Phy 381 (Lecture 2): Walaa. M.

Seif and Amal Refaei 2016/2017

Binding Energies
In addition to its correlation with the saturation nucleon density, the saturation
property of nuclear force also appears in the nuclear binding energies. The mass of a
certain nuclide ( ( ) in MeV) is less than the sum of the masses of the
corresponding Z free protons and N free neutrons ( ). The binding
energy (B) of a nucleus is defined as the negative of the difference between the
nuclear mass and the sum of the masses of the constituents,
( ) ( ) ( ) ( )
Here, B is a positive number. ( ) is the usual negative binding energy. While the
nuclear masses are of order A×1000 MeV, the electronic binding energies are of order
10-100 keV. Thus, we can neglect the electronic binding energies in calculating the
atomic masses, ( ) ( ) . When we deal with the atomic
masses, we replace the proton mass in Eq. (1) by the mass of hydrogen atom
(m(1H)=mp+me=1.0078u=938.78 MeV),
( ) ( )
( ) ( )

Displayed in Fig. 1(a) is the binding energy per nucleon (B/A) as a function of A.
The maxima of the binding energy per nucleon are drawn as function of A in Fig. 1(b).

Fig. 1(a): Binding energy per nucleon (B(A,Z)/A) as a function of A.

1
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

Figures 1(a) and 1(b) show that B/A increases with A in light nuclei. It reaches a
broad maximum around A= 56−60 for the even-Z isotopes of iron (26Fe) and nickel
(28Ni) where the nuclei are most tightly bound. Beyond this region, B/A decreases
slowly as a function of A. Generally, for A>12 we have ( )
.

Fig. 1(b): The binding energy per nucleon, for the maximally bound nuclei, as a function of A.

The shown behavior in Fig. 1 suggests that Light nuclei (A<56) can undergo
exothermic fusion into heavier nuclei. These reactions correspond to the various
stages of nuclear burning in stars. On the other hand, heavy nuclei (A>62) can release
energy in fission reactions or in α-decay, producing lighter nuclei. In practice, this is
observed mainly for very heavy nuclei A >150 .

The Semi-empirical mass formula:

An excellent parameterization of the binding energies of nuclei in their ground
state was proposed in 1935 by Bethe and Weizsäcker. In this formula, a few general
parameters are used to characterize the variation of the binding energy with A. This
formula relies on the liquid-drop model of nuclei but also incorporates two quantum
contributions, namely the asymmetry energy and the pairing energy. The liquid-drop
model treats some of the collective properties of nuclei in a similar way to the
calculation of the properties of a droplet of liquid. The Bethe-Weizsäcker mass
formula reads
( ) ( )
( ) 𝜹( ) ( )

2
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

The coefficients av, as, ac, asym and  are chosen so as to give a good approximation to
the observed binding energies. The A, Z and N dependence of each term reflects
simple physical properties. A good parameterization is,
, , , ,
( ) ( )
and 𝜹( ) { ( ) ( )
( )

The first term in Eq. 2 represents the volume term ( ). It reflects the nearest-
neighbor interactions. The linear dependence of B on A suggests that each nucleon
attracts only its closest neighbors, but not all of the other nucleons. From the
saturation property of nuclear density, each nucleon nearly has the same number of
neighbors. Thus, each nucleon contributes roughly the same amount to the binding
energy per nucleon, B/A ≃ 16 MeV.

The surface term ( ) lowers the binding energy. Internal nucleons feel
isotropic interactions. Nucleons near the surface of the nucleus are surrounded by
fewer neighbors coming only from the inside. Thus, they less tightly bound than
those in the internal region. Therefore this term is a surface tension term. It is
proportional to the surface area .

The third term is the Coulomb repulsion term of protons ( ( ) ).

It tends to make the nucleus less tightly bound. This term is calculable. Since each
proton repels all of the others, this term is proportional to Z(Z-1). An exact
calculation considering a uniformly charged sphere of radius R yields
( ) ( ) ( )
( ) ( )

This term favors a neutron excess rather than protons.

Conversely, the symmetry term ( ( ) ) favors equal numbers of

protons and neutrons. In the absence of electric forces, Z = N is energetically
favorable. The most stable light nuclei have N≈Z. The asymmetry term is then added
to take these effects into account. This term is very important for light nuclei. For
heavy nuclei, this term becomes less important ( ). This is because the rapid
increase in the Coulomb repulsion term requires additional neutrons for nuclear
stability.

Finally, the term δ(A) is a quantum pairing term. It favors configurations where
two identical nucleons are paired. It increases the binding energy for the even(Z)-
even(N) nuclei. Conversely, it reduces the binding energy for the odd(Z)-odd (N)
nuclei . When we have an odd number of nucleons (odd (Z)-even (N), or even(Z)-
odd(N) nuclei), this term does not contribute. We simply find evidence for this
pairing force by looking at the stable nuclei found in nature. While there are only four

3
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

nuclei with odd N and odd Z ( , , , ), we have more than 167 stable nuclei
with either even N or even Z.

We can use Eq. (1) to calculate the nuclear masses based on the binding energies
given by the semi-empirical mass formula (Eq. (2)),
( ) ( ) ( )

Fig. 2 shows the contributions of the various terms in the semi-empirical mass
formula to the binding energy per nucleon as functions of A. The predicted binding
energies for the maximally bound nuclei are also shown in Fig. 2. The figure only
presents the even–odd nuclei where the pairing term vanishes. Figure 2 shows that,
as A increases, the surface term loses its importance in favor of the Coulomb term.

Fig.2: The observed binding

energies, for odd-A nuclei, as a
function of A and the
predictions of the mass
formula given by Eq. (2).

For the odd-A nuclei of a given A, the binding energy follows a parabola in Z. An
example is shown in Fig. 3(a) for A=111. The minimum of the parabola (the
maximum binding energy) gives the value Z(A) for the most bound isobar.
Maximizing the binding energy (Eq. 2) with respect to Z ( ) yields,

( )

This value of Z is close to the value of Z that gives the stable isobar for a given A. To
find the exact value of Z, we minimize the atomic masses
( ) ( ) ( )

4
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

taking into accounts the neutron-proton mass difference to make sure of the stability
against -decay. This lead a slightly different form of ( ) for the stable nuclei,
( 𝜹 )
( )

Here, =0.75 MeV. This formula shows that light nuclei have a
slight preference for protons over neutrons because of their smaller mass. The
deviation of Z from Z=A/2 increases with increasing the mass number. Heavy nuclei
favor an excess of neutrons over protons because an extra amount of nuclear binding
must compensate for the Coulomb repulsion.

3(a) 3(b)

Fig.3: The atomic masses as a function of Z for A = 111 and A=112. The quantity plotted
is the difference between m(Z) and the mass of the lightest isobar. The dashed lines
show the predictions of the mass formula (Eq. 2) after being offset so as to pass through
the lowest mass isobars.

For even A, the binding energies follow two parabolas, one for even–even nuclei
and the other for odd–odd nuclei. An example is shown for A = 112 in Fig. 3. It can
happen that an unstable odd-odd nucleus lies between two β-stable even-even
isotopes. The more massive of the two β-stable nuclei can decay via 2β-decay to the
less massive. The lifetime for this process is generally large (≈ 1020 year). So, there
are often two stable isobars for even A. For A=112, there are two β-stable isobars
( and ). decays by 2β-decay to . The intermediate nucleus
can decay to both.

5
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

The Bethe– Weizsäcker formula can predict the maximum number of protons for
a given N and the maximum number of neutrons for a given Z. These are given by the
conditions that the last added proton and neutron must be bound,
( )
( ) ( ) ( )

( )
( ) ( ) ( )

The limits predicted by the mass formula are shown in Fig. 1 of Lecture 1. These limit
lines are called the proton and neutron drip-lines.

The Neutron and Proton Separation Energies:

The neutron separation energy ( ) is the amount of energy that is needed to
remove a neutron from a nucleus . It is given by the difference in binding energies
between and ,

( ) ( )
[ ( ) ( ) ( )]
[ ( ) ( ) ]

Similarly, the proton separation energy ( ) is the energy needed to remove a

proton from a nucleus,

( ) ( )
[ ( ) ( ) ( )]
[ ( ) ( ) ]

The neutron and proton separation energies are analogous to the ionization energies
in atomic physics. They tell us about the binding of the outermost (valence) nucleons.
Moreover, the separation energies show evidence for nuclear shell structure that is
similar to the atomic shell structure.

Quantum states of nuclei

A given nucleus of (A,Z) will have a large number of quantum states. They
correspond to different wavefunctions of the constituent nucleons. This is analogous
to an atom having lowest energy state (ground state) and a spectrum of excited
states. Figure 4 shows typical nuclear spectra. Transitions from the higher energy
states to the ground state occur rapidly. For an isolated nucleus the transition occurs
6
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

with the emission of photons (γ-rays) to conserve energy. An excited nucleus

surrounded by atomic electrons can also transfer its energy to an electron which is
subsequently ejected. This process is called internal conversion. The ejected
electrons are called conversion electrons.

Lifetimes of nuclear excited states are typically in the range 10−15 −10−10s. This is
why only nuclei in the ground state are present on Earth, with few exceptions. The
rare excited states with lifetimes greater than 1 s are called isomers. An example is
the first exited state of which has a lifetime of 1015 yr. Its ground state β-decay
has a lifetime of 8 hr. Isomeric states are specified by placing a m after A, ( ).
Excited states can be produced in collisions with energetic particles produced at
accelerators.

Fig. 4: Spectra of states of , , (scale on the left), and (scale on

the right). The spin-parities of the lowest levels are indicated.

Nuclear Angular Momentum and Parity:

A quantum state ( ) of a nucleus is defined by its nuclear spin (J) and parity (P),
as well as its energy ( ).The spin here is the total angular momentum (J) of
the constituent nucleons, including their spins. The parity is the sign by which the
total constituent wavefunction changes when the spatial coordinates of all nucleons
7
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

change sign. In the ground state, the quantum numbers are determined by unpaired
protons or neutrons. For even(Z)-even(N) nuclei, the ground state is
For odd–even (even-odd) nuclei, the quantum numbers are determined by the
unpaired proton (neutron),

( )
is the orbital angular-momentum quantum number of the unpaired nucleon. 1/2 is
the spin of nucleon. This unpaired spin can be either aligned (+) or anti-aligned ()
with the orbital angular momentum.

An important restriction on the allowed values of J comes from considering the

possible z components of the total angular momentum of the individual nucleons.
Each ( ) must be half-integral. Thus its only possible z
components are half-integral (±1/2 ℏ, ±3/2 ℏ, ±5/2 ℏ, …). This is the case when we
have one unpaired nucleon. For odd(Z)-odd(N) nuclei, we have an even number of
nucleons. There will be then an even number of half-integral components. Thus, the z
component of the total J takes only integral values. For odd(Z)-odd(N) nuclei, the
parity can be determined by multiplying together the parities of the two unpaired
nucleons, P=P1 P2.

Spins and parities are important in determining the allowed and forbidden decay
modes. The total angular momentum J has all of the usual properties of quantum
mechanical angular momentum vectors, ( )ℏ and ℏ (
). However for many applications involving angular momentum, the
nucleus behaves as a single entity with an intrinsic angular momentum (J).

As shown in Fig. 4., is a highly bound nucleus. There is a large gap between
its ground state and the first excited state. The first few excited states of have a
simple one-particle excitation spectrum due to a single neutron outside a stable
core. Both the and spectra are more complicated than the one-particle
spectrum of .

Collective vibrational and rotational excitations:

For heavy nuclei, collective vibrational and rotational excitations involving

many nucleons become more important. An example of a nucleus with vibrational
levels is in Fig. 4. has a set of excited states with energies

ℏ ( )

represents a frequency. The spectrum of exhibits collective vibrational states

of energy ℏ and 2ℏ .

8
 Phy 381 (Lecture 2): Walaa. M. Seif and Amal Refaei 2016/2017

An example of a nucleus with rotational levels is in Fig. 4. In this case, the

nucleus is a quantum rotor. The quantization of angular momentum then
implies a spectrum of states of energies
( )ℏ

Here, is the classical kinetic energy of a rigid rotor. L is the angular momentum.
I is the moment of inertia about the rotation axis. J is the angular momentum
quantum number. For even(Z)-even(N) nuclei, only even values of J are allowed
because of the symmetry of the nucleus. For example, the spectrum of has a
series of rotational states of J = 0 , 2 ,....,16 .
P + + +

Problems:
1. Exercises (1.2 and 1.3) in "Fundamentals in Nuclear Physics".
2. Use the semi-empirical mass formula to compute the total binding energy and the
binding energy per nucleon for 7Li, 20Ne, 56Fe and 235U, and 256Fm.
3. Evaluate the neutron separation energies of 7Li, 91Zr, and 236U
4. Evaluate the proton separation energies of 20Ne, 55Mn, and 197Au.
5. Starting from N=100, estimate the maximum number of neutrons can be obtained for tin
(50Sn).

------------------------------------------------------------------------------------------------------------
Reference:
1. "Fundamentals in Nuclear Physics", J. Basdevant, J. Rich and M. Spiro (Ch. 1 & Sec.
2.2).
2. "Introductory Nuclear Physics" by Kenneth S. Krane (Ch. 3)
------------------------------------------------------------------------------------------------------------------
These notes are just helping you to study the material presented in our text book that you must refer to it.