nuclear physics the fermi-gas model (8th...

1

1

Nuclear Physics

(8th lecture) Content

• Nuclear models #2: The Fermi-gas model (contd.)

The total kinetic energy

The asymmetry energy

The surface energy

• Nuclear models #3: The nuclear shell model

Experimental indication

What characterizes a shell?

Shells in atomic physics (reminder)

The nuclear single-particle shell model

The 1D and 3D harmonic oscillator case

2

The Fermi-gas model Since the nucleons are fermions, we can try to set up a model of

fermion „gas”.

Assumption: the fermions are „closed” in a spherical potential

well, but they move „freely” inside .

f

f

pp

pppn

if ,0

if ,1

p pf

1

The number of states in the phase-space:

ph

pp

d

dd2

2

dd2d

3

23

3

33

rpr

N

Vhp

p

hVf

f NN

8

3

3

42 33

3

3

From this follows:

The highest kinetic

energy at the Fermi-level:

MeV 332

2

M

pE

f

f

31

0

31

232

9

A

Z

r

hp fp

Ef

-40

33

7 R E

The depth of the

nuclear potential

is ~ 40 MeV for every

nucleus!

Approximating: 2

1

A

N

A

Z

But: ArRV 3

0

3

3

4

3

4

3

The total kinetic energy of the nucleons:

5

4d4

2

2dd

22

12

5

3

0

22

3

332

3

f

p

k

p

Mh

Vppp

Mh

Vpn

M

pE

f

pr

The total kinetic energy of the

protons and the neutrons:

35

232

23

10

3Z

MVE p

k

35

232

23

10

3N

MVE n

k

The asymmetry energy:

We note that , and a Taylor-expansion yields:

...2

1

9

5

2

1

3

5

2

1 312

32

35

353

5

35

35

A

ZN

A

ZNA

A

ZN

A

NAZ

2

1

A

Z

A

N

A

ZN

...2

1

9

5

2

1

3

5

2

1 312

32

35

353

5

35

35

A

ZN

A

ZNA

A

ZN

A

ZAN

31

8

3

Vhp f

N

4

In the second part only the second order

terms remain (the linear terms cancel):

A

ZN

A

ZNA

22

This way we get an asymmetry term, similar to the Weizsäcker

formula’s!

const.4

33

0

rV

A

3

53

523

223

10

3NZ

MVEEE n

k

p

kk

The total kinetic energy:

3

13

123

53

523

2

2

2

1

2

1

9

5

2

1

2

13

10

3

A

ZNA

MV

AEk

const.63,02

12

3

5

const.,522

2

12

3

1

The first part contributes to the volume term:

Note: it has positive sign, therefore it decreases the binding energy!

(unlike in the Weizsäcker formula)

A

Note: the asymmetry term also has a positive sign! This means

that it increases the kinetic energy, and therefore decreases the

binding energy (like in the Weizsäcker formula).

2

5

The surface energy:

For simplicity consider now a cube

(instead of a sphere)

zkykxkV

zyx sinsinsin23

rThe wave function:

,nL

kx

,m

Lk y

Lkz

and 2222

zyx kkk k

The number of states in the phase space :

33

33

4

8

12

3

4

22

LKKVAN

where K = kf

So the total number of nucleons: from where: 3KA 31

AK

Volume of the 1/8 sphere in the

space of (n, m, l) quantum numbers

where

3LV

6

For n = 0, or m = 0, or l = 0 there is no wave function!

However, they were also included in the volume of 1/8 sphere!

A correction needed for the 3 quart-circle surface:

23

4

3

3

4

8

12

KLKLAN

Obviously K’>K, i.e the kinetic energy

will be increased!

Since the number of the states

in the volume >> on the surface,

at first approximation → the correction term 31

AK 32

A

The „surface energy” term can also be explained – at least

qualitatively – in the Fermi-gas model.

Note: the attractive „nuclear forces”, however, are not included

and explained (they are introduced only as an „external”

potential, holding together the Fermi-gas)!

7

The main assumption of the Fermi-gas model was, that the

nucleons move „independently” – without interacting with each-

other – in an outside potential well.

Is this not a crazy assumption for strongly interacting particles?

f

f

pp

pppn

if ,0

if ,1

p pf

1

Answer:

In the ground state of the nucleus

every level is occupied up to the

Fermi-level. Particles could scatter

out only over the Fermi-level.

If a particle gets higher energy and momentum (outside the

Fermi-level) in a scattering process, energy and momentum

conservation would mean that the „other” particle should get

lower energy/momentum → no empty state there, forbidden!

This shows also the validity of the Fermi-gas model:

mainly the ground state properties 8

Nuclear shell model 1) Experimental indications

a) Neutron capture cross sections

b) Binding energy difference of

the last neutron

3

9

c) Excitation energy of the first excited states:

Conclusion: nuclei with the following Z and/or N numbers are

extremely stable (in comparison with their neighbours):

2, 8, 20, 28, 50, 82, 126 These are the „magic numbers”.

Nuclei with magic Z or N numbers are called „magic nuclei”

If Z and N both are magic numbers, they are „double magic nuclei”

The following nuclei are double magic:

He,4

2 O,16

8Ca,40

20 Ca,48

20Ni,48

28 Ni,78

28Pb208

82 10

2) What characterizes a „shell”?

„Shell is a set of quantum-mechanical states with the same main

quantum number” True?

Remember, in atomic physics:

The noble gases are

(experimental fact):

„Magic numbers” in atomic physics should be: 2, 10, 18, 36, 54, 86.

The nth „shell” can hold theoretically 2n2 electrons. Therefore the

theoretical magic numbers would be: n 2n2 Magic number

1 2 2 (☺)

2 8 10 (☺)

3 18 28 (??)

4 32 60 (??)

5 50 110 (??)

There is a discrepancy between

theory and experiment!

Conclusion: The magic numbers in

Nature are not defined by the rule

above!

2He, 10Ne, 18Ar, 36Kr, 54Xe, 86Rn

11

A quantum-mechanical system has a

set of states with different energies,

available for its constituents.

The excitation occurs if a particle goes

into a non-occupied state at higher

energy. If this state is „close” in

energy, then the excitation is easy

(requires small amount of energy).

If the next empty state is „far” in

energy, then the excitation is difficult.

Shell is a set of quantum-mechanical states with similar energy.

Shells are separated by larger energy gaps.

E

12

Shells in atomic physics (reminder):

22

0

2

4 1

8 nh

meE

H-atom (Bohr-model)

Wave function:

,,,,,

m

l

n

mln Yr

rRr r

r

What is the relation between n in the Bohr-model (defining the

energy-shells), and the 3 quantum numbers of the wave-function?

lml

l

nr

...3,2,1,0

...3,2,1,0

...3,2,1 n

2

2

2

0 nme

hr

The quantum-mechanical treatment (Schrödinger-equation):

,,,,2

22

rErrVm

This determines the

„energy-shells”!

(spherical coordinates)

4

13

1 lnn r

n nr l 2(2l+1)

1 0 0 1s 2

2 1 0 2s 2

0 1 2p 6

3 2 0 3s 2

1 1 3p 6

0 2 3d 10

4 3 0 4s 2

2 1 4p 6

1 2 4d 10

0 3 4f 14

This determines the „shells”!

Why are the shells different in the Periodic System?

2

10

28

60

18

36

Shells in atomic physics (reminder):

Bohr Quant. mech. No. of particles

But only in the H-atom!

(„True” Coulomb potential of

a point-like nucleus)

2

10

14

2

2

2

0 nme

hr

Bohr:

The „size” gets

larger!

Why are the shells different in the Periodic System?

The „screening”

of the occupied

inner states

deforms the

potential!

Splitting of the

states occur

according to l !

The shell parameters depend on the shape of the potential!

15

This is the „single-particle shell model”

Every particle moves independently in a mean potential

3) The nuclear shell model

„The shell parameters depend on the shape of the potential!”

How is the „nuclear potential”?

ji

jiji

A

i

i

i

Vm

H rr ,2

ˆ,

1

22

The nuclear Hamiltonian:

single-particle

operators

particle-particle

interactions A „trick”:

ji

A

i

ijiji

A

i

ii

i

VVVm

H1

,

1

22

~,

~

2ˆ rrrr

„mean” potential

(for single-particles) „residual” interaction

(will be neglected)

16

The shape of the „mean” nuclear potential.

• Central → depends only on the absolute value of ri: V(ri)

• Similar shape as the nuclear density

(since the nucleons create it)

The nuclear shell model (contd.)

0

0

rVrV

R

It is difficult to solve the

Schrödinger-equation for this shape.

Two different approximations:

square

well

harmonic oscillator

5

17


Case #1: square well (infinite)!

The Schrödinger equation in the interior region in spherical

coordinate system is: ,,,

20

2

rErVrM

rkjr nln ,, Solution: Bessel functions:

0 if ,

if ,0

0

V

Rr

R,rVrV

We get for the radial part:

0122

,202

,

2

,

2

n

nn

rEV

M

dr

d

rdr

d

where

,02,

2 2nn EV

Mk

Since the potential is central,

the wave-function can be separated:

m

lYr

rr

,

18

kR 2(2L+1) Sum

L= 0 (1s) 2 2

4,49 L= 1 (2p) 6 8

5,79 L= 2 (3d) 10 18

2 L= 0 (2s) 2 20

6,99 L= 3 (4f) 14 34

7,22 L= 1 (3p) 6 40 ?

2

8

20

The functions have to go to 0 at r = R

→ only roots of the Bessel functions

are allowed → kn,l → discrete values

→ En,l discrete values too.

Second column: angular momentum

quantum number (L) and the atomic

physics notation of the state.

The nuclear shell model (contd.) units 2

in 2

2

MR

L = 0

L = 1

L = 2

L = 0

L = 3

L = 1

EV 0

Note: inside a shell the

higher L states go lower!

19


Case #2 : the harmonic oscillator:

Wave function

(in one dimension):

xm

Hem

nx n

xm

nn

2

41 2

!2

1

where 22

d

d1 x

n

xn

n ex

exH

are the Hermite-polinomials

The energy spectrum:

2

1nEn

The 3D harmonic oscillator (in Descartes coordinates):

zm

Hym

Hxm

Hezyxzyxzyx nnn

zyxm

nnn

2

,,

222

,,

2

3

2

3nnnnE zyxn where zyx nnnn

,...2,1,0n

22

02

1rmVrV

Easiest solution is in Descartes coordinates

20


a) The parity of the states:

Since and

xm

Hexxx n

xm

n

2

2

xHxHx

x

x n

n

n 1

We get: xxx

x

x n

n

n 1

In 3D: zyxzyxzyx

zyx

zyx nnn

nnn

nnn ,,1,, ,,,,

rr n

n

n 1Finally:

b) The number of degeneration of the states

zyx nnnn How many different combinations of nx, ny, nz

delivers the same n?

n

n

x

x

nnnng

0 2

211

The possible number of particles in a state: 212 nng

6

21


The 3D harmonic oscillator (in spherical coordinates):

Wave function:

,,,,,

m

l

n

mln Yr

rRr r

r

lml

l

nr

...3,2,1,0

...3,2,1,0

,,,,2

22

rErrVm

The idea for finding the correspondence: functions should be

chosen to follow

• the same parity behaviour and

• the same number of degenerations!

Question:

What is the relation between n in the Descartes-states (defining the

energy-shells), and the 3 quantum numbers of the wave-function?

22

n (n+1)(n+2) nr l 2(2l+1)

0 2 0 0 1s 2

1 6 0 1 2p 6

2 12 1 0 2s 2

0 2 3d 10

3 20 1 1 3p 6

0 3 4f 14

4 30 2 0 3s 2

1 2 4d 10

0 4 5g 18

Remember:

rr n

n

n 1

212 nng


lnn r 2

Descartes spherical

2

8

20

40

70

Magic numbers:

2, 8, 20, 28, 50, 82, 126

The harmonic oscillator is a good „first” approximation,

but something more should be included!

nuclear physics the fermi-gas model (8th...

Documents