1) Einstein's mass-energy equivalence relation E=mc^2 relates mass and energy. One atomic mass unit (amu) is defined as 1/12 the mass of one carbon-12 atom, which is 1.6605x10-27 kg. 2) The binding energy of a nucleus is the energy required to break it into protons and neutrons. It is related to the mass defect based on differences in nuclear and nucleon masses. Binding energy per nucleon is highest for mid-sized nuclei like iron-56. 3) The binding energy curve shows binding energy per nucleon peaks around mass numbers 4, 8, 12, 16 and 20, which have even numbers of protons and neutrons.

1. Unit-I

2. Mass –energy equivalence relation
According to special theory of relativity , it is
necessary to treat mass as another form of
energy.
Einstein gave the famous energy mass
equivalence relation
E= mc2
Where ,c is velocity of light= 3x108
m/s

3. Atomic mass unit (a.m.u.)
• One atomic unit i.e. a.m.u. is defined as 1/12
mass of carbon atom
• The carbon 𝑐 has 6 protons and 6 neutrons
• It has the mass 1.992647x10−27
kg.
• According to the definition of a.m.u.
• 1a.m.u.=
𝑚𝑎𝑠𝑠 𝑜𝑓 𝑜𝑛𝑒 𝑐 𝑎𝑡𝑜𝑚
12
• =
1.992647𝑋10−27
12
= 1.6605x10−27
kg

4. • To convert mass into energy , we multiply it by
𝑐2
• Where C= 2.997𝑋108
m/s = velocity of light
• E = mx𝑐2
= 1.6605x10−27
x (2.997𝑋108
)2
= 1.4924x 10−10
Joules
=
1.4924x 10−10
1.602x 10−19 = 0.9315x 109
Mev
= 931.5 Mev

5. Binding Energy
The binding energy of a nucleus is defined as the
amount of energy required to break up the nucleus
into it’s constituents protons and neutrons.
binding energy is the energy required to separate
the nucleons from nucleus.
Thus , binding energy of nucleus is the energy
equivalent to the mass defect.
The binding energy of the nucleus 𝑍
𝐴
𝑋 is given by
B.E. = ∆ Mx𝐶2
= {[ Z𝑚𝑝 +(A-Z) 𝑚𝑛 ] − 𝑀𝑁} x 𝐶2

6. Average Binding Energy
• The average binding energy is defined as the
binding energy per nucleon.
• Average B.E. =
𝑡𝑜𝑡𝑎𝑙 𝑏𝑖𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑒𝑟𝑔𝑦
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑢𝑐𝑙𝑒𝑜𝑛𝑠
=
𝐵.𝐸.
𝐴
= {[ Z𝑚𝑝 +(A-Z) 𝑚𝑛 ] − 𝑀𝑁} x
𝐶2
𝐴

7. Binding Energy curve
• A graph between B.E. per nucleons and mass
number A for large number of nuclei is called
B.E. curve. It is as shown in fig.

8. • From Binding Energy curve , it is clear that
1)The B.E. for all stable nuclei is positive.
2) The B.E. per nucleon for very light nuclei such as 1
2
𝐻 is
very small.
3) The B.E. curve rises steeply at first and then more
gradually upto 8.8 Mev for most stable isotope of iron
26
56
𝐹𝑒.
4) The B.E.curve drops slowly to about 7.6 Mev for
Uranium.
5) The B.E. curve has sharps five peacks for He,Be,C ,O
and Ne.
6)The B.E. per nucleons is maximum in the range of A= 50
to A= 60

9. 7)The B.E. per nucleons is almost constant in
the range of A= 30 to A= 170
8)The B.E. per nucleons has lower value for the
light nuclei A<30 and heavy nuclei A>170.

10. Significance of B.E. Curve
The B.E. per nucleons is a measure of nucleus
stability .Greater the value of The B.E. per
nucleons , nucleus will be more stable.
From B.E. per nucleon curve , it is observed
that B.E. per nucleon value is less for light and
heavy nuclei.
B.E. per nucleon curve is most significant
because we concluded following two
important points.

11. If a heavy nuclei for A=240 is caused to break
down into two intermediate nuclei then the
resulting nuclei will be more stable than initial
two heavy nuclei . This process is called
nuclear fission.
If two light nuclei fuse together , then the
resulting nuclei is more stable than initial two
light nuclei. This process is known as nuclear
fusion.

12. Peaks in B.E. Curve
• There are five peaks in B.E. curve for the mass
number 4,8,12,16 and 20. The nuclei
corresponding to these peaks have even
number of protons and even number
neutrons.
• These nuclei are more stable than their
immediate neighbors and have higher value
of binding energy , give rise to peaks.

13. Packing fraction
• The mass number of nucleus is not whole
number but it may very close to whole
number
• The deviation of mass number from its
nearest whole number is called Packing
fraction.
• Packing Fraction
f =
𝑎𝑐𝑡𝑢𝑎𝑙 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑐𝑙𝑒𝑢𝑠(𝑀)−𝑀𝑎𝑠𝑠 𝑛𝑢𝑚𝑏𝑒𝑟(𝐴)
𝑚𝑎𝑠𝑠 𝑛𝑢𝑚𝑏𝑒𝑟(𝐴)

14. F =
𝑀−𝐴
𝐴
But , M-A = 𝛻𝑚= mass defect.
Hence
Packing fraction, f =
𝑀𝑎𝑠𝑠 𝑑𝑒𝑓𝑒𝑐𝑡
𝑚𝑎𝑠𝑠 𝑛𝑢𝑚𝑏𝑒𝑟
=
𝛻𝑚
𝐴
Hence ,the packing fraction is equal to mass
defect per nucleon of the nucleus.

15. Packing Fraction curve
• It is the graph between packing fraction(f) and
mass number (A)

17. • From packing fraction curve ,it I found that
For the light nuclei , the packing fraction (f) is
maximum and positive this indicates that
these nuclei are unstable.
The packing fraction is zero for oxygen
As the mass number (A) is increases beyond
16,the packing fraction (f) becomes negative
.This continue upto A= 180 and then again
become positive for heavy nuclei A= 180 and
these heavy nuclei are unstable.

18. The value of packing fraction for iron is
minimum so it is more stable.
The nuclei in middle range are stable

19. Spin Angular Momentum
• Protons and neutrons inside the nucleus
called as nucleons have spin about their axis .
• This spin motion is quantized
• Inside the nucleus each nucleon has spin
angular momentum and it’s magnitude is
given by
𝑆 = 𝑆(𝑆 + 1)ℏ
Where , ℏ =
ℎ
2𝜋

20. • Here ‘s’ is called spin angular momentum
quantum number. This momentum has two
components along z-axis
𝑚𝑠= +
1
2
and 𝑚𝑠= -
1
2
This represents the direction of spin.

21. Orbital angular momentum
• The orbital motion of nucleons is also
quantized .
• Inside the nucleus each nucleon has orbital
angular momentum and it’s magnitude is
given by
𝑙 = 𝑙(𝑙 + 1)ℏ
Where , ℏ =
ℎ
2𝜋

22. • Where , 𝑙 is orbital angular momentum
quantum number and it has only integral
values 0,1,2,3,……..
• The component of 𝑙 has along z –direction is
quantized 𝑚𝑠 is given by,
𝑚𝑠 = 𝑙,(𝑙 -1),(𝑙 -2),……. +1,0,-1,……- 𝑙(𝑙 -1),..- 𝑙

23. Total angular momentum
• The total angular momentum of nucleon (𝑗) is
the vector sum of orbital angular momentum
(𝑙) and spin angular momentum (𝑠) .
• (𝑗) = (𝑙) + (𝑠)
• The magnitude orbital angular momentum is
given by
𝑙 = 𝑙(𝑙 + 1)ℏ
Where , ℏ =
ℎ
2𝜋

24. • The spin angular momentum has two
components along z-axis
𝑚𝑠= +
1
2
and 𝑚𝑠= -
1
2
Hence total angular momentum has only two
values ,
j = 𝑙 +
1
2
and j = 𝑙 −
1
2
If 𝑙 = 0 then j = +
1
2
because j is always positive

25. Parity
• Parity describes a nuclear state. According to
quantum mechanics , the behavior of nuclear
particle can be describe in terms of wave
function (Ψ).
• The wave function (Ψ) depends upon the
coordinates x,y and z of particle.
• The nuclear parity is based on the symmetry
properties of wave function.

26. • If coordinate of particle is (x,y,z) and if they
are changed to (-x,-y,-z), which is mirror
image of the particle, then the behavior of
wave function Ψ under the inversion of
coordinates through the origin decides the
parity of the nuclear system.
• If the wave function satisfies the following eq.
Ψ (x,y,z) = Ψ (-x,-y,-z)
i.e. no change in the sign of Ψ then it is called
even parity or + parity.

27. • On other hand if Ψ (x,y,z) = − Ψ (-x,-y,-z)
i.e. the change in the sign of Ψ then it is odd
parity or –ve parity.
• in general Ψ (x,y,z) = 𝑃(-x,-y,-z)
where P is a quantum number and property
defined by it is called parity of the system.
P= +1 means even parity and
P= -1 means odd parity