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 photo-surface (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 I-V 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 (charge-carriers) 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

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

  2. The frequency of light influences the emitted electrons' energy.

  3. Electrons are emitted without a time delay.

  4. There is a minimum/threshold frequency, fc, below which no electrons are emitted.

 

If light was only a wave...

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

  2. Frequency should play a role in the energy of electrons

  3. 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 photo-surface, 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

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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)

  • Davisson-Germer 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 is the principal quantum number and represents the nth energy level.

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  • 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 double-slit 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.Single-slit diffraction: The uncertainty in position for beam going through a hole of diameter 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.

  • Single-slit diffraction: The uncertainty in position for beam going through a hole of diameter 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 

  • 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 anti-particle (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

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  • Pair annihilation: when a particle collides with its anti-particle, 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.

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  • 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 of the particles ​

    • The width of the barrier

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

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  • 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 alpha-particle.

  • If the alpha-particle 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 alpha-particle 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 alpha-particles 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.

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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 = λDUseful 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.

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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

Alpha-particles form as clusters of two protons and two neutros inside the nucleus well before they are emitted as alpha-particles. 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 alpha-particles 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 alpha-particles outside the nucleus (thanks to quantum tunneling).

  • Higher potential barriers and greater thickness to cross means a longer lifetime (e.g. polonium).

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According to the one-body model of alpha-decay, the alpha-particle is confined in the nucleus by the Coulomb potential barrier.

Classically, if the alpha particle doesn't have more kinetic energy than the potential energy of the barrier, the alpha-particle can't leave the nucleus

However, according to quantum mechanics there is a probability that the alpha-particle will leave the nucleus by tunneling through the barrier

We can calculate the alpha decay constant by multiplying

the "knocking frequency" (frequency of collisions with the barrier)

and the barrier penetration probability (PT)

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 anti-neutrino).

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 half-life: λ = ln(2)/Thalf-life

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

    • Units: becquerel (Bq)​

    • In a sample of 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.

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  • Measuring long half-lives: Not possible to measure using a G-M 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.

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