Nuclear Structure Theory

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Nuclear Structure Theory
Erich Ormand N-Division, Physics and Adv. Technol
ogies Directorate Lawrence Livermore National Lab
oratory
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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
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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?
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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)
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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
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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)
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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
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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
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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!
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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?
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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
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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
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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
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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
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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
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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
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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
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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!!!
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Coulomb-displacement energies
We apply the Wigner-Eckart theor
em and obtain the Isobaric-Mass-Multiplet
Equation (IMME)

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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
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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
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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
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Proton separation energiesdi-proton emission
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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
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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
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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
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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
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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)
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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
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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
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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.,
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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

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Second Quantization
Matrix elements for Slater determinants
Second quantization makes the computation of
expectation values for the many-body system
simpler
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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
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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!
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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
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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
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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
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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!
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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
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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
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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
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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)
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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?