Nuclear Radius

Rutherford Scattering Experiment

Rutherford first determined the radius of the nucleus in his famous alpha particle scattering experiment. The answer he arrived at was quite a surprise.

Change in energy

The alpha particles in Rutherford’s experiment had a kinetic energy of 5.3 MeV.
The gold nucleus in the experiment does work on the alpha particle to slow it down. Eventually, the alpha particle has zero kinetic energy (KE) and then is repelled.

Distance of closest approach

The closest point of approach is where the KE lost = electrostatic potential energy gained.
The formula for electrostatic potential energy is:
- PE $= \frac{Qq}{4\pi \epsilon_0r}$ where r is the distance of separation of charges Q and q.

Distance of closest approach 2

So:
- $(5.3 \times 10^6) \times (1.6 \times 10^{-19}) = \frac{(79 \times 1.6 \times 10^{-19})(2 \times 1.6 \times 10^{-19})}{4\pi \epsilon_0 r}$
Which gives:
- $r = \frac{79 \times 2 \times (1.6 \times 10^{-19})}{(5.3 \times 10^6) \times (4\pi \epsilon_0)} = 4.3 \times 10^{-14}$ m
This is an over-estimate for the radius of the nucleus. Rutherford was convinced that the alpha particles had not come into contact with the gold nucleus.

Electron Diffraction Experiment

Electron diffraction experiments are an alternative method of finding the radius of the nucleus.

De Broglie wavelength

The electron can behave like a wave, with de Broglie wavelength given by:
- $\lambda =\frac{h}{p}$
  - Where h is the Planck constant and p is the momentum of the electron.
The de Broglie wavelength is a wavelength associated with matter based on its mass and momentum.

Intensity of electrons

The graph of intensity of diffracted electrons against diffraction angle is shown here.

Explanation of graph

The angle θ, where the intensity is a minimum, can be explained by assuming that the electrons have diffracted around a spherical object (i.e. the nucleus).
The diameter of the nucleus, d, can be found using the equation:
- $\sin \theta = 1.22 \frac{\lambda}{d}$

Liquid Drop Model of the Nucleus

We assume the nucleus is of constant density and spherical in the liquid drop model.

Principle

The liquid drop model of the nucleus means that the volume of the nucleus is directly proportional to the number of nucleons present in that nucleus and that the nucleus is spherical.
$V = kA = \frac43 \pi r^3$
Where r is the radius of the nucleus with mass number A and volume V.

Radius

Using the previous equation:
- $r = \sqrt[3]{\frac{3kA}{4\pi}}$
Or more conveniently:
- $r=r_0A^{1/3}$
- Where r₀ is the radius of the proton.

Validity of model

The idea on the previous slide and the subsequent experimental confirmation is evidence that the liquid drop model is correct.
The consequence of this is that nuclear material has a constant density.

1Space, Time & Motion

1.1Motion

1.1.1Motion Basics

1.1.2Graphs of Motion

1.1.3Uniform Acceleration Equations

1.1.4Projectile Motion

1.1.5Bouncing Ball Example

1.2Forces

1.2.1Friction

1.2.2Terminal Speed

1.2.3Newton's Laws

1.2.4Exam-Style Question - Forces

1.2.5End of Topic Test - Newton's Third Law

1.3Momentum & Impulse

1.3.1Momentum

1.3.2Momentum 2

1.3.3End of Topic Test - Newton's Laws & Momentum

1.4Work, Energy & Power

1.4.1Work & Energy

1.4.2Conservation of Energy

1.4.3Power & Efficiency

1.4.4IB Multiple Choice- Work, Energy & Power

2The Particulate Nature of Matter

2.1Thermal Concepts

2.1.1Molecular Kinetic Theory Model

2.1.2Molecular Kinetic Theory Model 2

2.1.3Thermal Energy Transfer

2.1.4Thermal Energy Transfer Experiments

2.1.5Thermal Equilibrium

2.1.6Thermal Conductivity

2.1.7Heat Energy Transfer

2.2Modelling a Gas

2.2.1Ideal Gases

2.2.2Ideal Gases 2

2.2.3End of Topic Test - Thermal Energy & Ideal Gases

2.2.4Exam-Style Question - Ideal Gases

3Wave Behaviour

3.1Oscillations

3.1.1Progressive Waves

3.1.2Simple Harmonic Motion

3.2Travelling Waves

3.2.1Wave Speed & Phase Difference

3.2.2Longitudinal & Transverse Waves

3.2.3Electromagnetic Waves

3.2.4Electromagnetic Waves 2

3.2.5Sound Waves

3.3Wave Characteristics

3.3.1Polarisation

3.4Wave Behaviour

3.4.1Reflection & Absorption

3.4.2Reflection of Light

3.4.3Refraction

3.4.4Refractive index

3.5Standing Waves

3.5.1Standing Waves

3.5.2Standing Waves 2

3.5.3End of Topic Test - Standing Waves

3.5.4IB Multiple Choice - Waves

3.6Simple Harmonic Motion

3.6.1Simple Harmonic Motion

3.6.2Simple Harmonic Systems

3.6.3Energy in Simple Harmonic Motion

3.6.4End of Topic Test - Simple Harmonic Motion

3.6.5End of Topic Test - Simple Harmonic Motion

3.7Single Slit Diffraction

3.7.1Diffraction

3.8Interference

3.8.1Interference

3.8.2Interference 2

3.8.3Diffraction Gratings

3.8.4End of Topic Test - Diffraction

3.9Doppler Effect

3.9.1Doppler Effect

4Fields

4.1Circular Motion

4.1.1Circular Motion

4.1.2Centripetal Force

4.1.3IB Multiple Choice - Circular Motion

4.2Newton's Law of Gravitation

4.2.1Newton's Law

4.2.2Orbits of Planets & Satellites

4.2.3Escape Velocity & Synchronous Orbits

4.2.4End of Topic Test - Gravitational Fields

4.3Fields

4.3.1Fields

4.3.2Gravitational Field Strength

4.3.3Gravitational Potential

4.3.4Electric Field Strength

4.3.5Electrostatic & Gravitational Forces

4.3.6Electric Potential

4.3.7IB Multiple Choice - Gravitational Fields

4.4Fields at Work

4.4.1Radial Field Strength

4.4.2Orbits of Planets & Satellites

4.4.3Escape Velocity & Synchronous Orbits

4.4.4End of Topic Test - Gravitational Fields

4.4.5Uniform Electric Fields

4.4.6IB Multiple Choice - Electric Field & Potential

4.5Electric Fields

4.5.1Electric Charge

4.5.2Conservation of Charge

4.5.3Coulomb's Law

4.5.4Current

4.5.5Potential Difference

4.5.6IB Multiple Choice - Electric Charge

4.6Magnetic Effect of Electric Currents

4.6.1Magnetism

4.6.2Magnetic Field

4.6.3Magnetic Force

4.6.4Moving Charges in a Magnetic Field

4.6.5IB Multiple Choice - Magnetism

4.7Heating Effect of Currents

4.7.1Circuits

4.7.2Resistance

4.7.3Power and Conservation

4.7.4IB Multiple Choice - DC Circuits

4.8Electromagnetic Induction

4.8.1Electromagnetic Induction

4.8.2Electromagnetic Induction 2

4.8.3Magnetic Flux & Flux Linkage

4.8.4End of Topic Test - Electromagnetic Induction

4.9Power Generation & Transmission

4.9.1Alternating Currents

4.9.2Operation of a Transformer

4.10Capacitance

4.10.1Capacitance

4.10.2Parallel Plate Capacitor

4.10.3Energy Stored by a Capacitor

4.10.4Capacitor Discharge

4.10.5Capacitor Charge

4.10.6Magnetic Flux Density

4.10.7End of Topic Test - Capacitance & Flux Density

5Nuclear & Quantum Physics

5.1Discrete Energy & Radioactivity

5.1.1Atomic Spectra

5.1.2Atomic Structure

5.1.3Photons

5.1.4Discrete Energy Levels

5.1.5Alpha, Beta & Gamma Radiation

5.1.6Scientific Practices - Unstable Nuclei

5.1.7Radioactive Decay

5.1.8IB Multiple Choice - Atomic Physics

5.2Nuclear Reactions

5.2.1Mass & Energy

5.2.2Nuclear Fission

5.3The Interaction of Matter with Radiation

5.3.1Particles, Antiparticles & Photons

5.3.2Particle Interactions

5.3.3Classification of Particles

5.3.4Wave-Particle Duality

5.3.5Wave Functions & Probability

5.3.6IB Multiple Choice - Quantum Physics

5.4Nuclear Physics

5.4.1Nuclear Instability & Radioactive Decay

5.4.2Nuclear Radius

5.4.3IB Multiple Choice - Nuclear Physics

6Measurements

6.1Measurements & Errors

6.1.1Fundamental Units

6.1.2SI Prefixes, Standard Form & Converting Units

6.1.3End of Topic Test - Units & Prefixes

6.2Uncertainties & Errors

6.2.1Limitation of Physical Measurements

6.2.2Uncertainty

6.2.3Estimation

6.2.4End of Topic Test - Measurements & Errors

6.3Vectors & Scalars

6.3.1Scalars & Vectors

6.3.2Vector Problems

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