Topic 12: Quantum and Nuclear Physics
12.1 The Interaction of Matter with Radiation
The photoelectric effect
Definition: Phenomenon in which light (or other forms of electromagnetic radiation) incident on a metallic surface causes electron to be emitted from the surface.
Experiment: Evacuated tube, with a metallic photosurface (P), in which light passes through a small opening and causes electrons to be ejected. These electrons are collected by a collecting surface (C).

Since the collecting plate is connected to the negative terminal of the power supply, it will repel normally repel electrons and only absorb the energetic ones.

As the voltage is made more negative, there is a point at which the current ceases, called stopping voltage/potential (Vs).
Tip: The IV graph will change if the frequency of the light is increased. Each photon will have more energy, and hence, the stopping potential will be greater. The saturation current will depend upon the intensity of the light, but in the case of two lights with the same intensity, the saturation current for the higher frequency will be less. This follows logically from the photon nature of light: Same current means the same amount of electrons (chargecarriers) per second, but higher frequency means that electrons have more energy. Less photons per second means that fewer electrons are emitted, and so, smaller saturation current.
Observations

The intensity of the incident light neither affects the kinetic energy or the stopping voltage, solely the number of electrons emitted.

The frequency of light influences the emitted electrons' energy.

Electrons are emitted without a time delay.

There is a minimum/threshold frequency, fc, below which no electrons are emitted.
If light was only a wave...

Intense beams, which have more energy, should cause the emissions of electrons with higher kinetic energy.

Frequency should play a role in the energy of electrons

Low intensity beams should cause a time delay, since energy would need to accumulate before the emission of an electron.
Einstein's explanation: Light consists of photons, which are quanta or bundles of energy and momentum.

Photon's energy: E = hf = hc/λ

Planck's constant = h = 6.63 x 10^34

Photoelectric effect: Single photon of frequency f is absorbed by a single electron in the photosurface, so the electron's energy increases by hf. The electron will spend Ф Joules, called the work function, to free itself.

Electron's kinetic energy (after emission): E = e Vs = hf  Ф
Matter (or "de Broglie") Waves
As suggested by de Broglie, to any particle of momentum p, there corresponds a wave of wavelength given by the formula λ = h/p, something known as the duality of matter.

Electron diffraction: Electrons shot through or to a thin slice of crystal have a low probability of reaching a place where the path difference is not an integer number of wavelengths (constructive and destructive interference).

Electrons accelerated through a pd, they gain kinetic energy. Hence, we have eV = 1/2 mv = p²/2m.

λ = h/sqrt(2meV)


DavissonGermer experiment verifies de Broglie hypothesis (below).
The Bohr Model
Model proposed by Niels Bohr to interpret the scattering of alpha particles, which states that electrons are found at orbitals: fixed multiples of angular momentum that can be represented as a wave function.

Electrons in any atom have a definite/discrete energy (which explains the emission and absorption spectra).

Energy levels = electron wave = standing wave, since there is no energy transfer in standing waves

Hydrogen atom: Energy is given by E = 13.6/n², where n is the principal quantum number and represents the nth energy level.


Angular momentum (mvr): A vector product of the momentum of a particle and the radius of its orbit, of an electron in a stationary state is an integral value of h/2π. Hence, we have mvr = nh/2π.

Assumptions:

Electrons in an atom exist in stationary states, without emitting any electromagnetic radiation.

Electrons may move from one stationary state to another by absorbing or emitting a quantum of electromagnetic radiation, with difference in energy between stationary states given by ∆E = hf.


Limitations: Bohr's Model failed to explain...

Why some energy transitions are more likely to occur than others,

Predict behavior of other elements

Explain behaviors theoretically

Schrödinger's equation (wave function)
Describes the quantum state of the particles, where the square of the amplitude of the wave function │Ψ│² is proportional to the probability per unit volume of finding the particle at a distance r from the nucleus P(r) = │Ψ│²∆V or at (x,y), i.e. P(x,y) = │Ψ│²∆V.

Copenhagen interpretation: For doubleslit interference, the wave function is considered to be such that a single photon or electron passes through both slits and be everywhere on the screen until it is observed or measured.

Nothing is real unless it is observed. When observed, the wave function collapses.

Heisenberg's uncertainty principle
It is impossible to simultaneously measure the position and momentum of a particle with indefinite precision. The same applies to energy and time.

Uncertainty in position and momentum: ∆x∆p ≥ h/4π

Example: Since we know the wavelength of the electron and momentum and wavelength are related by p = h/λ, ∆x is infinite.Singleslit diffraction: The uncertainty in position for beam going through a hole of diameter b is approximately ∆x = b/2. When the opening is approximately of the same order as the de Broglie wavelength of the electrons, the wave will diffract. θ


Uncertainty in energy and time: ∆E∆t ≥ h/4π (where E is half the difference between the excited state and the ground state).

Useful to estimate the lifetime of an electron in excited state.


Singleslit diffraction: The uncertainty in position for beam going through a hole of diameter b is approximately ∆x = b/2. When the opening is approximately of the same order as the de Broglie wavelength of the electrons, the wave will diffract.

Formula: ∆x∆p = λp/2 = h/2.


Electron in a box: If an electron is confined to a region of length L where it can only move back and forth, the uncertainty in position is ∆x = L/2, and thus, ∆p = h/4π∆x = h/2πL.

Kinetic energy = p²/2m = h²/8π²mL²

Pair production and annihilation
Hydrogen atom 's electrons

Pair production: close to an atomic nucleus, where the electric field is very strong, a photon with minimum energy given by E = 2mc² can produce a particle and its antiparticle (e.g. e and e+), where m is the rest mass.

The atomic nucleus helps conserving energy and momentum.

Any excess energy (above 2mc²) will be converted into kinetic energy of the particles


Pair annihilation: when a particle collides with its antiparticle, producing 2 photons.

When they move in the opposite directions, the total energy of the system is ET = 2(mc² + EK) and the photons will travel in opposite direction.

Quantum Tunneling

Tunneling: A particle can effectively "borrow" energy from its surroundings, pass through a barrier and pay the energy back.

The energy required to go through a potential barrier is due to the uncertainty principle less than eV.

The wave function is continuous despite the fact that the particle requires more energy to "jump" the barrier, which is borrowed from surroundings

Energy level remains unchanged after barrier, but the amplitude decreases since it is proportional to P(r).

In order to increase P(r), one may reduce:

The mass m of the particles

The width w of the barrier

The difference ∆E between the energy barrier and that of the particles


Responsible for the relatively low temperature fusion that occurs in the Sun and useful in scanning tunneling microscopes (STM).
12.2 Nuclear Physics
Rutherford Scattering
Simple energy considerations can be used to calculate the distance of closest approach of an alphaparticle.

If the alphaparticle initially has kinetic energy upon approaching, when it stops close to the nucleus due to the electrostatic repulsion, the electrical potential energy of the alpha particle will be K(2e)(Ze)/d = 2KZe²/d, where Z is the atom's proton number.

An alphaparticle approaching with high kinetic energy will get closer to the nuclear (closest = nuclear radius).

R = Ro A^1/3, where R is the radius, Ro is the Fermi radius (1.2x10^15 m) and A is the mass number.

All nuclei have the same density, and so their volume is given by V = 4/3 πR^3 = 4/3 πARo^3.


Derivations from Rutherford scattering: When the alphaparticles have very high kinetic energy, and thus, the distance of closest approach is equal or less than 10^15m, deviations are observed, which is an evidence of the existence of the strong nuclear force, i.e. they are absorbed.
Electron diffraction
If the de Broglie wavelength λ of the electrons is about the same as the nuclear diameter D, then a minimum will be formed at sinθ = λD. Useful to measure nuclear radius.

More accurate than Rutherford scattering, because strong force does not affect electrons.

When electrons of much higher energy are used, the collisions are no longer elastic and energy is converted into mass, as several mesons are emitted from the nucleus

At high energies, the electrons penetrate into the nucleus and scatter off the quarks within protons and neutrons, something known as deep inelastic scattering, providing evidence for the quark model.
Nuclear energy levels
The emission of alpha and beta particles by radioactive decay often leaves the daughter nuclei in an excited discrete energy state (similar to electron energy levels). The state depends on the energy of the alpha or beta particle.
Alpha decay
Alphaparticles form as clusters of two protons and two neutros inside the nucleus well before they are emitted as alphaparticles. This is because the nucleons are in random motion within the nucleus but their kinetic energies are much smaller than those needed to escape.

The wave function │Ψ│ of the alphaparticles is not localized to the nucleus and allows overlaps with the potential energy barrier provided by the strong nuclear force, which means that there is a finite but very small probability of observing the alphaparticles outside the nucleus (thanks to quantum tunneling).

Higher potential barriers and greater thickness to cross means a longer lifetime (e.g. polonium).
Negative beta decay
Since the beta particles have a continuous energy spectrum, in order to conserve mass, energy, and momentum, the existence of the neutrino was suggested by Pauli (and later antineutrino).

Measuring long halflives: Not possible to measure using a GM tube. In these cases, a pure sample of the nuclide in a known chemical form needs to be separated, its mass measured and then a count rate taken. From this reading, the activity can be calculated by multiplying the count rate by the ratio.
The law of radioactive decay

Decay constant (λ): The probability that an individual nucleus will decay in a given time interval (e.g. 1 s)

Units: s^1, min^1, hour^1, day^1...

Relationship with halflife: λ = ln(2)/Thalflife


Activity (A): Number of nuclei decaying in a second in a sample

Units: becquerel (Bq)

In a sample of N undecayed nuclei, the activity will be given by ln(A) = λN.


Formulas: N = No e^λt; A = Ao e^λt; A = λNo e^λt, where Ao = λNo is the initial activity.