The short answer
A row of boxes can count available places without explaining why those places lie below the next group. The distinction is central to interpreting a model rather than merely matching a sequence.
Historical record and original analysis
Count and order answer different questions
Separate a counting rule from a choice about energetic arrangement.
| Question | Answer under the assumptions | Information still needed |
|---|---|---|
| How many places? | 18 | No energy ordering follows from that total. |
| Which level fills first? | A: 9/2; B: 7/2 | The ordering is stipulated, not measured. |
| Where do 13 go? | A: 10+3; B: 8+5 | An actual nuclear prediction needs more physics. |
Chronology
A timeline of the turning points
Lecture
Selected argument.
Teaching exercise
A stipulated ordering, not a measured spectrum.
What the shell argument adds
For l>0, Mayer’s lecture gives j=l±1/2 and 2j+1 states per level. Here l and j are orbital and total angular-momentum quantum numbers. Strong spin-orbit coupling changes level separation. Its example adds an eight-state j=7/2 level to a filled 20-state group, producing 28.
The arithmetic is one part of that explanation. It would not suffice to choose arbitrary capacities until their sum matched a desired number; a physical model must explain a separation in energy as well.
Sources for this section: [1]
Keep the counted population explicit
The lecture treats neutron and proton shell numbers separately. It also says detailed level order is guided by experiment.
For the exercise, count only one species. Every available state may hold one member of that species. Treat lower levels as filled before higher ones and ignore pairing, interactions between added particles and changes in ordering. These are declared classroom assumptions, not a complete nuclear calculation.
Sources for this section: [1]
An occupancy problem with two orderings
Use the real counting rule with an invented l=4 pair. Its j values are 9/2 and 7/2, giving capacities 10 and 8. Place 13 particles into this pair. Under ordering A, 9/2 lies lower; under ordering B, 7/2 lies lower. Find the occupancy in each case and decide whether capacity alone picks the ordering.
Reference answer: A contains 10 in the lower level and 3 in the upper; B contains 8 in the lower and 5 in the upper. Both have 18 available states and 13 occupants. The total capacity is the same, but the allocation between the two levels differs.
Now correct a mistaken label claiming the pair contains 36 states because spin must be multiplied in again. The rule already uses total angular momentum j; applying an additional factor of two would count the same spin information twice.
Sources for this section: [1]
What the example can establish
The exercise diagnoses a counting error and shows what changes when a stated ordering is reversed. It does not determine an actual ground-state spin, an excitation energy, an isotope’s stability or a new magic number. Those would require additional physical assumptions and evidence. An explanatory model must earn its ordering; a combinatorial total cannot supply it by itself.
Sources for this section: [1]
Continue the chronology
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Quick answers
Common questions
- Does the count determine a measured energy gap?
- No. The exercise explicitly supplies two possible orderings and contains no measured energies.
- Why is there no extra factor of two?
- The 2j+1 rule already includes spin.
Primary and institutional sources
How this history was checked
Selected original records are identified below. Invented inputs and calculations are labelled separately from historical observations. This article was reviewed on 31 October 2026.