Light behaves as a wave in interference and diffraction, yet knocks electrons out of metals as though it were a stream of particles — and matter, in turn, shows wave behaviour when it is small enough. This chapter is a reliable source of 2–3 NEET questions every year, usually numerical (stopping potential, threshold frequency, de Broglie wavelength) or graph-based, so both the concepts and the working formulas must be at your fingertips.
Electron Emission and the Work Function
Free electrons wander inside a metal but cannot leak out on their own: the positive ion lattice holds them back with an inward attractive pull at the surface. To escape, an electron must be supplied a minimum amount of energy called the work function of the metal, written $\phi_0$ (or $W$). It is a property of the metal and its surface condition, and is usually quoted in electron volts.
$$1\ \text{eV} = 1.602\times10^{-19}\ \text{J}$$
Typical values: caesium ≈ 2.14 eV, sodium ≈ 2.75 eV, zinc ≈ 4.3 eV, platinum ≈ 5.65 eV. Metals with low work functions (alkali metals) emit electrons even with visible light, which is why photocells use caesium-coated cathodes.
The required energy can be delivered in four ways:
- Thermionic emission — heating the metal so that thermal energy of electrons exceeds $\phi_0$ (used in cathode ray tubes, old valves).
- Field (cold) emission — applying a very strong external electric field ($\sim10^8$ V/m) that pulls electrons out.
- Photoelectric emission — illuminating the surface with light of suitable frequency; the emitted electrons are called photoelectrons.
- Secondary emission — bombarding the surface with fast-moving particles (usually electrons) that transfer energy on impact.
Only the third is developed in detail in this chapter, but the list itself is examinable.
The Photoelectric Effect: Experimental Laws
Hertz noticed that a spark gap discharged more easily when illuminated with ultraviolet light; Hallwachs and Lenard traced this to electron ejection from the illuminated metal. The standard laboratory arrangement uses an evacuated glass tube with a photosensitive plate (emitter, C) and a collector plate (A), a variable potential difference between them, and a microammeter to read the photocurrent. A quartz window is used because ordinary glass absorbs UV.
Key experimental behaviour:
- Effect of intensity (at fixed frequency above threshold): photocurrent is directly proportional to intensity. More intense light means more photons per second, hence more electrons per second — but not faster electrons.
- Effect of collector potential: making A more positive increases the current until it saturates — every emitted electron is being collected. Reversing the polarity (A negative) retards the electrons; at a particular negative value $V_0$, called the stopping potential or cut-off potential, the current becomes exactly zero. Then $$eV_0 = K_{max} = \tfrac12 m v_{max}^2$$
- Stopping potential is independent of intensity but increases linearly with frequency of the incident light. Curves for different intensities at the same frequency saturate at different heights but meet the voltage axis at the same $V_0$.
- Threshold frequency: for every metal there is a minimum frequency $\nu_0$ below which no emission occurs, no matter how intense the light or how long you wait. Equivalently there is a threshold wavelength $\lambda_0 = c/\nu_0$, and emission needs $\lambda \le \lambda_0$.
- Emission is instantaneous — the time lag is less than $10^{-9}$ s even for feeble light.
Classical wave theory fails on three counts: it predicts that energy of electrons should grow with intensity, that any frequency should work if you wait long enough, and that weak light should show a measurable time delay. None of these is observed.
Einstein's Photon Explanation
Einstein proposed that radiation of frequency $\nu$ is not a continuous wave as far as energy exchange is concerned, but a stream of quanta (photons), each carrying energy
$$E = h\nu = \frac{hc}{\lambda}, \qquad h = 6.63\times10^{-34}\ \text{J s}$$
One photon is absorbed by one electron — a one-shot, all-or-nothing transaction. Part of the photon's energy pays the escape cost $\phi_0$; the remainder appears as kinetic energy. For an electron emitted from the surface (which retains maximum energy):
$$K_{max} = h\nu - \phi_0 = h(\nu - \nu_0), \qquad \phi_0 = h\nu_0$$
and combining with the stopping potential,
$$eV_0 = h\nu - \phi_0 ;\Rightarrow; V_0 = \left(\frac{h}{e}\right)\nu - \frac{\phi_0}{e}$$
This single equation explains everything:
- Below $\nu_0$, $h\nu < \phi_0$ and no electron can escape — intensity is irrelevant because energy is not accumulated from many photons.
- Intensity means photons per second; doubling it doubles the number of electrons (current) without changing each electron's energy.
- Higher frequency means a more energetic photon, hence larger $K_{max}$ and larger $V_0$.
- Absorption is a single instantaneous event, so there is no time lag.
The $V_0$ versus $\nu$ graph is the workhorse of exam questions: a straight line whose slope is $h/e$ (same for all metals), whose intercept on the $\nu$-axis is $\nu_0$, and whose intercept on the $V_0$-axis is $-\phi_0/e$. Millikan's careful measurement of this slope gave an independent value of $h$, confirming Einstein's theory (1921 Nobel Prize).
Useful photon properties: a photon has zero rest mass, travels at $c$ in vacuum, is electrically neutral, carries momentum $p = h\nu/c = h/\lambda$, and is unaffected by electric or magnetic fields. In a photon–electron collision total energy and total momentum are conserved, but photon number need not be. A handy shortcut: $E(\text{eV}) = 1240/\lambda(\text{nm})$.
Wave Nature of Matter: de Broglie's Hypothesis
If radiation, long treated as a wave, has particle attributes, de Broglie argued that symmetry demands that material particles have wave attributes. He assigned to a particle of momentum $p$ a wavelength
$$\lambda = \frac{h}{p} = \frac{h}{mv}$$
called the de Broglie wavelength. Note that the expression contains no reference to charge — it applies to electrons, neutrons, atoms, cricket balls alike. It is precisely because $h$ is so tiny that macroscopic objects have unmeasurably small wavelengths: a 100 g ball at 10 m/s has $\lambda \sim 10^{-34}$ m, far below any detectable scale, whereas an electron easily reaches the 0.1 nm range comparable to crystal spacings.
Useful forms:
- In terms of kinetic energy: $\lambda = \dfrac{h}{\sqrt{2mK}}$
- For a charge $q$ accelerated from rest through potential $V$ (non-relativistic): $\lambda = \dfrac{h}{\sqrt{2mqV}}$
- For an electron: $\lambda \approx \dfrac{1.227}{\sqrt{V}}$ nm, with $V$ in volts. So a 100 V electron has $\lambda \approx 0.123$ nm.
- At temperature $T$, thermal particles have $K = \tfrac32 k_BT$, so $\lambda = h/\sqrt{3mk_BT}$.
For the same accelerating voltage, a heavier particle has a shorter wavelength ($\lambda \propto 1/\sqrt{m}$): a proton's wavelength is about 1/43 of an electron's. For the same speed, again $\lambda \propto 1/m$. For the same kinetic energy, $\lambda \propto 1/\sqrt{m}$.
De Broglie also used this idea to justify Bohr's angular momentum condition: an allowed orbit is one whose circumference contains a whole number of electron wavelengths, $2\pi r = n\lambda$, which rearranges to $mvr = nh/2\pi$.
Davisson–Germer Experiment and the Meaning of Duality
Electron waves were confirmed by Davisson and Germer (1927), who fired a beam of electrons from an electron gun at a single crystal of nickel and measured the intensity of electrons scattered at various angles using a movable detector.
- Electrons from a heated filament were accelerated through a variable potential (roughly 40–70 V).
- The regularly spaced nickel atoms acted as a natural diffraction grating.
- The scattered intensity showed a distinct maximum at a scattering angle of 50° for an accelerating voltage of 54 V — a pattern only explicable as constructive interference of waves.
- The measured wavelength, 0.165 nm, agreed closely with the de Broglie prediction $1.227/\sqrt{54} = 0.167$ nm.
Modern electron microscopes and electron/neutron diffraction studies of crystals are direct descendants of this result. Neutron diffraction, using thermal neutrons, is especially useful for locating light atoms in crystals.
The final picture: light and matter are neither classical waves nor classical particles. Which face shows up depends on the experiment — interference and diffraction reveal wave character, while photoelectric emission and Compton-type collisions reveal particle character. Crucially, the two faces are never displayed in the same measurement, and the wave associated with a particle is not a physical vibration in space but a probability amplitude: its intensity at a point tells you the probability of finding the particle there.
Common Mistakes and Exam Traps
- Confusing intensity with frequency. Increasing intensity raises the saturation photocurrent (number of electrons) and never changes the stopping potential or $K_{max}$; increasing frequency raises $K_{max}$ and $V_0$ but does not by itself raise the current. If the frequency is below $\nu_0$, no amount of intensity produces any current.
- Misreading the $V_0$–$\nu$ graph. The slope is universal ($h/e$), so lines for different metals are parallel; only the intercepts differ. Students often assume the metal with the larger threshold frequency has a steeper line.
- Using $\lambda = h/mv$ with the photon's "mass" or plugging $\phi_0$ in eV into an equation whose other terms are in joules. Keep units consistent: either convert $\phi_0$ to joules, or work entirely in eV using $E = 1240/\lambda(\text{nm})$ eV.
- Forgetting that $K_{max}$ is a maximum. Electrons emitted from below the surface lose extra energy, so photoelectrons emerge with all energies from 0 up to $K_{max}$; the stopping potential corresponds only to the fastest ones.
- Mass and wavelength dependence. For equal accelerating potentials, $\lambda \propto 1/\sqrt{m}$, not $\propto \sqrt{m}$ — and an uncharged particle like a neutron cannot be accelerated by a potential difference at all, so use $\lambda = h/\sqrt{2mK}$ there.