How One Straight-Line Graph Measures Planck's Constant

Light spreads and interferes like a wave, yet it knocks electrons out of metals in separate packets. Electrons, in turn, behave like particles in a wire but diffract like waves off a crystal. This double behaviour is the dual nature of radiation and matter.

This lesson covers the photoelectric effect, measuring Planck's constant, the de Broglie wavelength, and the Davisson-Germer experiment.

**Light arrives as photons of energy $hf$; a photon gives all its energy to one electron, which escapes only if $hf$ exceeds the work function ${\varphi }_0$, so $K_{\text{max}}=hf-{\varphi }_0$ and the threshold frequency is $f_0={\varphi }_0/h$.

What the wave picture could not explain:**

- Below $f_0$, no electrons are emitted however bright the light
- Maximum kinetic energy depends on frequency, not intensity
- Emission starts with no measurable time delay
- Brighter light gives more photoelectrons, and so a larger current

Worked example. Light of 400 nm falls on a metal with ${\varphi }_0=2.25$ eV:

$$E=\frac{hc}{\lambda }=\frac{6.63\times 10^{-34}\times 3.0\times 10^8}{4.0\times 10^{-7}}=4.97\times 10^{-19}\text{ J}=3.11\text{ eV}$$

$$K_{\text{max}}=3.11-2.25=0.86\text{ eV}{f}_0=\frac{2.25\times 1.6\times 10^{-19}}{6.63\times 10^{-34}}=5.43\times 10^{14}\text{ Hz}$$

The threshold wavelength is $c/f_0=552$ nm, so green light just fails to eject electrons.

An everyday example. Automatic street lights that switch on at dusk use a light sensor whose output depends on how much light reaches it.

The substance. Intensity controls how many electrons leave; frequency controls how fast the fastest ones move.

**Since $e{V}_0=hf-{\varphi }_0$, a graph of stopping potential against frequency is a straight line of slope $h/e$, cutting the frequency axis at the threshold frequency $f_0$.

The method.** Shine light of several known frequencies on the same photocell, and for each one raise the reverse voltage until the photocurrent just stops. Plot $V_0$ against f:

$$V_0=\frac{h}{e}f-\frac{{\varphi }_0}{e}$$

Worked example. A cell gives $V_0=0.50$ V at $6.0\times 10^{14}$ Hz and $V_0=1.33$ V at $8.0\times 10^{14}$ Hz.

$$\text{slope}=\frac{1.33-0.50}{2.0\times 10^{14}}=4.15\times 10^{-15}\text{ V s}h=4.15\times 10^{-15}\times 1.6\times 10^{-19}=6.64\times 10^{-34}\text{ J s}$$

The line meets the frequency axis at $f_0=6.0\times 10^{14}-\frac{0.50}{4.15\times 10^{-15}}=4.8\times 10^{14}$ Hz, giving ${\varphi }_0=h{f}_0=3.2\times 10^{-19}$ J, about 2.0 eV.

An everyday example. School physics labs use a photocell with coloured filters to collect exactly these readings, one filter for each frequency.

The substance. Every metal gives a line of the same slope — only the intercept changes, because h is universal but ${\varphi }_0$ is not.

**De Broglie proposed that every moving particle has a wavelength $\lambda =\frac{h}{p}=\frac{h}{mv}$, and for an electron accelerated through a potential V this becomes $\lambda =\frac{h}{\sqrt{2meV}}$, about $\frac{1.227}{\sqrt{V}}$ nm.

Why it follows.** A photon has momentum $p=h/\lambda$. De Broglie suggested the same relation should hold for matter, so particles would show wave effects whenever their wavelength is comparable to the size of what they meet.

Worked example 1 — an electron. Accelerated through 100 V:

$$\lambda =\frac{6.63\times 10^{-34}}{\sqrt{2\times 9.1\times 10^{-31}\times 1.6\times 10^{-19}\times 100}}=1.23\times 10^{-10}\text{ m}=0.123\text{ nm}$$

That is similar to the spacing of atoms in a crystal.

Worked example 2 — a cricket ball. A 0.16 kg ball at 30 m s${}^{-1}$:

$$\lambda =\frac{6.63\times 10^{-34}}{0.16\times 30}=1.4\times 10^{-34}\text{ m}$$

An everyday example. A fast bowler's delivery never diffracts past the batter, because its wavelength is unimaginably smaller than any gap it could pass through.

The substance. For the same kinetic energy, heavier particles have shorter wavelengths, since $\lambda =h/\sqrt{2mK}$ — a proton's wavelength is far shorter than an electron's.

The Davisson-Germer experiment fired a beam of electrons at a nickel crystal and found strong reflection at one particular angle, exactly as diffracted waves would give, with a wavelength matching de Broglie's prediction.

The set-up. An electron gun accelerates electrons through a chosen voltage and aims them at a nickel crystal. A movable detector measures how many electrons scatter at each angle.

The result. At 54 V, a sharp peak appeared at a scattering angle of $50^{\circ }$, which is a glancing angle of $65^{\circ }$ on the crystal planes.

Worked example — checking the wavelength.

$${\lambda }_{\text{de Broglie}}=\frac{1.227}{\sqrt{54}}=0.167\text{ nm}$$

From X-ray studies, the relevant crystal planes are 0.091 nm apart, so Bragg's condition gives

$$\lambda =2d\sin \theta =2\times 0.091\times \sin 65^{\circ }=0.165\text{ nm}$$

The two values agree closely.

Significance. Only waves produce a sharp peak at one angle, so the experiment confirmed that matter has a wave nature and that $\lambda =h/p$ is correct.

An everyday example. Materials scientists use electron diffraction to identify the crystal structure of new alloys and minerals.

The substance. Electrons show wave and particle behaviour, but never both in the same measurement — the experiment chosen decides which one appears.