Case Study 17.1 — Rutherford Scattering Revisited: From Gold Foil to Modern Nuclear Reactions

Historical Context

In 1909, Hans Geiger and Ernest Marsden, working under Ernest Rutherford's direction at the University of Manchester, performed what is now one of the most famous experiments in the history of physics. They directed a beam of alpha particles from a radium source ($E_\alpha \approx 5.5\,\text{MeV}$) at a thin gold foil and counted scintillations on a zinc sulfide screen at various angles. The expectation, based on J.J. Thomson's "plum pudding" model of the atom, was that all alpha particles would pass through with only small deflections. Instead, about 1 in 8000 alpha particles scattered at angles greater than $90°$ — some nearly straight back.

Rutherford's analysis of this data — published in 1911 — established that the atom contains a tiny, massive, positively charged nucleus. The analysis rested on two results derived in this chapter: the relationship between impact parameter and scattering angle ($b = a\cot(\theta/2)$) and the differential cross section ($d\sigma/d\Omega \propto 1/\sin^4(\theta/2)$).

The Original Data

Geiger and Marsden systematically verified the Rutherford prediction in 1913 with improved apparatus. Their data for 5.5 MeV alphas on gold at angles from $15°$ to $150°$ showed:

The product $N \sin^4(\theta/2)$ is approximately constant across two orders of magnitude in count rate — a striking confirmation of the $1/\sin^4(\theta/2)$ dependence.

Verifying the Rutherford Formula

Three specific predictions of the Rutherford formula were tested by Geiger and Marsden:

1. Angular dependence: $d\sigma/d\Omega \propto 1/\sin^4(\theta/2)$. Verified across the range $15°$--$150°$ as shown in the table above. The product $N\sin^4(\theta/2)$ was constant to within 10% across a factor of 4,000 in count rate.

2. Energy dependence: $d\sigma/d\Omega \propto 1/E^2$. Verified by using alpha sources of different energies (RaC' at 7.7 MeV, Ra at 4.8 MeV) and checking that the count rates scaled as $1/E^2$ at fixed angle.

3. Charge dependence: $d\sigma/d\Omega \propto Z^2$. Verified by comparing gold ($Z = 79$), silver ($Z = 47$), copper ($Z = 29$), and aluminum ($Z = 13$) foils. The measured ratios matched $Z^2$ to about 20%.

These three verifications, taken together, left no doubt that the scattering was governed by a $1/r$ Coulomb potential from a point-like charge concentrated in a region much smaller than the atom. Rutherford estimated the nuclear radius to be less than $3 \times 10^{-14}\,\text{m}$ — about 10,000 times smaller than the atom.

From Rutherford to Nuclear Size Measurements

The Rutherford formula assumes point-charge scattering. When the projectile penetrates inside the nuclear charge distribution, the scattering deviates from the Rutherford prediction. The ratio $\sigma_{\text{measured}}/\sigma_{\text{Ruth}}$ drops below unity at angles where the distance of closest approach is comparable to the nuclear radius.

For head-on scattering ($\theta = 180°$), the distance of closest approach is:

$$d_0 = \frac{2 k z_1 z_2 e^2}{E_{\text{CM}}} = \frac{z_1 z_2 e^2}{2\pi\epsilon_0 E_{\text{CM}}}$$

At $E_\alpha = 5.5\,\text{MeV}$:

$$d_0 = \frac{2 \times 79 \times 1.440\,\text{MeV}\cdot\text{fm}}{5.5 \times (197/201)} = \frac{227.5}{5.35} \approx 42.5\,\text{fm}$$

Since the gold nuclear radius is $R_{\text{Au}} \approx 1.2 \times 197^{1/3} \approx 7.0\,\text{fm}$, the 5.5 MeV alphas never come close to the nucleus — they are always in the Coulomb regime. This is why Rutherford's formula worked so perfectly.

To probe the nuclear surface, one needs $d_0 \approx R$, which requires:

$$E_{\text{CM}} \approx \frac{2 k z_1 z_2 e^2}{R} = \frac{2 \times 79 \times 1.440}{7.0} \approx 32.5\,\text{MeV}$$

corresponding to a lab alpha energy of about $33.2\,\text{MeV}$.

Modern Application: Rutherford Backscattering Spectrometry (RBS)

The Rutherford scattering formula is not merely of historical interest — it is the basis of Rutherford Backscattering Spectrometry (RBS), one of the most widely used techniques in materials science. In RBS:

1. A beam of light ions (typically 1--3 MeV ${}^{4}\text{He}^{+}$ or ${}^{1}\text{H}^{+}$) is directed at a sample.
2. Backscattered ions are detected at a large angle ($\theta \approx 160°$--$170°$) with an energy-resolving silicon detector.
3. The energy of the backscattered ion identifies the target element (via the kinematic factor), and the Rutherford cross section provides an absolute scale for composition analysis without external standards.

The kinematic factor for elastic backscattering at $\theta = 180°$ is:

$$K = \left(\frac{M_a - M_b}{M_a + M_b}\right)^2$$

For 2 MeV ${}^{4}\text{He}$ on various elements:

The clear separation in backscattered energy allows identification of different elements in a thin-film sample. The Rutherford cross section, known absolutely from first principles, allows quantitative composition determination without calibration standards — a powerful analytical advantage.

When Rutherford Fails: Nuclear Scattering

At sufficiently high energies or for sufficiently light targets, the projectile penetrates the Coulomb barrier and enters the nuclear interior. The elastic scattering cross section then deviates from the Rutherford prediction, showing:

• Interference oscillations: At intermediate angles, the Coulomb (long-range) and nuclear (short-range) scattering amplitudes interfere, producing oscillations in $\sigma/\sigma_{\text{Ruth}}$. The pattern depends sensitively on the nuclear potential.

• A diffraction pattern: At larger angles, the nuclear scattering dominates, producing Fraunhofer-like diffraction minima whose positions reveal the nuclear radius.

• Reaction cross sections: Above the barrier, nuclear reactions (inelastic scattering, transfer, fusion) remove flux from the elastic channel, reducing $\sigma_{\text{el}}$ below $\sigma_{\text{Ruth}}$.

These deviations from Rutherford scattering are precisely the data that constrain the optical model parameters discussed in Section 17.9.

Quantitative Analysis: The Coulomb-Nuclear Interference

The transition from Rutherford to nuclear scattering is quantified by the ratio $\sigma_{\text{exp}}/\sigma_{\text{Ruth}}$. For alpha particles on ${}^{208}\text{Pb}$, this ratio remains unity (within experimental precision) up to a critical angle called the grazing angle $\theta_{\text{gr}}$, beyond which it drops sharply. The grazing angle corresponds to the impact parameter equal to the sum of the projectile and target radii:

$$b_{\text{gr}} = R_a + R_b = r_0(A_a^{1/3} + A_b^{1/3})$$

For 40 MeV alphas on ${}^{208}\text{Pb}$: $b_{\text{gr}} \approx 1.25(208^{1/3} + 4^{1/3}) \approx 1.25(5.93 + 1.59) \approx 9.4\,\text{fm}$. Using $b = a\cot(\theta/2)$ with $a = 2 \times 82 \times 1.440 / (2 \times 40 \times 208/212) \approx 3.0\,\text{fm}$:

$$\theta_{\text{gr}} \approx 2\arctan(a/b_{\text{gr}}) \approx 2\arctan(3.0/9.4) \approx 35°$$

At angles beyond approximately $35°$, the alpha particle reaches the nuclear surface, nuclear absorption removes flux, and the elastic cross section falls below the Rutherford prediction. This "Rutherford-to-nuclear" transition is among the most cleanly observed phenomena in nuclear physics and provides an independent determination of nuclear radii.

At the Coulomb-nuclear interference angle, the nuclear scattering amplitude $f_N$ is comparable in magnitude to the Coulomb amplitude $f_C$. The two amplitudes interfere:

$$\frac{d\sigma}{d\Omega} = |f_C + f_N|^2 = |f_C|^2 + 2\text{Re}(f_C^* f_N) + |f_N|^2$$

The interference term $2\text{Re}(f_C^* f_N)$ is sensitive to the phase of the nuclear amplitude, which in turn depends on the optical potential. This makes Coulomb-nuclear interference measurements a sensitive probe of the real part of the nuclear potential at large radii — the nuclear "halo" region that is difficult to access by other means.

The Legacy: From Rutherford to FRIB

The intellectual line from Rutherford's gold-foil experiment to modern radioactive beam facilities is direct. Rutherford used scattering to reveal the existence of the nucleus; we now use the same technique — elastic scattering measured as deviations from the Rutherford formula — to study the structure of the most exotic nuclei produced at facilities like FRIB, RIBF (RIKEN), and FAIR (GSI).

The key innovation is inverse kinematics: instead of a light beam on a heavy target, one accelerates a heavy radioactive beam onto a light target (hydrogen or helium). The Rutherford cross section is the same (it depends only on the product $z_1 z_2$ and the CM energy), but the kinematics are different — the heavy ejectile is focused into a narrow forward cone, requiring specialized detection systems.

A striking recent example is the elastic scattering of ${}^{11}\text{Li}$ (a halo nucleus with a diffuse neutron distribution extending far beyond the normal nuclear radius) from protons at TRIUMF. The measured angular distribution shows deviations from the optical model prediction for a normal nucleus at much smaller angles than expected, directly revealing the anomalously large matter radius of this exotic system.

Putting It Together: What Rutherford Scattering Measures and What It Cannot

Rutherford scattering, for all its power, provides specific and limited information:

What Rutherford scattering measures well: - The product $z_1 z_2$ (from the absolute cross section at known energy and angle) - The charge of an unknown target (RBS, when the projectile is known) - An upper limit on nuclear size (the distance of closest approach at the energy where deviations begin) - A normalization standard for other scattering measurements (since the Rutherford cross section is known absolutely from first principles, with no adjustable parameters)

What Rutherford scattering cannot measure: - Nuclear structure beyond the charge distribution (neutron distributions are invisible to Coulomb scattering) - Nuclear excited states (purely elastic — no internal degrees of freedom are excited) - Short-range nuclear force properties (the Coulomb potential dominates at all distances accessible to the projectile, until the nuclear surface is reached) - Spin-dependent effects (the Rutherford formula is spin-independent)

The experimental program of nuclear physics over the past century has been, in many ways, the story of going beyond Rutherford — using nuclear reactions (transfer, knockout, charge-exchange, inelastic scattering) to access the information that elastic Coulomb scattering cannot provide. Yet Rutherford scattering remains indispensable as a calibration standard, a diagnostic tool, and the starting point for understanding every nuclear scattering experiment.

Key Numbers to Remember

Discussion Questions

1. Geiger and Marsden counted scintillations by eye through a microscope in a darkened room. Each measurement took hours. What systematic errors would you worry about in their experiment?

2. RBS uses the Rutherford cross section as an absolute standard. Under what conditions might the Rutherford formula be unreliable for RBS analysis? (Hint: consider light elements and/or high beam energies.)

3. The Rutherford formula predicts $d\sigma/d\Omega \to \infty$ as $\theta \to 0$. In practice, what limits the cross section at small angles? Discuss both atomic (electron screening) and experimental (finite beam size, minimum detector angle) effects.

4. Modern electron scattering experiments at facilities like Jefferson Lab (USA) and MAMI (Germany) use electrons rather than alpha particles to probe nuclear structure. What advantages do electrons have over alpha particles for measuring nuclear charge distributions?