You know right away that there is something fishy going on, since
in a system with any nontrivial complexity, there is no such thing
as "the" stability. For example, a marble on a saddle-surface
exhibits positive stability in some directions and negative
stability in other directions. Similarly, for an ideal egg on
a tabletop the "nodding" mode has positive stability whereas
the "rolling" mode has zero stability i.e. neutral stability.
A) The usual "curve of binding energy" is not the total energy
of the nuclide or atom but rather the energy /per nucleon/. This
way of looking at things is relevant for a very particular mode
of transformation, such as might sometimes be encountered inside
a star: Imagine you have many, many such nuclides, and the
transformation makes them 1% less numerous while making each one
1% heavier.
B) In contrast, the "curve of binding energy" is highly misleading
if you want to discuss the stability of a given nuclide with
respect to things like emission of an alpha ... or capture of
a proton.
In particular, in connection with the recent so-called "news",
the claim was made that you could not possibly obtain energy
from a nuclear fusion reaction starting with nickel and hydrogen,
because nickel is already at the peak of the "curve of binding
energy", and adding a proton would move the energy the wrong
way. Alas this claim is quite wrong.
The fact is that nickel (and every other element you've ever seen)
*will* bind a neutron and release energy in the process, and if
you can catalyze the Coulomb barrier it *will* bind a proton and
release energy in the process. For elements like iron and nickel
and all heavier elements, the newly-added nucleon will contribute
less than its fair share to the binding energy, so the /average/
binding energy on a per-nucleon basis will go down, but the
total binding energy is what counts, and that goes *up*.
Clearly the nickel has no fuel value in the scenario discussed
in item (A) above ... but the hydrogen does have fuel value.
To repeat: Turning 59 copies of 58Ni into 58 copies of 59Cu is
one thing (case A, energetically unfavorable) ... but stability
with respect to turning one hydrogen and one 58Ni into one 59Cu
is something else entirely (case B, energetically favorable).
Stability in one direction is not the same as stability in
another direction.
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As always, keep in mind that binding energy is measured in the
downward direction, so that an object with more binding energy
has less overall energy. This is standard terminology, but it
is very confusing to non-experts.
=============
While we're on the subject, here's another point that is a
notorious source of confusion:
The zero for calculating the nuclear "binding energy" is based
on constructing a given atom from hydrogen atoms *and* neutrons
*without* any weak decays. That's important, because it is
perfectly possible to construct any atom you want starting
from just hydrogen, since you can always make a neutron by
inverse beta decay (typically electron capture). The two
ways of making the atom would give you the same atom but a
different measure of the binding energy, different by roughly
the sum of the beta-decay energies.
You can see this instantly in the lower corner of the chart
of the nuclides, where both H and n have zero binding energy.
This stands in contrast to the _mass deficit_, which is another
measure of energy, and plays by the opposite rules. The mass
deficit is measured with everybody starting from hydrogen (or
rather from carbon-12). You can see instantly that the mass
deficit for the neutron is different from the mass deficit
for hydrogen, different by exactly the beta-decay energy.
The _excess mass_ is the negative of the mass deficit.