DP IB Physics: SL

E. Nuclear and Quantum Physics

E.1 Structure of the atom

DP IB Physics: SL E. Nuclear and Quantum Physics E.1 Structure of the atom Understandings Standard level and higher level: 8 hours
a)	The Geiger–Marsden–Rutherford experiment and the discovery of the nucleus
b)	Nuclear notation [math]{}^{A}_{Z}X[/math] where A is the nucleon number Z is the proton number and X is the chemical symbol
c)	That emission and absorption spectra provide evidence for discrete atomic energy levels
d)	That photons are emitted and absorbed during atomic transitions
e)	That the frequency of the photon released during an atomic transition depends on the difference in energy level as given by [math]E = hf[/math]
f)	That emission and absorption spectra provide information on the chemical composition.

Additional higher level: 3 hours Understandings Standard level and higher level: 8 hours
⇒	The relationship between the radius and the nucleon number for a nucleus as given by [math]R = R_0 A^{\frac{1}{3}}[/math] and implications for nuclear densities
⇒	Deviations from Rutherford scattering at high energies
⇒	The distance of closest approach in head-on scattering experiments
⇒	The discrete energy levels in the Bohr model for hydrogen as given by [math]E = -\frac{13.6}{n^2} \text{ eV}[/math]
⇒	That the existence of quantized energy and orbits arise from the quantization of angular momentum in the Bohr model for hydrogen as given by [math]mvr = \frac{nh}{2π}[/math]

a) The Geiger–Marsden–Rutherford Experiment and the Discovery of the Nucleus
In the early 1900s, the plum pudding model suggested that atoms were made of a diffuse cloud of positive charge with electrons scattered inside.
The Experiment (1909):
Conducted by Hans Geiger and Ernest Marsden, supervised by Ernest Rutherford.
⇒ Setup:
A source of alpha particles (positively charged, helium nuclei) was directed at a very thin sheet of gold foil (~ a few atoms thick).
A fluorescent screen (zinc sulfide) surrounded the foil to detect scattered alpha particles.
Figure 1 Discover of nucleus
⇒ Observations:
1. Most alpha particles passed straight through → the atom is mostly empty space.
2. Some were deflected at small angles → alpha particles were slightly repelled.
3. A few bounced back at large angles (>90°) → a dense, positively charged region existed in the atom.
⇒ Conclusion:
– Atom has a small, dense, positively charged nucleus at the center.
– The nucleus contains most of the atom’s mass, but is tiny compared to the size of the atom.
⇒ Significance:
This overturned the plum pudding model and led to the Rutherford nuclear model, where:
– Electrons orbit a central nucleus.
– The atom is mostly empty space.
b) Nuclear Notation:
This is a shorthand to describe the identity of a nucleus (isotope).
[math]{}^{A}_{Z}X[/math]
Where:
– X = Chemical symbol (e.g. H, He, C, U)
– A = Mass number (total number of nucleons = protons + neutrons)
– Z = Atomic number (number of protons)
Figure 2 Nuclear Notation
Calculations:
– Neutrons = A−Z
– The chemical properties are determined by Z, not A.
⇒ Example:
Carbon-14: [math]{}^{14}_{6}\text{C}[/math] ⇒ 6 protons, 8 neutrons
c) Emission and Absorption Spectra – Evidence for Discrete Atomic Energy Levels
Emission Spectrum: The bright lines emitted by an atom when electrons fall to lower energy levels.
Absorption Spectrum: The dark lines where specific wavelengths are absorbed as electrons move to higher energy levels.
⇒ Explanation:
In atoms:
– Electrons exist in discrete energy levels (quantized).
– When an electron absorbs energy (e.g. from heat or light), it jumps to a higher energy level → absorption.
– When it falls back to a lower level, it releases energy as a photon → emission.
[math]E_{\text{photon}} = E_{\text{high}} – E_{\text{low}} \\
E_{\text{photon}} = hf \\
E_{\text{photon}} = \frac{hc}{\lambda}[/math]
Where:
– h = Planck’s constant
– f = frequency
– λ = wavelength
– c = speed of light
Figure 3 Formation of spectral lines
⇒ Experimental Evidence:
Hydrogen emission spectrum shows distinct colored lines (e.g. Balmer series).
Each line corresponds to a specific transition between energy levels.
This proves that electrons do not have continuous energy, but only certain allowed states.
d) That photons are emitted and absorbed during atomic transitions
⇒ Atomic transitions:
Atoms consist of a nucleus surrounded by electrons that occupy discrete energy levels or orbitals. Electrons can move between these levels by absorbing or emitting energy.
Figure 4 Photons are emitted and absorbed during atomic transitions
⇒ Emission:
– When an electron falls from a higher energy level to a lower one, it releases energy.
– This energy is emitted in the form of a photon (a quantum of light).
⇒ Absorption:
When an electron absorbs energy from an external source (e.g., light or heat), it jumps to a higher energy level.
The energy absorbed must exactly match the difference between the two levels.
These processes are:
– Quantized: Only specific energy changes (not continuous).
– Governed by quantum physics, particularly the Bohr model of the atom.
e) The frequency of the photon released during an atomic transition depends on the difference in energy level, as given by:
[math]E = hf[/math]
Where:
E = Energy of the photon
h = Planck’s constant (6.626×10⁻³⁴ J.s)
f = Frequency of the photon
Also related to wavelength:
Since [math]c = fλ[/math], where ccc is the speed of light, we can also write:
[math]E = \frac{hc}{\lambda}[/math]
Figure 5 Electron transmission
This equation shows:
Larger energy differences produce higher frequency (shorter wavelength) photons (e.g. ultraviolet).
Smaller energy differences produce lower frequency (longer wavelength) photons (e.g. infrared).
f) Emission and absorption spectra provide information on chemical composition
Emission and absorption spectra:
Emission Spectrum: Produced when excited atoms emit light at specific wavelengths.
– Appears as bright lines on a dark background.
Absorption Spectrum: Produced when light passes through a cool gas.
– The gas absorbs specific wavelengths, creating dark lines in the continuous spectrum.
How does this identify chemical elements?
Each element has unique energy level spacings, so:
The frequencies of absorbed or emitted photons are unique.
These frequencies show up as lines at specific positions in the spectrum.
Like a “fingerprint”, the pattern of lines tells us which element is present.
⇒ Applications:
Astronomy: Identify elements in stars and galaxies from their spectra.
Chemistry and Forensics: Analyze unknown substances.
Environmental science: Detect pollutants.
Additional higher level: 3 hours
a) Relationship between the radius and the nucleon number for a nucleus
[math]R = R_0 A^{\frac{1}{3}}[/math]
Where:
– R = radius of the nucleus
– [math]R_0 ≈ 1.2 × 10^{-15} m[/math] = empirical constant
– A = nucleon number (total number of protons + neutrons)
This formula shows that the radius of a nucleus increases with the cube root of the number of nucleons.
For example:
A nucleus with A=8 (like oxygen) has a radius:
[math]R = R_0 A^{\frac{1}{3}} \\
R = R_0 (8)^{\frac{1}{3}} \\
R = R_0 (2)[/math]
A nucleus with A=64 (like zinc) has:
[math]R = R_0 A^{\frac{1}{3}} \\
R = R_0 (64)^{\frac{1}{3}} \\
R = R_0 (4)[/math]
So, doubling the radius increases the volume ~8x, which is consistent with a roughly constant nuclear density.
⇒ Implications for nuclear density:
Since:
Volume of a nucleus
[math]V = \frac{4}{3} \pi R^{3} \\
V = \frac{4}{3} \pi \left( R_0 A^{\frac{1}{3}} \right)^{3} \\
V = \frac{4}{3} \pi R_0^{3} A[/math]
We can say that:
Volume ∝ A
Mass ∝ A (since each nucleon ~ same mass)
Then:
[math]\text{Density} = \frac{\text{Mass}}{\text{Volume}} \\
\text{Density} \approx \frac{A}{A} \\
\text{Density} \approx \text{constant}[/math]
Conclusion: All nuclei, regardless of size, have roughly the same density, about:
[math]2.3 × 10^{17} kg/m^3[/math]
This is incredibly dense — much denser than any ordinary matter (like steel or lead).
b) Deviations from Rutherford scattering at high energies
Rutherford’s experiment recap:
– Alpha particles were fired at thin gold foil.
– Most passed through → atom is mostly empty space.
– Some deflected at large angles → small, dense, positively charged nucleus exists.
⇒ Predictions of Rutherford scattering:
Rutherford’s formula accurately predicted the number of alpha particles deflected at various angles at low and moderate energies.
At high energies:
As the energy of incoming particles increases:
– Particles approach the nucleus more closely.
– They begin to interact with the strong nuclear force (at distances < ~1 femtometer).
⇒ Deviations observed:
1. Scattering pattern changes:
– More particles deviate from Rutherford’s predictions.
– Scattering no longer follows the classic [math]\frac{1}{\sin^4\left(\frac{\theta}{2}\right)}[/math] distribution
2. Evidence of internal nuclear structure:
– At very high energies (like GeV), particles can probe inside the nucleus.
– Reveals non-uniform charge distribution, quarks, or nucleon substructure.
3. Nuclear reactions can occur:
At extreme energies, the incoming particle might penetrate the nucleus and cause:
– Nuclear disintegration
– Particle emission
– Creation of new particles
Figure 6 Rutherford Scattering
c) Distance of Closest Approach in Head-On Scattering
Concept:
In Rutherford-type scattering experiments, charged particles (e.g. alpha particles) are directed head-on toward a target nucleus (like gold).
If an alpha particle moves directly toward the nucleus, it slows down due to electrostatic repulsion. The distance of closest approach is the point at which its kinetic energy is completely converted into electrostatic potential energy — the particle stops momentarily before reversing direction.
[math]\frac{1}{4 \pi \varepsilon_0} \cdot \frac{Z_1 Z_2 e^2}{r} = \frac{1}{2} m v^2[/math]
Where:
– [math]Z_1, Z_2[/math] = atomic numbers of alpha particle and target nucleus (e.g., 2 for alpha and 79 for gold),
– e = elementary charge,
– r = distance of closest approach,
– v = speed of the alpha particle,
– [math]\frac{1}{2} m v^2[/math] = initial kinetic energy,
– [math]\frac{1}{4 \pi \varepsilon_0} \cdot \frac{Z_1 Z_2 e^2}{r}[/math] = electrostatic potential energy at closest approach.
This distance gives an upper limit to the nuclear radius.
A smaller distance of closest approach implies the nucleus is very compact and massive.
d) Bohr Model of the Hydrogen Atom
The Bohr model explains the discrete spectral lines of hydrogen and introduced the idea of quantized energy levels.
Formula for Energy Levels:
[math]E_n = -\frac{13.6}{n^2} \text{ eV}[/math]
Where:
– [math]n^2[/math] is the energy of the electron in the n-th orbit,
– n is the principal quantum number (1, 2, 3…),
– 13.6 eV is the ionization energy of hydrogen (energy required to remove the electron from n=1n = 1n=1 to infinity).
Figure 7 Bohr’s model of an atom
Negative sign: energy is bound (electron is trapped).
As n→∞, E→0: electron is free.
The difference between levels gives the energy of emitted/absorbed photons:
[math]\Delta E = E_{\text{high}} – E_{\text{low}} = hf[/math]
e) Quantization of Angular Momentum
Bohr’s model introduced the idea that only certain orbits are allowed for the electron — those where its angular momentum is quantized.
[math]m v r = \frac{n h}{2 \pi}[/math]
Where:
– m = mass of the electron,
– v = orbital speed of the electron,
– r = radius of the orbit,
– h = Planck’s constant,
– n = positive integer (quantum number).
⇒ Implications:
Only orbits that satisfy this condition are allowed — this restricts the electron to certain radii.
This leads to quantized energy levels (explains why electrons don’t spiral into the nucleus).
Transitions between these orbits release or absorb photons, creating the hydrogen spectral lines.
Figure 8 Quantization of angular momentum