Tunnelling in Nature: Decay, Fusion, and the STM
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You should be comfortable with the time-independent Schrödinger equation and the concept of wave number \( k \) and decay constant \( \kappa \). You must also understand the transmission coefficient \( T \) as a probability ratio from the previous lecture.
Why this matters
Tunnelling is not just a mathematical curiosity; it is the mechanism that allows the Sun to shine and enables us to image atoms.
In the previous lecture, we derived the transmission probability for a particle encountering a rectangular potential barrier. We found that even when the particle’s energy \( E \) is less than the barrier height \( V_0 \), the probability \( T \) is non-zero, decaying exponentially with the barrier width \( a \). This lecture moves from the abstract derivation to concrete physical phenomena. We will see how this single quantum effect explains alpha decay in radioactive nuclei, the fusion reactions powering stars, and the operating principle of the Scanning Tunnelling Microscope (STM), which allows us to visualise the surface of materials at the atomic scale.
The Physical Picture
Imagine a ball rolling towards a hill. In classical mechanics, if the ball does not have enough kinetic energy to reach the top, it rolls back down. It cannot pass through the hill. In quantum mechanics, the particle is described by a wave function. When this wave encounters a region where its energy is insufficient to sustain a propagating wave (the barrier), the wave function does not abruptly stop. Instead, it penetrates into the barrier and decays exponentially. If the barrier is not too wide, the wave function on the far side is non-zero. This means there is a finite probability of finding the particle on the other side. This is tunnelling. The key insight is that the probability depends exponentially on the width of the barrier and the square root of the energy deficit. A small change in width or energy can lead to a massive change in the tunnelling rate.
Alpha Decay
Alpha decay is the emission of a helium-4 nucleus (two protons and two neutrons) from a heavier nucleus. Classically, the alpha particle is trapped inside the nucleus by the strong nuclear force, which creates a deep potential well. However, the Coulomb repulsion between the positively charged alpha particle and the remaining protons creates a potential barrier outside the nuclear radius. The alpha particle has a fixed energy \( E \) determined by the nuclear binding energies, which is typically less than the height of the Coulomb barrier \( V_0 \). Therefore, classically, it should never escape. Quantum mechanically, it tunnels through the Coulomb barrier.
The Gamow factor, which governs the decay rate, is derived from the transmission probability. For a Coulomb barrier, the integral of the decay constant \( \kappa \) over the barrier width leads to an exponential dependence on the inverse square root of the energy. This explains the Geiger–Nuttall law: the half-life of alpha emitters varies dramatically with the energy of the emitted alpha particle. A slight increase in energy results in a much shorter half-life because the exponential term becomes less suppressive.
where \( \lambda \) is the decay constant, \( R \) is the nuclear radius, and \( b \) is the classical turning point where \( E = V(b) \).
Nuclear Fusion in Stars
Fusion is the process where two light nuclei combine to form a heavier nucleus, releasing energy. In the core of the Sun, the temperature is approximately \( 1.5 \times 10^7 \) K. The typical thermal energy \( k_B T \) at this temperature is about 1.3 keV (mean kinetic energy \( \tfrac{3}{2}k_B T \approx 1.9 \) keV). However, the Coulomb barrier between two protons is on the order of hundreds of keV. Classically, protons do not have enough energy to overcome this repulsion and fuse. The Sun would not shine.
Quantum tunnelling allows protons to penetrate the Coulomb barrier. The probability of tunnelling is extremely small for any single pair of protons, but the density of protons in the solar core is enormous. The reaction rate is proportional to the product of the density and the tunnelling probability. This process, known as the proton-proton chain, is the primary source of energy in the Sun. Without tunnelling, stellar nucleosynthesis would be impossible at the observed temperatures.
The Scanning Tunnelling Microscope
The Scanning Tunnelling Microscope (STM) is a tool used to image surfaces with atomic resolution. It consists of a very sharp metallic tip and a conducting sample. A small bias voltage \( V \) is applied between the tip and the sample. The vacuum gap between the tip and the sample acts as a potential barrier of width \( d \) and height \( \phi \) (the work function of the metal).
Electrons from the tip tunnel through the vacuum gap to the sample (or vice versa). The tunnelling current \( I \) is proportional to the transmission probability \( T \). From the rectangular barrier approximation, for a wide barrier where \( \kappa d \gg 1 \), the transmission coefficient is approximately:
Since \( \kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}} \), the current depends exponentially on the distance \( d \):
This exponential sensitivity means that a change in distance of just 0.1 nm can change the current by an order of magnitude. By keeping the current constant and adjusting the tip height as it scans across the surface, the STM can map the topography of the surface with atomic resolution (height changes of a few picometres are detectable).
The derivation
Tunnelling Through a Rectangular Barrier
Derives the transmission coefficient for a particle incident on a rectangular barrier, exhibiting quantum tunnelling and the thick-barrier exponential suppression.
In the STM, the barrier is not a perfect rectangle; it is a trapezoid because the bias voltage tilts the potential. The simple exponential law \( I \propto e^{-2\kappa d} \) is an approximation valid for low bias voltages. At higher biases, the energy dependence of the density of states and the barrier shape become significant, and the current-distance relationship deviates from a simple exponential. C
Worked example
Problem. An electron in an STM tunnels through a vacuum gap of width \( d = 0.5 \) nm. The work function of the metal is \( \phi = 5.0 \) eV. Assume the electron energy \( E \) is negligible compared to \( \phi \) (so \( V_0 - E \approx \phi \)). Calculate the decay constant \( \kappa \) and the factor by which the tunnelling current decreases if the gap is increased to \( d' = 0.6 \) nm.
- Convert units to SI: \( d = 0.5 \times 10^{-9} \) m, \( \phi = 5.0 \times 1.602 \times 10^{-19} \) J = \( 8.01 \times 10^{-19} \) J. Electron mass \( m = 9.11 \times 10^{-31} \) kg, \( \hbar = 1.055 \times 10^{-34} \) J s.
- Calculate \( \kappa \): \[ \kappa = \sqrt{\frac{2m\phi}{\hbar^2}} = \sqrt{\frac{2(9.11 \times 10^{-31})(8.01 \times 10^{-19})}{(1.055 \times 10^{-34})^2}} \] \[ \kappa = \sqrt{\frac{1.459 \times 10^{-48}}{1.113 \times 10^{-68}}} = \sqrt{1.311 \times 10^{20}} \approx 1.145 \times 10^{10} \ \mathrm{m^{-1}} \]
- The current ratio is \( \frac{I'}{I} = \frac{e^{-2\kappa d'}}{e^{-2\kappa d}} = e^{-2\kappa(d' - d)} \). \[ \Delta d = 0.1 \times 10^{-9} \ \mathrm{m} \] \[ \frac{I'}{I} = e^{-2(1.145 \times 10^{10})(0.1 \times 10^{-9})} = e^{-2.29} \]
- Calculate the numerical value: \( e^{-2.29} \approx 0.101 \).
Units check. \( \kappa \) is in \( \mathrm{m^{-1}} \), \( d \) is in \( \mathrm{m} \), so the exponent \( \kappa d \) is dimensionless. The ratio is dimensionless.
Three quick tests
- Why does the tunnelling probability depend exponentially on the barrier width?
Answer
The wave function decays as \( e^{-\kappa x} \) inside the barrier. The probability is proportional to the square of the wave function, \( e^{-2\kappa x} \). Thus, the probability decays exponentially with width. - In alpha decay, why do higher energy alpha particles have shorter half-lives?
Answer
Higher energy means a smaller energy deficit \( (V_0 - E) \), which leads to a smaller \( \kappa \). A smaller \( \kappa \) means slower decay of the wave function inside the barrier, resulting in a higher transmission probability and thus a faster decay rate. - If the gap in an STM is halved, by what approximate factor does the current increase?
Answer
The current is proportional to \( e^{-2\kappa d} \). If \( d \) becomes \( d/2 \), the new current is proportional to \( e^{-\kappa d} \). The ratio is \( e^{-\kappa d} / e^{-2\kappa d} = e^{\kappa d} \). This is a large increase, dependent on the specific \( \kappa d \) value.
Problem set for PU-202 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.