sum_{\ell=0}^\infty (2\ell+1)(\eta_\ell – 1) P_\ell(\cos\theta),
\end{equation}
where \(\eta_\ell = e^{2i\delta_\ell}\) is the phase factor for the \(\ell\)-th partial wave.

Elastic Scattering Cross Section

The differential elastic scattering cross section for the \(el\)-th partial wave is
\(dv{\sigma_\ell}{\Omega} = \left| f_\ell(\theta) \right|^2
= \left| \frac{1}{2ik}(2\ell+1)(\eta_\ell – 1) P_\ell(\cos\theta) \right|^2\)

Integrating over all solid angles, we obtain the total elastic scattering cross section for the \(ell\)-th partial wave:
\(sigma_\ell^{\text{sc}} = \int \left| f_\ell(\theta) \right|^2 d\Omega.\)

Using the orthogonality relation of Legendre polynomials,
\(int_0^\pi P_\ell(\cos\theta)^2 \sin\theta \, d\theta = \frac{2}{2\ell+1},\)
we get
\(sigma_\ell^{\text{sc}} = \frac{\pi}{k^2}(2\ell+1)\, |1-\eta_\ell|^2.\)