Nuclear Structure Theory

Erich Ormand N-Division, Physics and Adv. Technol

ogies Directorate Lawrence Livermore National Lab

oratory

Nuclear Structure Theory

Of course, I cant cover everything in just three

hours At least we speak a common language. It cou

ld be worse

I could have to explain the rules to baseball!

Homework Explain the infield-fly rule

Nuclear Structure in the future

Nuclear physics is something of a mature field,

but there are still many unanswered questions

about nuclei Do we really know how they are put t

ogether? This is a fundamental question in nuclea

r physics and we are now getting some interesting

answers for example, three-nucleon forces are

important for structure Atomic nuclei make up the

vast majority of matter that we can see (and

touch). How did they (and we) get there?

Nucleosynthesis during supernovae (r-process)

Of particular importance are the limits of

nuclear existence How do we address these questio

ns with theory?

Before we get started Some useful reference

materials

General references for nuclear-structure physics

Angular Momentum in Quantum Mechanics, A.R.

Edmonds, (Princetion Univ. Press, Princeton,

1968) Structure of the Nucleus, M.A. Preston and

R.K. Bhaduri, (Addison-Wesley,Reading, MA, 1975)

Nuclear Models, W. Greiner and J.A. Maruhn,

(Springer Verlag, Berlin, 1996)

Basic Ideas and Concepts in Nuclear Physics, K.

Heyde (IoP Publishing, Bristol, 1999)

Nuclear Structure vols. I II, A. Bohr and B.

Mottelson, (W.A. Benjamin, New York, 1969)

Nuclear Theory, vols. I-III, J.M. Eisenberg and

W. Greiner, (North Holland, Amsterdam, 1987)

Before we get started Some useful reference

materials

References for many-body problem

Shell Model Applications in Nuclear Spectroscopy,

P.J. Brussaard and P.W.M. Glaudemans, (North

Holland, Amsterdam) The Nuclear Many-Body problem

, P. Ring and P. Schuck, (Springer Verlag,

Berlin, 1980) Theory of the Nuclear Shell Model,

R.D. Lawson, (Clarendon Press, Oxford, 1980)

A Shell Model Description of Light Nuclei, I.S.

Towner, (Clarendon Press, Oxford, 1977)

The Nuclear Shell Model Towards the Drip Line,

B.A. Brown, Progress in Particle and Nuclear

Physics 47, 517 (2001)

Review for applying the shell model near the drip

lines

Nuclear masses, what nuclei exist?

Lets start with the semi-empirical mass formula,

Bethe-Weizsäker formual, or also the liquid-drop

model. There are global Volume, Surface,

Symmetry, and Coulomb terms And specific correct

ions for each nucleus due to pairing and shell

structure

A main goal in theory is to accurately describe

the Binding energy

Values for the parameters, A.H. Wapstra and N.B.

Gove, Nulc. Data Tables 9, 267 (1971)

Nuclear masses, what nuclei exist?

For each energy term, there are also shape

factors dependent on the quadrupole deformation

parameters b and g Note the

liquid-drop always has a minimum for spherical

shapes, deformed ground states are a consequence

of shell corrections

Nuclear masses, what nuclei exist?

Pairing Shell correction In general, the li

quid drop does a good job on the bulk properties

But we need to put in corrections due to shell

structure Strutinsky averaging difference betwe

en the energy of the discrete spectrum and the

averaged, smoothed spectrum

Mean-field single-particle spectrum

Discrete spectrum

Smoothed spectrum

How well does a mass formula work?

Liquid-drop parameters are fit to known masses

(or even calculated) There are several variants

Most formulae reproduce the known masses at the

level of 600 keV heavier nuclei, and 1 MeV for

light nuclei

The location of the neutron drip-line in rather

uncertain!

Mass Formulae

Homework

Use the Bethe-Weizsäker fromula to calculate

masses, and determine the line of stability

(ignore pairing and shell corrections)

Show that the most stable Z0 value is

More advanced homework

Assume symmetric fission and calculate the energy

released. Approximately at what A value is the

energy released 0?

Inter-nucleon interactions

Look at the simplest case Two nucleons

NN-scattering The deuteron From these we infer t

he form of the nucleon-nucleon interaction

The starting point is, of course, the Yukawa

hypothesis of meson exchange Pion, rho, sigma, tw

o pion, etc. However, it is also largely phenomen

ological Deuteron binding energy 2.224 MeV Deut

eron quadrupole moment 0.282 fm2

Scattering lengths and ranges for pp, nn, and

analog pn channels Unbound! Note that

Vpp ? Vnn ? Vpn Some of the most salient features

are the Tensor force and a strong repulsive core

at short distances

NN-interactions

Argonne potentials R.B. Wiringa, V.G.J. Stoks, R.

Schiavilla, PRC51, 38 (1995) Coulomb One pion

exchange intermediate- and short-range

Bonn potential R. Machleidt, PRC63, 024001 (200

1) Based on meson-exchange Non-local Effectiv

e field theory C. Ordóñez, L. Ray, U. van Kolck,

PRC53, 2086 (1996) E. Epelbaoum, W. Glöckle,

Ulf-G. Meißner, NPA637, 107 (1998)

Based on Chiral Lagrangians Expansion in momentum

relative to a cutoff parameter ( 1 GeV)

Generally has a soft core All are designed to r

eproduce the deuteron and NN-scattering

NN-interactions

Pion exchange is an integral part of

NN-interactions Elastic scattering in momentum sp

ace Or, through a Fourier transform, co

ordinate space ( )

Off-shell component present in the Bonn

potentials Non-local (depends on the ene

rgies of the initial and final states)

Plays a role in many-body applications and

provides more binding

Tensor operator

Three-Nucleon interactions

First evidence for three-nucleon forces comes

from exact calculations for t and 3He

Two-nucleon interactions under bind

Note CD-Bonn has a little more binding due to

non-local terms Further evidence is provided by a

b initio calculations for 10B NN-interactions giv

e the wrong ground-state spin! 1 instead of 3

Recent first-principles calculation for 10B

Interactions in real-world applications

Ideally, we would like to use these fundamental

interactions in our theory calculations

In most cases this is not really practical as the

the NN-interaction has a very strong repulsive

core at short distances This means that in many-b

ody applications an infinite number of states are

needed as states can be scattered to high-energy

intermediate states This means we need to use eff

ective interactions Derived from some formal theo

ry This is in principle possible but is also very

difficult and is becoming practical only now for

light nuclei Assume they exist as the formal theo

ry stipulates and determine it empirically to

reproduce data This has permitted many studies in

nuclear structure to go forward

Isospin

Isospin is a spectroscopic tool that is based on

the similarity between the proton and neutron

Nearly the same mass, qp1, qn0

Heisenberg introduced a spin-like quantity with

the z-component defining the electric charge

Protons and neutrons from an isospin doublet

Add isospin using angular momentum algebra,

e.g., two particles T1 T0

With T0, symmetry under p ? n

Isospin

For Z protons and N neutrons Even-even NZ

T0 Odd A TTz Odd-odd NZ T0 or T1 (Above

A22, essentially degenerate) I

f VppVnnVpn, isospin-multiplets have the same

energy and isospin is a good quantum number

Isospin

Of course, Vpp ? Vnn ? Vpn NN-int

eraction has scalar, vector, and tensor

components in isospin space Note th

at the Coulomb interaction contributes to each

component and is the largest!!!

Coulomb-displacement energies

We apply the Wigner-Eckart theor

em and obtain the Isobaric-Mass-Multiplet

Equation (IMME)

Coulomb-displacement energies

Can we use the IMME to predict the proton

drip-line? Binding energy difference between mirr

or nuclei In a T-multiplet, often the bindi

ng energy of the neutron-rich mirror is measured

Calculated with theory

Simple estimate from a charged sphere with radius

r1.2A1/2

Coulomb-displacement energies

Map the proton drip-line up to A71 using Coulomb

displacement with an error of 100-200 keV on

the absolute value B.A. Brown, PRC42, 1513 (1991)

W.E. Ormand, PRC53, 214 (1996) B.J. Cole,

PRC54, 1240 (1996) W.E. Ormand, PRC55, 2407

(1997) B.A. Brown et al., PRC65, 045802 (2002)

Use nature nature to give us the strong

interaction part, i.e., the a-coefficient by

adding the Coulomb-displacement to the

experimental binding energy of the neutron-rich

mirror

Yes! Coulomb displacement energies provide an ac

curate method to map the proton drip line up to

A71

Proton separation energiesdi-proton emission

Separation energies

Candidates for two-proton emission

Lifetime is very sensitive to the separation

energy (R-matrix or WKB approximation)

Primary competition is beta decay

Proton separation energiesdi-proton emission

Many-body Hamiltonian

Start with the many-body Hamiltonian

Introduce a mean-field U to yield basis

The mean field determines the shell stru

cture In effect, nuclear-structure calculations r

ely on perturbation theory

Residual interaction

Single-particle wave functions

With the mean-field, we have the basis for

building many-body states This starts with the si

ngle-particle, radial wave functions, defined by

the radial quantum number n, orbital angular

momentum l, and z-projection m

Now include the spin wave function Two cho

ices, jj-coupling or ls-coupling

Ls-coupling jj-coupling is very convenient wh

en we have a spin-orbit (l?s) force

Multiple-particle wave functions

Total angular momentum, and isospin

Anti-symmetrized, two particle, jj-coupled wave

function Note JTodd if the particles

occupy the same orbits Anti-symmetrized, two part

icle, LS-coupled wave function

Two-particle wave functions

Of course, the two pictures describe the same

physics, so there is a way to connect them

Recoupling coefficients Note that t

he wave functions have been defined in terms of

and , but often we need them in terms of the

relative coordinate We can do this in two ways

Transform the operator

Two-particle wave functions in relative coordinate

Use Harmonic-oscillator wave functions and

decompose in terms of the relative and

center-of-mass coordinates, i.e.,

Harmonic oscillator wave functions are a ver

y good approximation to the single-particle wave

functions We have the useful transformation

2n1l22n2l22nl2NL Where the M(nlNLn1l1

n2l2) is known as the Moshinksy bracket

Note this is where we use the jj to LS coupling

transformation For some detailed applications loo

k in Theory of the Nuclear Shell Model, R.D.

Lawson, (Clarendon Press, Oxford, 1980)

Many-particle wave function

To add more particles, we just continue along the

same lines To build states with good angular mome

ntum, we can bootstrap up from the two-particle

case, being careful to denote the distinct

states This method uses Coefficients of Fractiona

l Parentage (CFP) Or we can make a many-b

ody Slater determinant that has only a specified

Jz and Tz and project J and T

The Slater determinant is very convenient

especially in second quantization formalism

Second Quantization

Second quantization is one of the most useful

representations in many-body theory

Creation and annhilation operators

Denote 0? as the state with no particles (the

vacuum) ai creates a particle in state i ai a

nnhilates a particle in state i

Anticommuntation relations Many-body Slate

r determinant

Second Quantization

Operators in second-qunatization formalism

Take any one-body operator O, say quadrupole E2

transition operator er2Y2m, the operator is

represented as where ?jOi? is the single

-particle matrix element of the operator O

The same formalism exists for any n-body

operator, e.g., for the NN-interaction

Here, Ive written the two-body matrix e

lement with an implicit assumption that it is

anti-symmetrized, i.e.,

Second Quantization

Angular momentum tensors Creation operators rotat

e as tensors of rank j Not so for annihilation op

erators Anti-symmetrized, two-body state

Second Quantization

Matrix elements for Slater determinants

Second quantization makes the computation of

expectation values for the many-body system

simpler

The mean field

One place to start for the mean field is the

harmonic oscillator Specifically, we add the cent

er-of-mass potential The Good Provides

a convenient basis to build the many-body Slater

determinants Does not affect the intrinsic motion

Exact separation between intrinsic and center-of

-mass motion The Bad Radial behavior is not rig

ht for large r Provides a confining potential, s

o all states are effectively bound

Hartree-Fock

There are many choices for the mean field, and

Hartree-Fock is one optimal choice

We want to find the best single Slater

determinant F0 so that Thouless theorem

Any other Slater determinant F not orthogonal to

F0 may be written as Where i is a state occ

upied in F0 and m is unoccupied Then

Hartree-Fock

Let i,j,k,l denote occupied states and m,n,o,p

unoccupied states After substituting back we get

This leads directly to the Hartree-Fock

single-particle Hamiltonian h with matrix

elements between any two states a and b

Hartree-Fock

We now have a mechanism for defining a

mean-field It does depend on the occupied states

Also the matrix elements with unoccupied states

are zero, so the first order 1p-1h corrections do

not contribute We obtain an eigenvaule equa

tion (more on this later) Energies of A1

and A-1 nuclei relative to A

Hartree-Fock Eigenvalue equation

Two ways to approach the eigenvalue problem

Coordinate space where we solve a

Schrödinger-like equation Expand in terms of a ba

sis, e.g., harmonic-oscillator wave function

Exapnsion Denote basis states by Greek letters, e

.g., a From the variational principle,

we obtain the eigenvalue equation

Hartree-Fock Solving the eigenvalue equation

As I have written the eigenvalue equation, it

doesnt look to useful because we need to know

what states are occupied We use three steps Make

an initial guess of the occupied states and the

expansion coefficients Cia For example the lowest

Harmonic-oscillator states, or a Woods-Saxon and

Ciadia With this ansatz, set up the eigenvalue e

quations and solve them Use the eigenstates i? f

rom step 2 to make the Slater determinant F0, go

back to step 2 until the coefficients Cia are

unchanged

The Hartree-Fock equations are solved

self-consistently

Hartree-Fock Coordinate space

Here, we denote the single-particle wave

functions as fi(r) These equations are

solved the same way as the matrix eigenvalue

problem before Make a guess for the wave function

s fi(r) and Slater determinant F0

Solve the Hartree-Fock differential equation to

obtain new states fi(r) With these go back to st

ep 2 and repeat until fi(r) are unchanged

Exchange or Fock term UF

Direct or Hartree term UH

Again the Hartree-Fock equations are solved

self-consistently

Hartree-Fock with the Skyrme interaction

In general, there are serious problems trying to

apply Hartree-Fock with realistic NN-interactions

(for one the saturation of nuclear matter is

incorrect) Use an effective interaction, in parti

cular a force proposed by Skyrme

Ps is the spin-exchange operator

The three-nucleon interaction is actually a

density dependent two-body, so replace with a

more general form, where a determines the

incompressibility of nuclear matter

Hartree-Fock with the Skyrme interaction

One of the first references D. Vautherin and

D.M. Brink, PRC5, 626 (1972) Solve a Shrödinger-l

ike equation Note the effective mass m

Typically, m have to, and is determined by the parameters t1

and t2 The effective mass influences the spacing

of the single-particle states The bias in the pas

t was for m/m 0.7 because of earlier

calculations with realistic interactions

tz labels protons or neutrons

Hartree-Fock calculations

The nice thing about the Skyrme interaction is

that it leads to a computationally tractable

problem Spherical (one-dimension) Deformed Axia

l symmetry (two-dimensions) No symmetries (full t

hree-dimensional) There are also many different c

hoices for the Skyrme parameters

They all do some things right, and some things

wrong, and to a large degree it depends on what

you want to do with them Some of the leading (or

modern) choices are M, M. Bartel et al., NPA386

, 79 (1982) SkP includes pairing, J. Dobaczewsk

i and H. Flocard, NPA422, 103 (1984)

SkX, B.A. Brown, W.A. Richter, and R. Lindsay,

PLB483, 49 (2000) Apologies to those not mentione

d! There is also a finite-range potential based o

n Gaussians due to D. Gogny, D1S, J. Dechargé and

D. Gogny, PRC21, 1568 (1980). Take a look at J. D

obaczewski et al., PRC53, 2809 (1996) for a nice

study near the neutron drip-line and the effects

of unbound states

Hartree-Fock calculations

Picture of the single-particle potential and the

effective mass for 208Pb D. Vautherin and D.M.

Brink, PRC5, 626 (1972)

Proton and neutron single-particle states

Hartree-Fock calculations

Permits a study of a wide-range of nuclei, in

particular, those far from stability and with

exotic properties, halo nuclei

The tail of the radial density depends on the

separation energy S. Mizutori et al. PRC61, 04432

6 (2000)

H. Sagawa, PRC65, 064314 (2002)

Drip-line studies J. Dobaczewski et al., PRC53, 2

809 (1996)

Hartree-Fock calculations

Shell structure Because of the self-consistency,

the shell structure can change from nucleus to

nucleus

As we add neutrons, traditional shell closures

are changed, and may even disappear!

This is THE challenge in trying to predict the

structure of nuclei at the drip lines!

J. Dobaczewski et al., PRC53, 2809 (1996)

Beyond mean field

Hartee-Fock is a good starting approximation

There are no particle-hole corrections to the HF

ground state The first correction is

However, this doesnt make a lot of sense for

Skyrme potentials They are fit to closed-shell nu

clei, so they effectively have all these

higher-order corrections in them!

We can try to estimate the excitation spectrum of

one-particle-one-hole states Giant resonances

Tamm-Dancoff approximation (TDA)

Random-Phase approximation (RPA)

You should look these up! A Shell Model Descripti

on of Light Nuclei, I.S. Towner

The Nuclear Many-Body Problem, Ring Schuck

Low-lying structure The interacting Shell Model

The interacting shell model is one of the most

powerful tools available too us to describe the

low-lying structure of light nuclei

We start at the usual place Construct many

-body states fi? so that Calculate Hamiltoni

an matrix Hij?fjHfi? Diagonalize to obtain eig

envalues

Computational Limit is about 108

Nuclear structure with NN-interaction

This is not practical because of the short-range

repulsion in VNN

V(r) in 1S0 channel

Strong repulsion at 0.5 fm ??jH?i? large

Problem Repulsion in strong interaction ?

Infinite space!

Can we get around this problem?Effective

interactions

Choose subspace of for a calculation

(P-space) Include most of the relevant physics

Q -space (excluded - infinite)

Effective interaction Two approaches

Bloch-Horowitz Lee-Suzuki

Q

P

Energy dependent

HeffPXHX-1P

Effective interactions permit first-principles

shell-model applications

Impossible problem ? Difficult problem

Two, three, four, A-body operators

Compromise between size of P

-space and number of clusters Three-body clusters

?

Effective interactions can make the problem

tractable

The general idea behind effective interactions

Effective interaction to produce exact

eigenvalues with P-space

We choose the P-space as a compromise betwee

n computational complexity and physics

H ? e1, e2, ,enP, , e? Heff ? e1, e2, ,enP

Find Heff with the decoupling Condition

QXHX-1P0

or HeffPXHX-1P

Effective interactions in the real world

A lot of progress has been recently (No-core

shell model) using effective interactions derived

from realistic NN-interactions

There is still a LOT of work yet to be done and

they may never be practical beyond A 20

The practical Shell Model Choose a model space to

be used for a range of nuclei

E.g., the 0d and 1s orbits (sd-shell) for 16O to

40Ca or the 0f and 1p orbits for 40Ca to 120Nd

We start from the premise that the effective

interaction exists We use effective interaction t

heory to make a first approximation (G-matrix)

Then tune specific matrix elements to reproduce

known experimental levels With this empirical int

eraction, then extrapolate to all nuclei within

the chosen model space

The empirical shell model works well!

But be careful to know the limitations!

The Shell Model

We write the Hamiltonian as Start with

closed inner core, e.g., for 24Mg, close the

p-shell Single-particle energy ei two-body inter

actions with particles in the core

Active valence particles in a computationally

viable model space, e.g., the 0d5/2, 0d3/2, 1s1/2

orbits for 24Mg Two-body matrix elements Excitat

ions outside the valence space are not included

except renormalizations in the matrix elements

Shell-model basis states

Need to construct the many-body basis states to

calculate matrix elements of H

Two choices Impose symmetry quantum numbers, such

as parity, angular momentum, isospin, etc.

Limit oneself to just parity, Jz and Tz and let

the Hamiltonian do the rest A very useful approac

h is a bit-representation known as the M-scheme

Phase operations are reduced to

counting set bits

Basis states, continued

Counting the number of basis states

Order-of-magnitude estimate n particles, and Nsp

s single-particle states Nsps in the sd-shell 1

2 (0d5/26, 0d3/24, 1s1/22) Nsps in the fp-shel

l 20 (0f7/28, 0f5/26, 1p3/24, 1p1/22)

Includes states of all J and Jz Number of J

z0 divide by a factor of ten Number of states wi

th a given J

Getting the eigenvalues and wavefunctions

Setup Hamiltonian matrix ?jHi? and diagonalize

Lanczos algorithm Bring matrix to tri-diagonal fo

rm nth iteration computes 2nth momen

t But you cant find eigenvalues with calculated

moments Eigenvalues converge to extreme (largest

and smallest) values 100-200 iterations needed

for 10 eigenvalues (even for 108 states)

Application of the shell model

A18, two-particle problem with 16O core

Two protons 18Ne (T1) One Proton and one neutro

n 18F (T0 and T1)

Two neutrons 18O (T1)

Homework

Part 1

How many states for each Jz? How many states of

each J and T?

Part 2

What are the energies of the three 0 states in

18O?