Dual Nature of Radiation and Matter

A metal contains a large number of free (conduction) electrons that move about randomly inside the metal but cannot normally escape its surface. This is because the surface behaves like a potential barrier — an electron trying to leave is pulled back by the net attraction of the positive ions left behind. To escape, an electron at the metal's surface must be given a minimum amount of extra energy. This minimum energy required to just eject an electron from a metal surface, without giving it any extra kinetic energy, is called the work function of that metal, denoted W₀.

Work function is a property of the metal and its surface condition, and is usually expressed in electron volts (eV), where 1 eV is the energy gained by an electron accelerated through a potential difference of 1 volt (1 eV = 1.6×10^-19 J). Typical work functions range from about 2 eV to 6 eV; alkali metals such as caesium, potassium and sodium have particularly low work functions (around 2-2.5 eV), which is why they are used as coatings in photosensitive devices.

Depending on how the extra energy is supplied, electron emission from a metal is classified into four types: thermionic emission, where suitable heating supplies enough thermal energy to free electrons (used in vacuum tubes and electron guns); field emission, where a very strong external electric field (~10⁸ V/m) pulls electrons out of the metal; photoelectric emission, where light of suitable frequency shines on the surface and supplies the energy; and secondary emission, where high-speed electrons or other particles striking the surface knock out additional electrons.

Since the work function is essentially a threshold energy, it can also be expressed in terms of a minimum, or threshold frequency ν₀, of electromagnetic radiation needed to eject an electron: W₀ = hν₀, where h is Planck's constant.

Worked Example: The work function of sodium is 2.3 eV. Find its threshold frequency and threshold wavelength.
W₀ = 2.3 eV = 2.3 × 1.6×10^-19 J = 3.68×10^-19 J.
ν₀ = W₀/h = 3.68×10^-19 / 6.63×10^-34 = 5.55×10¹⁴ Hz.
λ₀ = c/ν₀ = 3×10⁸ / 5.55×10¹⁴ = 5.41×10^-7 m = 541 nm.
So light of wavelength longer than 541 nm (lower frequency) cannot eject photoelectrons from sodium no matter how intense it is.

Heinrich Hertz first noticed (while studying electromagnetic waves) that sparking across a gap became easier when the electrodes were illuminated by ultraviolet light. Hallwachs and Lenard studied this effect systematically: they found that when ultraviolet light fell on a negatively charged zinc plate, the plate lost its charge, whereas a positively charged plate was unaffected — showing that negatively charged particles (electrons) were being ejected by the light.

A typical experimental set-up uses an evacuated glass/quartz tube containing a photosensitive emitter plate and a collector plate, connected to a battery and a sensitive ammeter (to measure photocurrent) with a variable, reversible potential difference applied between the plates.

Effect of intensity of light: for light of a fixed frequency (above threshold) and a fixed accelerating potential, the photoelectric current increases linearly with the intensity of incident light, eventually reaching a maximum value called the saturation current (when all emitted electrons reach the collector).

Effect of potential: if the potential of the collector is made increasingly negative with respect to the emitter, the photocurrent decreases and eventually becomes zero at a particular negative potential called the stopping potential (or cut-off potential) V₀. At this potential even the most energetic photoelectrons are just turned back, so eV₀ equals the maximum kinetic energy of the emitted electrons.

Effect of frequency: for a given material, the stopping potential is found to increase linearly with the frequency of incident light, but is completely independent of its intensity. Below a certain frequency, called the threshold frequency ν₀ (characteristic of the material), no photoelectrons are emitted at all, however intense the light. Also, photoelectric emission is found to be an instantaneous process — electrons appear within about 10^-9 s of illumination, with no measurable time lag, even for very dim light.

These observations are summarised as the laws of photoelectric effect: (i) photocurrent is directly proportional to intensity, for a fixed frequency above threshold; (ii) there exists a threshold frequency below which no emission occurs regardless of intensity; (iii) above the threshold, the maximum kinetic energy (and hence stopping potential) of photoelectrons increases linearly with frequency and is independent of intensity; (iv) emission is instantaneous, with no observable time delay.

Classical wave theory predicts that a more intense wave should simply supply more energy to each electron, so it cannot explain why K_max is independent of intensity, why there is a threshold frequency, or why emission is instantaneous for arbitrarily weak light (energy should need time to accumulate). This failure prompted Einstein's photon explanation.

Worked Example: In an experiment, the stopping potential was found to be 0.5 V for light of frequency 5×10¹⁴ Hz and 1.5 V for frequency 7.5×10¹⁴ Hz. Find the slope of the V₀ versus ν graph and hence estimate Planck's constant.
Slope = ΔV₀/Δν = (1.5 - 0.5) / (7.5×10¹⁴ - 5×10¹⁴) = 1.0 / (2.5×10¹⁴) = 4.0×10^-15 V·s.
Since slope = h/e, h = 4.0×10^-15 × 1.6×10^-19 = 6.4×10^-34 J s — close to the accepted value, confirming Einstein's linear relationship.

Einstein proposed that light of frequency ν consists of discrete packets of energy called photons, each carrying energy E = hν, and travelling at speed c. In the photoelectric effect, a photon is absorbed entirely by a single electron in a one-to-one interaction — an electron cannot collect energy from multiple photons over time. If the photon's energy hν is less than the work function W₀, the electron simply cannot escape, however many such (insufficiently energetic) photons arrive — this explains the threshold frequency. If hν ≥ W₀, the electron is emitted immediately (explaining the absence of any time lag) with the leftover energy appearing as kinetic energy. Increasing the intensity of light simply increases the number of photons striking the surface per second, which increases the number of photoelectrons (current) but not the energy each one carries — this explains why K_max is independent of intensity.

This reasoning gives Einstein's photoelectric equation: the maximum kinetic energy of an emitted photoelectron equals the photon energy minus the work function: K_max = hν - W₀ = h(ν - ν₀). Since eV₀ = K_max, this can also be written as eV₀ = hν - W₀, which is exactly the linear relationship between stopping potential and frequency observed experimentally, with slope h/e and intercept -W₀/e.

The photon picture also assigns light quanta a momentum, even though photons are massless: since energy and momentum for light are related by E = pc (as in electromagnetic theory), a photon of frequency ν (wavelength λ) has momentum p = E/c = hν/c = h/λ. This particle-like momentum is what makes light exert radiation pressure and is essential in explaining phenomena like the Compton effect.

Worked Example 1: Light of wavelength 400 nm is incident on a metal of work function 2.0 eV. Find the maximum kinetic energy of the photoelectrons and the stopping potential.
Photon energy E = hc/λ = (6.63×10^-34 × 3×10⁸) / (400×10^-9) = 4.9725×10^-19 J = 3.11 eV (dividing by 1.6×10^-19).
K_max = E - W₀ = 3.11 - 2.0 = 1.11 eV.
Stopping potential V₀ = K_max/e = 1.11 V.

Worked Example 2: Find the momentum of a photon of wavelength 500 nm.
p = h/λ = 6.63×10^-34 / (500×10^-9) = 1.326×10^-27 kg m/s.

Since light, traditionally thought of as a wave, was shown by the photoelectric effect to also have particle-like (photon) properties with momentum p = h/λ, de Broglie proposed the converse: material particles, traditionally thought of as particles, should also possess wave-like properties. He hypothesised that a particle of momentum p has an associated wavelength, called the de Broglie wavelength, given by λ = h/p = h/(mv), where m is the mass and v the speed of the particle.

Because Planck's constant h is extremely small (6.63×10^-34 J s), the de Broglie wavelength of everyday macroscopic objects is immeasurably tiny and has no observable consequence, while for microscopic particles like electrons — which have very small mass — the associated wavelength can be comparable to atomic dimensions or X-ray wavelengths, making wave effects like diffraction observable.

For a charged particle of charge q and mass m accelerated from rest through a potential difference V, its kinetic energy equals qV, so p = √(2mqV), giving λ = h/√(2mqV). For an electron (q = e), this reduces to the widely used practical formula λ = 1.227/√V nm, where V is in volts.

The de Broglie hypothesis also gave a physical explanation for Bohr's quantization of angular momentum in the hydrogen atom: an electron in a stable orbit was interpreted as a standing de Broglie wave that fits exactly around the circumference, so 2πr = nλ (n = 1, 2, 3, ...). Substituting λ = h/(mv) into this condition immediately gives the Bohr quantization rule for angular momentum, L = mvr = nh/2π — showing that quantization is a natural consequence of the wave nature of the electron.

Worked Example 1: Find the de Broglie wavelength of an electron accelerated through a potential difference of 100 V.
λ = 1.227/√100 = 1.227/10 = 0.1227 nm ≈ 1.23×10^-10 m.
(Check via λ = h/√(2meV): 2meV = 2 × 9.11×10^-31 × 1.6×10^-19 × 100 = 2.915×10^-47; √(2meV) = 5.40×10^-24 kg m/s; λ = 6.63×10^-34/5.40×10^-24 = 1.23×10^-10 m = 0.123 nm — matches.)

Worked Example 2: Find the de Broglie wavelength of a 0.15 kg ball moving at 30 m/s, to see why wave effects are unobservable for macroscopic objects.
λ = h/(mv) = 6.63×10^-34 / (0.15 × 30) = 6.63×10^-34/4.5 = 1.47×10^-34 m — vastly smaller than any measurable length, so no wave behaviour is ever observed for the ball.

Clinton Davisson and Lester Germer accelerated a beam of electrons from an electron gun through a known potential difference and directed it onto the surface of a nickel crystal inside a vacuum chamber. A movable detector measured the intensity of electrons scattered at different angles from the crystal surface.

If electrons behaved purely as classical particles, the scattered intensity would vary smoothly with angle. Instead, Davisson and Germer observed a pronounced maximum in the scattered electron intensity at a specific angle for a given accelerating voltage — a diffraction pattern strikingly similar to the pattern produced when X-rays are diffracted by a crystal lattice, where the regularly spaced atomic planes act as a diffraction grating. This is analogous to the Bragg diffraction condition for waves reflecting off crystal planes, nλ = d sinθ, where d is the interplanar spacing and θ is measured from the crystal planes.

Crucially, the wavelength that had to be assigned to the electrons to explain the observed diffraction angle matched almost exactly the de Broglie wavelength λ = h/p calculated from the electron's momentum at that accelerating voltage. This quantitative agreement, obtained independently from a completely different phenomenon (diffraction, rather than any energy measurement), was the first direct experimental confirmation that matter has an intrinsic wave nature, exactly as de Broglie had hypothesised. Davisson and Germer received the Nobel Prize in Physics for this discovery.

This wave nature of fast electrons is also the physical basis of the electron microscope: since accelerated electrons can be given a de Broglie wavelength thousands of times shorter than the wavelength of visible light, electron microscopes achieve far higher resolving power than optical microscopes, allowing scientists to image structures as small as individual molecules.

Worked Example: In the Davisson-Germer experiment, electrons were accelerated through 54 V. Calculate their de Broglie wavelength and compare with the experimentally observed diffraction wavelength of about 0.165 nm.
Using λ = 1.227/√V nm: λ = 1.227/√54 = 1.227/7.348 = 0.167 nm.
This theoretical value (0.167 nm) is in excellent agreement with the wavelength (~0.165 nm) obtained independently from the electron diffraction pattern, confirming the de Broglie relation.