The Shell Model of the Nucleus

2. The primitive model

[Sec. 5.3 and 5.4 Dunlap]

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

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.

Atomic Shell Model

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

Principle Quantum No = 1)( lnr

n=1

n=2

n=3

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”.

En

Z

mcn

Z

nc

mceZEn

2

222

2

2220

242

2

1

)()4(2

1

Principle Quantum No =

Radial node counter = nr

1)( lnr

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

Nuclear

nl

We must now , however, use the shape of the nuclear potential – in which nucleons move – this is the Woods-Saxon potential, which follows the shape of the nuclear density (i.e. number of bonds).

aRr

VrV

/)(exp1)( 0

Nuclear Shell ModelFor 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.

),().(

)()()(),,(

lmnl YrR

rRr

where as in the atom the ),( lmY are the spherical harmonics

im

lmlm ePY2

1).(,

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.

)(lmP

RERrmc

cllrV

dr

dRr

dr

d

rmc

cnl

22

22

22

2

.2

))(1()(

1

2

)(

Nuclear Shell Model

nlnlnlnlnl

nlnlnlnl

RERrmc

llcrV

dr

dR

rdr

Rd

mc

c

RERrmc

llcrV

dr

dRr

dr

d

rmc

c

22

2

2

2

2

2

22

22

22

2

.2

)1(.)()(

2

2

)(

.2

)1(.)()(

1

2

)(

Solving the Radial Wave Equation

Now make the substitution which is known as “linearization”

nlnlnlnl UEU

rmc

llcrV

dr

Ud

mc

c

22

2

2

2

2

2

.2

)1()()(

2

)(

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 l=0, the above equation reduces to the famous 1D form. So what we really need to do is now to solve is:

nlnlnlnl UEU

rmc

llc

aRr

V

dr

dU

mc

c

22

20

22

2

.2

)1(.)(

/)(exp12

)(

[Eq. 5.7]

rRU nlnl .

Looking at the Centrifugal Barrier

nlnlnlnl UEU

rm

llrV

dr

Ud

mc

c

2

2

2

2

2

2

.2

)1()(

2

)(

The diagram shows the effect of the centrifugal barrier for a perfectly square well nucleus. The effect of angular momentum is to force the particle’s wave Unl(r) outwards.

2

22

21

2

.

mr

LmE

mr

L

rmL

Centrifugal potential

0l 1l 2ls p d

Solutions to the Infinite Square Well

nlnlnlnlnl RER

mr

llrV

dr

dR

rdr

Rd

m

2

2

2

22

2

)1()(

2

2

]/)exp[(1)(

sSaxon Wood

2

11)(

Oscillator Harmonic

for 0

for )(

WellSquare Finite

for

for )(

WellSquare Infinite

0

2

0

0

0

aRr

VrV

R

rVrV

Rr

RrVrV

Rr

RrVrV

Solutions to the Infinite Square Well

nlnlnlnlnl RER

rmc

llcV

dr

dR

rdr

Rd

mc

c

22

2

02

2

2

2

2

)1()(2

2

)(

The solutions to this equation are the Spherical Bessel Functions

3

2

2

)(

)sin()()cos()(3)sin(3 2

)(

))cos((-)sin( 1

)(sin 0

kr

krkrkrkrkr

kr

krkrkrkr

kr

l )(krjl

2

2

)(

.2

c

Emck

Solutions to the Infinite Square WellThe zero crossings of the Spherical Bessel Functions occur at the following arguments for knl r

So that the wavenumber knl is given by:

2

2

02

2

2

2

.2

)(

nlnl

nl

nlnl

nlnl

xE

R

x

mc

cE

R

xk

xRk

And the energy of the state as:

Rkx nlnl

Coulomb Infinite Square Well Harmonic Oscillator

p1

f1

COMPARISON OF SCHRODINGER EQN SOLUTIONS

2

8

20

34

40

58Apart from 2,8 and 20 all the other numbers predicted by the primitive shell model are WRONG.

Note that the energy sequence is effective the same in all potential wells