Microscopic Nuclear Theory
RIA Summer School
August 1-6, 2005
Lawrence Berkeley National Laboratory
Lectures by
James P. Vary
ISU, LLNL, SLAC
“Introduction and the Phenomenological
Extreme Single Particle Shell Model (ESPSM)”
Plan for the 3 lectures
 Phenomenology - Extreme Single Particle Shell Model (ESPSM)
 Theory of Effective Operators
 Ab-initio No-Core Shell Model (NCSM)
 Applications with realistic NN interactions
 Applications with realistic NN + NNN interactions
 Reactions and Scattering with extensions of the NCSM
 Conclusions and Outlook
1
The Extreme Single Particle Shell Model
Illustrated with the 197 Au Example
Since the Nobel Prize awarded to Mayer, Haxel, Jenssen and Suess for the
phenomenological single-particle shell model, ever improving
parameterizations of the single-particle model have emerged.
Basic starting point is to assume nucleons move independently in their
average or “mean” field - generated by averaging over all the pairwise and
triplet, etc., interactions. [Big ? = How to derive this picture from NN+NNN’s]
Much of the body of low-energy nuclear structure and nuclear reaction data are
interpreted, to first approximation, with this 1-particle quantum mechanical
picture of independent Fermions in a 3-D potential well.
Hands-on activity - web-based tool for calculating properties of nuclei in this
Extreme Single-Particle Shell Model (ESPM)
http://www.physics.iastate.edu/jvary/NPC.htm
h"# = e#"# $ 1- particle problem in QM
h = t + u + uC
t=
p2
2m
u(r) =
U0
(r%R )
a
+ uSO $ Saxon - Woods form
1+ e
U 0 & %50MeV
R = r0 A1/ 3 ; r0 & 1.2 fm
a = diffuseness parameter & 0.5 fm
uC (r) = Coulomb potential of uniformly charged sphere
for the case of protons
Ze 2 '
r2 *
=
) 3 % 2 ,; r - R
2R (
R +
=
Ze 2
; r.R
r
!
2
Definition of the Spin-Orbit Potential
*
2
# h &
1 d, 1 /
uSO (r) = %
U SO S • L
r) R
2(
/
r dr ,
$ m" c '
+1+ e a SO .
2
!
!
!
# h &
= 2.0 fm 2
%
2(
$ m" c '
1#
3&
lj S • L lj = % j( j + 1) ) l(l + 1) ) (
2$
4'
Total Nuclear Wavefunctions = Slater Determinants =
Antisymmetrized product of these single-particle solutions
Total Nuclear Energies = simple sum of single-particle
energies of the occupied orbits
The Extreme Single Particle Shell Model
Illustrated with the 197 Au Example
Potentials are those of C.M. Perey and F.G. Perey,
Nuclear Data Tables 10, 540 (1972)
Note that, for the purposes of this illustrative study, the potential parameter
set chosen is that of the Protons from Perey and Perey. The Neutron potential
is developed from the Proton potential simply by changing the sign of the
(N-Z) term and dropping the Z/A1/3 term.
The Coulomb potential is that of a uniformly charged sphere with radius
equal to the radius parameter R = r0A 1/3 of the central potential
3
Techniques employed - Results generated
 Numerical evaluation of <H> in oscillator basis - matrix
elements by Gauss quadrature, one (l,j) channel at a time
hij = " i h " j ; where " i are harmonic oscillator states
!
 Matrix diagonalization of H by Jacobi method to obtain
eigenvalues and eigenvectors
 Numerical evaluation of eigenvectors from their expansion in
the oscillator basis:
"# = $ a#i% i
i
 Sorting of all eigenvalues to occupy states starting from the
most deeply bound until all nucleons used (ESPSM ground state)
 Evaluation of density distributions by summation over squares
of wavefunctions!of occupied orbits (ESPSM ground state)
 Evaluation of rms radius (ESPSM ground state)
http://www.physics.iastate.edu/jvary/NPC.htm
Rule: leaving a parameter unchanged, e.g. “0”, selects the default value of that parameter
Simplest test: insert your email and hit submit button - generates default case.
Next test: insert email, proton number, neutron number, select neutron or protons, & hit submit
4
Caution: This is a beta version of the web site & code.
It takes some practice to avoid exceeding code’s limits.
Choosing default values should be OK through 238-U,
possibly beyond. Output tells you when you exceed
allowed parameter Values.
Questions to consider
What are the asymptotic properties of the single-particle
wavefunctions and how do they follow from elementary
quantum mechanical (QM) considerations?
Where is the Fermi surface for the Neutrons and the Protons and
what experimental information would you use to verify that
these are at their correct positions?
Can one observe the progress towards classical behavior in the
wavefunctions with increasing orbital angular momentum?
What features does the spectra exhibit that reflect the details of
the magnitude and shape of the spin-orbit potential?
In what ways does the density distribution reflect the features of
the central potential?
Can one map out the behavior of the single particle states with
atomic number, A? What happens to the shell structure?
5
Key Parameters for Neutrons in 197Au
The “Default” case at the web site
-47.95482
1.25000
0.65000
15.00000
0.47000
0.00000
118.00000
79.00000
48
12
7.50000
600
0.05000
U0 - strength of Saxon-Woods in MeV
r0 - range parameter in fm (multiplies A**(1/3))
a - diffuseness parameter in fm
USO - strength of spin-orbit in MeV
aSO - diffuseness of spin-orbit in fm
STRC=(0,1) => Coulomb of uniform charged sphere (off,on)
Controls whether one is calculating the neutrons or protons
Neutron number
Proton number
Number of gauss points
Maximum principal Q# of HO basis
hbar*omega of HO basis in MeV
NB: current default value=7.0 MeV => enter 7.5 to reproduce
the tables used to generate the HO functions shown here)
Number of r-points on uniform grid
Grid size in fm
Key Parameters for Protons in 197Au
-64.07599
1.25000
0.65000
15.00000
0.47000
1.00000
118.00000
79.00000
48
12
7.50000
600
0.05000
U0 - strength of Saxon-Woods in MeV
r0 - range parameter in fm (multiplies A**(1/3))
a - diffuseness parameter in fm
USO - strength of spin-orbit in MeV
aSO - diffuseness of spin-orbit in fm
STRC=(0,1) => Coulomb of uniform charged sphere (off,on)
Controls whether one is calculating the neutrons or protons
Neutron number
Proton number
Number of gauss points
Maximum principal Q# of HO basis
hbar*omega of HO basis in MeV
NB: current default value=7.0 MeV => enter 7.5 to reproduce
the tables used to generate the HO functions shown here)
Number of r-points on uniform grid
Grid size in fm
Fundamental constants employed:
Nucleon mass: mc2 = 938.9185 MeV
i.e. the average of neutron and proton mass (estimate
the importance of this approximation)
6
7
8
9
Note that the Experimental
gs spin is 3/2+ and there is
a 1/2+ at 77 keV and
an 11/2- isomer at 409 keV.
Why are the neutrons not
easily excited?
Survey of the single-particle wavefunctions
Rnlj (r)
[Ylm l ($ ,% ) & sms ] jm j
r
' degeneracy in m j
"# = " nljm j =
e# = enlj
Examine these plots of Rnlj (r) /r and comment
on the QM properties seen:
!
 Harmonic oscillator
versus Saxon-Woods
!
 As a function of binding energy
 As a function of angular momentum
 Near the origin and in the tail region
 Other features anticipated/observed
10
11
12
13
14
15
16
17
Additional questions to ponder
• How does one measure radial charge
densities and radial mass densities?
• Are the oscillations in the ESPSM densities
reasonable and how would residual
interactions be likely to change them?
• How does the measured charge density of
197-Au compare with the ESPSM result.
• How do the experimental spectra of 197-Au
compare with the excitations of the ESPSM
near the Fermi Surface?
18
Planned Extensions
• Density of nucleons in the Harmonic
Oscillator model to compare with the Perey
and Perey model densities
• Simple spin-orbit potential added to
Harmonic Oscillator and matched to
experimental single-particle spectra
• Single particle electromagnetic transition
matrix elements: B(E1), B(M1), etc.,
• Newer phenomenological shell model
potentials - A.J. Koning & J.P. Delaroche,
Nucl. Phys. A 713, 231(2003).
Beyond the ESPSM:
Interactions among the nucleons
 Shell model with phenomenological (fit) interactions - discuss
recent review articles of Alex Brown, Andres Zuker, et al.
 Shell model with derived (G-matrix) interactions - discuss
recent review article of Morten Hjorth-Jenssen, et al.
 Ab-initio No-Core Shell Model (NCSM) = Our focus here
based on the Theory of Effective Operators - Goal is to derive
the nuclear shell model from “first principles” = “ab-initio”
19
Appendix A
Behavior of Resonances When Solved
in a Harmonic Oscillator Basis
n+19Ne Scattering
Central Potential Case
20
21
22
23
24