From the Universe to the Atom

Module 8 is worth about a quarter of the paper, but it is taught last, so it often gets less practice. The quantum mechanical nature of the atom is the most examined part (over 5 marks a year), followed by the structure of the atom and properties of the nucleus.

• Origins of the elements: the Big Bang, evidence for an expanding universe, stellar spectra and the Hertzsprung–Russell diagram, nucleosynthesis in stars, $E=m{c}^2$.
• Structure of the atom: cathode rays, Thomson's charge-to-mass experiment, Millikan's oil drop, Rutherford's nuclear model, Chadwick's neutron.
• Quantum mechanical nature of the atom: the Bohr model and hydrogen spectrum, de Broglie's matter waves, Schrödinger's contribution.
• Properties of the nucleus: alpha, beta and gamma decay, half-life, mass defect and binding energy, fission and fusion.
• Deep inside the atom: the Standard Model of matter, evidence from particle accelerators.

As in Module 7, many questions are about how experimental evidence changed a model. Learn each experiment as: what was done, what was observed, what was concluded.

The Big Bang and expansion

The universe began about 13.8 billion years ago in an extremely hot, dense state and has been expanding and cooling since. As it cooled, energy converted into matter ($E=m{c}^2$): quarks and leptons, then protons and neutrons, then (within minutes) hydrogen and helium nuclei. About 380 000 years later, electrons joined nuclei to form neutral atoms and the universe became transparent. Stars and galaxies formed later by gravitational collapse.

Evidence for the Big Bang:

• Redshift and Hubble's law: Hubble (1929), combining his distances with Slipher's redshifts, found that recession speed is proportional to distance, $v=H_{0}d$. This shows the universe is expanding.
• Cosmic microwave background (CMB): discovered by Penzias and Wilson in 1965, it is the cooled radiation from when atoms formed, now a near-perfect black body at about 2.7 K.
• Abundance of light elements: about 75% hydrogen and 25% helium by mass, matching Big Bang predictions.

Example. A galaxy is receding at $5600{\text{km s}}^{-1}$. With $H_0=70{\text{km s}}^{-1}{\text{Mpc}}^{-1}$, its distance is $d=v/H_0=5600/70=80$ Mpc.

Stars and the H–R diagram

Stellar spectra give a star's temperature and composition (see Module 7). Stars are grouped into spectral classes O, B, A, F, G, K, M from hottest (blue) to coolest (red). The Hertzsprung–Russell diagram plots luminosity against surface temperature (temperature increasing to the left).

A star's hot, dense interior emits a continuous spectrum close to a black-body curve, whose peak gives the surface temperature. Cooler gas in the star's outer atmosphere absorbs particular wavelengths, producing dark absorption lines that identify the elements present. Hot, low-density gas, such as a glowing nebula, gives bright emission lines at those same wavelengths.

The main regions of the H–R diagram:

• Main sequence: a diagonal band where stars fuse hydrogen in their cores, from hot, bright, massive stars at top left to cool, dim, low-mass stars at bottom right.
• Red giants and supergiants: top right, cool but very luminous because they are huge.
• White dwarfs: bottom left, hot but dim because they are small.

Nucleosynthesis and $E=m{c}^2$

In a main-sequence star, hydrogen nuclei fuse to make helium. Stars up to about 1.3 times the Sun's mass do this mainly by the proton–proton chain. More massive stars, with hotter cores, mainly use the CNO cycle, in which carbon, nitrogen and oxygen act as catalysts. After the core hydrogen runs out, stars become giants and fuse helium into carbon and oxygen; the most massive stars go on to make elements up to iron. Elements heavier than iron are made mainly by neutron capture, in giant stars and in supernovae and neutron-star mergers.

The final step of the main branch of the proton–proton chain is

The energy comes from mass: the products have less mass than the reactants (here about 0.0138 u less, releasing 12.9 MeV), and $E=m{c}^2$ gives the energy released.

Example. The Sun emits $3.85\times 10^{26}$ W. Each second its mass decreases by

Cathode rays

Cathode rays are emitted from the negative electrode of an evacuated discharge tube. Scientists debated whether they were waves or particles. Their deflection by electric and magnetic fields showed they carry negative charge, and their ability to turn a small paddle wheel was taken at the time as evidence of momentum, supporting particles.

Thomson's charge-to-mass experiment (1897)

Thomson balanced electric and magnetic forces on the cathode rays so they passed straight through, giving their speed $v=E/B$. He then used the deflection in one field alone (textbooks usually use the radius $r=mv/(qB)$ in the magnetic field). Combining these gives

The ratio was over 1000 times larger (about 1840 by modern values) than for a hydrogen ion and the same whatever metal or gas was used, so cathode rays were tiny, universal negative particles: electrons. This led to Thomson's "plum pudding" model of the atom.

Example. The fields balance with $E=2.4\times 10^4{\text{V m}}^{-1}$ and $B=1.2\times 10^{-3}$ T, so $v=2.0\times 10^7{\text{m s}}^{-1}$. In the magnetic field alone the radius is 9.5 cm, so

This is close to the accepted value for the electron.

Millikan's oil drop experiment (1909)

Tiny charged oil drops were held stationary between horizontal plates by adjusting the voltage so that the electric force balanced the weight: $qE=mg$ with $E=V/d$. Every charge measured was a whole-number multiple of about $1.6\times 10^{-19}$ C, showing that charge is quantised and giving the value of $e$. Combined with Thomson's $q/m$, this gave the electron's mass.

Example. A drop of mass $4.9\times 10^{-15}$ kg is held still between plates 1.2 cm apart at 1200 V:

which is $3e$.

Rutherford and Chadwick

Geiger and Marsden (1909) fired alpha particles at thin gold foil. Most went straight through, but a very small number bounced back through large angles. Rutherford (1911) concluded that the atom is mostly empty space with almost all its mass and all its positive charge in a tiny, dense nucleus, with electrons around it. The model could not explain why orbiting electrons do not radiate energy and spiral in, or why atoms give line spectra.

Chadwick (1932) studied the neutral radiation produced when alpha particles hit beryllium. It knocked protons out of paraffin wax. Using conservation of momentum and energy, he showed it could not be gamma rays and must be a neutral particle with about the mass of a proton: the neutron. This explained why atomic mass is greater than the proton count alone.

The Bohr model (1913)

Bohr kept Rutherford's nucleus and added quantum ideas:

1. Electrons move only in certain allowed orbits (stationary states) without radiating energy.
2. The angular momentum of an allowed orbit is quantised: $mvr=\frac{nh}{2\pi }$.
3. An electron emits or absorbs a photon only when it moves between orbits, with $E=hf=E_i-E_f$.

This explained the line spectrum of hydrogen. For hydrogen the energy levels are $E_n=-13.6/n^2$ eV, and the wavelengths are given by the Rydberg equation (data sheet):

$R=1.097\times 10^7{\text{m}}^{-1}$, $n_f$ is the final level and $n_i$ the initial level. Transitions down to $n_f=2$ give the visible Balmer series; to $n_f=1$, the ultraviolet Lyman series.

Example. An electron in hydrogen falls from $n=5$ to $n=2$.

so $\lambda =434$ nm (violet). The photon energy is $hc/\lambda =4.58\times 10^{-19}$ J, or 2.86 eV, matching the energy levels:

Limitations of the Bohr model: it only works for hydrogen and one-electron ions; it cannot explain the different intensities of spectral lines, the splitting of lines into close pairs (fine structure) or in magnetic fields (Zeeman effect); and its postulates are not explained, just assumed.

de Broglie's matter waves (1924)

de Broglie proposed that if light has particle properties, particles should have wave properties, with wavelength

An electron in an allowed orbit forms a standing wave, so a whole number of wavelengths must fit around the orbit: $2\pi r=n\lambda$. Substituting $\lambda =h/mv$ gives Bohr's quantisation condition, so de Broglie explained why only certain orbits are allowed. Davisson and Germer (1927) confirmed matter waves by diffracting electrons off a nickel crystal.

Example. An electron accelerated from rest through 150 V has $\frac{1}{2}m{v}^2=qV$, so $v=7.26\times 10^6{\text{m s}}^{-1}$ and

This is similar to the spacing of atoms in a crystal, which is why crystals diffract electrons.

Schrödinger

Schrödinger (1926) described electrons with a wave equation. Fixed orbits are replaced by orbitals, regions where the electron is likely to be found, and the model works for many-electron atoms.

Radioactive decay

In decay equations, mass number and atomic number are both conserved. Examples:

The (anti)neutrino carries away some energy, so beta particles have a range of energies.

Half-life (data sheet)

$N_t$ is the number of undecayed nuclei (or activity) after time $t$, $N_0$ the starting amount, $\lambda$ the decay constant and $t_{1/2}$ the half-life. Use the same time unit throughout.

Example. Iodine-131 has a half-life of 8.0 days. A sample has an activity of 6.0 MBq.

To fall to 10% of the original, $0.10=e^{-\lambda t}$, so

Mass defect and binding energy

The mass of a nucleus is less than the total mass of its separate protons and neutrons. This mass defect $\Delta m$ is the mass equivalent of the binding energy, the energy needed to pull the nucleus apart into separate nucleons ($E=\Delta m{c}^2$, with $1\text{u}=931.5\text{MeV}/c^2$). The strong nuclear force, a short-range attraction between nucleons, holds the nucleus together against the electrostatic repulsion between protons.

Example. Carbon-12 (atomic mass exactly 12.000000 u) contains 6 protons, 6 neutrons and 6 electrons. Using hydrogen atoms (1.007825 u, which includes the electrons) and neutrons (1.008665 u):

Binding energy $=0.098940\times 931.5=92.2$ MeV, or 7.68 MeV per nucleon.

Binding energy per nucleon rises steeply for light nuclei, peaks at about 8.8 MeV near iron-56 and nickel, then falls slowly for heavy nuclei. Moving towards the peak releases energy: fusion of light nuclei and fission of heavy nuclei both produce products that are more tightly bound per nucleon, so mass decreases and energy is released.

Fission and chain reactions

When uranium-235 absorbs a slow neutron it splits into two medium-sized nuclei plus two or three neutrons. For example,

Using masses U-235 235.043930 u, Ba-141 140.914411 u, Kr-92 91.926156 u and n 1.008665 u, the mass lost is 0.186033 u, so the energy released is $0.186033\times 931.5=173$ MeV per fission.

The released neutrons can cause further fissions: a chain reaction. If each fission causes more than one more on average, the reaction rate grows rapidly (supercritical); an uncontrolled chain reaction is the basis of a nuclear weapon. In a reactor, control rods absorb neutrons so that on average exactly one neutron from each fission causes another. A moderator slows neutrons so they are more likely to cause fission in U-235. A minimum (critical) mass is needed so that not too many neutrons escape.

The Standard Model

The Standard Model sorts all known fundamental particles into three groups:

The Higgs boson (found at the Large Hadron Collider in 2012) is associated with the field that gives particles mass. Gravity is not included in the Standard Model. Every particle has an antiparticle (some, like the photon, are their own).

Quarks are never found alone. Hadrons are made of quarks: baryons have three quarks (proton uud, charge $+1$; neutron udd, charge $0$) and mesons have a quark and an antiquark (such as pions).

Interactions and conservation laws

In every particle interaction, charge, baryon number and lepton number are conserved. In beta-minus decay, a down quark in a neutron changes to an up quark by emitting a W⁻ boson, which then decays into an electron and an electron antineutrino. Check $\text{n}\to \text{p}+{\text{e}}^{-}+{\bar{\nu }}_e$: charge $0=+1-1+0$; baryon number $1=1$; lepton number $0=+1-1$.

Evidence from accelerators

Particle accelerators use electric fields to speed up charged particles and magnetic fields to steer them into high-energy collisions. Deep inelastic scattering of electrons off protons (SLAC, late 1960s) revealed point-like quarks inside protons; the W and Z bosons (CERN, 1983) and the Higgs boson (2012) confirmed Standard Model predictions.

• Using $v=H_{0}d$ with inconsistent units. Keep km s⁻¹ and Mpc together.
• Swapping $n_i$ and $n_f$ in the Rydberg equation and getting a negative wavelength.
• Forgetting the neutrino or antineutrino in beta decay equations, or not balancing both mass and atomic numbers.
• Saying that binding energy is energy stored in the nucleus that is released. It is the energy needed to separate it; more binding energy per nucleon means more stable.
• Mixing time units (days and seconds) inside $e^{-\lambda t}$.
• Calling the neutron or proton a fundamental particle. They are baryons made of quarks.

• Explain how an experiment changed the model: method, observation, conclusion, and the limitation of the old model that it fixed.
• Assess the Bohr model or the Big Bang theory: give strengths with evidence, limitations, and a judgement.
• Binding energy calculations: show the mass defect in u to at least 5 decimal places before converting to MeV; give per-nucleon values when comparing nuclei.
• Decay and half-life questions: show conservation of $A$ and $Z$ explicitly; for $N=N_0{e}^{-\lambda t}$ show rearranging with logarithms.
• Round to the significant figures of the data and give units such as MeV, ${\text{C kg}}^{-1}$, nm or Mpc.

• Module 5: circular motion and centripetal force are behind Bohr orbits and the circular paths in Thomson's experiment and accelerators.
• Module 6: charges in electric and magnetic fields ($F=qE$, $F=qvB$, $v=E/B$) are the physics of Thomson's and Millikan's experiments.
• Module 7: spectra and Wien's law classify stars; $E=hf$ and photons underpin the Bohr model; $E=m{c}^2$ gives nuclear energies; wave–particle duality of light leads to de Broglie's matter waves.