Higher Born and Form Factors

Second Born Approximation

The first Born approximation replaces $\psi$ by $\phi$ in the Lippmann-Schwinger equation. To go to second order, we iterate once more, substituting the first-order solution back into the integral equation:

In position representation, the second-order correction to the scattering amplitude is:

The second-order term represents double scattering: the particle interacts with the potential at $\vec{r}_2$, propagates via the free Green's function to $\vec{r}_1$, and scatters again. Each higher order adds an additional scattering event.

The Born Series

Iterating the Lippmann-Schwinger equation to all orders generates the Born series. It is most elegantly expressed using the transition operator $T$, defined by$V|\psi\rangle = T|\phi\rangle$:

This is a geometric series in the operator $G_0 V$, which formally sums to:

The scattering amplitude is the matrix element of $T$ between plane-wave states:

The Born series converges when the operator norm of $G_0 V$ is less than unity, which corresponds to the weak-potential condition. For strong potentials, the series diverges and alternative methods (partial waves, variational, numerical) are required.

The Optical Theorem

One of the most fundamental results in scattering theory connects the total cross section to the forward scattering amplitude. The optical theorem states:

Derivation from unitarity: Conservation of probability requires that the total flux removed from the incident beam equals the total scattered flux. The incident plane wave and forward-scattered wave interfere, and this interference is responsible for the attenuation of the beam. Formally, from the $S$-matrix unitarity$S^\dagger S = 1$:

The optical theorem provides a powerful consistency check: any approximate scattering amplitude should satisfy it (at least approximately). The first Born approximation gives a real$f(\theta)$ for a real potential, so $\text{Im}\,f(0) = 0$ at first order—the optical theorem is satisfied only to zeroth order. The second Born approximation produces an imaginary part that restores consistency to first order in$V$.

Nuclear Form Factors

For scattering from an extended target with charge distribution $\rho(\vec{r})$, the Born amplitude factorizes into a point-particle amplitude times a form factor:

The form factor is the Fourier transform of the charge distribution, normalized so that$F(0) = 1$ (total charge). The measured cross section becomes:

By measuring the differential cross section at various angles (and hence various $q$), one maps out $|F(q)|^2$ and can invert the Fourier transform to determine the charge distribution $\rho(r)$.

Electron-Nucleus Scattering

Electron scattering experiments pioneered by Hofstadter in the 1950s revealed nuclear charge distributions. For a uniform sphere of radius $R$:

The form factor has zeros at $qR = 4.49, 7.73, \ldots$, producing characteristic diffraction minima. From the positions of these minima, one extracts the nuclear radius. The empirical result is:

where $A$ is the mass number. At small $q$, the form factor can be expanded to extract the mean-square charge radius:

Inelastic Form Factors and Deep Inelastic Scattering

The form factor concept extends to inelastic scattering. If the target changes from initial state $|i\rangle$ to final state $|f\rangle$, the inelastic form factor is the matrix element:

where the sum runs over all charged constituents. In deep inelastic electron-proton scattering, the proton breaks apart and one observes structure functions$W_1(q^2,\nu)$ and $W_2(q^2,\nu)$ that depend on both the momentum transfer and the energy transfer $\nu$. Bjorken scaling of these structure functions provided the first evidence for point-like constituents (quarks) inside the proton.

Mott Scattering

For relativistic electrons scattering from a nucleus, the Rutherford formula requires a correction due to the electron's spin. The Mott cross section for an electron (spin-1/2, velocity $v$) from a point nucleus of charge $Ze$ is:

The $\cos^2(\theta/2)$ factor arises from the spin-orbit interaction. In the non-relativistic limit ($v/c \to 0$), this reduces to the Rutherford formula. The full cross section for a finite nucleus combines Mott with the form factor:

Key Concepts Summary