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!