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The Shell Model of the Nucleus 2. The primitive model

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Reason for Nuclear Shells Type of particles Fermions Fermions Indentity of particles ... felt by the electron is the Coulomb potential Nuclear ... – PowerPoint PPT presentation

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Title: The Shell Model of the Nucleus 2. The primitive model


1
The Shell Model of the Nucleus2. The primitive
model
  • Sec. 5.3 and 5.4 Dunlap

2
Reason for Nuclear Shells
ATOM
NUCLEUS
Type of particles Fermions Fermions Ind
entity of particles
electrons neutrons protons Charges
all charged some charged Occupancy
considerations PEP
PEP Interactions
EM Strong
EM Shape
Spherical Approximately
spherical
The atom and nucleus have some differences but
in some essential features (those underlined)
they are similar and we would expect similar
quantum phenomenon
ATOM SPECIAL NUMBERS 2, 10, 18, 36, 54,
86 NUCLEUS SPECIAL NUMBERS 2, 8, 20, 28, 50,
82, 126
where there is extra strong binding.
3
Atomic Shell Model
n1
n2
n3
4
Atomic Shell Model
The amazing thing about the 1/r potential is that
certain DEGENERGIES (same energies) occur for
different principal quantum no n and l.
Principle Quantum No Radial node counter nr
5
Atomic Shell Model
Starting with the Solution of the Schrodinger
Equation for the HYDROGEN ATOM
The natural coordinate system to use is spherical
coordinates (r, ?, ?) in which the Laplacian
operator is
6
Nuclear Shell Model
For a spherically symmetric potential which we
have if the nucleus is spherical (like the atom)
then the wavefunction of a nucleon is separable
into angular and radial components.
where as in the atom the
are the spherical harmonics
where the are Associated Legendre
Polynomials made up from cos? and sin? terms.
THE RADIAL EQUATION is most important because it
gives the energy eigenvalues.
7
Nuclear Shell Model
Solving the Radial Wave Equation
Eq. 5.7
Now make the substitution which
is known as linearization
The similarity with the 1D Schrodinger equation
becomes obvious. The additional potential
terms is an effective potential term due to
centrifugal energy. In the case of l0, the
above equation reduces to the famous 1D form. So
what we really need to do is now to solve is
8
Looking at the Centrifugal Barrier
s p d
The diagram shows the effect of the centrifugal
barrier for a perfectly square well nucleus. The
effect of angular momentum is to force the
particles wave Unl(r) outwards.
Centrifugal potential
9
Solutions to the Infinite Square Well
10
Solutions to the Infinite Square Well
The solutions to this equation are the Spherical
Bessel Functions
l
11
Solutions to the Infinite Square Well
The zero crossings of the Spherical Bessel
Functions occur at the following arguments for
knl r
So that the wavenumber knl is given by
And the energy of the state as
12
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13
COMPARISON OF SCHRODINGER EQN SOLUTIONS
Coulomb
Infinite Square Well
Harmonic Oscillator
14
58
Apart from 2,8 and 20 all the other numbers
predicted by the primitive shell model are WRONG.
40
34
20
Note that the energy sequence is effective the
same in all potential wells
8
2
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