8.1.8

Nuclear Radius

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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

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

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

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

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

Intensity of electrons

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

Explanation of graph

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

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

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

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.

1

Measurements & Errors

1.1

Measurements & Errors

1.1.1SI Base Units 1.1.2Combining SI Units 1.1.3Prefixes of Units 1.1.4Standard Form 1.1.5Converting Units 1.1.6End of Topic Test - Units & Prefixes 1.1.7Limitation of Physical Measurements 1.1.8Uncertainty 1.1.9Estimation 1.1.10End of Topic Test - Measurements & Errors 1.1.11Diagnostic Misconceptions - Converting between SI 1.1.12Diagnostic Misconceptions - SI & Area/Volume

2

Particles & Radiation

2.1

Particles

2.1.1Atomic Model 2.1.2Specific Charge, Protons & Neutron Numbers 2.1.3End of Topic Test - Atomic Model 2.1.4Isotopes 2.1.5Stable & Unstable Nuclei 2.1.6End of Topic Test - Isotopes & Nuclei 2.1.7A-A* (AO3/4) - Stable & Unstable Nuclei 2.1.8Particles, Antiparticles & Photons 2.1.9Particle Interactions 2.1.10Classification of Particles 2.1.11End of Topic Test - Particles & Interactions 2.1.12Quarks & Antiquarks 2.1.13Application of Conservation Laws 2.1.14End of Topic Test - Leptons & Quarks 2.1.15Exam-Style Question - Radioactive Decay

2.2

Electromagnetic Radiation & Quantum Phenomena

2.2.1The Photoelectric Effect 2.2.2The Photoelectric Effect Explanation 2.2.3End of Topic Test - The Photoelectric Effect 2.2.4Collisions of Electrons with Atoms 2.2.5Energy Levels & Photon Emission 2.2.6Wave-Particle Duality 2.2.7End of Topic Test - Absorption & Emission 2.2.8Diagnostic Misconceptions - Electron Volts 2.2.9Diagnostic Misconceptions - Converting eV & Joules

3

Waves

3.1

Progressive & Stationary Waves

3.1.1Progressive Waves 3.1.2Wave Speed & Phase Difference 3.1.3Longitudinal & Transverse Waves 3.1.4End of Topic Test - Progressive Waves 3.1.5Polarisation 3.1.6Stationary Waves 3.1.7Stationary Waves 2 3.1.8End of Topic Test - Polarisation & Stationary Wave 3.1.9A-A* (AO3/4) - Stationary Waves

3.2

Refraction, Diffraction & Interference

3.2.1Interference 3.2.2Interference 2 3.2.3End of Topic Test - Interference 3.2.4Diffraction 3.2.5Diffraction Gratings 3.2.6End of Topic Test - Diffraction 3.2.7Refraction at a Plane Surface 3.2.8Internal Reflection & Fibre Optics 3.2.9End of Topic Test - Refraction 3.2.10Exam-Style Question - Waves

4

Mechanics & Materials

4.1

Force, Energy & Momentum

4.1.1Scalars & Vectors 4.1.2Vector Problems 4.1.3End of Topic Test - Scalars & Vectors 4.1.4Moments 4.1.5Centre of Mass 4.1.6End of Topic Test - Moments & Centre of Mass 4.1.7Motion in a Straight Line 4.1.8Graphs of Motion 4.1.9Bouncing Ball Example 4.1.10End of Topic Test - Motion in a Straight Line 4.1.11Acceleration Due to Gravity 4.1.12Projectile Motion 4.1.13Friction 4.1.14Terminal Speed 4.1.15End of Topic Test - Acceleration Due to Gravity 4.1.16Newton's Laws 4.1.17Momentum 4.1.18Momentum 2 4.1.19End of Topic Test - Newton's Laws & Momentum 4.1.20A-A* (AO3/4) - Newton's Third Law 4.1.21Work & Energy 4.1.22Power & Efficiency 4.1.23Conservation of Energy 4.1.24End of Topic Test - Work, Energy & Power 4.1.25Exam-Style Question - Forces 4.1.26Diagnostic Misconceptions - Perpendicular Vectors 4.1.27Diagnostic Misconceptions - Weight Acts Downwards 4.1.28Diagnostic Misconceptions - Acceleration Direction 4.1.29Diagnostic Misconceptions - Stationary Objects 4.1.30Diagnostic Misconceptions - Action & Reaction 4.1.31Diagnostic Misconceptions - RF Direction 4.1.32Diagnostic Misconceptions - RF Acceleration 4.1.33Diagnostic Misconceptions - Change in Momentum

4.2

Materials

4.2.1Density 4.2.2Bulk Properties of Solids 4.2.3Energy in Materials 4.2.4Young Modulus 4.2.5End of Topic Test - Materials

5

Electricity

5.1

Current Electricity

5.1.1Basics of Electricity 5.1.2Current-Voltage Characteristics 5.1.3End of Topic Test - Basics of Electricity 5.1.4Resistivity 5.1.5Superconductivity 5.1.6A-A* (AO3/4) - Superconductivity 5.1.7End of Topic Test - Resistivity & Superconductors 5.1.8Circuits 5.1.9Power and Conservation 5.1.10Potential Divider 5.1.11Emf & Internal Resistance 5.1.12End of Topic Test - Power & Potential 5.1.13Exam-Style Question - Resistance 5.1.14Diagnostic Misconceptions - Constant Current 5.1.15Diagnostic Misconceptions - Potential Difference

6

Further Mechanics & Thermal Physics (A2 only)

7

Fields & Their Consequences (A2 only)

7.1

Fields (A2 only)

7.2

Gravitational Fields (A2 only)

7.2.1Newton's Law 7.2.2Gravitational Field Strength 7.2.3Gravitational Potential 7.2.4Orbits of Planets & Satellites 7.2.5Escape Velocity & Synchronous Orbits 7.2.6End of Topic Test - Gravitational Fields

7.3

Electric Fields (A2 only)

7.3.1Coulomb's Law 7.3.2Electric Field Strength 7.3.3Electric Field Strength 2 7.3.4Electric Potential 7.3.5End of Topic Test - Electric Fields 7.3.6A-A* (AO3/4) - Electric and Gravitational Field

7.4

Capacitance (A2 only)

7.4.1Capacitance 7.4.2Parallel Plate Capacitor 7.4.3Energy Stored by a Capacitor 7.4.4Capacitor Discharge 7.4.5Capacitor Charge

7.5

Magnetic Fields (A2 only)

7.5.1Magnetic Flux Density 7.5.2End of Topic Test - Capacitance & Flux Density 7.5.3Moving Charges in a Magnetic Field 7.5.4Magnetic Flux & Flux Linkage 7.5.5Electromagnetic Induction 7.5.6Electromagnetic Induction 2 7.5.7Alternating Currents 7.5.8Operation of a Transformer 7.5.9Magnetic Flux Density 7.5.10End of Topic Test - Electromagnetic Induction

8

Nuclear Physics (A2 only)

8.1

Radioactivity (A2 only)

8.1.1Rutherford Scattering 8.1.2Alpha & Beta Radiation 8.1.3Gamma Radiation 8.1.4Radioactive Decay 8.1.5Half Life 8.1.6End of Topic Test - Radioactivity 8.1.7Nuclear Instability 8.1.8Nuclear Radius 8.1.9Mass & Energy 8.1.10Binding Energy 8.1.11Induced Fission 8.1.12Safety Aspects of Nuclear Reactors 8.1.13End of Topic Test - Nuclear Physics 8.1.14A-A* (AO3/4) - Nuclear Fusion

9

Option: Astrophysics (A2 only)

9.1

Telescopes (A2 only)

9.1.1Astronomical Telescopes 9.1.2Reflecting Telescopes 9.1.3Single Dish Radio Telescopes 9.1.4Large Diameter Telescopes

9.2

Classification of Stars (A2 only)

9.2.1Classification by Luminosity 9.2.2Absolute Magnitude 9.2.3Black Body Radiation 9.2.4Stellar Spectral Classes 9.2.5Hertzsprung-Russell Diagrams 9.2.6Astronomical Objects

9.3

Cosmology (A2 only)

9.3.1Doppler Effect 9.3.2Hubble's Law 9.3.3Quasars 9.3.4Detecting Exoplanets

10

Option: Medical Physics (A2 only)

10.1

Physics of the Eye (A2 only)

10.1.1Physics of Vision 10.1.2Defects of Vision 10.1.3Lenses 10.1.4Correcting Defects of Vision

10.2

Physics of the Ear (A2 only)

10.2.1Structure of the Ear 10.2.2Sensitivity of the Ear 10.2.3Hearing Defects

10.3

Biological Measurement (A2 only)

10.3.1Electrocardiography (ECG)

10.4

Non-Ionising Imaging (A2 only)

10.4.1Ultrasound Imaging 10.4.2Ultrasound Imaging 2 10.4.3Fibre Optics & Endoscopy 10.4.4Magnetic Resonance Scanning

10.5

X-Ray Imaging (A2 only)

10.5.1Diagnostic X-Rays 10.5.2X-Ray Image Processing 10.5.3Absorption of X-Rays 10.5.4CT Scanners

10.6

Radionuclide Imaging & Therapy (A2 only)

10.6.1Imaging Techniques 10.6.2Half Life 10.6.3Gamma Camera 10.6.4High Energy X-Rays 10.6.5Radioactive Implants 10.6.6Imaging Comparisons

11

Option: Engineering Physics (A2 only)

11.1

Rotational Dynamics (A2 only)

11.1.1Moment of Inertia 11.1.2Rotational Kinetic Energy 11.1.3Rotational Motion 11.1.4Torque & Angular Acceleration 11.1.5Angular Momentum 11.1.6Angular Work & Power

11.2

Thermodynamics & Engines (A2 only)

11.2.1First Law of Thermodynamics 11.2.2Non-Flow Processes 11.2.3p-V Diagrams 11.2.4Engine Cycles 11.2.5Second Law & Engines 11.2.6Reversed Heat Engines

12

Option: Turning Points in Physics (A2 only)

12.1

Discovery of the Electron (A2 only)

12.1.1Cathode Rays 12.1.2Thermionic Electron Emission 12.1.3Electron Specific Charge 12.1.4Millikan's Experiment

12.2

Wave-Particle Duality (A2 only)

12.2.1Newton's & Huygen's Theories of Light 12.2.2Electromagnetic Waves 12.2.3Photoelectricity 12.2.4Wave-Particle Duality 12.2.5Electron Microscopes

12.3

Special Relativity (A2 only)

12.3.1Michelson-Morley Experiment 12.3.2Einstein's Theory of Special Relativity 12.3.3Time Dilation 12.3.4Length Contraction 12.3.5Mass & Energy

Jump to other topics

1

Measurements & Errors

1.1

Measurements & Errors

1.1.1SI Base Units 1.1.2Combining SI Units 1.1.3Prefixes of Units 1.1.4Standard Form 1.1.5Converting Units 1.1.6End of Topic Test - Units & Prefixes 1.1.7Limitation of Physical Measurements 1.1.8Uncertainty 1.1.9Estimation 1.1.10End of Topic Test - Measurements & Errors 1.1.11Diagnostic Misconceptions - Converting between SI 1.1.12Diagnostic Misconceptions - SI & Area/Volume

2

Particles & Radiation

2.1

Particles

2.1.1Atomic Model 2.1.2Specific Charge, Protons & Neutron Numbers 2.1.3End of Topic Test - Atomic Model 2.1.4Isotopes 2.1.5Stable & Unstable Nuclei 2.1.6End of Topic Test - Isotopes & Nuclei 2.1.7A-A* (AO3/4) - Stable & Unstable Nuclei 2.1.8Particles, Antiparticles & Photons 2.1.9Particle Interactions 2.1.10Classification of Particles 2.1.11End of Topic Test - Particles & Interactions 2.1.12Quarks & Antiquarks 2.1.13Application of Conservation Laws 2.1.14End of Topic Test - Leptons & Quarks 2.1.15Exam-Style Question - Radioactive Decay

2.2

Electromagnetic Radiation & Quantum Phenomena

2.2.1The Photoelectric Effect 2.2.2The Photoelectric Effect Explanation 2.2.3End of Topic Test - The Photoelectric Effect 2.2.4Collisions of Electrons with Atoms 2.2.5Energy Levels & Photon Emission 2.2.6Wave-Particle Duality 2.2.7End of Topic Test - Absorption & Emission 2.2.8Diagnostic Misconceptions - Electron Volts 2.2.9Diagnostic Misconceptions - Converting eV & Joules

3

Waves

3.1

Progressive & Stationary Waves

3.1.1Progressive Waves 3.1.2Wave Speed & Phase Difference 3.1.3Longitudinal & Transverse Waves 3.1.4End of Topic Test - Progressive Waves 3.1.5Polarisation 3.1.6Stationary Waves 3.1.7Stationary Waves 2 3.1.8End of Topic Test - Polarisation & Stationary Wave 3.1.9A-A* (AO3/4) - Stationary Waves

3.2

Refraction, Diffraction & Interference

3.2.1Interference 3.2.2Interference 2 3.2.3End of Topic Test - Interference 3.2.4Diffraction 3.2.5Diffraction Gratings 3.2.6End of Topic Test - Diffraction 3.2.7Refraction at a Plane Surface 3.2.8Internal Reflection & Fibre Optics 3.2.9End of Topic Test - Refraction 3.2.10Exam-Style Question - Waves

4

Mechanics & Materials

4.1

Force, Energy & Momentum

4.1.1Scalars & Vectors 4.1.2Vector Problems 4.1.3End of Topic Test - Scalars & Vectors 4.1.4Moments 4.1.5Centre of Mass 4.1.6End of Topic Test - Moments & Centre of Mass 4.1.7Motion in a Straight Line 4.1.8Graphs of Motion 4.1.9Bouncing Ball Example 4.1.10End of Topic Test - Motion in a Straight Line 4.1.11Acceleration Due to Gravity 4.1.12Projectile Motion 4.1.13Friction 4.1.14Terminal Speed 4.1.15End of Topic Test - Acceleration Due to Gravity 4.1.16Newton's Laws 4.1.17Momentum 4.1.18Momentum 2 4.1.19End of Topic Test - Newton's Laws & Momentum 4.1.20A-A* (AO3/4) - Newton's Third Law 4.1.21Work & Energy 4.1.22Power & Efficiency 4.1.23Conservation of Energy 4.1.24End of Topic Test - Work, Energy & Power 4.1.25Exam-Style Question - Forces 4.1.26Diagnostic Misconceptions - Perpendicular Vectors 4.1.27Diagnostic Misconceptions - Weight Acts Downwards 4.1.28Diagnostic Misconceptions - Acceleration Direction 4.1.29Diagnostic Misconceptions - Stationary Objects 4.1.30Diagnostic Misconceptions - Action & Reaction 4.1.31Diagnostic Misconceptions - RF Direction 4.1.32Diagnostic Misconceptions - RF Acceleration 4.1.33Diagnostic Misconceptions - Change in Momentum

4.2

Materials

4.2.1Density 4.2.2Bulk Properties of Solids 4.2.3Energy in Materials 4.2.4Young Modulus 4.2.5End of Topic Test - Materials

5

Electricity

5.1

Current Electricity

5.1.1Basics of Electricity 5.1.2Current-Voltage Characteristics 5.1.3End of Topic Test - Basics of Electricity 5.1.4Resistivity 5.1.5Superconductivity 5.1.6A-A* (AO3/4) - Superconductivity 5.1.7End of Topic Test - Resistivity & Superconductors 5.1.8Circuits 5.1.9Power and Conservation 5.1.10Potential Divider 5.1.11Emf & Internal Resistance 5.1.12End of Topic Test - Power & Potential 5.1.13Exam-Style Question - Resistance 5.1.14Diagnostic Misconceptions - Constant Current 5.1.15Diagnostic Misconceptions - Potential Difference

6

Further Mechanics & Thermal Physics (A2 only)

7

Fields & Their Consequences (A2 only)

7.1

Fields (A2 only)

7.2

Gravitational Fields (A2 only)

7.2.1Newton's Law 7.2.2Gravitational Field Strength 7.2.3Gravitational Potential 7.2.4Orbits of Planets & Satellites 7.2.5Escape Velocity & Synchronous Orbits 7.2.6End of Topic Test - Gravitational Fields

7.3

Electric Fields (A2 only)

7.3.1Coulomb's Law 7.3.2Electric Field Strength 7.3.3Electric Field Strength 2 7.3.4Electric Potential 7.3.5End of Topic Test - Electric Fields 7.3.6A-A* (AO3/4) - Electric and Gravitational Field

7.4

Capacitance (A2 only)

7.4.1Capacitance 7.4.2Parallel Plate Capacitor 7.4.3Energy Stored by a Capacitor 7.4.4Capacitor Discharge 7.4.5Capacitor Charge

7.5

Magnetic Fields (A2 only)

7.5.1Magnetic Flux Density 7.5.2End of Topic Test - Capacitance & Flux Density 7.5.3Moving Charges in a Magnetic Field 7.5.4Magnetic Flux & Flux Linkage 7.5.5Electromagnetic Induction 7.5.6Electromagnetic Induction 2 7.5.7Alternating Currents 7.5.8Operation of a Transformer 7.5.9Magnetic Flux Density 7.5.10End of Topic Test - Electromagnetic Induction

8

Nuclear Physics (A2 only)

8.1

Radioactivity (A2 only)

8.1.1Rutherford Scattering 8.1.2Alpha & Beta Radiation 8.1.3Gamma Radiation 8.1.4Radioactive Decay 8.1.5Half Life 8.1.6End of Topic Test - Radioactivity 8.1.7Nuclear Instability 8.1.8Nuclear Radius 8.1.9Mass & Energy 8.1.10Binding Energy 8.1.11Induced Fission 8.1.12Safety Aspects of Nuclear Reactors 8.1.13End of Topic Test - Nuclear Physics 8.1.14A-A* (AO3/4) - Nuclear Fusion

9

Option: Astrophysics (A2 only)

9.1

Telescopes (A2 only)

9.1.1Astronomical Telescopes 9.1.2Reflecting Telescopes 9.1.3Single Dish Radio Telescopes 9.1.4Large Diameter Telescopes

9.2

Classification of Stars (A2 only)

9.2.1Classification by Luminosity 9.2.2Absolute Magnitude 9.2.3Black Body Radiation 9.2.4Stellar Spectral Classes 9.2.5Hertzsprung-Russell Diagrams 9.2.6Astronomical Objects

9.3

Cosmology (A2 only)

9.3.1Doppler Effect 9.3.2Hubble's Law 9.3.3Quasars 9.3.4Detecting Exoplanets

10

Option: Medical Physics (A2 only)

10.1

Physics of the Eye (A2 only)

10.1.1Physics of Vision 10.1.2Defects of Vision 10.1.3Lenses 10.1.4Correcting Defects of Vision

10.2

Physics of the Ear (A2 only)

10.2.1Structure of the Ear 10.2.2Sensitivity of the Ear 10.2.3Hearing Defects

10.3

Biological Measurement (A2 only)

10.3.1Electrocardiography (ECG)

10.4

Non-Ionising Imaging (A2 only)

10.4.1Ultrasound Imaging 10.4.2Ultrasound Imaging 2 10.4.3Fibre Optics & Endoscopy 10.4.4Magnetic Resonance Scanning

10.5

X-Ray Imaging (A2 only)

10.5.1Diagnostic X-Rays 10.5.2X-Ray Image Processing 10.5.3Absorption of X-Rays 10.5.4CT Scanners

10.6

Radionuclide Imaging & Therapy (A2 only)

10.6.1Imaging Techniques 10.6.2Half Life 10.6.3Gamma Camera 10.6.4High Energy X-Rays 10.6.5Radioactive Implants 10.6.6Imaging Comparisons

11

Option: Engineering Physics (A2 only)

11.1

Rotational Dynamics (A2 only)

11.1.1Moment of Inertia 11.1.2Rotational Kinetic Energy 11.1.3Rotational Motion 11.1.4Torque & Angular Acceleration 11.1.5Angular Momentum 11.1.6Angular Work & Power

11.2

Thermodynamics & Engines (A2 only)

11.2.1First Law of Thermodynamics 11.2.2Non-Flow Processes 11.2.3p-V Diagrams 11.2.4Engine Cycles 11.2.5Second Law & Engines 11.2.6Reversed Heat Engines

12

Option: Turning Points in Physics (A2 only)

12.1

Discovery of the Electron (A2 only)

12.1.1Cathode Rays 12.1.2Thermionic Electron Emission 12.1.3Electron Specific Charge 12.1.4Millikan's Experiment

12.2

Wave-Particle Duality (A2 only)

12.2.1Newton's & Huygen's Theories of Light 12.2.2Electromagnetic Waves 12.2.3Photoelectricity 12.2.4Wave-Particle Duality 12.2.5Electron Microscopes

12.3

Special Relativity (A2 only)

12.3.1Michelson-Morley Experiment 12.3.2Einstein's Theory of Special Relativity 12.3.3Time Dilation 12.3.4Length Contraction 12.3.5Mass & Energy

Practice questions on Nuclear Radius

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1
Which of these statements is true at the distance of closest approach?Multiple choice
2
What is the equation for the De Broglie wavelength?Multiple choice
3
What is the equation for the angle of diffraction?Multiple choice
4
What can we conclude from the liquid drop model of the nucleus?Multiple choice
5
According to the liquid drop model, the volume of a nucleus is proportional to what?Multiple choice

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Nuclear Instability

Mass & Energy