Theme E — Nuclear and Quantum Physics
E.1.5 Rutherford Scattering and Nuclear Radius — IB Physics HL Revision Notes & Topical Questions
5 min read 9 parts Lesson 7 of 45 1 quiz
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
E.1.5 Rutherford Scattering and Nuclear Radius — IB PHYSICS HL notes: Use R = R0 A1/3), nuclear density and the closest approach method to estimate nuclear size.Dependence — the radius depends on the nucleon number A because more nucleons need more spaceRelation — R = R0A1/3, fou…
- Fermi radius
- Doubling A
- Data booklet
- What the symbols mean
- Much denser
- Initial energy
Part 1 of 7Nuclear radius
Dependence
- Dependence the radius depends on the nucleon number A because more nucleons need more space
Relation
- R = R0A1/3, found from experimental data such as scattering
Fermi radius
- R0 = 1.20 × 10−15 m (1.20 fm), the radius of a hydrogen nucleus with A = 1
Graph
- R against A1/3 is a straight line through the origin with gradient R0
Doubling A
- Doubling A raises the radius by only a factor 21/3 = 1.26
Data booklet
- Data booklet the value of R0 is given
Part 2 of 7Nuclear density
Volume
- V = (4/3)πR3 = (4/3)πR03A for a spherical nucleus
Mass
- Mass m = Au where u is the atomic mass unit
Density
- ρ = m/V = 3u/(4πR03), since A cancelsWhat the symbols mean: ρ = density (kg m−3 or g cm−3) · m = mass (kg) · V = volume (m3)
Result
- Result the density is the same for all nuclei and independent of radius
Value
- Value of the order 1017 kg m−3, about 2.3 × 1017 kg m−3
Meaning
- Meaning nucleons are evenly spaced throughout the nucleus whatever its size
Part 3 of 7Comparison with the atom
Much denser
- Much denser nuclear density is far larger than atomic density
Mass
- Mass most of the atom's mass is in the nucleus
Size
- Size the nucleus is very small compared with the atom
Space
- Space atoms are mostly empty space
Part 4 of 7Rutherford scattering and closest approach
Initial energy
- Initial energy the alpha particle starts with kinetic energy Ek = ½mv2
Back-scattering
- Back-scattering some alpha particles return, showing electrostatic repulsion by the nucleus
Closest approach
- Closest approach the repulsion reduces the speed to zero momentarily at distance d before the direction reverses
Energy balance
- Energy balance at that point the initial kinetic energy equals the electric potential energy, Ek = Ep
Part 5 of 7Closest approach equation
Potential energy
- Ep = kQq/d with Q = 2e for the alpha particle and q = Ze for the target
Result
- Ep = k(2e)(Ze)/d = 2kZe2/d
Distance
- Distance d = 2kZe2/Ek
Charge
- Charge an alpha particle has the charge of two protons
Units
- Units convert MeV to joules by multiplying by 1.60 × 10−13
Part 6 of 7Interpretation
Upper limit
- Upper limit d is an upper limit on the nuclear radius if the particle feels only the electrostatic force
Higher energy
- Higher energy brings the particle closer to the nucleus
Strong force
- Strong force when the energy is large enough to reach the strong-force region, the scattering departs from the electrostatic prediction
Angle dependence
- Angle dependence scattering to large angles is greater for smaller impact parameter, larger Z and smaller energy
Scattering law
- Scattering law the number at angle θ is proportional to 1/sin4(θ/2)
Shortcut
- Shortcut kZe2 = 1.44 Z MeV fm, so d = 2 × 1.44Z/Ek fm with Ek in MeV
Head-on only
- Head-on only the equation applies to head-on collisions
Part 7 of 7More from the specification
Rutherford Alpha Scattering
- Directing alpha particles at thin gold foil proved that atoms possess a dense, positively charged nucleus containing most atomic mass, surrounded by orbiting electrons.
Discrete Atomic Energy Levels
- Bound electrons occupy quantized energy levels;
- de-excitation between orbits causes the emission of light with energy equal to the energy gap Delta E = E2 - E1.
Photon Emission and Planck's Constant
- The wavelength of an emitted photon is determined by the de-excitation energy via Delta E = hf = hc / lambda,
- where h represents Planck's constant.
Atomic Line Emission Spectrum
- Because energy level separations are unique to each chemical element, the resulting emission and absorption spectra display characteristic discrete spectral lines.
Emission spectrum
- The set of discrete wavelengths emitted by excited atoms of an element.
Photon
- A quantum of electromagnetic energy, E = hf
Wavelength
- Distance between a point on one wave and the same point on the next wave
Spectrum
- Range of colours in white light.
Constant
- Not changing.
Absorption
- The conversion of wave energy to internal energy of the medium.
Energy level
- A discrete energy an electron in an atom can have
Photon energy
- E = hf = hc/λ, the energy carried by one photon.
Angular momentum
- L = Iω for a rigid body; L = mvr⊥ for a particle.
∆E
∆E hf = E2 − E1
What the symbols mean: E = energy of one photon (J) · h = Planck constant (6.63 × 10−34 J s) · f = frequency (Hz)
Key terms
| Term | Meaning |
|---|---|
| Fermi radius | R0 = 1.20 fm, the constant in R = R0A1/3. |
| Nuclear density | The same for all nuclei, ρ = 3u/(4πR03), about 2 × 1017 kg m−3. |
| Distance of closest approach | The distance at which the incident particle stops momentarily, d = 2kZe2/Ek. |
| Femtometre | 10−15 m. |
| Collimated beam | A narrow parallel beam. |
Avoid these mistakes
1
Using the charge e for the alpha particle — it is 2e.
2
Treating the distance of closest approach as the exact nuclear radius.
3
Taking R proportional to A instead of A1/3.
4
Forgetting to convert MeV to joules.
5
Applying the head-on formula to glancing collisions.
Marked practiceQuizzes for this lesson
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