# Nuclide Stability

## Stable and Unstable Nuclides

Unstable nuclides under go radioactive decay or fission, and they are often called radioactive nuclides. Some unstable nuclides occur naturally, and many have been made artificially. The process of making (synthesizing) them involve nuclear reactions.

On this page, we discuss the stability of nuclides in terms of their masses. The discussion applies to both stable and unstable nuclides.

## Masses of Nuclides

Energy drives all changes including nuclear reactions and radioactive decays. Energy and mass are equivalent. Thus, the masses of nuclides are related to their stability and radioactivity.

In order to discuss the instability of nuclides, we look at their masses. There are several ways to look at their masses. Masses of nuclides can be compared to either their components (hydrogen atoms and neutrons) or some other standard such as 12C.

The following are some of the ways we look at the masses of nuclides.

• Binding Energy
The concept of binding energy comes from the consideration of making a nuclide from its components: hydrogen atoms and neutrons. Thus, the binding energy, BE, of a nuclide AEZ is the energy released when the atom is synthesized from the appropriate numbers of hydrogen atoms (Z H) and neutrons (N n),

Binding energy (MeV)
of some nuclides
NuclideBE MeV
2D 2.226
4He 28.296
14N 104.659
16O 127.619
40Ca 342.052
58Fe 509.945
206Pb 1622.340
238U 1822.693

Z H + N n ® AEZ + BE

If mH, mn, and mE are masses of hydrogen atoms, neutrons, and the nuclide respectively, then

BE = Z mH + N mn - mE

Thus, binding energy is zero for hydrogen, as it is one of the standard. All other nuclides have positive BE. the binding energy for D, He and 238U are calculated below:

2D1 - deterium
BE = (1.007825 + 1.008665 - 2.0141) 931.481 MeV = 2.226 MeV

4He2
BE = (2*1.007825 + 2*1.008665 - 4.002603) 931.481 MeV = 28.30 MeV

238U92
BE = (92*1.007825 + 146*1.008665 - 238.0289) 931.481 MeV = 1822.06 MeV

These examples and the binding energy given in the Table support a general statement. The more nucleons packed into a nucleus, the more energy is released, and thus the higher the binding energy. Therefore binding energy is not a good indicator of nuclide stability.

• Average Binding Energy
In order to find a good indicator of stability of nuclides, we take a look at the average binding energy per nucleon, BEave. This turns out to be a useful criterion for the stability of nuclides.

Average binding energy, BEave
(MeV) of some nuclides
Nuclide BEave
MeV
BE
MeV
2D 1.113 2.226
4He 7.074 28.296
14N 7.476 104.659
16O 7.976 127.619
19F 7.779 147.801
40Ca 8.551 342.052
55Mn 8.765 482.070
58Fe 8.792 509.945
62Ni 8.795 545.259
206Pb 7.875 1622.340
238U 7.658 1822.693
For stable nuclides, the plot of average binding energy against mass number shows the effect of nucleon pairing, magic number, and atomic mass. Furthermore, the average binding energy is the highest for isotopes of iron, Fe, and nickel, Ni. Some typical values are given in the Table.

• Fission and fusion nuclear energy
In general, nuclides with masses lighter than 58 have lower average binding energies. Thus, combining light nuclides into heavier nuclides (fusion) results in releasing energy. This type of energy is known as fusion energy. Very light nuclides do fuse in stars providing the energy to power the star.

On the other hand, heavier nuclides than 58 also have lower average binding energies. When nuclides such as 235U and 239Pu split into two pieces (fission) of lighter nuclides, fission energy is also released.

The binding energies of both light and heavy nuclides are lower than nuclides with mass number around 58-60, as shown in a plot of the binding energy as a function of mass number. Fission and fusion energy comes from the difference in binding energy of nuclides of different masses. The above plot is taken from Nuclear Binding Energy.

• Mass excesses
Comparison of Masses for Some Nuclides
Nuclide Mass, amu Mass excess
amu
-BEave
amu
-BE
amu
H 1.007825 0.007825 0 0
n 1.008665 0.008665 0 0
3He 3.01603  0.016030 -0.00276 0.00828
4He 4.002600 0.002600 -0.00760 0.03040
12C 12.000000 0 -0.00825 0.09894
16O 15.994915 -0.005085 -0.00857 0.1369
40Ca 39.96259  -0.037410 -0.00917 0.3669
54Fe 53.939612 -0.060388 -0.00938 0.5065
56Fe 55.934939 -0.065061 -0.00944 0.52851
208Pb82 207.976627 -0.023373 -0.00845 1.757
238U92 238.050784 0.050784 -0.00813 1.934
The mass of 12C is defined to be 12 atomic mass units (amu) exactly, and thus the average mass of a nucleon in 12C is exactly 1 amu. The mass of 12C is the same as its mass number A. The difference between mass and the mass number A is the mass excess. The mass excess of 12C is zero.

The mass excesses of some nuclides are compared with the negative average binding energy (-BEave), and binding energy (-BE) in the table here.

Since hydrogen and neutron are used as the standard for binding energy, and 12C is the standard for mass excess, binding energy cannot be converted directly to mass excess. However, the two are related. Nuclides with low mass excesses also have low binding energy.

Mass excess can be used to evaluate the energy of decay. The mass excesses of 40Sc21 and 40Ca20 are -20.527 and -34.847 MeV respectively. Thus the energy of the decay process

40Sc21 ® 40Ca20 + b+
or 40Sc21 + e- ® 40Ca20
is Edecay = -20.527 - (-34.847) = 14.32 MeV

Of course, 1.02 MeV (2 times the rest mass of an electron) has to be spent in producing the positron-electron pair in positron decay, but not in electron capture. This example illustrates the fact that the mass excess not only vary as mass number A changes, it also vary as the atomic number Z changes for isobars.

• Mass excesses of isobars
Mass excess of isobars
with mass number 123
Nuclide Mass
excess
(amu)
Mass
excess
(MeV)
In49 -0.0896 -83.5
Sn50 -0.0943 -87.8
Sb51 -0.0958 -89.2
Te52 -0.0967 -90.1
I53 -0.0944 -87.9
Xe54 -0.0915 -85.2
Cs55 -0.0870 -81.0
Ba56 -0.0808 -75.3
Mass excesses for isobars vary as the atomic number Z changes. Isobars convert to each other by the beta decay process. As an example, the mass excesses of isobars with mass number 123 are given in the Table here.

Since Te52 and Te52 have the lowest mass excesses, these are stable isobars among them.

For unstable nuclides, handbooks usually give mass excesses rather than their masses.

### Internet Resource

Masses of nuclides
Binding energy
Nuclear structure - binding energy
Subatomic physics detectors

E-mail: cchieh@uwaterloo.ca