原子 Atomic Physics · Laboratory experiment

The Franck–Hertz experiment: quantised atomic energy levels

Aim: to accelerate electrons through mercury vapour (or neon), record the anode current against the accelerating voltage, and show from the equally spaced current maxima that atoms absorb energy only in discrete quanta. Determine the first excitation energy and the wavelength of the light it emits.

Franck–Hertz tube & control unit

X–Y recorder: anode current I_A against U₂

Turn the U₂ knob to trace points by hand, or record a full ramp. Click a maximum to add it to the table.

Observation table: current maxima

Graph: U_peak against order n

Theory

Electrons emitted by a hot cathode are accelerated by $U_2$ toward a grid, then must overcome a small retarding voltage $U_3$ to reach the anode. Collisions with atoms are elastic (negligible energy loss for $m_e \ll M$) until the electron energy reaches the first excitation energy $E_1$. An inelastic collision then removes exactly $E_1$, and the electron can no longer climb $U_3$, so the current drops. Raising $U_2$ by another $E_1/e$ lets electrons excite a second time, and so on:

$$ U_{peak}(n) = U_0 + n\,\frac{E_1}{e},\qquad \lambda = \frac{hc}{E_1} .$$

For Hg, $E_1$ = 4.89 eV (6¹S₀ → 6³P₁), so λ = 253.7 nm (UV). For Ne, the 3p levels lie near 18.7 eV. They decay by visible 3p → 3s emission, so glowing layers appear in the tube, one more for every 18.7 V. Watch them form in the apparatus view.

The simulation is Monte-Carlo electron transport

For each voltage, hundreds of electrons are tracked across the gap. Each gains energy from the field and excites an atom with probability $1 - e^{-n\sigma(E)\,dx}$ once above $E_1$. The gas density n (from the Hg vapour pressure at the oven temperature) sets the mean free path. No curve is drawn by hand: the peaks, their spacing, the smearing at low temperature and the ionisation onset all emerge from the model.

Procedure

Heat the Hg oven to 170–190 °C. Set U₁ ≈ 1.5 V and U₃ ≈ 1.5–2 V.
Increase U₂ slowly from 0 to 30 V while recording I_A (or record the full ramp).
Mark each maximum and fit U_peak against n. The slope is E₁/e and the intercept includes the contact potential.
Repeat at 150 °C and observe how the curve changes. Then switch to neon (U₃ ≈ 6 V, U₂ up to 80 V).

Things to notice

The first maximum is not at 4.9 V: the contact potential between the cathode and grid shifts all peaks. The spacing is what matters.
At low oven temperature, electrons travel far between collisions, overshoot E₁, the peaks smear, and ionisation (10.4 eV) makes the current rise steeply.

Viva questions

Why does the current not fall to zero at a minimum?

Electrons start with a spread of energies and collide at random positions, so some always reach the anode.

Why use a retarding voltage?

It rejects electrons that have just lost E₁ in an inelastic collision near the grid, which makes the dips sharp.

What did Franck and Hertz prove (Nobel Prize 1925)?

Atoms accept energy from electrons only in discrete amounts, which directly confirmed Bohr's quantised energy levels.