when look at the table of the nuclear world, why there are some nucleus more stable then the other? which mean, why some will decay while some are not?

OK, this basically the ultimate question that nuclear physics want to answer.

so, the very fundamental reason, no one know.

but in the elementary level, or by experimental fact and some assumption. we have Binding Energy to estimate or predict the stability of a nucleus. when the Binding Energy is larger then Zero, it must be unstable and will decay under conservation laws. if it is less then zero, it may be stable or not, it depends on whether it reach the bottom of energy level.

Binding Energy can also be though as the energy required to break the nucleus.

In liquid drop model, we imagine the nucleus is like a liquid. and nucleons inside just like liquid molecules. experiments show that nucleus is a spherical object. and it density is a constant. and the interaction range of nuclear force is short, few fm. thus, it likes a incompressible liquid drop. the radius of it is related to the mass number:

R^3 = A

the Binding Energy ( = \Delta M(A,Z,N) = mass deficit) is given by theoretical assumption and experimental fact.

\Delta M(A,Z,N) = - a_1 A + a_2 A^{\frac {2}{3} } + a_3 Z^2 / A^{ \frac{1}{3}} + a_4 (N-Z)^2 /A \pm a_5 A^{- \frac{3}{4} }

the first 3 terms are theoretical assumption and the lat 2 terms are from experimental fact. All coefficients are given by experimental measurement.

The first term is the “volume energy” by the nuclear force, which is proportional to the number of nucleons.

the 2nd term is the “surface tension” from the “liquid”. we can see its dimension is area.  (why this term is + ? ) it  explained why smaller nucleus has less Binding energy.

the 3rd term is the Coulomb potential term.

the 4th term is the balance term.  if the number of neutron and proton is no balance,

the 5th term is the “Symmetry term“. for even-even of neutron and proton number, the nucleus is more stable, thus, we choose minus sign for it. for odd-odd combination, nucleus are more unstable, thus, plus sign for it. for other, like ood – even or even-odd combination, this term is zero.

the value of the coefficients are:

a_1 \simeq 15.6 MeV

a_2 \simeq 16.8 MeV

a_3 \simeq 0.72 MeV

a_4 \simeq 23.3 MeV

a_5 \simeq 34 MeV

The below plot is the Binding Energy per nucleon in  Z against N.

Lets use the liquid drop model and Binding Energy to look the β-decay. the β-decay conserved the mass number A. there are 2 β-decays.

\beta_- : n \rightarrow p + e^- + \bar{\nu_e}

\beta_+ : p \rightarrow n + e^+ + \nu_e

so, the β+ decay decrease the number of proton while β– decay increase the number of proton.

The below diagram show the β-decay for A = 22. we can see the 22Ne is stable, since no more β-decay can help to reach a lower energy level.

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