nuclear shell model

1. Submitted To: DR Naveed sb.
Submitted by: Usama Yousaf
BSF1800372
BS PHY VII (Eve)
Division of Science and Technology , University of Education
Township , Lahore.

2. Nuclear Shell Model
(Introduction)
●This model was developed in 1949 following independent work by several
physicists, most notably Eugene Paul Wigner, Maria Goeppert Mayer and J.
Hans D. Jensen, who shared the 1963 Nobel Prize in Physics for their
contributions.
●In this model the protons and neutrons occupy separate systems of
shells, analogous to the shells in which electrons are found outside the
nucleus.
●The shell model describes how much energy is required to move nucleons
from one orbit to another
● The nucleons move under the action of a particular potential well as
electrons move in K , L, M shells due to Coloumb’s potential.

3. Why we need to develop Shell Model
The liquid drop model can explain the observed variation of the nuclear
binding energy with mass no (A) and fission of heavy nuclei but could not
explain various nuclear properties such as
⇨ Magic Numbers
⇨ Quadrupole Moment
⇨ Spin
⇨ parity

4. Explanation:
⇨ Atoms having 2,10,18,36,54,86 electrons have their outermost shells
completely filled
⇨ As in atomic physics , there are inert gases / noble gases where the
outermost shell is completely filled and electrons are tightly bound so, we can
say these gases are stable.

6. ⇨ Similarly , the nuclei that have their proton or neutron equal to one of
magic numbers show high stability and are more abundant than other nuclei.
⇨ The magic numbers are
Whereas some nuclei contain magic number of both protons and neutrons
such as Helium-(z=2 , N=2) , Oxygen-(z=8 , N=8)
Calcium-(z=20 , N=20) . So, these are called doubly magic numbers and show
exceptionally high stability.

7. Evidences to show Existence of Shell
Structure in Nuclei
⇨ Nuclei containing magic number of protons and neutrons show very high
stability.
⇨ Measurements show that separation energy of a neutron from a nucleus
containing a magic number of neutrons is large as compared to that for a
nucleus which doesn’t have magic number of neutron.
⇨ The stable end product of all the three natural radioactive series (Uranium
series , Actinium series , Thorium series) are three isotopes of lead
where all have magic number z=82 of protons.

8. Continue…
⇨The naturally occurring isotopes whose nuclei contain magic number of
neutron or proton have generally greater relative abundance. For example
Whereas Strontium an Barium have 50 and 81 neutrons respectively.
⇨ The number of stable isotopes of an element containing a magic number
of protons is large as compared to those for other elements. For example
Calcium (z=20) have six stable isotopes as compared to Argon(z=18) and
Titanium(z=22) having three and five isotopes respectively.

9. Assumptions
1)The existence of spherically symmetric central field of force
governing the motion of individual nucleon inside the nucleus.
2) The central field of force in this case is assumed to be an
average field due to all the nucleons in the nucleus so, force on
a nucleon will be same as on the other nucleon.
3) This model assumed that L-S Coupling only hold for light
nuclei
4) The heavier nuclei hold J-J Coupling
5) Nucleons move in its orbits within nucleus independently

10. Quadrupole Moment
The nuclear electric quadrupole moment is a parameter which describes the
effective shape of the ellipsoid of nuclear charge distribution. A non-zero
quadrupole moment Q indicates that the charge distribution is not spherically
symmetric.

11. Limitations of Shell Model
(1) It cannot explain the schmidt lines .
schmidt line is basically a line of magnetic dipole moment as a function of
nuclear spin.
(2) It cannot explain the large electric quadrupole moment of the non-
spherical nucleus.
(3) It cannot explain the higher energy state of the nucleus( which are
explained by collective model).
4)Fails to explain The stability of four stable nuclei H, Li, B, N