Wave-particle duality (3.12.2)

In response to the limitations of Newton’s corpuscular theory of light, Christiaan Huygens developed the wave theory of light in the late 1600s.

According to Huygens, light behaves as a wave rather than as a stream of particles. So he introduced a concept now known as Huygens’ principle:

• Every point on a wavefront acts as a source of secondary wavelets that spread out in all directions at the same speed as the original wave.

• The new wavefront at any later time is found by drawing a surface tangential to these wavelets.

This principle enables the prediction of how waves propagate, including diffraction and refraction.

Advantages of the wave theory:

• Huygens’ wave model explained diffraction and interference, phenomena that Newton’s corpuscular theory could not account for.
• For example, Young’s double-slit experiment (1801) provided experimental confirmation of Huygens’ ideas, showing that light produces an interference pattern when passing through two slits; behaviour that is characteristic of waves.

In Young’s double-slit experiment, light from a single source passes through two narrow slits, producing two coherent waves with a constant phase difference. The waves then interfere either constructively or destructively:

• Constructive interference occurs when they arrive in phase (path difference = where is an integer), producing bright fringes.

• Destructive interference occurs when they arrive out of phase (path difference = where is an integer), producing dark fringes.

For particles, we would expect two bright fringes adjacent to each of the slits with no central bright fringe.

The regular spacing of bright and dark fringes is direct evidence of wave behaviour, as only waves can superimpose in this way.

Electromagnetic (EM) waves are composed of an oscillating electric and magnetic field adjacent to one another:

• The two fields oscillate perpendicular to each other and to the direction of energy transfer (they are transverse waves).
• The electric and magnetic fields are in phase. Their peaks and troughs occur simultaneously.
  In a vacuum, all EM waves travel at the speed of light

All waves in the electromagnetic spectrum are fundamentally the same type of wave. The divisions into categories (such as infrared or ultraviolet) are human-made for convenience, based on typical sources, uses, and effects.

In 1887, Heinrich Hertz provided the first experimental proof that radio waves and, therefore, electromagnetic waves exist, confirming Maxwell’s theory:

• He used a spark-gap transmitter, which consisted of two metal spheres connected to an induction coil and a capacitor, to produce high-voltage sparks. The sparks emitted radio-frequency electromagnetic waves.
• A receiver loop, made of a single wire with a small spark gap, received the incoming radio waves. The radio waves then induced oscillating currents in the wire, causing visible sparks to appear.

The transmitter and receiver sparks appeared almost simultaneously. This was evidence that the waves travelled at extremely high speeds, supporting Maxwell’s prediction of electromagnetic wave propagation.

Hertz showed that electromagnetic waves share the same behaviours as light. They can be reflected, refracted, polarised, and produce interference patterns. He calculated the wave speed using the following experimental setup:

1. In Hertz’s experiment, a reflecting metal sheet was placed opposite a wave transmitter to reflect the waves. This created standing waves between the transmitter and the reflector.
2. A receiver loop was moved along the wave path, detecting positions where sparks were weakest (nodes) and strongest (antinodes).
3. The distance between two adjacent nodes was equal to half a wavelength,
4. The frequency of the waves was known and determined from the transmitter. Hertz used the wave equation, to calculate the speed of the wave. Hertz discovered that it was equal to the speed of light, confirming that light is an electromagnetic wave.

In 1849, Fizeau used mechanical timing to estimate the speed of light, employing a spinning cogwheel and a distant mirror. The diagram below shows the setup used:

1. A beam of light strikes a partially reflecting mirror and is directed through a gap in the cogwheel towards a distant mirror.
2. The light is reflected by the distant mirror and travels back through the same gap in the cogwheel (if it is stationary or rotating at a low frequency) to a detector.
3. The rotation frequency of the cogwheel was steadily raised until the light reflected from the distant mirror was intercepted by the subsequent tooth on the cogwheel. This resulted in a complete absence of detected light.

A black body radiator is a theoretical object that absorbs and emits radiation at all wavelengths. While true black bodies are ideal and do not exist in reality, stars provide the closest real-world approximation.

The spectrum of radiation emitted by a black body is determined solely by its temperature.

The intensity–wavelength graph for black bodies shows the relationship between the temperature and the peak wavelength of emitted radiation for different objects. As the temperature in kelvin rises, the peak wavelength reduces, and the intensity increases.

The classical Rayleigh-Jeans law treats radiation as continuous waves. The law predicts that the intensity is inversely proportional to the fourth power of the wavelength:

This is in agreement with experimental data for long wavelengths. However, it predicts that the intensity will approach infinity as the wavelength approaches zero.

Experimental results show that the intensity reaches a maximum before declining in the ultraviolet region, where the classical model fails dramatically.

This failure is known as the ultraviolet catastrophe.

The photoelectric effect is the phenomenon where electrons are emitted from a metal surface when light with a specific minimum frequency is incident upon it.

According to classical wave theory, the energy of the emitted electrons should depend on the intensity of the incident light. Therefore, low-frequency light should eventually eject electrons with enough exposure time. However, observations showed that:

• there is no electron emission below a threshold frequency regardless of intensity,
• emission is instantaneous above , and that
• electron energy depends on frequency, not intensity.

In photoelectric experiments, the kinetic energy of the electrons is not measured directly. Instead, the voltage required to just stop most of the energetic electrons is measured: this is known as the stopping potential,

At this voltage, the electric field does work on each electron to bring it to rest. Therefore:

A plot of the stopping potential against the frequency is a straight line. This is because Einstein’s photoelectric equation may be rewritten as:​

• The gradient of the line is
• The intercept on the frequency axis (x-intercept) is the threshold frequency,
• The intercept on the stopping potential axis (y-intercept) is

De Broglie proposed that all matter particles exhibit wave-like properties. He formulated an equation relating a particle’s wavelength to its momentum given by:

De Broglie’s hypothesis was confirmed when electron beams passing through thin metal foils produced concentric diffraction rings, similar to those observed in X-ray diffraction patterns. This gave clear evidence that electrons have wave-like behaviour.

Increasing the speed of the electrons increased the diffraction angle. This supported the wave model because increasing particle speed (and momentum) would decrease The diffraction grating equation:

This suggests that a larger produces larger diffraction angles. Therefore, reducing by increasing the speed results in smaller diffraction angles: this is what was observed in experiments.

The workings of a transmission electron microscope (TEM) is outlined below:

1. Electrons from an electron gun are accelerated by a high voltage and pass through a condenser lens, which bends them into a wide beam that hits the sample.
2. Electrons travelling through the middle of the lens travel undisturbed; those near the edges are bent inwards by a magnetic field towards the axis.
3. An objective lens and an intermediate lens produce the first image of the sample.
4. A projector lens magnifies this image and projects it onto a fluorescent screen.

• The resolution depends on the electron wavelength; faster electrons (shorter wavelength) provide more detail.
• Because the electrons are emitted from a hot cathode, they have a range of speeds (dependent on their temperature). This causes them to take slightly different paths through the lens, producing lens aberrations that blur the image.
• Thicker samples slow electrons down, increasing their wavelength and reducing resolution.

Electrons exhibit wave-like behaviour similar to light. Light is able to pass through a very thin metal film because its wave amplitude is not completely attenuated. Similarly, if a barrier is sufficiently thin, the amplitude of an electron’s matter wave is also not reduced to zero.

As a result, electrons can penetrate the barrier even though they do not have enough energy to overcome it; this phenomenon is known as quantum tunnelling.

The workings of a scanning tunnelling microscope (STM) are outlined below:

1. A sharp tip is positioned approximately 1 nm above a conductive surface, controlled with extreme precision (to 0.001 nm) by piezoelectric devices.
2. Electrons at the surface of the target sample are confined by a Coulomb (potential) barrier, which they normally cannot cross.
3. Due to their wave-like nature, a small fraction of electrons can quantum tunnel across the gap, creating a tunnelling current.

The tunnelling probability, and therefore the current, decreases exponentially with the distance between the tip and surface.

A STM has two modes of operation:

1. In constant-current mode, a feedback circuit moves the tip up or down to keep the current the same, creating a map of the surface’s shape at the scale of atoms.
2. In constant-height mode, the height stays fixed, and changes in the tunnelling current are used to form the image.

It is important to know and understand the differences between a transmission electron microscope (TEM) and a scanning tunnelling microscope (STM):