Q026 · Electrons losing energy to atoms

Franck–Hertz Excitation

Why can a higher voltage make the current drop? Once the electrons have enough energy to excite the gas atoms, they lose it in inelastic collisions. Too slow to climb the small retarding voltage in front of the collector, they no longer reach it, and the current drops. The dips repeat each time the voltage rises by one more excitation energy, about every 4.9 volts for mercury.

Accelerate electrons through a gas and count how many reach the collector. Look at a single collision, scan the voltage and measure the spacing between the dips in current.

Recorded particles $\text{—}$

Collected fraction $\text{—}$

01 / Follow the energy

One collision can stop an electron reaching the collector.

Scan and save. Remove the gas and repeat. Then shift the contact voltage: does the whole pattern move, or does its spacing change?

Cutaway tube · cathode → accelerating grid → retarding gap → collector · microscopic collision markers are enlarged

Accelerating voltage

U/\mathrm V

11

Each scan voltage gets 200 independent electrons.

Try a comparison

Gas collision strength

L/\lambda

40

Retarding voltage

U_r/\mathrm V

1.5

Source energy scale

k_BT_s/\mathrm{eV}

0.12

Preparation & sampling

Contact offset

U_c/\mathrm V

1

Voltage scan spacing

\Delta U/\mathrm V

0.25

Sampling seed

s_0

7261

The probe only selects an exposure or voltage. Other changes start a fresh sample; save first. Nothing runs by itself.

Electrons fly forward in one dimension, and the gas has a single excitation channel at 4.9 eV. Above that threshold, the chance of a collision per unit distance is a fixed constant. Counts are simulated. The model leaves out elastic deflection, electrons caught by the grid, ionization, space charge and calibrated mercury cross sections.

02 / What did the particles do?

The current curve is built from counted electrons.

$U/\mathrm V$ · Blue: current sample. Amber: the most recently saved other sample. Drag to choose the probe point.

Show zero-flight-length limit

At the selected probe point

\text{—}

\text{—}

For a fixed injected flux, the recorded fraction equals the size of the normalized collector current. Whiskers are pointwise 95% Wilson intervals.

03 / Measure the spacing

Mark two valleys in the recorded current.

Click a recorded scan point in the chart below. Choose neighboring valleys yourself; the cursors do not consult the excitation energy.

A B

\text{—}

Only cursors on recorded scan voltages produce a spacing. This manual difference has grid and valley-selection uncertainty; it is not an automatic excitation-energy fit.

Inside the model · separate from the readout

The electron regains energy after a collision.

$x/L,\quad K/\mathrm{eV}$ · Last simulated flight at this voltage. Drops are actual collision events from its saved record; this diagnostic is not measured by the collector.

\text{—}

04 / Save the evidence

Save before changing the experiment.

Up to eight saved samples. Restoring one brings back its original settings and records; acquiring again starts a new run. The CSV also holds the simulated collision or recoil diagnostics.

Model, units & primary sources

K_g=K_0+e\max(0,U-U_c)-N\varepsilon,\quad\varepsilon=4.9\,\mathrm{eV}

The 1D accelerating region has uniform field; x runs from cathode to grid. Emission energies follow an exponential flux distribution. Excitation has a constant hazard per unit forward distance above threshold; each event removes one energy quantum. The collector accepts energies above the retarding barrier. Below the contact offset we clamp the accelerating field to zero; we do not model reversed extraction. The ideal reference assumes vanishing excitation flight length, not the finite-hazard data model. Neither curve is calibrated mercury data. Gas density is controlled by optical depth, independently of the source energy scale.

MIT Junior Lab · Franck–Hertz · Leybold · Hg tube geometry

Start with the essentials

Focus question: Why can a higher voltage make the current drop?
One-sentence intuition: The collector responds to the kinetic energy the electron has left when it arrives, not to the total energy it picked up along the way.

Core mathematical model

Kinetic energy at the grid

K_g=K_0+e\max(0,U-U_c)-N\varepsilon

The accelerating field adds energy at a steady rate, and each excitation takes away the same fixed quantum. Below threshold there are no inelastic losses.

Collection condition

K_g>eU_r,\qquad I/I_0=N_{\rm collected}/N_{\rm injected}

The current is normalized to a fixed injected flux. With the source unchanged, raising the retarding barrier can only turn back more electrons.

Spacing between dips in the ideal case

e\Delta U\simeq\varepsilon=4.9\,\mathrm{eV}

The estimate assumes an electron excites an atom almost as soon as it reaches threshold. A finite path before each collision, the spread of source energies and the discrete voltage steps all shift the observed minima.

Common difficulties

What was actually recorded?

Typical misconceptionThe collector measured every collision along the way.

Better mental modelThe collector only records whether each electron arrived. Collision positions and energies are shown separately as model diagnostics.

What the model covers

Typical misconceptionIf the simulation works, a real instrument has been calibrated.

Better mental modelElectrons fly forward in one dimension, and the gas has a single excitation channel at 4.9 eV. Above that threshold, the chance of a collision per unit distance is a fixed constant. Counts are simulated. The model leaves out elastic deflection, electrons caught by the grid, ionization, space charge and calibrated mercury cross sections.

Run the experiment

01
Make a prediction
Before scanning, predict whether raising the voltage always increases the collected current.
What to observe: The data panel starts empty. Record some data before you interpret anything.
02
Record a comparison
Scan the voltage and save the result, then repeat with the gas removed. Look at one electron flight at a current dip and another where the current has recovered.
What to observe: An electron can lose energy exciting an atom and then pick up energy again before it reaches the grid.
03
Try to break the explanation
Increase the contact offset and mark neighboring valleys before and after the shift. Then broaden the source energy spread.
What to observe: The voltage where the dips begin and the spacing between them are different quantities. Choose broad or coarsely sampled valleys with care.

Franck–Hertz Excitation

The current curve is built from counted electrons.

Mark two valleys in the recorded current.

The electron regains energy after a collision.

Save before changing the experiment.