Estimating Nuclear Size Using Scattering Data ⚛️✨

The size of the nucleus can be estimated using alpha particle scattering experiments, particularly by analyzing:

1. The kinetic energy of alpha particles as they approach the nucleus.
2. The Coulomb force between the alpha particle and the nucleus.
3. The distance of closest approach, where the alpha particle momentarily stops before being repelled.

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1. Concept: Distance of Closest Approach

When a high-energy alpha particle (

$\alpha$

) approaches a nucleus, it slows down due to Coulomb repulsion and eventually stops at a minimum distance, before being scattered back.

At this turning point, the initial kinetic energy (

$KE$

) of the alpha particle is completely converted into electrostatic potential energy (

$PE$

):

$KE = PE$

$\frac{1}{4 \pi \epsilon_0} \frac{Z e \cdot 2e}{r_{\text{min}}} = KE$

where:

• $KE$ = Initial kinetic energy of the alpha particle.
• $Z e$ = Charge of the nucleus ($Ze$, where$Z$ is the atomic number).
• $2e$ = Charge of the alpha particle ($2 \times 1.6 \times 10^{-19}$ C).
• $r_{\text{min}}$​ = Distance of closest approach (estimate of nuclear size).
• $\frac{1}{4\pi\epsilon_0} = 9.0 \times 10^9$ (Coulomb’s constant, in N·m²/C²).

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2. Example Calculation: Estimating Nuclear Size for Gold (

$^{197}_{79}Au$

​Au)

🔹 Given Data:

• Alpha particle energy:$KE = 5.0$KE=5.0 MeV =$8.0 \times 10^{-13}$ J.
• Gold nucleus charge:$Z = 79$, so$Ze = 79 \times (1.6 \times 10^{-19})$ C.

🔹 Step 1: Solve for

$r_{\text{min}}$

​

$r_{\text{min}} = \frac{1}{4 \pi \epsilon_0} \frac{(79e) (2e)}{KE}$

​

$r_{\text{min}} = \frac{(9.0 \times 10^9) (79 \times 1.6 \times 10^{-19}) (2 \times 1.6 \times 10^{-19})}{8.0 \times 10^{-13}}$

​

$r_{\text{min}} \approx \frac{(9.0 \times 10^9) (2.53 \times 10^{-35})}{8.0 \times 10^{-13}}$

$r_{\text{min}} \approx 2.8 \times 10^{-15} \text{ m} = 2.8 \text{ fm}$

✅ Result: The estimated nuclear size for gold is

$r_{\text{min}} \approx 2.8$

 femtometers (fm).

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3. Implications of the Calculation

• This confirms that nuclei are extremely small (on the femtometer scale).

• The actual nuclear radius follows the empirical formula:

  $R = R_0 A^{1/3}$
  

  where

  $R_0 \approx 1.2$

   fm.

• For gold (

  $A = 197$

  ):

  $R_{\text{gold}} = 1.2 \times (197)^{1/3} \approx 7.0 \text{ fm}$

🔹 Comparing with our result

$r_{\text{min}} = 2.8$

 fm:

• The actual nuclear radius is slightly larger because the distance of closest approach is an upper limit, not an exact radius.
• The strong nuclear force affects the exact boundary of the nucleus.

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4. Summary: Key Findings

Concept	Explanation
Scattering Experiment	Alpha particles are repelled by the Coulomb force.
Distance of Closest Approach	Where KE is fully converted to electric potential energy.
Gold Nucleus Estimate	$r_{\text{min}} \approx 2.8$ fm, actual radius$\approx 7.0$ fm.
General Nuclear Size	Estimated using$R = R_0 A^{1/3}$, confirming nuclear femtometer scale.

 

Key Takeaways

✅ Scattering experiments provide a direct estimate of nuclear size.
✅ The nucleus is incredibly small (~1–10 fm), confirming its dense nature.
✅ These findings laid the foundation for nuclear physics, including fission and fusion reactions.

By analyzing alpha scattering, scientists uncovered the true size and structure of atomic nuclei, shaping modern nuclear science! ⚛️🚀✨

Activities:

• Numerical problems on closest approach
• Class activity: Derivation of the formula

Assessment:

• Worksheet on nuclear size calculations

Bohr Model

In the Bohr model, electrons orbit the nucleus in circular paths at specific quantized distances known as energy levels or shells. The energy of these levels is determined by the balance between the electrostatic attraction between the electron and the nucleus and the centrifugal force of the electron's motion. The formula to calculate the energy levels in the Bohr model for hydrogen is given by:

$E_n = -\dfrac{R_H}{n^2}$

where

$E_n$

 is the energy of the electron in the

$n$

nth energy level,

$R_H$

​ is the Rydberg constant for hydrogen, and

$n$

 is the principal quantum number representing the energy level (with

$n = 1, 2, 3, ...$

).

By plugging in the values for the Rydberg constant and the principal quantum number, one can calculate the discrete energy levels in the Bohr model for the hydrogen atom. These energy levels correspond to the different orbits or shells in which electrons can reside around the nucleus, each with a specific energy associated with it.

The Bohr model provides a simplified yet insightful framework for understanding the quantized nature of atomic energy levels and the spectral lines observed in the hydrogen atom. The discrete energy levels calculated in this model offer a glimpse into the cosmic symphony of electrons and nuclei, where quantum principles govern the radiant dance of particles in the celestial realm of atomic physics.

Angular Momentum

The existence of quantized energy levels and orbits in the Bohr model arises from the elegant quantization of angular momentum, a fundamental principle that governs the dynamics of particles in circular motion and shapes the ethereal landscape of atomic physics.

In the Bohr model, electrons orbit the nucleus in circular paths, akin to planets orbiting the sun, with specific quantized angular momenta determined by the integer multiples of Planck's constant divided by

$2\pi$

 (the reduced Planck constant, denoted as

$\hbar$

). The quantization of angular momentum in the Bohr model is expressed by the formula:

$L = n\hbar$

where

$L$

 is the angular momentum of the electron in its orbit,

$n$

 is an integer representing the quantum number, and

$\hbar$

 is the reduced Planck constant. This quantization condition imposes discrete values of angular momentum on the electron, leading to the discreet orbits and energy levels observed in the hydrogen atom.

The connection between angular momentum quantization and the existence of quantized energy levels in the Bohr model is profound. The discreet values of angular momentum correspond to specific orbits or shells in which electrons can reside around the nucleus, influencing the energy levels and spectral lines observed in atomic spectra. The quantized nature of angular momentum shapes the ethereal dance of electrons in the cosmic symphony of atomic structure, revealing the intricate interplay between quantum mechanics and classical dynamics in the radiant realm of the Bohr model.