__CHAPTER 2: NUCLEAR MASSES AND
BINDING ENERGY__

__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**

(http://fys246.nuclear.lu.se/topics.asp)

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 > 10^{12}
years) and are, therefore, found in nature. The ENSDF (Evaluated Nuclear
Structure Data Files) data base at www.nndc.bnl.gov 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

_{},_{}, _{},_{} and _{}

The 4 unstable magic nuclei are

_{},_{}, _{}, and _{}

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.

**ATOMIC AND
NUCLEAR MASSES**

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 (http://amdc.in2p3.fr/). 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 M_{atom}
by using the relation,

M(Z, N) = M_{atom}
{Zm_{e} B_{e} (Z)}.

Here Z m_{e} is the mass of Z electrons and B_{e}
(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/C^{2}. In simple calculations, it is
sufficient to use a value of 931.5 MeV. One amu is defined as 1u = _{}x M (* ^{12}C*) i.e. 1/12

**BINDING
ENERGY**

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=mc^{2}.

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)= ZM_{p}C^{2} + NM_{n}C^{2}
M(Z,N)C^{2} ,

where M_{p} and M_{n} are the masses
of the proton and the neutron respectively. In terms of the neutral atomic
masses,

BE (Z,N) = (Z M_{p} + Z m_{e})C^{2}
+ N M_{n}C^{2} (M + Z m_{e})C^{2}

= ZM_{H}C^{2} + NM_{n}C^{2}
M_{atom}C^{2},

where M_{H} and M_{atom} 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∆_{H}C^{2}
+ N∆_{n}C^{2} - ∆(Z,N)C^{2},

where
∆_{H}C^{2} = 7.2890 MeV and ∆_{n}C^{2}
= 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 ^{56}Fe 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.

Fig.3: Plot of BE/A vs. A

Although it is generally mentioned in the text books
that the maximum in BE/A occurs for ^{56}Fe (8.790 MeV/A) but as has
been pointed out by Shurtleff and Derringh (Am. J. of Phys. 57,552, 1989), it
is ^{62}Ni (8.795 MeV/A) that is most stable although it is not as
abundant as ^{56}Fe 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 = (M_{a} + M_{b})
C^{2} (M_{c} + M_{d}) C^{2}

= (BE_{c} + BE_{d}) (BE_{a}
+ BE_{b})

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

S_{n} = -Q_{n} = BE (N, t) BE (N-1,
Z)

S_{p} = -Q_{p} = BE (N,t) BE (N,
Z-1)

Similarly S_{2n} = -Q_{2n} and S_{2p}
= -Q_{2p}.

When S_{n} or S_{p} = 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^{-20}sec.

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.

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**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.