Alpha Particle Scattering — Revision Notes

⚡ 30-Second Revision

Alpha Particles: — $^4_2\text{He}^{2+}$, charge $+2e$, mass $approx 4,\text{amu}$.
Key Observations:

* Most pass undeflected ($>99.8$). * Few deflected at small angles. * Very few (1 in 8000-20000) deflected at large angles ($>90^circ$, some $180^circ$).

Conclusions:

* Atom mostly empty space. * Dense, positively charged nucleus at center. * Nucleus contains almost all mass.

Distance of Closest Approach ($r_0$): — Minimum distance in head-on collision.

$$r_0 = \frac{1}{4piepsilon_0} \frac{2Ze^2}{K}$$

where $K$ is initial kinetic energy, $Z$ is atomic number of target. $r_0 propto \frac{1}{K}$, $r_0 propto Z$.

Impact Parameter ($b$): — Perpendicular distance from nucleus to initial velocity vector.

$$b = \frac{kZe^2 cot(\theta/2)}{K}$$

where $k = \frac{1}{4piepsilon_0}$. Smaller $b implies$ larger $heta$.

Scattering Formula (Qualitative): — Number of scattered particles $N(\theta) propto \frac{1}{sin^4(\theta/2)}$.
Rutherford Model Limitations:

* Cannot explain atomic stability (electron spiral). * Cannot explain discrete line spectra (predicts continuous).

2-Minute Revision

Alpha Particle Scattering, also known as Rutherford's Gold Foil experiment, was a landmark experiment that revealed the nuclear structure of the atom. High-energy alpha particles (helium nuclei, $+2e$ charge) were directed at a thin gold foil.

The key observations were: most particles passed straight through, indicating atoms are mostly empty space; a small fraction were deflected at small angles, suggesting a positive charge within; and a very rare few were deflected at large angles, even bouncing back, which proved the existence of a tiny, dense, positively charged nucleus at the atom's center.

This led to Rutherford's nuclear model, where electrons orbit the nucleus. Quantitatively, the distance of closest approach ($r_0$) for a head-on collision is given by $r_0 = \frac{1}{4piepsilon_0} \frac{2Ze^2}{K}$, showing $r_0$ is inversely proportional to kinetic energy ($K$) and directly proportional to the atomic number ($Z$).

The impact parameter ($b$) determines the scattering angle ($heta$), with smaller $b$ leading to larger $heta$. While revolutionary, Rutherford's model had limitations: it couldn't explain atomic stability (electrons should spiral into the nucleus) or the discrete line spectra of atoms (it predicted continuous spectra).

These limitations paved the way for Bohr's quantum model.

5-Minute Revision

The Alpha Particle Scattering experiment, conducted by Rutherford, Geiger, and Marsden, fundamentally changed our understanding of the atom. Before this, Thomson's 'plum pudding' model was prevalent, suggesting a uniformly positive atom with embedded electrons. Rutherford's experiment used high-speed alpha particles (positively charged helium nuclei, $2e$) as probes, directed at an ultra-thin gold foil. The observations were critical:

1
Most alpha particles passed straight through undeflected — (over 99.8%), implying atoms are largely empty space.
2
A small fraction were deflected through small angles, indicating some positive charge within the atom.
3
A very small fraction (about 1 in 8000) were deflected through large angles, some even bouncing back, which was the most surprising. This led to the conclusion that the entire positive charge and almost all the mass of the atom are concentrated in an extremely tiny, dense central region called the nucleus. Electrons were proposed to orbit this nucleus.

Key quantitative aspects include:

Distance of Closest Approach ($r_0$): — For a head-on collision, the alpha particle's initial kinetic energy ($K$) is fully converted into electrostatic potential energy at $r_0$. Thus, $K = \frac{1}{4piepsilon_0} \frac{(2e)(Ze)}{r_0}$, which gives $r_0 = \frac{1}{4piepsilon_0} \frac{2Ze^2}{K}$. This formula shows $r_0 propto 1/K$ and $r_0 propto Z$. For example, if an alpha particle with $5,\text{MeV}$ kinetic energy approaches a gold nucleus ($Z=79$), $r_0 approx 4.5 \times 10^{-14},\text{m}$. This provides an upper limit for the nuclear radius.
Impact Parameter ($b$): — This is the perpendicular distance from the center of the nucleus to the initial velocity vector of the alpha particle. It determines the scattering angle ($heta$). A smaller $b$ means a closer approach, stronger repulsion, and thus a larger $heta$. The relationship is $b propto cot(\theta/2)$.
Rutherford Scattering Formula (qualitative): — The number of alpha particles scattered at an angle $heta$ is proportional to $1/sin^4(\theta/2)$, explaining the rapid decrease in particles at larger angles.

Despite its success, Rutherford's model had two major limitations: it couldn't explain the stability of atoms (orbiting electrons should radiate energy and spiral into the nucleus) and it couldn't account for the discrete line spectra observed from atoms (it predicted a continuous spectrum). These limitations were addressed by Bohr's quantum model.

Prelims Revision Notes

Alpha Particle Scattering (Rutherford's Experiment)

1. Experimental Setup:

Source: — Radioactive source (e.g., Radium) emitting high-energy alpha particles ($^4_2\text{He}^{2+}$, charge $+2e$).
Target: — Very thin gold foil ($approx 10^{-7},\text{m}$ thick, high $Z=79$).
Detector: — Movable ZnS screen (scintillation counter) to detect scattered alpha particles.

2. Key Observations:

Most (>$99.8%$): — Passed straight through undeflected. (Atom is mostly empty space).
Few: — Deflected through small angles. (Positive charge present, causing repulsion).
Very few (1 in 8000-20000): — Deflected through large angles ($>90^circ$, some $180^circ$). (Positive charge and mass concentrated in a tiny, dense nucleus).

3. Rutherford's Nuclear Model:

Atom has a tiny, dense, positively charged nucleus at its center.
Almost all mass of the atom is concentrated in the nucleus.
Electrons orbit the nucleus in a vast empty space.
Atom is electrically neutral.

4. Quantitative Concepts:

Distance of Closest Approach ($r_0$): — For a head-on collision ($b=0$, $heta=180^circ$), initial kinetic energy ($K$) is converted to electrostatic potential energy ($U$).

$$K = U = \frac{1}{4piepsilon_0} \frac{(2e)(Ze)}{r_0}$$

$$r_0 = \frac{1}{4piepsilon_0} \frac{2Ze^2}{K}$$

* $r_0 propto \frac{1}{K}$ (Higher energy $implies$ smaller $r_0$). * $r_0 propto Z$ (Higher atomic number $implies$ larger $r_0$). * Order of magnitude: $r_0 approx 10^{-14}$ to $10^{-15},\text{m}$ (nuclear size).

Impact Parameter ($b$): — Perpendicular distance of alpha particle's initial velocity vector from nucleus center.

$$b = \frac{kZe^2 cot(\theta/2)}{K}$$

where $k = \frac{1}{4piepsilon_0}$. * Large $b implies$ small $heta$ (weak interaction). * Small $b implies$ large $heta$ (strong interaction). * $b=0 implies \theta=180^circ$ (head-on collision).

**Number of Scattered Particles ($N(\theta)$):**

$$N(\theta) propto \frac{1}{sin^4(\theta/2)}$$

(Rapid decrease in particles at larger angles).

5. Limitations of Rutherford's Model (Classical Physics):

Atomic Stability: — Orbiting electrons are accelerating, should continuously radiate energy, spiral into nucleus, making atom unstable. (Contradicts stable atoms).
Line Spectra: — Continuous radiation implies continuous spectrum. (Contradicts discrete line spectra observed from atoms).

6. Comparison with Thomson's Model:

Thomson: — Uniform positive sphere, electrons embedded, mass/charge distributed. Predicted only small deflections.
Rutherford: — Concentrated positive nucleus, electrons orbit, mass/charge concentrated. Explained large deflections.

Vyyuha Quick Recall

To remember Rutherford's observations and conclusions: Most Straight, Some Small, Very Few Back.

Most Straight: Most alpha particles passed straight through (atom is Mostly Space).
Some Small: Some deflected at Small angles (positive charge is Somewhere).
Very Few Back: Very Few bounced Back (dense, positive Nucleus at center).