2.3.18 · D5Modern Physics
Question bank — Nuclear structure — protons, neutrons, nuclear forces
Before we start, every symbol used on this page is defined here — nothing below leans on you remembering it from elsewhere.
Two pictures carry the whole page. Look at them before answering — the questions keep pointing back to them.
Figure 1 — the nucleon–nucleon potential. This is the single most useful graph in nuclear physics: it shows the energy between two nucleons as you slide them from far away to touching. A downward dip means "they'd rather be here" (attraction); an upward wall means "keep out" (repulsion).
Figure 2 — saturation vs. packing. The left cartoon is the wrong picture (everyone pulls everyone); the right is the true picture (each ball only velcros to the few balls touching it). This one difference is why binding energy grows like , not .
True or false — justify
True or false: Two protons a metre apart feel the strong nuclear force pulling them together.
False. In Figure 1 the dip has completely flattened to zero by fm; a metre is fm out, so only Coulomb repulsion acts, pushing them apart.
True or false: A neutron and a proton attract each other via the strong force about as strongly as two protons do.
True. The nuclear force is charge independent — –, –, and – strong attractions are nearly equal, because the force ignores electric charge entirely.
True or false: Because it is bound, a helium nucleus weighs more than its 2 protons + 2 neutrons weighed separately.
False. Binding releases energy, and that lost energy carries away mass (). The bound nucleus is lighter (this is just in reverse).
True or false: A larger nucleus (bigger , i.e. more balls) is denser than a small one.
False. Both mass () and volume () grow linearly in , so they cancel: density is the same for every nucleus ( kg/m³ — the estimate is worked below).
True or false: The strong nuclear force is always attractive.
False. In Figure 1 the curve dips (attraction, – fm) but rises into a wall ==below fm== (a repulsive "hard core") that stops the nucleus from collapsing to a point.
True or false: Doubling the number of nucleons roughly doubles the total binding energy.
True. Because the force is saturated (Figure 2, right) — each nucleon bonds only to nearest neighbours — , not .
True or false: Neutrons make a heavy nucleus less stable because they add mass.
False. Neutrons add strong-force attraction with no extra Coulomb repulsion, so extra neutrons help heavy nuclei stay bound. That's why the stability line bends toward .
True or false: The radius formula assumes nucleons are packed at roughly constant density.
True. Constant density means volume number of nucleons (), and gives the cube root directly; is the size of one ball's share.
True or false: Isotopes of an element differ in the number of protons ().
False. ==Isotopes share == (same proton count, same element) and differ in , hence in . It's isobars that differ in while sharing .
Spot the error
"Protons all repel each other, so a nucleus with many protons simply cannot exist." — find the flaw.
The reasoning forgets the second, stronger force. Coulomb repulsion is real, but inside fm the strong force (the deep dip in Figure 1) is larger and wins. Repulsion doesn't vanish — it just loses.
"The strong force pulls every nucleon toward every other nucleon, like gravity does." — find the flaw.
That's the left cartoon in Figure 2. If it were true, binding energy would grow like the number of pairs . Experiment shows , forcing the right cartoon: saturation, each nucleon bonds only to its immediate neighbours.
"Density must rise for heavy nuclei because you cram in more nucleons." — find the flaw.
You cram in more nucleons and the volume grows to fit them (). Mass and volume scale together, so their ratio (density) stays constant.
"Yukawa's pion has mass, which is why the strong force reaches infinitely far." — find the flaw.
Backwards. A massive mediator can only be "borrowed" briefly (Heisenberg uncertainty principle, ), so it travels only fm — a finite range, which is exactly the width of the dip in Figure 1. A massless mediator (the photon) is what gives infinite range.
"To find binding energy, subtract the parts' mass from the nucleus's mass: ." — find the flaw.
The subtraction is reversed. The nucleus is lighter, so (a positive number), and .
" and are isotopes because they have the same mass number." — find the flaw.
Same but different makes them isobars, not isotopes. Isotopes must share (be the same element).
"The nuclear force is stronger between two protons than between a proton and a neutron, because protons also feel electricity." — find the flaw.
Electric force is a separate force, not part of the nuclear force. The nuclear attraction is charge-independent and nearly equal for –, –, –; the protons' extra Coulomb repulsion actually makes the net – situation weaker.
Why questions
Why can two positive protons stay glued together at all, when like charges repel?
At fm the strong nuclear force (the deep dip in Figure 1, stronger than EM) overwhelms Coulomb repulsion; the protons are simply too close for repulsion to win.
Why does the strong force have a short range, while gravity and electromagnetism reach forever?
Its mediator (the pion) has mass; borrowing energy to make a massive particle can only last a short time, limiting its travel to fm. Massless mediators (graviton, photon) impose no such time limit, so those forces are infinite-ranged.
Why do heavy stable nuclei have more neutrons than protons?
Coulomb repulsion grows like (every proton pushes every other proton — that's the "everyone-to-everyone" left cartoon), but the saturated strong attraction grows only . Extra neutrons add attraction without adding repulsion, restoring the balance.
Why is the total binding energy roughly proportional to rather than to ?
Because of saturation (Figure 2, right): each nucleon bonds to a fixed small number of neighbours, so the count of bonds grows like the number of nucleons (), not like the number of all possible pairs ().
Why does a nucleus weigh less than the sum of its free nucleons?
Assembling it releases the binding energy, and by that departing energy carries away mass. The missing mass is called the mass defect.
Why must the strong force be repulsive at very short range and not just attractive?
If it were purely attractive, nucleons would collapse onto each other with no lower size limit. The repulsive wall below fm in Figure 1 sets a minimum spacing, giving nuclei their roughly constant density.
Edge cases
What is the strong force between two nucleons separated by exactly fm?
Essentially zero — well past the flat tail of Figure 1's dip (– fm). Only the (weaker, long-range) Coulomb push survives between charged nucleons.
For a single free neutron (no other nucleons around), what strong-force binding does it feel?
None — the strong force needs a neighbour within a few fm. A lone neutron has nothing to velcro to; consistently, free neutrons are unstable and beta-decay with a -minute half-life.
Does the constant-density result hold for (a single proton)?
The scaling law is a smooth approximation for many-nucleon nuclei; with there are no neighbours and no "packing," so treating a lone proton as an -radius sphere is only a rough limiting statement, not a precise fit.
As two protons are brought from fm inward to fm, describe how the net force changes sign.
Follow Figure 1 right-to-left: near – fm the attraction switches on and the curve dips (net attractive); inside fm the wall takes over and the net force flips to strongly repulsive.
What happens to the "density is independent of " argument for a very light nucleus like ?
It weakens: with only two nucleons there is a large "surface" and no interior, so the packed-marbles picture (interior + saturation) is a poor approximation — the constant-density law is asymptotic, best for large .
If the pion were massless, what would happen to the nuclear force's range?
It would become infinite-ranged (like electromagnetism), since as — and nuclear physics would look nothing like reality.
Where does kg/m³ come from? (a 4-line estimate)
Recall One-line summary of every trap
Strong force = short-range (Fig 1 dip), charge-independent, saturated (Fig 2 right), attractive-then-repulsive; binding removes mass; density is constant; extra neutrons buy stability against Coulomb growth.