Nuclear Masses and Binding Energies

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Nuclear Chart

Nuclear chart is a plot between Z vs. N, where Z denotes the proton number and N denotes the neutron number. A typical plot is shown below where proton number (Z) is plotted along the horizontal axis and neutron number (N) on the vertical axis. This is a very useful and very common chart which tells so much about nuclei.

Fig. 1: Z vs. N plot for all known nuclides


Every nuclide on this chart is assigned a little square, the naturally abundant nuclide being shown by black filled squares; these black squares constitute the line of beta stability or simply the line of stability. It may be noted that the line of stability coincides with the N=Z line in the light mass region. It, however, begins to deviate from it significantly in the higher mass region. Stable nuclei begin to have more number of neutrons than protons. On the left of the beta stable nuclei are the neutron deficient nuclei (or, proton rich nuclei) which are prone to positive beta decay; they try to convert the extra protons into neutrons and reach the stable region. On the right of the beta stable nuclei are the neutron rich nuclei which are prone to negative beta decay; they try to convert the extra neutrons into protons so as to reach the line of stability. As we move to heavier mass regions and cross the Pb isotopes (Z=82), alpha decay starts. This is the region of unstable nuclei only. Alpha decay and spontaneous fission dominate in this region.

Also, shown by vertical and horizontal red color lines on the plot are the magic numbers for both neutrons (N) and protons (Z). The nuclides with these Z and/or N values are more stable than the neighbors. Also shown are the magic numbers in the super-heavy region (Z=114, 126, N=184), which are still not discovered and are being speculated upon even now.

As we cross the beta-decaying nuclei on either side of the stability line, there is a big region of undiscovered nuclei defined on both sides by the proton drip line and the neutron drip line. These nuclei, often called exotic nuclei, are the focus of current researches in nuclear physics. The neutron/proton drip line corresponds to the boundary after which the neutron/proton binding energy becomes negative and no more nuclei are possible once these limits are reached. The proton drip line is known much better than the neutron drip line. We also note that the neutron drip line is much further from the line of stability.

There are about 277 stable or, very long-lived nuclei known in the chart of nuclides. About 26 have a half-life comparable to or, larger than the age of the earth (Half-life > 1012 years) and are, therefore, found in nature. The ENSDF (Evaluated Nuclear Structure Data Files) data base at ( lists about 3063 nuclides, most of them artificially produced in the laboratories around the world. These exhibit various decay modes as listed below:

(i) α – decay

(ii) β + – decay or Electron Capture

(iii) β - decay

(iv) Spontaneous Fission

(v) one – proton decay

(vi) two – proton decay

(vii) Exotic decays – (like cluster decay of C, O, F, Ne, Mg, Si)


At present, 9 doubly magic nuclei are known of which 5 are stable, namely

The 4 unstable magic nuclei are

It is interesting to note that A=5 and 8 mass numbers have no known stable nuclei. Similarly, there are no stable isotopes for Z=43 (Technetium) and Z = 61 (Promethium) proton numbers. Physicists are still trying to find the reasons behind many of these observations.


The information about the stability of nuclides primarily comes from the mass data. The atomic masses comprise one of the most precisely measured data sets. The most recent compilation of atomic masses has been presented by

Audi, Wapstra and Thibault in two papers in the journal Nuclear Physics A 729 (2003)129 and 337. There is also a website dedicated to atomic masses where most updated information may be found ( The atomic masses are measured by experimental techniques like mass spectroscopy, nuclear reactions (Q-values), and decay data like alpha decay etc.

The bare nuclear mass M(Z, N), where an atom has been stripped of all the electrons, can be obtained from the atomic mass Matom by using the relation,

                                     M(Z, N) = Matom  {Zme Be (Z)}.

Here Z me is the mass of Z electrons and Be (Z) is the electronic binding energy of Z electrons in the given atom.

The atomic masses are tabulated in terms of a quantity called the mass excess defined by

                                     ∆ (Z, N) ≡ M (Z, N) A u,

where A is the mass number and u is the atomic mass unit (amu) given by u = 931.49386 MeV/C2. In simple calculations, it is sufficient to use a value of 931.5 MeV. One amu is defined as 1u = 1/12 x M (12C) i.e. 1/12th the atomic mass of a Carbon atom. The Carbon atom is, therefore, used like a standard measure of mass for measuring the masses of other atoms.


What is binding energy? When two bodies stick together, we need to provide a minimum energy to separate the two bodies. As examples when two magnets stick together or, an electron gets bound to a nucleus as in H-atom, or the moon gets bound to the earth, the two can be separated only by providing a definite amount of energy. In each case, we need to supply a minimum energy to separate the two objects. On the other hand, if we take two objects which are free, and get bound as they are brought close together, a minimum amount of energy will be released after the bound state is formed. This energy is called the binding energy. The mass of the bound system decreases by the amount of energy released by the system following the energy mass relationship E=mc2.

This decrease in mass becomes quite significant in nuclei where the binding energies are quite large. It becomes about 0.1 to 0.8 percent of the total mass, a significant fraction of the total mass. It is, therefore, very important to take into account this quantity in all the calculations in nuclei. We, however, generally ignore this change in bound systems such as two magnet, earth-moon or sun-earth systems where the binding energy is almost an insignificant portion of the total mass the system.

The binding energy BE of a nucleus with Z protons and N neutrons is defined as

                                   BE (Z,N)= ZMpC2 + NMnC2  M(Z,N)C2 ,

where Mp and Mn are the masses of the proton and the neutron respectively. In terms of the neutral atomic masses,

                                   BE (Z,N) = (Z Mp + Z me)C2 + N MnC2 (M + Z me)C2
                                            = ZMHC2 + NMnC2 MatomC2,

where MH and Matom are masses of hydrogen atom and the neutral atom for the (Z,N) nuclide respectively. In terms of the mass excess ∆,

                                  BE (Z,N) = Z∆HC2 + N∆nC2 - ∆(Z,N)C2

where ∆HC2 = 7.2890 MeV and ∆nC2 = 8.0713 MeV.

A plot of the binding energy B(Z,N) vs A is shown in Fig.2. It is observed that BE (Z, N) is proportional to A with near linear behaviour. The behaviour of BE (Z, N) implies saturation of nuclear forces as well as its short range nature. It also suggests the strong nature of the nuclear force and its charge independence. These topics will be discussed in detail in the later chapters.

Fig. 2: Plot of total binding energy BE vs. mass number A

Another quantity discussed so often is the binding energy per nucleon defined as B.E./A. The plot of B.E./A vs. A exhibits a maximum near 56Fe with a nearly constant value in a band of 7.5 8.5 MeV per nucleon. The nuclei near the maximum are most stable and the most abundant in nature.

Although it is generally mentioned in the text books that the maximum in BE/A occurs for 56Fe (8.790 MeV/A) but as has been pointed out by Shurtleff and Derringh (Am. J. of Phys. 57,552, 1989), it is 62Ni (8.795 MeV/A) that is most stable although it is not as abundant as 56Fe because of difficulties in its formation/synthesis.

Q-value and n/p separation energies

Knowing the nuclear masses, it is easy to calculate the Q-value for a reaction or a decay process represented by

                                         a + b → c + d

The left hand side will have only one term say a in a spontaneous decay process and Q>0. The Q-value is defined as

                                         Q = (Ma + Mb) C2 (Mc + Md) C2
                                           = (BEc + BEd) (BEa + BEb)

Q-values for the removal of 1 or 2 nucleons are defined as:

                                         Sn = -Qn = BE (N, t) BE (N-1, Z)
                                         Sp = -Qp = BE (N,t) BE (N, Z-1)

Similarly S2n = -Q2n and S2p = -Q2p.

When Sn or Sp = 0, the last neutron or the last proton is unbound and drips out from the nuclei like drops of water dripping from a wet sponge. This defines the neutron or, the proton drip line. The table of Audi, Wapstra and Thibault (2003) contains a listing of the Q-values, separation energies etc. It is a very useful source of information and other data, when studying a nucleus.

One-proton decay is expected to occur near the proton drip line but the Coulomb barrier hinders the decay giving rise to life-times of proton decay comparable to β-decay. This is also the reason that proton decay has been observed. There are also discussions now about 2-proton decays.

One neutron decay is however very difficult to observe. There is no coulomb barrier that will hinder the decay of neutron. One must, therefore, depend on the angular momentum barrier. The nuclei just inside the n-drip line will have a life-time of the order of ms while those outside of the n-drip line will have life-times of the order of 10-20sec.

For light nuclei, the table of Audi-Wapstra-Thibault gives masses of nuclei obtained from resonances in nuclear reactions. The 1p decay gives information about the proton drip line in the heavier nuclei. But this information is also not complete. The information about n-drip line (from experiments) is much poorer. The proton and neutron drip lines have experimentally been defined only up to A=24.

Line of beta stability Nuclides which do not undergo beta decay, constitute the line of beta stability or simply the line of stability.

Beta decay Negative (Positive) beta decay is a spontaneous decay process by which nuclei convert their excess neutrons (protons) into protons (neutrons) and emit an electron (positron) along with an anti-neutrino (neutrino).

Alpha decay Alpha decay is a spontaneous decay process by which nuclei shed a cluster of 2 protons and 2 neutrons (a He nucleus) in order to get rid of electric charge (so that Coulomb repulsion decreases) and the nuclei reach stable isotopes of Pb/Bi.

Spontaneous fission Spontaneous fission is a spontaneous decay process by which heavy nuclei (like Uranium isotopes) break apart mostly into two fragments. They do so because of the excessive Coulomb repulsion.

Magic numbers are the proton/neutron numbers 2, 8, 20, 28, 50, 82, 126(only neutron). Nuclides with these many number of protons and/or neutrons display extra stability.

Half-life is the time taken by a sample of radioactive isotope to decay to half its initial strength or number.

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