The Shell Model of the Nucleus

1. Evidences

[Sec. 5.1 and 5.2 Dunlap]

Alpha Particle Decay QWhat is causing this bump around Z=82, N=126, A=208

The fission barrier on the SEMFTo calculate the height of the fission barrier using the SEMF is fairly complex, but can be done as seen in this study – Fig12.3 Dunlap.

The dotted lines show variations that are understood on the shell model.

Note that the barrier is only small ~3MeV for A>250.

Atomic Shell ModelBy the end of the 1920s, the laws of quantum mechanics had been worked out. They had been applied to the hydrogen atom. They had also been extended to the MULTI-ELECTRON ATOM. This gave the first full understanding of the PERIODIC TABLE OF THE ELEMENTS.

Atomic Shell ModelStarting with the Solution of the Schrodinger Equation for the HYDROGEN ATOM

EVmc

c

)(

22

2

22

r

The natural coordinate system to use is spherical coordinates (r, , ) – in which the Laplacian operator is

2

2

2222

22

sin

1sin

sin

11

rrr

rrr

and the central potential being “felt” by the electron is the Coulomb potential

r

erV

)4()(

0

2

Atomic Shell ModelAngular solutions of the 3D Schrodinger Eqn. are the spherical harmonic functions Yl,m(,). l is the angular momentum quantum number, m is called the magnetic quantum number. l=0 l=1 l=2 l=3

+3 +2 +1 0 -1 -2 -3

Every l state has (2l+1) magnetic substates

m

Atomic Shell ModelRadial solutions of the hydrogen atom wavefunction are complicated functions involving the associated Laguerre function )(xLab

0

12

0

2/1

3

3

0,

2.

2.

)!(2

! )1(2)( 0

na

ZrL

na

Zre

lnn

ln

na

ZrR l

lnna

Zr

ln

rn0

1

0

2

1

0

n

1

2

3

Principle Quantum No = 1)( lnr

Atomic Shell ModelThe amazing thing about the 1/r potential is that certain DEGENERGIES (same energies) occur for different principal quantum no “n” and “l”.

i.e. when n=2, l=0 and l=1 have the same energies

When n=3, l=0,1 and 2 have the same energy.

En

ZEn

2

Atomic Shell ModelHowever when we extend the model to MULTI-ELECTRON atoms the degeneracy is lost.

The potential each electron moves in is now more complicated.

Occupancy

r

ZerV

2

)(

r

erV

2

)(

The potential seen by the electron changes from these two extremes as it moves about the nucleus.

Atomic Shell Model

2

10

18

36

54

86

Atomic Shell ModelSingle electron separation energy

Atomic Shell ModelAtomic Shell ModelCovalent Radius

Atomic Radius

Evidence for Nuclear Shells

Single neutron separation energy

Evidence for Nuclear Shells

The famous binding energy per nucleon (B/A) as predicted by the SEMF – does not get it quite right. There are ripples and bumps which occur at the nuclear MAGIC NUMBERS, 28, 50, 82 and 126

Evidence for Nuclear Shells

Another evidence for EXTRA STRONG NUCLEAR BINDING at the special “MAGIC NUMBERS” is that the frequency of ISOTONES is greatest when N=20, 28, 50 and 82.

THE NUCLEAR MAGIC NUMBERS are: 2, 8, 20, 28, 50, 82, 128, 184

Reason for Nuclear Shells

Type of particles Fermions Fermions

Indentity of particles electrons neutrons + protons

Charges all charged some charged

Occupancy considerations PEP PEP

Interactions EM Strong + EM

Shape Spherical Approximately spherical

ATOM NUCLEUS

The atom and nucleus have some differences – but in some essential features (those underlined) they are similar and we would expect similar quantum phenomenon - i.e. some kind of SHELL STRUCTURE.