Z X with A=Z+N N neutron number 44 N A atomic/weight number 1 - - PowerPoint PPT Presentation

INTRODUCTION TO NUCLEAR MODELS

NUCLEUS BASICS

size:

3 1

0 A

R R ≅

fm 2 . 1

0 ≅

R

fm 1 . 5 ≅ R

mass:

n p

m N m Z m + < ) X (

(that’s why it holds together) binding energy:

[ ]

2

) , ( ) , ( c N Z m m N m Z N Z B

n p

⋅ − + =

[ ]

MeV 6 . 661 ) Ge ( 44 36 ) Ge (

2 76 76

= ⋅ − + = c m m m B

n p

A

50 100 150 200 250 2 4 6 8 Fe U He fusion fission

B/A [MeV/nucleon]

Daniel Kollar – Friday Physics Session

N A Z X

with A=Z+N

44 76 32Ge

Z – proton number N – neutron number A – atomic/weight number

NUCLEAR FORCES

YUKAWA THEORY OF NUCLEAR FORCES short range, spin-orbital character based on exchange of π0, π+, and π–

N

N + N → N + N

N N N

π 0 p + n → p + n

p p p p n n n n

π 0 π +

p p n n

π – Fermi

p p n n

e – ν

too weak

p p n n

e – Heisenberg

violates angular momentum conservation consider range of 1.4 fm ⇒ from uncertainty principle m c r ≈ → mπ ≈ 140 MeV

NUCLEAR MODELS

• total wavefunction of the nucleus is far too complicated to be useful even if it

was possible to calculate it (only possible for the lightest nuclei) ⇒ we make use of models and use simple analogies

Collective Independent particle Quantum mechanical Semiclassical Types of nuclear models

Fermi gas Shell Liquid drop Rotational Vibrational

FERMI GAS MODEL Built on analogy between nucleus and ideal gas

• particles don’t interact
• particles move independently in the mean field of the nucleus

Ground state → particles occupy lowest energy states allowed by the Pauli principle

Fermi energy EF

–R R R –R V V –V0 –V0

neutrons protons

Fermi sea

FERMI MODEL

Distribution of nucleon momentum states:

( )

3 3

2 2

• π

p Vd dn =

m EF 1 const ⋅ =

equal for all nuclei → for Z = A – Z = A/2

) MeV 40 ( MeV 30

0 ≈

≈ V EF

∫

=

F

E

dn n

• total number of states up to EF:

– momentum → energy – volume ⇒

A r V

3

3 4 π =

– np = Z ; nn = A – Z

3 2

4 9 2

2 2

      ⋅ = A Z mr E p

F

π

• Fermi energy:

3 2

4 9 2

2 2

      − ⋅ = A Z A mr E n

F

π

NUCLEAR LIQUID DROP MODEL

= ) , ( Z A B

Weizsäcker formula for the binding energy (A ≥ 30)

3 1

A R ∝

A aV ⋅

condensation energy ∝ V holding nucleus together

3 2

A aS ⋅ −

surface tension ∝ S near-surface nucleons are bound less

3 1

) 1 (

−

⋅ − ⋅ − A Z Z aC

Coulomb potential

1 2

) 2 (

−

⋅ − ⋅ − A Z A aA

asymmetry

∆ −

     + − = ∆ δ δ

2 1

−

∝ A δ

even-even even-odd

• dd-odd

pairing energy

NUCLEAR LIQUID DROP MODEL

( ) ( ) ( )

∆ + − + − + + − − + =

− − 1 2

2 1 ) , (

3 1 3 2

A Z A a A Z Z a A a A a m Z A Zm Z A m

A C S V n p

Weizsäcker formula for the mass of the nucleus

     + − = ∆

• e

ee δ δ

2 1

−

∝ A δ

( ) ( )

Z A B m Z A Zm Z A m

n p

, ) , ( − − + = for constant A ⇒ m(A,Z) is quadratic in Z

m(A,Z) Z ee

• β+, EC

β– even A m(A,Z) Z

• dd A

β– β+, EC

76As 76Se 76Ge

β– β+ 2β–

NUCLEAR LIQUID DROP MODEL

( ) ( ) ( )

∆ + − + − + + − − + =

− − 1 2

2 1 ) , (

3 1 3 2

A Z A a A Z Z a A a A a m Z A Zm Z A m

A C S V n p

Valley of stability

NEUTRON NUMBER N PROTON NUMBER Z

For fixed A the most stable Z is obtained by differentiating m(A,Z)

3 2

0075 . 1 1 2 A A Z + ⋅ ≅ N Z =

2 8 20 28 50 82 136 2 8 20 28 50 82

MAGIC NUMBERS:

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

not explained by Fermi gas model nor liquid drop model

NUCLEAR SHELL MODEL – WHY?

New model needed to explain discontinuities of several nuclear properties

A

50 100 150 200 250 2 4 6 8 Fe U He fusion fission

B/A [MeV/nucleon]

?

semi-empirical experimental

→ binding energy → high relative abundances → low n-capture cross section → high excitation energies → …

MAGIC NUMBERS

Magic numbers indicate similarity of nucleus to electron shells of atom, BUT still different from “Atomic magic numbers” (2, 10, 18, 36, 54, 86)

Magic number = closed shell

NUCLEAR SHELL MODEL

AIM → Explain the magic numbers ASSUMPTION → Interactions between nucleons are neglected → Each nucleon can move independently in the nuclear potential STEPS → Find the potential well that resembles the nuclear density → Consider the spin-orbit coupling

( )

, ,

( ) ( ) ( )

i i ij i i i j i j i

H T V r v r V r λ

≠

  = + + −    

∑ ∑ ∑

NUCLEAR SHELL MODEL

Hamiltonian of a nucleus: Potential well

central potential residual potential

• 2. Harmonic

Oscillator

• 3. Woods-Saxon

Potential

Central potential Residual potential ⇒ λ → 0

• 1. Square Well

R r

V(r) V0

Potential well candidates ( )

2 2 2 2 2

1 2 ( ) 2

nl nl nl

l l d M R E V r R dr r Mr   − + − − =      

• Solve Schrödinger equation

Closed Shell

≠

Magic Number

NUCLEAR SHELL MODEL

Square well potential

2 2 1s 8 6 1p 18 10 1d 20 2 2s 34 14 1f 40 6 2p 58 18 1g Total Occupation

V(r) R r

• V0

( )

V r R V r r R − ≤  =  > 

⇒ no analytical solution

Closed Shell

≠

Magic Number

NUCLEAR SHELL MODEL

Harmonic potential

2 2 1s 8 6 1p 20 12 1d,2s 40 20 1f,2p 70 30 1g,2d,3s 112 42 1h,2f,3p 168 56 1i,2g,3d,4s Total Occupation

V(r) r

• V0

( )

2 2

1 2 V r V M r ω = − +

⇒ analytical solution possible

Closed Shell

≠

Magic Number

NUCLEAR SHELL MODEL

Woods-Saxon potential

2 2 1s 8 6 1p 20 12 1d,2s 40 20 1f,2p 70 30 1g,2d,3s 112 42 1h,2f,3p 168 56 1i,2g,3d,4s Total Occupation

V(r) r

• V0

( ) ( )

1 exp

r R a

V V r

−

= − +

⇒ no analytical solution

R

resembles the nuclear density from scattering measurements

NUCLEAR SHELL MODEL

Maria Mayer (Physical Review 78 (1950), 16) suggested:

• 1. There should be a non-central component
• 2. It should have a magnitude which depends on S & L

Spin-orbit coupling contribution

( ) ( ) ( )

s

V r V r V r = + ⋅ L s

( ) ( )

1

s s

d V r V f r r dr =

with

Woods-Saxon shape

non-central potential

Results in energy splitting of individual levels for given J (angular momentum)

j = l +/- ½ j = l – ½ j = l + ½ ∆E for l > 0 e.g. ⇒ 1d 1d 5/2 1d 3/2

NUCLEAR SHELL MODEL

1s 1p 1d 2s 1f 2p 1g 2d 1h 3s 2f 1i 3p 2g 3d 4s 1s 1p 1d 2s 1f 2p 1g 2d 1h 3s 2f 1i 3p 2g 3d 4s

20 28 126 82 50 8 2

3d 3s 2d

2 3 2 1 5 2

4s

2 1

2g 2s

2 1 7 2

2d

2 3

1g

5 2

1p

2 3

1p

2 1 72

2f 1i 1d 5 2 1d

2 3 9 2

1g 1f 5 2

7 2

1f 2p

2 3

2p

2 1 92

1h

2 11

1h

2 11

1i

2 13

2f 5 2 2g 72

92

3p

2 3

3d 3p

2 1

1s

2 1

2 1 3 4 6 5

square well harmonic

• scillator

Woods-Saxon potential plus spin-orbit coupling

Level splitting

The essential features are given by any potential of the form

( ) ( ) ( )

s

V r V r V r = + ⋅ L σ

Energies of levels are parameter dependent Shell model fails when dealing with deformed nuclei, i.e., nuclei far from magic numbers REMARKS Collective models: rotational, vibrational

Other models

Close to CLOSED-SHELL nuclei well described by shell model However, most of the nuclear properties are indeed determined by nucleons outside the closed shells Collective models → treating the closed shells as inert and only dealing with the rest Models not mentioned (but used):

• 1. rotational model

→ rotations of permanently deformed nuclei

• 2. vibrational model

→ excitations within shell – multipole account

• 3. Nilsson model

→ shell model with deformed potential

• 4. α-particle model

→ α-particle clusters inside the nucleus

• 5. interacting boson model

→ considering pairs of nucleons as bosons