Microscopic Analysis of Excitation of Giant Resonances by Alpha Scattering and Nuclear Compressibili

1
Microscopic Analysis of Excitation of Giant
Resonances by Alpha Scattering and Nuclear
Compressibility

• Shalom Shlomo
• (Graduate Student Au Kim Vuong)
• Texas AM University

Hanoi, August 2005
2
Abstract

• Studies of isoscalar giant monopole and dipole
  resonances are of particular current interest
  since their strength distributions, S(E), are
  sensitive to the value of nuclear matter (NM)
  incompressibility coefficient, K, which is
  directly related to the curvature of the NM
  equation of state (EOS). The EOS is very
  important physical quantity in the study of
  properties of nuclei, supernova collapse, neutron
  starts and heavy-ion collisions. The main
  experimental tool for studying isoscalar giant
  resonance (GR) is inelastic alpha-particle
  scattering since alpha-particles are selective as
  to exciting isoscalar modes and the angular
  distributions of the excitation cross sections at
  small angles are characteristic for some of the
  multipolar modes. We emphasize that it is common
  in theoretical work on GR to calculate S(E) for a
  simple scattering operator F, within the
  mean-field based random phase approximation
  (RPA), whereas in the analysis of the cross
  sections one carries out distorted wave Born
  approximation (DWBA) calculations with a certain
  transition potential.
• We will review the elements of DWBA theory for
  calculating the excitation cross section, and the
  Hartree-Fock (HF) based RPA theory for the
  calculation of S(E), centroid energies, and
  transition densities of GR, with an emphasize on
  fully self-consistent calculations. We will
  present results of microscopic calculations of
  S(E), and of excitation cross sections. The
  implications of these results will be discussed.

3
Outline
I. Introduction Background Basic properties of
nuclei Classical and Quantum Description,
Mean-Field Giant Resonances (GR) Modes of
Oscillations, Classical Description Equation
of State and Compressibility

• Hadron excitation of giant resonance
• Experimental and Theoretical Approaches
• Folding Model Distorted Wave Born Approximation
  (FM-DWBA)

• Microscopic theory of giant resonance
• Hartree-Fock (HF) with Skyrme Interaction
• Random Phase Approximation (RPA)

• Results and Conclusions
• HF-RPA calculations of strength functions and
  energies of GR
• FM-DWBA calculations of cross-sections
• Conclusions

4
Background
Nuclear physics
Study of structure, interactions and properties
of nuclei. Aim to quantitatively understand and
relate vast amount of properties of nuclei in
terms of few constituents, elementary laws and
processes.
Current situation
Active area of research. We have a certain
picture (understanding) obtained through 70 years
of phenomenological research, qualitative
consideration and application of laws quantum
mechanics. Q. M. is very successful in describing
properties of nuclei. There is no evidence
contradicting Q. M.
Recent emphasize
Properties of nuclei under extreme conditions of
excitation energy (temperature), angular momentum
and N-Z (asymmetry).
Relation to other areas
Astrophysics source of energy stars, structure
and evolution of stars, origin of the elements,
neutron stars and supernova. Other systems
atomic clusters, metal clusters, trapped ions,
and mesoscopic systems.
5
Constituents
- Nuclei can be described in terms of neutrons
and protons degrees of freedoms (DOF) or in a
finer scale, in terms of quarks and gluons DOF. -
We will concentrate on the description in terms
of nucleons (hadrons) and consider low energy
phenomena.
Example Pb nucleus, Z 82 protons (charge
Ze), N 126 neutrons (charge 0), A Z N
208 nucleons Isotopes, fix Z and vary
N Isotones, fix N and vary Z Isobars, fix A and
vary Z and N
Interactions
- Strong interaction short range, 1-2 fm
(1E-15m), keep nucleons together, particle
decay. - Electromagnetic interaction coulomb
repulsion between protons electric and
magnetic properties, electromagnetic decay. -
Weak interaction beta decay. - Gravitation
usually ignored.
Nuclei exhibit fascinating phenomena of
single-particle motion, a correlated few body
motion and collective motion.
6
Basic properties
Size of nuclei 2 10 fm 10-12 cm.
Characteristic time (single particle motion)
R/c 10-21 sec.
7
Matter and charge density distributions
(a) Charge density distribution ?c(r) for doubly
magic nuclei 16O, 40Ca, 90Zr, 132Sn, and 208Pb.
The theoretical curves are compared with the
experimental data points (units are ?c(efm-3)
and r(fm)). (b) Nuclear matter density
distributions ?m(fm-3) for the magic nuclei.
8
Nuclear Binding (Weizsacker formula)
MeV (Volume)
MeV (Surface)
MeV (Coulomb)
MeV (Symmetry)
MeV (odd-odd)
(pairing)
(even-odd)
The contribution to B/A. Note that the surface,
asymmetry and Coulomb terms all subtract from
the bulk term.
MeV (even-even)
9
Map of the existing nuclei. The black squares in
the central zone are stable nuclei, the broken
inner lines show the status of known unstable
nuclei as of 1986 and the outer lines are the
assessed proton and neutron drip lines (Hansen
1991).
10
Description of many-body system
Classical description
Phase space coordinates of constituents
q1, , qA (position) p1, , pA (momenta)
A number of particles
System hamiltonian kinetic energy potential
energy
H T(p1,,pA) U(q1,
, qA)
For two-body interaction
Hamiltons equations of motion
These are coupled differential equations, solved
with initial (boundary) conditions.
Phase space distribution function
Matter density
11
Quantum description
Write Hamiltonian in Cartesian coordinates
Then put
?
The Schrödinger Equation
is solved
with boundary conditions. ?n
eigenfunction, En eigenenergy
Knowledge of ? is needed to calculate a physical
quantity associated with operator Ô
Transition matrix element
Example
Matter density
12
Example 1D harmonic oscillator
Classical
Hamilton equations of motion
?
Solution
continuous
Quantum
Solution
quantized
13
Mean-field approximation
The many-body Schrödinger equation H? E ? is
difficult to solve. In the mean-field
approximation each particle moves independently
from other nucleons in a single particle
potential, representing its interactions with all
other nucleons.
Approximation
Antisymmetrization operator (fermions) or
symmetrization operator (bosons)
14
Spherical symmetry
Wood-Saxon potential popular
MeV,
Spherical symmetry
is reduced to
where
15
Numerical solution of
Mesh points
For numerical approximation of derivative use
Taylor expansion
then
Now
?
?
?
etc.
?
Knowledge of
?
16
The spin-orbit coupling describe real nuclei
For the harmonic oscillation spin orbit
interaction, the energy eigenvalues become
Where,
d -l or l 1 if j l 1/2 or l-1/2
Single-particle spectrum up to N5. the various
contributions to the full orbital and spin orbit
splitting are presented. Partial and accumulated
nucleon numbers are also given.
17
Regularity of nuclear spectra
1. For all even-even nuclei, the ground state is
(always) Jp 0 (pairing)

• First excited state of even-even nuclei is Jp
  2 except
• 4He (0), 16O(0), 40Ca(0), 90Zr (0), 208Pb
  (3-).

3. Islands of isomers near closed shells (long
live excited states)
4. Regions of large deformations (large
quadrupole moments)
150 A 180 220 A 250
5. Giant resonances
collective motion
18
The isovector giant dipole resonance
The total photoabsorption cross-section for
197Au, illustrating the absorption of photons on
a giant resonating electric dipole state. The
solid curve show a Breit-Wigner shape. (Bohr and
Mottelson, Nuclear Structure, vol. 2, 1975).
19
Macroscopic picture of giant resonance
L 0
L 2
L 1
20
A Classical Picture of the Breathing Mode

• In the classical description of the breathing
  mode, the nucleus is modeled after a drop of
  liquid that oscillates by expanding and
  contracting about its spherical shape.
• We consider the isoscalar breathing mode in
  which the neutrons and protons move in phase
  (?T0, ?S0).

21
In the scaling model, we have the matter density
oscillates as
We consider small oscillations, so ? is very
close to zero ( 0.1). Performing a Taylor
expansion of the density
we obtain,
22
We have,
Where
is equal to

• This nicely agrees with the

transition density obtained from microscopic (HF
based RPA) calculations.
23
Equation of state and nuclear matter
compressibility

• The nuclear matter (NZ and no Coulomb
  interaction) incompressibility coefficient, K, is
  a very important physical quantity in the study
  of nuclei, supernova collapse, neutron stars, and
  heavy-ion collisions, since it is directly
  related to the curvature of the nuclear matter
  (NM) equation of state (EOS), E E(?).

E/A MeV
? 0.16 fm-3
? fm-3
E/A -16 MeV
24
(No Transcript)
25
History

• Isoscalar Giant Monopole Resonance (ISGMR)
• 1977 Dicovery of the Centroid energy of the
  ISGMR in 208Pb
• E0 13.5 MeV (TAMU)
• This led to modification of commonly used
  effective nucleon-nucleon interactions.
  Hartree-Fock (HF) plus Random Phase Approximation
  (RPA) calculations, with effective interactions
  (Skyrme and others) which reproduce data on
  masses, radii and the ISGMR energies have
• K 210 20 MeV (J.P. BLAIZOT, 1980).
• Isoscalar Giant Dipole Resonance (ISGDR)
• 1980 Experimental Centroid Energy in 208Pb at
• E1 21.3 MeV (Jülich), PRL 45 (1980) 337
• 19 MeV, PRC 63 (2001) 031301.
• HFRPA with interactions reproducing E0
  predicted E1 25 MeV.
• K 170 MeV from ISGDR ?
• More recently E1 22 MeV D. H. Youngblood, PRC
  69 (2004) 034611.
• Relativistic mean field (RMF) plus RPA
• The NL3 interaction predict K 270 MeV from the
  ISGMR
• N. Van Giai et al., NPA 687 (2001) 449.

26
Recent ExperimentsExperimental improvement in
alpha excitation of giant resonances signal
background from 1/3 to 20/1

• For Isoscalar Giant Monopole Resonance
• - Locating the ISGMR in light and medium nuclei.
• - Improved data for the ISGMR in heavy nuclei.
• - happy with systematic, i.e., good agreement of
  data with HFRPA results for interactions
  associated with K 230 MeV.
• Discovery of Transition Distribution Strength for
  the ISGDR.
• - The experimental data for the centroid
  energies are smaller than
• HFRPA prediction by 1-2 MeV for various
  nuclei.
• Investigation
• - Microscopic calculation of cross sections
  accuracy!
• - Modified interaction.
• - Effects of self-consistency violations.

27
Distorted wave Born approximation
A. Statement of the problem
The full wave function that describes a direct
nuclear reaction a A ? b B,
The full wave function obeys Schodinger equation
and the boundary condition which is typical for a
scattering problem
The first term describes the plane wave in the
incident channel a.
The second term is the sum of outgoing spherical
waves in all possible reaction channels ß.
28
The differential cross section dsaß for the
transition from the channel a to the channel ß is
defined as the RATIO between the outgoing flux
per unit time and the incoming flux per unit time
per unit area.
The probability flux in terms of the wave
function
is
For
For
The differential cross section
29
B. The formal solution of the scattering problem
The total hamiltonian H can be partitioned for a
specific reaction channel (ß)
Multiplying by from the left and
integrating over the internal variables of the
projectile and the target
with
Introducing a spherically symmetric distorting
potential , we have
30
The solution of this equation is the sum of a
particular solution and the
solution of the homogeneous
equation
We impose the boundary conditions
outgoing spherical wave
1)
4) particular solution is regular at
The solution of the homogeneous equation that
satisfies the given boundary conditions 1) and 2)
is well-known from the scattering theory
where is the phase shift.
31
is the regular solution of
with the asymptotic form
Using
We can get the elastic scattering amplitude
? is the scattering angles (between and
).
Therefore the elastic scattering cross section
is given by
32
The particular solution is given by
where is the Greens function
satisfying
and the boundary conditions
4) particular solution is regular at
Making a multipole expansion
where satisfies
33
The Greens function can be written
- regular solution,
- irregular solution,
is the smaller (greater) of
and
The incoming flux exists only in the incident
channel a. Therefore, the solution of the
homogeneous equation is added to the particular
solution only if ß a.
We have
34
When
and
Asymptotically
with
The scattering amplitude
is
35
C. Distorted Wave Approach to Scattering Problem.
The distorted wave Born approximation (DWBA)
hinges on the following two approximations
1) We assume that is small is
made, hence, it can be treated as
Therefore, the terms
are also small and we can retain only the elastic
term
2) We can choose so that the elastic
cross sections fit the experimentally measured
one at the given energy E. In this case
We thus obtain the approximate expression for the
scattering amplitude
36
D. Application of DWBA to Inelastic Scattering.
The typical inelastic scattering experiment can
be written in form
a denotes a projectile nucleus in its ground
state.
A and A denote the target nucleus in the ground
and excited state, respectively.
The inelastic scattering amplitude becomes
fit the elastic scattering cross
section
Orthonormality of
for a ? ß
Denoting for A, ß f, and for A ,
a i , we finally have
37
Hadron excitation of giant resonances
Theorists calculate transition strength S(E)
within HFRPA using a simple scattering operator
F rLYLM
Experimentalists calculate cross section within
Distorted Wave Born Approximation (DWBA)
or using folding model.
38
The Scattering Operator
can be simplified in certain cases to have a
simple form
A) Born Approximation a-wave functions are plane
wave
B)
C) Coulomb excitation by a classical particle
39
Thus the cross section is directly related to
the matrix element
One defines
S(E) strength function, response function,
structure function.
Example inelastic electron scattering in Born
approximation
General transition operator F
Energy weighted sum rule (EWSR)
Giant resonance contributes 30 - 100 to EWSR.
40
DWBA-Folding model description
41
EWSR energy weight sum rule
42
Elastic angular distributions for 240 MeV alpha
particle. Filled squares represent the
experimental data. Solid lines are fit to the
experimental data using the folding model DWBA
with nucleon-alpha interaction.
43
Since, a particles are S 0, T 0, they are
ideal for studying electric (?S 0) isoscalar
(?T 0) Giant Resonances.
44
Hartree-Fock Method

• Within the Hartree-Fock (HF) approximation, the
  ground state wave function
• of the nucleus with A nucleons is a Slater
  determinant built from the single-
• particle wave functions
• with the assumption that each nucleon moves in
  the mean-field created by all
• the nucleons and the ground state wave function
  gives the lowest possible expectation value of
  the total Hamiltonian.
• In the spherical case, the single-particle wave
  function is written in terms of the
• radial spherical harmonic
  , and isospin functions.

45

• The total Hamiltonian of the nucleus is written
  as a sum of the kinetic T and potential V
  energies
• Where
• The total energy E

46
Now we apply the variation principle to derive
the Hartree-Fock equations. We minimize
with the constraint of particle number
conservation,
i.e.
We obtain the Hartree-Fock equations
47
Hartree-Fock with Skyrme Interaction.
For the two-body nuclear potential Vij, we take a
Skyrme type effective NN interaction given by,
Where is the spin exchange operator,
and
are the right and left arrows indicate that the
momentum operators act on the right and on the
left, respectively.

ti, xi, a, W0 are Skyrme parameters. We have 10
parameters.
48
The total energy
where
49
where
Now we apply the variation principle to derive
the HF equations. We minimize
50
We have
with
51
Substitution of all the variations and the
single-particle wave function into the variation
equation and making the coefficients of each
independent variation
vanish , we find the HF equations
With an initial guess of the single-particle wave
functions (example harmonic oscillator wave
functions), we can determine m, U(r), and W(r)
and solve the HF equation to get a set of new
single-particle wave functions then one can
proceed in this way until reaching convergence.
52
Coulomb energy The contribution to the energy
density functional arising from the Coulomb
interaction is given by
The direct Coulomb potential
The exchange Coulomb potential
(Slater approximation)
53
Center of mass (CM) correction We must subtract
the contributions of the center of mass motion to
the total binding energy and the charge rms radii
rch.
a). Correction to the total binding energy
The total kinetic energy operator is
The center of mass kinetic energy operator is
Now
For the harmonic oscillator one has
54
b). Correction to the charge rms radii rch.
The mean-square radius for the point proton
distribution corrected for the CM motion is
(harmonic oscillator approximation)
The charge mean-square radius to be fitted to the
experimental data is obtained as
where are the
charge mean square radius of the proton and
neutron, respectively. The last term is the
correction due to spin-orbit effect.
55
Results for the total binding energy
56
Results for the charge rms radii
57
The values of the Skyrme parameters
58
Nuclear matter properties
59
Single-Particle Energies (in MeV) for 40Ca
TAMU Skyrme Interaction B. K. Agrawal, S.
Shlomo and V. Kim Au, Phys. Rev. C 72, 014310
(2005).
60
Hartree-Fock (HF) Random Phase Approximation
(RPA) in fully self-consistent calculation
1) Assume a form of Skyrme interaction (delta
type)
2) Carry out HF calculations for ground states
and determine the Skyrme parameters by a fit to
binding energies and radii.
3) Determine the particle-hole interaction,
4) Carry out RPA calculations of the strength
function, transition density, etc.
The RPA particle-hole (p-h) Greens function is
given by the integral equation
Where G0 is the non-interacting (p-h) Greens
function and Vph is the residual
p-h interaction. For self-consistency, G0 and Vph
are deduced from the same
two-body interaction.
61
For the scattering operator
(strength function)
(transition density)
is associated with the strength in the region of
E?E/2 and is consistent
with
We add that using Skyrme type interaction, G0 can
be evaluated from
Where H0 is the HF hamiltonian and eh and ?h are
the single-particle and wave function of the
occupied state, respectively. Moreover, continuum
effects, such as particle escape width, can be
taken into account using
where rlt and rgt are the lesser and greater of r1
and r2 respectively, U and V are the regular and
irregular solution of (H0-Z)? 0, with the
appropriate boundary conditions, and W is the
Wronskian.
62

• Are mean-field RPA calculations fully
  self-consistent ?
• NO ! In practice, one makes approximations.
• Mean field and Vph determined independently
• ? NO information on K.
• In HFRPA one
• 1. neglects the Coulomb part in Vph
  
  2. neglects the two-body spin-orbit
• 3. uses limited upper energy for s.p. states
  (e.g. Eph(max) 60 MeV)
  
• 4. introduces smearing parameters.
• Main effects

• change in the moments of S(E), of the order of
  0.5-1 MeV

Note that
- spurious state mixing in the ISGDR
- inaccuracy of transition densities.
63
Self-consistent calculation within constrained HF
64
Commonly used scattering operators
In fully self-consistent HF-RPA calculations the
(T0, L1) spurious state (associated with the
center-of-mass motion) appears at E0 and no
spurious state mixing (SSM) in the ISGDR occurs.
In practice SSM takes place and we have to
correct for it.
(prescriptions for ? discussion in the
literature)
NUMERICS Rmax 90 fm
?r 0.1 fm (continuum RPA) Ephmax
500 MeV ?1 ?2 Experimental range
65
Continuum RPA calculation from S. Shlomo and G.
Bertsch, Nucl. Phys. A243 (1975) 507.
The L 3 response for ?-absorption in 16O, 40Ca,
208Pb obtained in the RPA model. The strength is
given in s. p. u./MeV.
66
Continuum RPA calculation from S. Shlomo and G.
Bertsch, Nucl. Phys. A243 (1975) 507.
The L 3 response for ?-absorption in 16O, 40Ca,
208Pb obtained in the RPA model. The strength is
given in s. p. u./MeV.
67
Isoscalar Monopole Strength Functions
90Zr
116Sn
S(E) fm4/MeV
144Sm
208Pb
E MeV
68
Strength function for the spurious state and
ISGDR calculated using a smearing parameter ?/2
1 MeV in continuum RPA (CRPA). The transition
strength S1, S3 and S? correspond to the
scattering operators f1, f3 and f ?,
respectively. The spurious state mixing (SSM)
caused due to the long tail of the spurious state
is projected out using the operator f ?.
69
Isoscalar strength functions of 208Pb for L 0
- 3 multipolarities are displayed. SC (full
line) corresponds to the fully self-consistent
calculation where LS (dashed line) and CO (open
circle) represent the calculations without the ph
spin-orbit and Coulomb interaction in the RPA,
respectively. The Skyrme interaction SGII Phys.
Lett. B 106, 379 (1981) was used.
70
Isovector strength functions of 208Pb for L 0
- 3 multipolarities are displayed. SC (full
line) corresponds to the fully self-consistent
calculation where LS (dashed line) and CO (open
circle) represent the calculations without the ph
spin-orbit and Coulomb interaction in the RPA,
respectively. The Skyrme interaction SGII Phys.
Lett. B 106, 379 (1981) was used.
71
S. Shlomo et al.,
Phys. Rev. Lett. 59, 1054 (1987).
The dotted curve is for the ISGMR. The dashed
curve is for 100 of the IVGDR with constructive
Coulomb interference (N C). Their sum (solid
curve) agrees nicely with the data for the ISGMR.
The dashed-dotted curve is the IVGDR with
destructive interference (N C).
72

• Kolomiets, O. Pochivalov, and
• S. Shlomo, PRC 61 (2000) 034312

ISGMR, f(r) r2Y00, Ea 240 MeV SL1
interaction, K 230 MeV.
Reconstruction of the ISGMR EWSR in 116Sn from
the inelastic a-particle cross sections. The
middle panel maximum (00) double differential
cross section obtained from ?t (RPA). The lower
panel maximum cross section obtained with ?coll
(dashed line) and ?t (solid line) normalized to
100 of the EWSR. Upper panel The solid line
(calculated using RPA) and the dashed line are
the ratios of the middle panel curve with the
solid and dashed lines of the lower panel,
respectively.
73
S. Shlomo and A.I. Sanzhur, Phys. Rev. C 65,
044310 (2002)
ISGDR
SL1 interaction, K 230 MeV, Ea 240 MeV
Reconstruction of the ISGDR EWSR in 116Sn from
the inelastic a-particle cross sections. The
middle panel maximum double differential cross
section obtained from ?t (RPA). The lower panel
maximum cross section obtained with ?coll
(dashed line) and ?t (solid line) normalized to
100 of the EWSR. Upper panel The solid line
(calculated using RPA) and the dashed line are
the ratios of the middle panel curve with the
solid and dashed lines of the lower panel,
respectively.
74
Fully Self-Consistent HF Based RPA Results For
Breathing Mode Energy (in MeV)

• TAMU Data D. H. Youngblood et al, Phys. Rev. C
  69, 034315 (2004) C 69, 054312(2004).
• Nguyen Van Giai and H. Sagawa, Phys. Lett. B106,
  379 (1981).
• c) TAMU Interaction B. K. Agrawal, S. Shlomo
  and V. Kim Au, Phys. Rev. C 72, 014310 (2005).

75
Conclusions
Fully self-consistent calculations of the ISGMR
using Skyrme forces lead to K 230-240 MeV.
ISGDR At high excitation energy, the maximum
cross section for the ISGDR drops below the
experimental sensitivity. There remain some
problems in the experimental analysis.
It is possible to build bona fide Skyrme forces
so that the incompressibility is close to the
relativistic value. Recent relativistic mean
field (RMF) plus RPA lower limit for K equal to
250 MeV.
? K 240 20 MeV.
sensitivity to symmetry energy.
76
References
1 A. Bohr and B. Mottelson, Nuclear Structure,
Vol. II, Benjamin, London (1975).
2 A. deShalit and H. Feshbach, Theoretical
Nuclear Physics, Vol. I Nuclear Structure,

John Wiley Sons, Inc. New York (1974).
3 D. J. Rowe Nuclear Collective Motion Models
and Theory, Methuen and Co. Ltd.
(1970).
4 P. Ring and P. Schuck, The nuclear many-body
problems, Springer, New York-
Heidelerg-Berlin (1980).
5 G. F. Bertsch and R. A. Broglia, Oscilations
In Finite Quantum Systems, Cambridge
University Press (1994).
6 G. R. Satchler, Direct Nuclear Reactions,
Oxford University Press, Oxford (1983).
7 S. Shlomo and G. F Bertsch, Nucl. Phys.
A243, 507 (1975).
77
8 S. Shlomo and D. H. Youngblood, Phys. Rev. C
47, 529 (1993).
9 A. Kolomiets, O. Pochivalov and S. Shlomo,
Phys. Rev. C 61, 034312 (2000).
10 S. Shlomo and A. I. Sanzhur, Phys. Rec. C
65, 044310 (2002).
11 B. K. Agrawal, S. Shlomo and A. I. Sanzhur,
Phys. Rev. C 67, 0343314 (2003).
12 B. K. Agrawal, S. Shlomo and V. Kim Au,
Phys. Rev. C 68, 031304(R) (2003).
13 B. K. Agrawal, S. Shlomo and V. Kim Au,
Phys. Rev. C 70, 057302 (2004).
14 N. K. Glendenning, Phys. Rev. C 37, 2733
(1988).
15 B. K. Agrawal, S. Shlomo and V. Kim Au,
Phys. Rev. C 72, 014310 (2005).
78
Acknowledgments
Work done at
Work supported by
Grant number PHY-0355200
Grant number DOE-FG03-93ER40773