Nuclear ModelsBasic Concepts in Nuclear Theory

Joachim A. Maruhn

Literature

Foundations

Collective models

Single-particle models: phenomenological and self-consistent

The Fermi-gas model

Topics

W. Greiner and J. A. Maruhn, „Nuclear Models“, Springer

P. Ring and P. Schuck, „The Nuclear Many-body Problem“, Springer.

J. A. Maruhn, P.-G. Reinhard, and E. Suraud, „Simple Models of Many-Fermion

Systems“, Springer

Basic facts for nuclear models

Constant binding: saturating interaction

Decrease for heavy nuclei indicates

increasing importance of Coulomb

force.

Nuclear radii go as

showing constant density on the average

independent of A.

3

0 0, 1.2 fmR r A r

30.15 0.17 fm

Basic facts (2)

The density falls off „rapidly“ at the

surface: some models assume a

sharp surface.

Heavier nuclei deviate from N=Z:

again effect of Coulomb. Otherwise

there is a tendency to symmetry.

The liquid-drop model

The Bethe-Weizsäcker formula describes the binding energy of spherical

nuclei with mass number A, charge number Z, and neutron number N.

omitting smaller corrections.

This simple formula already provides an understanding of

the binding properties going from light to heavy nuclei

the location of the b-stable line

the possibility of fusion and fission

the potential energy involved in nuclear deformation (unfortunately not the

kinetic part)

2 2 2

3

3

volume term surface termSymmetry termCoulomb term

( )· · · ·B V S C Sym

Z N ZE a A a A a a

AA

16 20 0.75 21V S C Syma MeV a MeV a MeV a MeV

Fission in the liquid drop

Angular momentum

Two wave functions (or operators) with angular momentum eigenvalues

can becoupled to a resulting angular momentum using

where

and is a Clebsch-Gordan coefficient.

Most important case: coupling to a scalar of two operators:

with the usual Condon-Shortlex phase convention

1 2

1 2 1 2 1 1 2 2( | )m m

J M j j J m m M j m j m

1 2 1 2( | )j j J m m M1 2 1 2 1 2, | |M m m j j J j j

( 1)( 0 | 0)

2 1

j m

jj m mj

*( 1) 1( 0 | 0)

2 1 2 1

j m

jm j m jm j m jm jm

m m m

jj m m T T T T T Tj j

*( 1) j m

j m jmT T

Nuclear deformation

Assuming a time-dependent sharp surface located at

it can be expanded as

with the the collective deformation coordinates.

The parity is given by and

The lowest values of describe the simplest deformation

modes:

0: monopole or breathing mode: higher energy

1: translation, not an excitation

2: quadrupole, most important deformation

3: octupole, asymmetric deformation

4: hexadecupole: important for heavy nuclei

( , )R

*

0

,

( , ) (1 ( ) ( , ))R R t Y

( )t

( 1)*

, ( 1)

The spherical vibrator model

Assume a spherical ground state and a harmonic potential andconsider the quadrupole only

For the kinetic energy use collective velocitiesand a harmonic form

leading to the Lagrangian

This corresponds to five harmonic oscillators forall with an energy quantum of

Second quantization can be done as usual leading to boson operators

and a total energy

Thus an equidistant spectrum is generated. N is the phonon number.

21

2 22( )V C

2

21

2 22T B

2 2

1 12 2 2 22 2

L B C

2, 1,0, 1, 2

2 2 2C B

2 2 2 2 2ˆ ˆ ˆ ˆˆ, , n b b b b 5

2E N

Spherical vibrator spectrum

The multiplets for a given phonon number are determined by angular

momentum coupling and Bose symmetry. Each phonon carries an angular

momentum of 2.

Example: 114 Cd

Transition probabilities

via gamma emission can

also be calculated.

The model describes a

limiting case that is of

very limited applicability.

The rigid rotor

In the surface deformation model, a nucleus cannot rotate around an axis

of symmetry z, so that

For a deformed axially symmetric nucleus the Hamiltonian is

with the angular momenta in the comoving frame (see classical mechanics).

Now we have

so that the rotational energy with the quantized lab-frame angular

momentum becomes

leading to the famous

rotational band structure.

Because of symmetry

only even angular

momenta occur.

2 2ˆ ˆ' 'ˆ

2

x yJ JH

' 0zJ

2 2 2 2ˆ ˆ ˆ ˆ' ' 'x yJ J J J

2 ( 1)

2J

J JE

The principal-axes frame

The quadrupole deformations because of

contain five real parameters

Expressing the trigonometric functions through Cartesian coordinates

we get Cartesian deformation coefficients via

Inserting the definitions of the spherical harmonics leads to

Selecting the axes along the principal axes requires

*

2, 2( 1)

cos sin cos sin sinz x y

r r r

*

0 2 2

2 2 2

0

(1 )

(1 2 2 2 )

R R Y

R

20 2 1 2 2

1 8 8 1 8(2 ), ( ), ( 2 )

15 15 2 156i i

0

The coordinates b and

This leads to

The five degrees of freedom are now the intrinsic deformation coordinates

a0 and a2 plus the Euler angles giving the orientation.

Bohr and Mottelson used

chosen such that

The elongation becomes

2 1 20 0 2 2 2 0 20, , with a and a reala a

0 2

1 8 1 8(2 ) ( )

15 2 156a a

0 2

1cos , sin

2a ab b

22

2

b

5 5 2 5 44 4 3 4 3

cos , cos( ), cos( )

b b

Symmetries in the intrinsic frame

The arbitrariness of choosing

the intrinsic axes leads to

symmetry requirements for the

collective wave function that

restricts certain quantum

numbers.

prolate oblate

Types of collective behavior

The rotation-vibration model

We assume a well-deformed axially-symmetric nucleus executing small

vibrations around its equilibrium shape at

Assume dynamic deviations given by

and a harmonic potential

To lowest order, a Hamiltonian can then be set up as

Note that the dynamic deformation makes rotations about the z-axis

possible.

This contains harmonic b-vibrations in , -vibrations in modified by the

dynamic coupling to the rotation, plus rotational excitations.

Originally proposed by Bohr and Mottelson.

0 0 2, 0a ab

0 0 2,a ab

2 210 22

( , )V C C

2 2 2 22 2 22 2

2 2 2

ˆ ˆ ˆ1 1ˆ2 2 2 2 16

z zJ J JH C C

B B

The spectrum

The resulting energy formula is

yielding the rotational-vibrational spectrum. Note the coupling of the -

vibrations to rotation via the eigenvalue K of J‘z

221 1

2 | | 1 ( 1)2 2 2

E n n K J J Kb b

This structure is quite well

realized in nature, but the

interpretation of bands

higher than the b-band

remains controversial.

The selection of even J only

is caused by the symmetries

for some bands.

-unstable nuclei (Wilets-Jean)

42 20

02( , ) 2

8

CV D

bb b b

b

Generalizations

Using a potential with higher powers of the allows for describing more

complicated behavior (Gneuss & Greiner)

Contains a harmonic kinetic

energy plus polynomial to sixth

order of coupled to scalar.

All of these models are purely

phenomenological: their

parameters have to be fitted

for each nucleus.

proton-neutron „scissor mode“

using p, n

(V. Maruhn-Rezwani

et al., 1975)

2

2

The interacting boson approximation

proposed by Arima and Iachello

Constructs the collective excitations through two types of bosons:

scalar s-bosons and J=2 d-bosons: operators

These are interpreted as combinations of valence nucleons and their total

number should be constant:

A Hamiltonian is set up consisting of the bosons energies times their

number plus couplings between the bosons, again all with fitted coefficients.

Here (...)L means coupling to angular momentum L

,, ,s s d d

s dN n n s s d d

0

02 2 2 2

2

00 0 0 0

0

02 2

2 0

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

L L

s s d d L

L

H n n C d d dd

V d d ds d s dd

V d d ss s s dd

U d s ds U s s ss

The IBA (2)

An attractive feature is that several limiting cases can be solved exactly

using group theory: SU(3), SU(5), O(6).

A transition between different structures in neighboring nuclei appears

more clearly.

The fixed boson number should lead to a cutoff in angular momenta, which

is not seen. The model had to add additional bosons.

The types of spectra described now appears to be quite similar to that of

the „geometric model“.

Other difference to geometric model: kinetic energy is as complicated as

potential one; transition operators are not constrained by the geometric

interpretation.

Odd nuclei can be described in the Boson-Fermion model, where single-

particle operators are added.

IBA Applications

Single-particle models: Motivation

Experimental evidence for shell structure:

for magic numbers of protons or neutrons

larger total binding energy

larger separation energy for removing single nucleon

larger energy of lowest excited state

larger number of isotones or isotopes, respectively

nuclei tend to be spherical

Magic numbers are:

2, 8, 20, 28, 50, and 82 for bith protons and neutrons

126 for neutrons

114 or 126 for protons (predicted for superheavy elements)

Analogous to noble gases in atomic structure? Shell structure!

These effects describe small perturbations on the liquid-drop binding

energy but are crucial for the description of excitations

Magic numbers

Extrapolation to superheavies

Single-particle potentials

There is no external potential like in atoms: the single-particle structure:

it must be mitvated as a mean field by Hartree-Fock theory

The magic numbers can be explained by phenomenological potentials

Woods-Saxon:

Harmonic oscillator

Square well

0

( )( )

1 r R a

VV r

e

3

0 50MeV, 1.2 fm, 0,5fmV R A a

2 21( )

2V r m r

41MeV

0( )

V r RV r

r R

Woods-Saxon is most realistic, but harmonic

oscillator is often preferred because of

analytic eigenfunctions

Shell structure for simple potentials

Higher magic numbers

are not correct with

any of these potentials.

The spin-orbit coupling

Suggestion (Mayer & Jensen): add a spin-orbit force

with l and s now coupling to

Since

are good quantum numbers, the effect of te coupling can be calculated via

and the splitting becomes

Note that C turns out to be negative.

Question: origin of the spin-orbit force?

·V V Cl s

12

j l

2 2 2 2( ) , ,j l s l s

2 2 2 21 1· ( ) ( 1) ( 1) ( 1)

2 2l s j l s j j l l s s

2 2

1 1

2 2

1 1 3 1 1 1( )( ) ( )( ) ( )

2 2 2 2 2 2j l j lE E C l l l l C l

Shells with spin-orbit coupling

The magic numbers now come

out correct.

In many cases, the angular

momentum of the single-particle

states also explains the nuclear

angular momentum near magic

nuclei.

There are large deviations for

nuclei between magic shells:

nuclear deformation has to be

added.

The spin-orbit force can now be

seen as a relativistic effect (see

relativistic mean-field model)

Deformed nuclei: the Nilsson model

The oscillator potential can simply be generalized as

and the single-particle Hamiltonian becomes (maintaining axial symmetry)

The deformation parameter is b0

It turns out that, contrary to

expectations, gaps in the spectrum

appear also for deformed shapes,

leading to deformed ground states

and fission isomers

The behavior of the total energy

is wrong, however: needs

shell corrections

2 2 2 2 2 2( ) ( )2

x y z

mV r x y z

22 2 2 2 2

0 0 0 20

1ˆ ( , ) (2 )2 2

h m r m r Y l s lm

b

The Nilsson single-particle level scheme

Strutinsky shell corrections

The total energy in a single-particle model should be given by

This leads to huge fluctuations. The shell correction idea is based on

where dU is calculated by subtracting a „smoothed“ part from the sum ofsingle-particle energies. The resulting shell correction is typically a few MeV, negative near shell closures, positive otherwise.

This is the base of the modern microscopic-macroscopic (mic-mac) model, which uses more advanced versions of the phenomenological single-particlemodel and the droplet model. It is very successful for describing bindingenergies and fission barriers.

Note that there are still conceptual problems: neutrons and protons aredecoupled largely and the parameters are fitted, making extrapolation risky.

Hartree-Fock models do not have a problem with total energy but are not asaccurate at present.

occupied

( ) ( ) Coulombk

k

E b b

LDM( ) ( ) ( )E E Ub b d b

Two-center models

The Nilsson model does not correctly describe the transition to fission (it

always produces ellipsoids, not two fragments).

Two-center models try to cure this and are therefore better for the

description of fission and heavy-ion interactions.

Example: the two-center oscillator

It uses two oscillator potentials

centered in the fragments and

interpolated smoothly

Shape parameters:

The breakup of the neck poses prolems

2 1

2 1

2 1

Separation

Asymmetry

Neck

Deformations i i

z z

A A

A A

d

b a

d

Hierarchy of Models

Models with prescribed potentials: square-well, harmonic oscillator, Woods-Saxon,

Yukawa+Exponential (YPE), two-center models based on these.

In their latest versions these are still quantitatively superior.

Self-consistent models based on Hartree or Hartree-Fock. Most widespread are:

Skyrme-force, zero-range nonrelativistic

Gogny force, finite range nonrelativistic

Relativistic Meson-Field Theory (a.k.a. Walecka model). Interaction mediated by

relativistic mesons

Point-coupling model. Interaction through relativistic point interaction terms.

Density functionals: appear as intermediate step in self-consistent models

but can be more general since they need not derive from a force model.

Recently source of hype.

The number of parameters is generally similar: 6 - 12

The coexistence of many approaches shows the richness of nuclear theory

Hartree-Fock

We start with a general Hamiltonian with 2-body forces

(dependence on spin, isospin, and momenta can be added).

The many-body state can be expanded in Slater determinant states

with the ki a selction from a complete set of single-particle states.

The Hartree-Fock approximation consists in replacing this expansion by a

single Slater determinant (SSD), but with the single-particle wave functions

determined from a variational principle

The SSD is varied by changing the occupation of states

This is a particle-hole (ph) excitation.

2

1

1( )

2 2

Ai

i j

i i j

pH v r r

m

1 1 2

1

† † †ˆ ˆ ˆ... |0A A

A

k k k k k

k k

c a a a

ˆ ˆ| | 0 | | 0H Hd d

1 2

† † † †ˆ ˆ ˆ ˆ ˆ... | 0 , with ,A ima a a a a m A i Ad

The Hartree-Fock conditions

Using the second-quantized version of the Hamiltonian

with

the variational equation

leads to the Hartree-Fock conditions

which imply that the single-particle Hamiltonian

must have vanishing ph matrix elements

1 2 3 4 1 2 4 3

1 2 3 4

† † †1ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ,2

ij i j k k k k k k k k

ij k k k k

H t a a v a a a a T V

1 2 3 4 1 2 3 4

2*

2 3 3 3 * * and ( ) ( ) ( ) ( )2

kl l k k k k k k k kk

t d r v d r d r r r v r rm

† *ˆ ˆˆ ˆ0 | | | | .i mH a a c Hd d

1

( ) 0, with and A

mi mjij mjji

j

t v v m A i A

1

ˆ ( ), wit unrestricteh an dd kl

A

kl kjlj kjjl

j

h t v v k l

The Hartree-Fock equations

This can be achieved by choosing the single-particle states as eigenstates

defining also the single-particle energies.

In coordinate space the HF equations take the form

with the mean field

This is the simplest case without spin and isospin dependence.

The HF equations are a self-consistent problem that has to solved

iteratively.

Practical solution for heavy nuclei still requires simplified interactions like

the Skyrme or Gogny forces.

1

( )A

kl kl kjlj kjjl k kl

j

h t v v d

3 *( ) ' ( ') ( ) ( ')j j

j

U r d r v r r r r

22 3 *

1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).2

A

k k j j k k k

j

r U r r r d r v r r r r rm

Features of Hartree-Fock

Skyrme force: a typical effective force which simulates many-body effects

through density dependence introduced via a three-body force. It is easy to

handle because of zero range.

The interactions are normally fitted only to a very small number of

spherical nuclei and provide impressive predicitive power.

Hartree-Fock calculations generally describe bulk properties of nuclei quite

well, but details of the spectra do not come out as well.

Constraints are needed to calculate properties for non-equilibrium

deformations.

2 2112 0 0 1 2 1 1 2 1 22

2 1 2 0 1 2 1 2

123 3 1 2 2 3

1 2 1 2

(1 ) ( ) ( ) ' ( )

( ) i ( ) ' ( )

( ) ( )

with k=( ) / 2i and k'=-( ) / 2i

v t x P r r t r r k k r r

t k r r k W k r r k

v t r r r r

d d d

d d

d d

The Skyrme Energy Functional

0 01 1

3

Coulomb pair cm

222

2

2 2

1 1 2 2 1 2

32 24

4

2 2 2

1 1

16 1

2 3

6

2

q q q

q q

q

q

q q q

q q

q q

q

E d x x E E E

xm

J

J

b b

t x t x J

b b

b

t t J

b bb

b

E

E

0 1 2

1 2

3

1 10 0 0 1 22 2 4 2 2

1 1 1 11 1 2 2 1 24 2 2 8 2 2

1 1 1 11 1 2 2 38 2 2 4 2

1 1

0 0 1

1 2

2 3

31

3 3 4, 44 42 24

1 , , 1 1 ,

, 3 1 1 ,

3 , 1 ,

, (standard choice)

x x x

x x

x

b b b

b b

b b

b b

t t x t t

t x t x t t

t x t x t

t x t b b

Note: b4´ was introduced to

make spin-orbit coupling

similar to that in RMFT

The RMFT Lagrangian (most common variant)

nucleons mesons coupling

nucleons

2 2 21 1mesons 2 2

21 12 2

3 41 1coupling 3 4

1 102 2

( ) ( )

( ) (1 )

w

B

m m

m

g b d g

g

i m

R R R R A A

R A e

L L L L

L

L

L

ith A A A

Fitting Strategies

2-fit to experimental data:

binding energies (all forces)

diffraction radii (NL-Z2, NL-3, P-F1, SkIx)

surface thicknesses (NL-Z2, NL-3, P-F1, SkIx)

r.m.s. radii (NL3, NL-Z2, P-LA, P-F1)

spin-orbit splitting (Skyrme forces, P-LA)

isotope shift in Pb (SkIx)

neutron radii (NL3)

nuclear matter properties (NL3)

The selected (semi-) magic nuclei are: 16O, 40,48Ca, 56,58Ni, 88Sr, 90Zr, 100,112,120,124,132Sn, 136Xe, 144Sm, 202,208,214Pb

These are highly unrepresentative nuclei!

Earlier Predictions for the Next Magic

Superheavy magic proton number Nilsson Model (oscillator) Z=114

J. Grumann, U. Mosel, B. Fink, W. Greiner Z. Physik 228 (1969) 1

Nilsson-Strutinksy Z=114S. G. Nilsson, C. F. Tsang, A. Sobiczewski, P. Möller, Nucl. Phys. A131 (1969) 1.

Skyrme III Z=114, 120, 138M. Beiner, H. Flocard, M. Vénéroni, P. Quentin, Phys. Scripta 10A (1974) 84.

YPE and Folded Yukawa Z=114P. Möller, J. R. Nix, G. A. Leander, Z. Physik A323 (1986) 41.

YPE + Woods-Saxon Z=114Z. Patyk, A. Sobiczewski, Nucl. Phys. A533 (1991) 132.

RMF (NL-SH) Z=114C. A. Lalazissis, M. M. Sharma, P. Ring, Y. K. Gambhir, Nucl. Phys. A608 (1996) 202.

Skyrme SkP, Sly7 Z=126S. Cwiok, J. Dobaczewski, P. H. Heenen, P. Magierski, W. Nazarewicz, Nucl. Phys. A611 (1996) 211.

Magic Numbers for 3 Forces

Single-Particle Levels for 114X184

Single-Particle Levels for 120X172

Density of 120

Central depression discovered independently by J. Déchargé, J. F. Berger, K. Dietrich, and M. S. Weiss,

Phys. Lett. B451, 275 (1999).

Systematics of density distributions

Fission barriers

Uncertainties are still quite

large. In any case, the barriers

are quite narrow and lead to

short fission liftimes.

In experiment, the main

problem is still how to get

sufficiently many neutrons into

the system: radioactive beams?

Collectivity from single-particle models

Based on phenomenological or self-consistent models, collective vibrations

can be calculated as coherent excitations of many nuclei (RPA, TDHF). This

works well for higher-lying states.

It is also possible to calculate potentialsV(b,) to predict surface vibrations

in a nonlinear description. This it was long possible to predict

deformations, and moments of inertia, but not the vibrational excitations.

The problem is the calculation of the kinetic energy. Only recently a

reasonable description of the lowest vibrational states was achieved.

The same holds for very large-scale motion like fission and heavy-ion

reactions. For the latter a transition from the Hartree-Fock regime to a

collisional system is expected.

The Fermi gas model (1)

The nuclear potential is approximated by an infinite well of cubic shape

The eigenfunctions are

and eigenenergies (occuopied up to the Fermi energy

Going to spherical coordinates

we get (for degeneraxy factor g)

0 , 0 , ,( , , )

, otherwise

V x y z aV x y z

( , , ) · · · , , , 1,2,x y z

yx zn n n x y z

nn nx y z N sin x sin y sin z n n n

a a a

222 , ( , , )

2x y z x y zn n n n n n n nm a

2 2

2

FF

k

m

3 2d d d d d dx y zn n n n n n

333

22

22

2

32F

a mN g

The Fermi gas model (2)

The density of particles is

leading to the relation between the Fermi momentum and the density

(relatively constant throughout the periodic table.)

The total energy density (excluding potential energy) becomes

mean per particle

For different Z and N it behaves as

giving one reason for

the symmetry energy

33

22

3 22

2

32F

N N m

V a

g

21

36

1.41fmFkg

35

22

22

1 2

52F

me g

3

5F

ee

53

53

533

~5

N Ze

A

100, 0 100A Z

Further Developments

Treatment of correlations „beyond the mean field“

Derivation from underlying QCD

Special topics not addressed above, e.g.:

pairing!

giant resonances

high-spin states

b decay

cluster models

nuclear matter theory

Nuclear reactions

low-energy: models f many different kinds: TDHF, coupled channels, trajectorymethods, hydrodynamics

high energy: mostly thermodynamics and statistical mechanics

Nuclear Theory remains a very rich field with two fundamental directions:

small Fermion systems and the interplay of collectivity and single-particle aspects

the search for the underlying interaction

Many methods and models of quite different levels of sophistication coexist!