1

Lecture 3Lecture 3

Nuclear models:Nuclear models:

Shell model Shell model

WS2012/13WS2012/13: : ‚‚Introduction to Nuclear and Particle PhysicsIntroduction to Nuclear and Particle Physics‘‘, Part I, Part I

2

Nuclear modelsNuclear models

Models with strong interaction between

the nucleons

Nuclear modelsNuclear models

Models of non-interacting

nucleons

� Fermi gas model

� Optical model

�…

� Liquid drop model

� αααα-particle model

� Shell model

�…

Nucleons interact with the nearest

neighbors and practically don‘t move:

mean free path λ λ λ λ << RA nuclear radius

Nucleons move freely inside the nucleus:

mean free path λ λ λ λ ~ RA nuclear radius

3

III. Shell modelIII. Shell model

4

Shell modelShell model

Magic numbers: Nuclides with certain proton and/or neutron numbers are found

to be exceptionally stable. These so-called magic numbers are

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

— The doubly magic nuclei:

— Nuclei with magic proton or neutron number have an unusually large number

of stable or long lived nuclides.

— A nucleus with a magic neutron (proton) number requires a lot of energy to

separate a neutron (proton) from it.

— A nucleus with one more neutron (proton) than a magic number is very easy to

separate.

— The first exitation level is very high: a lot of energy is needed to excite such

nuclei

— The doubly magic nuclei have a spherical form

�Nucleons are arranged into complete shells within the atomic nucleus

5

Excitation energy for mExcitation energy for magic nuclei

6

Nuclear potentialNuclear potential

Rr

RrrU

≥≥≥≥∞∞∞∞

<<<<

====,

,0)(

U(r)

r

∞∞∞∞

R0

� The energy spectrum is defined by the nuclear potential

���� solution of Schrödinger equation for a realistic potential

� The nuclear force is very short-ranged => the form of the potential follows the

density distribution of the nucleons within the nucleus:

� for very light nuclei (A < 7), the nucleon distribution has Gaussian form

(corresponding to a harmonic oscillator potential)

� for heavier nuclei it can be parameterised by a Fermi distribution. The latter

corresponds to the Woods-Saxon potential

1) Woods-Saxon potential:

a

Rr

e

UrU

−−−−

++++

−−−−====

1

)( 0

Rr

RrUrU

≥≥≥≥

<<<<−−−−

====,0

,)(

0

2) a����0:

approximation by the

rectangular potential

well :

a=0

a>0

e.g. 3) approximation by the

rectangular potential well

with infinite barrier energy :

7

SchrSchröödinger equationdinger equation

ΨΨΨΨ====ΨΨΨΨ EH

)(2

ˆ2

rUM

H ++++∇∇∇∇

−−−−====h

(((( ))))2

2

222

2

2

2

sin

1sin

sin

11

ϕϕϕϕθθθθθθθθ

θθθθθθθθ ∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂====∇∇∇∇

rrrr

rr

(((( ))))2

2

2sin

1sin

sin

1ˆϕϕϕϕθθθθ

θθθθθθθθθθθθ

λλλλ∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂====

22 ˆˆ L====−−−− λλλλh

(((( )))) (((( ))))ϕϕϕϕθθθθϕϕϕϕθθθθ ,)1(,ˆ 22

lmlm llL ΥΥΥΥ++++====ΥΥΥΥ h

2

22

2

2

2ˆ11

h

L

rrr

rr−−−−

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂====∇∇∇∇

Schrödinger equation:

Single-particle

Hamiltonian operator:

•Eigenstates: ΨΨΨΨ(r) - wave function

•Eigenvalues: E - energy

•U(r) is a nuclear potential – spherically symmetric ����

Angular part:

L – operator for the orbital angular momentum

Eigenstates: Ylm – spherical harmonics

(1)

(2)

(3)

Rr

RrrU

≥≥≥≥∞∞∞∞

<<<<

====,

,0)(

� Consider U(r) -

rectangular potential

well with infinite

barrier energy

8

Radial partRadial part

(((( )))) (((( ))))ϕϕϕϕθθθθϕϕϕϕθθθθ ,)(,, 1 lmrr ΥΥΥΥ⋅⋅⋅⋅ΨΨΨΨ====ΨΨΨΨ

(((( )))) (((( )))) (((( )))) (((( ))))ϕϕϕϕθθθθϕϕϕϕθθθθ ,,2

ˆ11

211

2

2

2

2

2

lmlm rErM

L

rrr

rrMΥΥΥΥΨΨΨΨ====ΥΥΥΥΨΨΨΨ

++++

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂−−−−

h

(((( )))) (((( ))))rErM

ll

rrr

rrM112

22

2

2

2

)1(1

2ΨΨΨΨ====ΨΨΨΨ

++++++++

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂−−−−

hh

The wave function of the particles in the nuclear potential can be decomposed into

two parts: a radial one ΨΨΨΨ 1111(r), which only depends on the radius r, and an angular

part Ylm(θ,ϕϕϕϕ) which only depends on the orientation (this decomposition is possible

for all spherically symmetric potentials):

From (4) and (1) =>

(4)

=> eq. for the radial part:

(((( ))))r

)r(r1

ℜℜℜℜ====ΨΨΨΨ

Substitute in (5):

(5)

)r(E)r(M2

)1l(l

rdr

)r(d

M2 2

2

2

22

ℜℜℜℜ====ℜℜℜℜ++++

++++ℜℜℜℜ

−−−−hh (6)

9

Constraints on E Constraints on E

0)r(M2

)1l(l

rE

dr

)r(d

M2 2

2

2

22

====ℜℜℜℜ

++++−−−−++++

ℜℜℜℜ hhEq. for the radial part:

From (7) ����

1) Energy eigenvalues for orbital angular momentum l:

E:

l=0 s

l=1 p states

l=2 d

l=3 f

…

(7)

2) For each l: -l < m < l => (2l+1) projections m of angular momentum.

The energy is independent of the m quantum number, which can be any integer

value between ± l. Since nucleons also have two possible spin directions, this

means that the l levels are 2·(2l+1) times degenerate if a spin-orbit interaction is

neglected.

3) The parity of the wave function is fixed by the spherical wave function Yml and

reads (−1)l: (((( )))) (((( )))) (((( ))))ϕϕϕϕθθθθϕϕϕϕθθθθϕϕϕϕθθθθ ,)()1(,)(,,ˆ11 lm

l

lm rrPrP ΥΥΥΥ⋅⋅⋅⋅ΨΨΨΨ−−−−====ΥΥΥΥ⋅⋅⋅⋅ΨΨΨΨ====ΨΨΨΨ

⇒ s,d,.. - even states; p,f,… - odd states

10

Main quantum number Main quantum number nn

0)r(M2

)1l(l

rE

dr

)r(d

M2 2

2

2

22

====ℜℜℜℜ

++++−−−−++++

ℜℜℜℜ hh

0)r()r()r()r( ====ℜℜℜℜ++++′′′′′′′′ℜℜℜℜ λλλλ

Eq. for the radial part:

Solution of differential eq:

(7)

Spherical Bessel functions jl(x):

�Boundary condition for the surface, i.e. at r =R: Ψ(Ψ(Ψ(Ψ(R,θ,ϕ),θ,ϕ),θ,ϕ),θ,ϕ) =0

���� restrictions on k in Bessel functions:

⇒ main quantum number n – corresponds to nodes of the Bessel function : Xnl

0)kR(jl ====

2

nl

2

2

2

nl

22

nl XRME2

XRkXkR ====⇒⇒⇒⇒====⇒⇒⇒⇒====h

(((( )))) (((( ))))ϕϕϕϕθθθθϕϕϕϕθθθθ ,)(,, lml krjr

Ar ΥΥΥΥ⋅⋅⋅⋅====ΨΨΨΨ

2

2 2

h

MEk ====

(8)

U(r)

r

∞∞∞∞

R0

)kr(jr

A)r( l====ℜℜℜℜ

����The wave function:

11

Shell model with Shell model with rectangularrectangular potentialpotential wellwell

2

22

2MR

XE

nl

nl

h====

2

nlnl XConstE ⋅⋅⋅⋅====Thus, according to Eq. (8) : (9)

Energy states are quantized ���� structure of energy states Enl

Nodes of Bessel function

42.93

28.62

14.31

0

30

20

10

0

========

========

========

−−−−====

Xn

Xn

Xn

jstatessl

72.72

49.41

1

21

11

1

========

========

−−−−====

Xn

Xn

jstatespl

76.51

2

12

2

========

−−−−====

Xn

jstatesdl

99.61

3

13

3

========

−−−−====

Xn

jstatesfl

1.81

4

14

4

========

−−−−====

Xn

jstatesgl

18102.33CEd1

862.20CEp1

2286.9CEs1

EEwithstatesdegeneracyXCEstate

d1

p1

s1

nl

2

nlnl

⋅⋅⋅⋅====

⋅⋅⋅⋅====

⋅⋅⋅⋅====

≤≤≤≤⋅⋅⋅⋅====

581864CEg1

4067.59CEp2

34148.48CEf1

2025.39CEs2

g1

p2

f1

s2

⋅⋅⋅⋅====

⋅⋅⋅⋅====

⋅⋅⋅⋅====

⋅⋅⋅⋅====

� First 3 magic numbers are reproduced, higher – not!

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

Note: this result corresponds to for U(r) = rectangular

potential well with infinite barrier energy

2·(2l+1),l=0

2·(2l+1),l=1

12

Shell model with WoodsShell model with Woods--Saxon potentialSaxon potential

� Woods-Saxon potential:

a

Rr

e

UrU

−−−−

++++

−−−−====

1

)( 0

�The first three magic numbers (2, 8 and 20) can then be understood as

nucleon numbers for full shells.

�Thus, this simple model does not work for the higher magic numbers. For

them it is necessary to include spin-orbit coupling effects which further split

the nl shells.

13

SpinSpin--orbit interactionorbit interaction

lsVrUM

H ˆ)(2

ˆ2

++++++++∇∇∇∇

−−−−====h

),,(),,(ˆ)(2

2

ϕϕϕϕθθθθϕϕϕϕθθθθ rErVrUM

ls ΨΨΨΨ====ΨΨΨΨ

++++++++

∇∇∇∇−−−−h

(((( )))) ),,(),,()(2

2

ϕϕϕϕθθθθϕϕϕϕθθθθ rVErrUM

ls ΨΨΨΨ−−−−====ΨΨΨΨ

++++

∇∇∇∇−−−−h

),,(),,(ˆ ϕϕϕϕθθθθϕϕϕϕθθθθ rVrV lsls ΨΨΨΨ====ΨΨΨΨ

)s,l(CV lsls ====

� Introduce the spin-orbit interaction Vls – a coupling of the spin and the

orbital angular momentum:

where

eigenvalues

•spin-orbit interaction:

total angular momentum:

slslslsljj 2))((22

++++++++====++++++++====⋅⋅⋅⋅

)(2

1 222

sljsl −−−−−−−−====⋅⋅⋅⋅

Eigenstates

slj ++++====

- scalar product of sandl

14

SpinSpin--orbit interactionorbit interaction

),,(),,()(2

1 222

ϕϕϕϕθθθθϕϕϕϕθθθθ rVrsljC lsls ΨΨΨΨ====ΨΨΨΨ−−−−−−−−⋅⋅⋅⋅

[[[[ ]]]])1s(s)1l(l)1j(j2

CV2

lsls ++++−−−−++++−−−−++++====h

lCllllCVlj lslsls22

3

2

1)

2

1)(

2

1(

2:

2

1 22

2hh

====

⋅⋅⋅⋅−−−−−−−−−−−−++++++++====++++====

)1(22

3

2

1)

2

1)(

2

1(

2:

2

1 22

2

++++−−−−====

⋅⋅⋅⋅−−−−−−−−−−−−++++−−−−====−−−−==== lCllllCVlj lslsls

hh

�Consider:

This leads to an energy splitting ∆Els which linearly increases with the angular

momentum as

lsls Vl

E2

12 ++++====∆∆∆∆

� It is found experimentally that Vls is negative, which means that the state

with j =l+ 1/2 is always energetically below the j = l − 1/2 level.

Eigenvalues:

15

�The total angular momentum quantum

number j = l±1/2 of the nucleon is denoted by

an extra index j: nlj

e.g., the 1f state splits into a 1f7/2 and a 1f5/2

state

� The nlj level is (2j + 1) times degenerate

���� Spin-orbit interaction leads to a sizeable

splitting of the energy states which are indeed

comparable with the gaps between the nl

shells themselves.

� Magic numbers appear when the gaps

between successive energy shells are

particularly large:

SpinSpin--orbit interactionorbit interaction

1f7/2

1f5/21f

Single particle energy levels:

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

j (2j+1)

(2)

(2)

(4)

(2)

(6)

(4)

(8)

(10)

(2)(6)(4)

16

CollectiveCollective Nuclear ModelsNuclear Models

17

Collective excitations of nucleiCollective excitations of nuclei

�The single-particle shell model can not properly describe the excited states of

nuclei: the excitation spectra of even-even nuclei show characteristic band

structures which can be interpreted as vibrations and rotations of the nuclear

surface

low energy excitations have a collective origin !

�The liquid drop model is used for the description of collective excitations of

nuclei: the interior structure, i.e., the existence of individual nucleons, is neglected

in favor of the picture of a homogeneous fluid-like nuclear matter.

� The moving nuclear surface may be described quite generally by an expansion

in spherical harmonics with time-dependent shape parameters as coefficients:

where R(θθθθ,φφφφ,t) denotes the nuclear radius in the direction (θθθθ,φφφφ) at time t, and R0 is

the radius of the spherical nucleus, which is realized when all ααααλµ λµ λµ λµ =0.

The time-dependent amplitudes ααααλµλµλµλµ(t) describe the vibrations of the nucleus with

different multipolarity around the ground state and thus serve as collective

coordinates (tensor).

(1)

18

Collective excitations of nucleiCollective excitations of nuclei

vibrationsvibrations rotationsrotations

19

Properties of the coefficients ααααλµ λµ λµ λµ ::::

- Complex conjugation: the nuclear radius must be real, i.e.,

R(θθθθ,φφφφ,t)=R*(θθθθ,φφφφ,t).

Applying (2) to (1) and using the property of the spherical harmonics

one finds that the ααααλµλµλµλµ have to fulfill the condition:

Collective coordinates

(2)

(3)

(4)

- The dynamical collective coordinates ααααλµλµλµλµ (tensors!) define the distortion -

vibrations - of the nuclear surface relative to the groundstate.

- The general expansion of the nuclear surface in (1) allows for arbitrary

distortions: λλλλ=0,1,2,….

(1)

20

Types of Multipole Deformations Types of Multipole Deformations

�The monopole mode, , , , λλλλ = 0.

The spherical harmonic Y00 is constant, so that

a nonvanishing value of αααα00 corresponds to a change of the radius of the sphere.

The associated excitation is the so-called breathing mode of the nucleus. Because

of the large amount of energy needed for the compression of nuclear matter, this

mode is far too high in energy to be important for the low-energy spectra

discussed here. The deformation parameter αααα00 can be used to cancel the overall

density change present as a side effect in the other multipole deformations.

ππππ4

100 ====Y

Groundstate

�The dipole mode, λλλλ = 1.

Dipole deformations, λλλλ = 1 to lowest order, really do not

correspond to a deformation of the nucleus but rather to a

shift of the center of mass, i.e. a translation of the nucleus, and

should be disregarded for nuclear excitations since

translational shifts are spurious.

θθθθcos10 ≈≈≈≈Y

)4

1(R)Y1(R)t,,(R 00000000 ππππ

ααααααααφφφφϑϑϑϑ ++++====++++====

Monopole

mode λλλλ=0

21

Types of Multipole Deformations Types of Multipole Deformations

�The quadrupole mode, , , , λλλλ = 2

The quadrupole deformations - the most important

collective low energy excitations of the nucleus.

�The octupole mode, , , , λλλλ = 3

The octupole deformations are the principal asymmetric

modes of the nucleus associated with negative-parity bands.

�The hexadecupole mode, , , , λλλλ = 4

The hexadecupole deformations: this is the highest angular

momentum that has been of any importance in nuclear

theory. While there is no evidence for pure hexadecupole

excitations in the spectra, it seems to play an important role

as an admixture to quadrupole excitations and for the

groundstate shape of heavy nuclei.

22

Types of Multipole Deformations Types of Multipole Deformations

23

Quadrupole deformations

(5)

� The quadrupole deformations are the most important vibrational degrees of

freedom of the nucleus.

For the case of pure quadrupole deformation (λλλλ = 2) the nuclear surface is given by

� Consider the different components of the quadrupole deformation tensor αααα2µ2µ2µ2µ

The parameters αααα2µ 2µ 2µ 2µ are not independent - cf. (4):

From (4):

(6) => αααα20202020 is real (since αααα20202020==== αααα∗∗∗∗20202020)))) ; and we are left with five independent real

degrees of freedom: αααα20202020 and the real and imaginary parts of αααα21212121 and αααα22222222

(6)

�To investigate the actual form of the nucleus, it is best to express this in

cartesian coordinates by rewriting the spherical harmonics in terms of the

cartesian components of the unit vector in the direction (θθθθ,φφφφ) :

(7)

24

From spherical to cartesian coordinates

Spherical coordinates (r,θ,φθ,φθ,φθ,φ) Cartesian coordinates (x,y,z)=>((((ζ,ξ,η)ζ,ξ,η)ζ,ξ,η)ζ,ξ,η)

r

The invention of Cartesian coordinates in

the 17th century by René Descartes

(Latinized name: Cartesius)

25

Cartesian coordinates

(8)Cartesian coordinates fulfil subsidiary conditions

(9)

Substitute (9) in (5):

where the cartesian components of the deformation are related to the spherical

ones by

(10)

(11)

26

Subs. (10) into (12) and accounting that

we obtain:

In (11) six independent cartesian components appear (all real) , compared to the

five degrees of freedom contained in the spherical components. However, the

function R(θθθθ,φφφφ) fulfills

Cartesian coordinates

(12)

(12)

���� 5 independent cartesian components

As the cartesian deformations are directly related to the streching (or

contraction) of the nucleus in the appropriate direction, we can read off that:

� αααα20202020 describes a stretching of the z axis with respect to the у and x axes,

� αααα22222222, , , , αααα2222−−−−2222 describes the relative length of the x axis compared to the у axis (real

part), as well as an oblique deformation in the x-y plane,

� αααα21212121, , , , αααα2222−−−−1111 indicate an oblique deformation of the z axis.