How To Make a Particle

Concept: Turning energy into matter has sharp thresholds: creating a particle of mass m always creates its antiparticle too, so the photon must bring at least 2mc². Near a heavy nucleus (which absorbs the recoil momentum but almost none of the energy) a photon converts once E_γ ≥ 2mc²(1 + m/M_N) ≈ 2mc², measured in the chamber (lab) frame. Any surplus becomes kinetic energy T = E_γ − 2mc² — visible as longer, straighter tracks, since the radius in the chamber field is r = p/qB. Climbing the beam opens rungs at 1.022 MeV (e⁺e⁻), 0.2113 GeV (μ⁺μ⁻), 0.2791 GeV (π⁺π⁻), 0.9874 GeV (K⁺K⁻), 1.877 GeV (p p̅), 3.554 GeV (τ⁺τ⁻), 3.739 GeV (D⁺D⁻) and 10.56 GeV (B⁺B⁻) — eight real, observable particle–antiparticle channels.
Serves: Modern physics / intro particle physics: pair production, antimatter, E = mc² energy budgeting, why colliders chase higher energy.
Inquiry: The beam starts parked below every rung with the chamber empty and the clock paused. Predict whether 1 GeV can make p p̅, then fire the beam and climb the ladder: each species switches on at exactly 2mc². "Rung above / below" parks the beam exactly on a threshold, where the pair is born at rest (T = 0).
Misconception: "E > mc² is enough." Creation is pairwise — charge (and, for the proton, baryon number) conservation doubles the bill: threshold 2mc², not mc². The faint thin ticks on the ladder mark mc²; the beam marker sails past them without lighting anything. A second trap: "protons can't be created" — they can, as p p̅. And a lone photon in empty space can never convert at any energy.
Caveats: Rates are schematic: the lightest open pair overwhelmingly dominates real photoproduction, so heavier channels are shown occasionally and tagged "rare channel". The photon's drawn wavelength is log-compressed, not to scale. √s for γ + Pb is ≈ 194 GeV at every setting of the beam knob — it is shown in The Physics sheet only, never as the control.

Principle	Where it appears
Predict, then check	The prediction panel sits first in the sidebar and the sim opens paused, below every rung, with an empty chamber — nothing on screen answers "is 1 GeV enough for p p̅?" until the student drives the beam there.
Cognitive conflict	The "Yes — 1 GeV > 0.938 GeV" chip encodes the forgotten-antiparticle error; its rebuttal appears only if chosen, and sends the student to the ladder rather than handing over the number.
Invariant vs varying	Threshold rungs are fixed at 2mc²(1+m/M_N) — the exact lab-frame threshold with the nuclear recoil; only the beam marker moves. Rungs stay unlabeled until the student's beam has actually reached them.
Operational quantities	Open pairs, heaviest species, its threshold, spare kinetic energy T and the next threshold are read off live — each row appearing only once a prompt needs it. Track radius is r = p/qB with the field and length scale declared on the canvas.
Formal layer on demand	Threshold, invariant mass, momentum, T and r = p/qB stay behind the ⚛︎ The Physics button, which opens the revision sheet.

Teach me Hide Text

Speed

Pair-production chamber

Guided Inquiry

1 · A photon and a nucleus

Let’s set the scene. A γ ray flies toward a lead nucleus in a 1 T chamber. New particles can appear only at sharp energies, and you’ll find them marked on the ladder. Right now the beam starts below every rung, so the chamber is empty.

2 · Protons from light?

Let’s aim high. A proton’s rest energy is 0.938 GeV. If the photon carried 1 GeV, do you think proton–antiproton pairs would appear?

3 · Surplus energy

Now let’s go above a rung. At the electron rung the pair is born at rest. If the photon energy rose well above that rung, what do you think the two tracks would do?

4 · Price a tau pair

Time to do the sums yourself. The τ lepton’s mass is 1.777 GeV/c², and pair creation pays two rest energies. So what is 2mc² for a τ⁺τ⁻ pair? Have a go before you pick.

5 · The rung in a PET scanner

Finally, a rung you might meet in real life. Hospital PET scanners detect 511 keV photons from e⁺e⁻ annihilation. Yet creating that pair costs 1.022 MeV, not 511 keV. Why do you think that is?

Listen

Photon energy E_γ, chamber frame

▾

0.5 MeVlog scale12 GeV

"Rung above / below" parks the beam exactly on a threshold — there the pair is born at rest (T = 0).

Open pair channels

▾

Eγ = 0.8 MeV

Pairs open—

Heaviest pair—

Threshold 2mc²(1+m/M)—

Kinetic energy T (heaviest pair)—

Next threshold—

Pair threshold (chamber frame)

Invariant mass of γ + nucleus

Momentum of each twin (chamber frame, M_N → ∞)

Kinetic energy T of the pair (chamber frame, M_N → ∞)

Track radius in the chamber field

√s is frame-invariant and is not what the beam knob sets: for γ + Pb it is ≈ 194 GeV even when E_γ is 1 MeV, because almost all of it is the nucleus' own rest energy. The threshold that matters here is the chamber-frame photon energy E_γ. A lone photon in empty vacuum cannot convert at any energy — energy and momentum cannot both be conserved; the heavy nucleus takes up the recoil momentum while absorbing almost none of the energy.

The Physics

Pair production · what this sim shows, the physics behind it, and every control

At a glance

A gamma ray passing a lead nucleus can turn into a particle and its antiparticle. This sim lets you set the photon's energy and watch which particles appear, how they move in a magnetic field, and why only certain energies work.

What you'll learn

How the energy needed to make a particle depends on its mass and its charge.
Where the energy above that cost goes, and how it shapes the tracks.
What role the nucleus plays.
How to read the energy ladder, and how to read a track's curvature in a magnetic field.

The set-up: a photon, a lead nucleus and a 1 T field, with the energy ladder alongside.

What's on screen

Photon γ (teal wave): flies in from the left along the dashed beam axis. Its drawn wavelength is schematic.
Pb nucleus (gold disc): the target, at rest.
Tracks: each particle is drawn in its own colour and ends in a dot labelled with its name. The pill under the chamber names the pair and its T.
Ladder (right): a log energy axis with one bar per pair. A bar fills in once the beam reaches it, and the shaded band reaches up to the beam energy.
Open pair channels (sidebar): which pairs the beam can make, appearing after the first pair.

See itFire a 30 MeV photon. Watch for: two tracks leaving the nucleus and curling in opposite directions, each ending in a labelled dot.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Photon energy E_γ	The photon's energy in the chamber frame, on a log scale. Moving it clears the tracks on screen.	0.5 MeV–12 GeV opens at 0.8 MeV
⚡ Fire γ	Sends a photon at the nucleus, then keeps sending one every few seconds. It also starts the clock.	button
↓ Rung below / ↑ Rung above	Parks the beam exactly on the next threshold below or above the current energy.	8 thresholds
Open pair channels	Which pairs are open, the heaviest one, its threshold, the T row (kinetic energy of the heaviest pair) and the next threshold. It stays hidden until a pair is open, and the last two rows appear when the Guided Inquiry or a See-it brings them in.	panel
▶ Play / ⏸ Pause	Runs or freezes the clock. The sim opens paused.	top bar
Speed	Plays the motion faster or slower on screen. The energies and tracks stay the same.	0.25×–4× opens at 0.5×
↻ Reset	Returns to 0.8 MeV with the chamber empty and the beam off.	top bar

Fixed: the target is a ²⁰⁸Pb nucleus (M_Nc² = 193.687 GeV), the chamber is 0.40 m wide, and B = 1 T into the page. When several pairs are open the sim draws the lightest most often and tags the others "rare channel".

A photon makes a pair, never a single particle

A photon has no electric charge, so the charges it leaves behind must add to zero. A particle can therefore appear only together with its antiparticle.

Charge conservation for a conversion near a nucleus N.

Each bar on the ladder is a pair, so the energy has to pay for two rest energies: roughly E_γ ≥ 2mc², not mc².
For the electron, mc² = 0.511 MeV, so the cheapest pair needs about 1.022 MeV. The sim opens at 0.8 MeV, above mc² but below 2mc².
For the proton, charge is only half the story. Baryon number must balance too, so the pair is p p̄.

See itFire the 0.8 MeV beam. Watch for: the photon reaching the nucleus and carrying on, with the chamber left empty.

Each pair switches on at a sharp threshold

The threshold in the chamber frame, where the nucleus is at rest.

Below the threshold nothing is made, however long you wait. At it, the pair appears with no energy to spare. The small factor (1 + m/M_N) is a surcharge for the nucleus's recoil (section 07).

pair	2mc²	threshold	surcharge m/M_N
e⁺e⁻	1.022 MeV	1.022 MeV	0.00026%
μ⁺μ⁻	0.2113 GeV	0.2114 GeV	0.055%
τ⁺τ⁻	3.554 GeV	3.586 GeV	0.92%
B⁺B⁻	10.56 GeV	10.85 GeV	2.7%

All eight thresholds on the sim's log axis. The beam marker sits at the opening 0.8 MeV.

See itPark the beam exactly on the first rung. Watch for: the first pair appearing, with the Threshold row equal to the photon energy.

Energy above the threshold becomes motion

Whatever the photon carries beyond the threshold is left over as the pair's kinetic energy, T = E_γ − 2mc². The sim's T readout subtracts the full threshold, so it differs only by the small surcharge.

The sim shares E_γ equally between the two particles and lets the heavy nucleus take up the momentum balance with no energy of its own:

Momentum of each particle, with the nucleus treated as infinitely heavy.

On the threshold p = 0: the pair is born at rest. As T grows, pc approaches E_γ/2 and the particles approach the speed of light. For e⁺e⁻, β = 0.94 at 3 MeV and 0.995 at 10 MeV.

See itFire a 10 MeV photon. Watch for: the T row reading about 9 MeV (10 MeV minus the 1.022 MeV threshold), and the tracks leaving the nucleus.

Track radius grows with momentum

The field bends the two particles in opposite directions, because their charges are opposite. Each follows a circle of radius

In practical units r (m) = p (GeV/c) ÷ (0.2998 · B (T)) for a charge of one unit.

Radius is proportional to momentum. More T means more p, a larger r, and a straighter track.
For e⁺e⁻: r = 4.7 mm at 3 MeV, 1.7 cm at 10 MeV, 17 cm at 100 MeV, and 1.7 m at 1 GeV, more than four chamber widths.
The two tracks leave the nucleus about mc²/pc either side of the beam axis, never wider than 45°. That is 20° at 3 MeV and 0.6° at 100 MeV.

E_γ

These controls move the sim too. The numbers are for the heaviest open pair, which the sim does not always draw.

See itFire 3 MeV, then 100 MeV. Watch for: the 3 MeV tracks curling tightly beside the nucleus, and the 100 MeV tracks sweeping out in wide arcs.

The nucleus takes the recoil momentum

In empty space a photon of energy E carries momentum E/c. A pair with that same total momentum always has more energy than E, because the pair has mass. No photon energy fixes this, so conversion in empty space is impossible.

A nearby nucleus absorbs the missing momentum. Being so heavy, it takes almost none of the energy. That small energy is the surcharge (1 + m/M_N).
The invariant mass √s of γ + Pb is not what the beam knob sets. It is 193.69 GeV at 1 MeV and only 205.34 GeV at 12 GeV, almost all of it the nucleus's own rest energy.

The invariant mass of photon plus nucleus, and the threshold written in terms of it.

The sim always has the nucleus in place, so it cannot show this section. The sheet gives only the reasoning.

Where it matters

A PET scan reads the first rung backwards: a pair annihilates into two photons.

PET scanners. A positron from the tracer annihilates with an electron into two 511 keV photons that leave back to back. Together they carry the same 1.022 MeV that the first rung charges.
Stopping gamma rays in lead. Above about 5 MeV, pair production is the main way lead absorbs gamma rays. This is why heavy-element shielding works at high photon energies.
Gamma-ray telescopes. The Fermi Large Area Telescope detects gamma rays by seeing them convert to e⁺e⁻ pairs in thin tungsten foils. The curve of the tracks gives the particles' momenta.

Go deeper: where the threshold comes from

The photon and the nucleus have invariant mass squared s = (M_Nc²)² + 2E_γM_Nc². The pair and the nucleus can all end up at rest in the centre-of-mass frame, so the threshold is √s = (M_N + 2m)c². Solving for E_γ gives exactly E_γ = 2mc²(1 + m/M_N).
If an atomic electron takes the recoil instead (triplet production), M_N becomes m_e and the threshold rises to 4m_ec² = 2.044 MeV. The sim models only the nucleus.

Summary

In one line. Near a nucleus, a photon makes a particle and its antiparticle once E_γ reaches 2mc²(1 + m/M_N). The energy above that becomes the pair's motion.

Key points

A photon is neutral, so charge (and baryon number) must balance: particles come in pairs and the bill is 2mc².
Each pair has a sharp threshold. The surcharge m/M_N is tiny for an electron and 2.7% for a B meson.
T = E_γ − 2mc² becomes momentum, so r = p/qB grows and the tracks straighten.
The nucleus takes the recoil momentum. E_γ in the chamber frame is the control, not √s.

Common mix-ups

"A photon with more than mc² can make the particle." It makes a pair, and the pair costs 2mc².
"A high enough photon energy will convert anywhere." Not in empty space, at any energy. A heavy body must take up the recoil.
"More energy means tighter curls." It is the other way round. Tight curls mean small momentum, as on the threshold.

Things to try on your own

Set E_γ to 1 GeV. Work out the radius of an electron track from pc ≈ 0.5 GeV, then compare with the live panel in section 06.
Predict the D⁺D⁻ threshold from 2mc²(1 + m/M_N), then park the beam on it with ↑ Rung above and compare.
Step up the ladder one rung at a time and watch the Open pair channels panel grow.