Nuclide Stability

Stable and Unstable Nuclides

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

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 ^AE^Z 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
  Nuclide	BE 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 ® ^AE^Z + BE
  If m_H, m_n, and m_E are masses of hydrogen atoms, neutrons, and the nuclide respectively, then
  BE = Z m_H + N m_n - m_E

  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, BE_ave. 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 (-BE_ave), 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 E_decay = -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.

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