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Nucleons & Nuclei
a quick guide to the real essentials in the subject
which particle and nuclear physicists won’t tell you
The Paradox
Nucleon - (www.jlab.org)
3 basic quarks plus a sea of
gluons and quark-antiquark pairs
~ 1 fm
Do electron scattering on nuclei and deep inelastic scattering and find that bare nucleons have
a radius ~ 1 fm. Therefore, in a big nucleus like 208Pb the nucleons should be overlapping and
you would think that the structure and dynamics of the system would depend on quark and gluon degrees
of freedom. It does not. In fact, nuclei behave as if the nucleons in them are non-interacting little spheres,
bound in a collective potential well. This “single particle” nature of nuclei makes them easy to treat.
particle physicist’s nucleus - overlapping nucleons
-nearly a bag of quark gluon plasma!
nuclear physicist’s nucleus - independent, nearly
non-interacting nucleons moving in a
collective potential well
Nucleon-Nucleon Potential
Experimental data can be fit up to energies ~ 300 MeV with a set of short range potentials
where the coordinates of the two nucleons and the relative coordinates are
and where the potentials have Yukawa forms, e.g.,
-
Independent Particle Behavior in Nuclear Matter and Finite Nuclei
Sum strong, attractive nucleon-nucleon potential with a repulsive “hard core” at rC ~ 0.4 fm
to all orders in one type of interaction (ladder graphs) . . .
n
n

n
n
+
n
 n

n
n
+
n
n

n


+
. . .
n
Bethe-Salpeter sum shows that strongly interacting systems of nucleons
at nuclear matter density behave like a system of non-interacting
(i.e., “independent”) quasi-particles, with quantum numbers (mass, charge, spin)
close to those of bare nucleons, and moving in a collective potential well.
(Infinite) Nuclear Matter
Consider a nucleus of mass number A, volume V, and equal numbers of neutrons and protons.
Let A tend toward infinity . . .
Binding energy per nucleon is measured to be
Treat as a degenerate, zero temperature Fermi gas of spin-1/2 particles . . .
Fermi wave number (at the top of Fermi sea) is
This corresponds to a (e.g., proton) Fermi energy
So use a spherical square well or spherical harmonic oscillator potential
and use the Schroedinger equation to solve for the single particle wave function
Spherically symmetric, central potential
so orbital angular momentum a good quantum number
Radial wave function satisfies:
Maria Goeppert Mayer figured it out - add in a spin-orbit coupling
Experiment shows that certain numbers of nucleons
(2, 8, 20, 28, 50, 82, 126 , . . .) confer tighter binding.
The central potential on the previous page does not explain this.
Adding in a spin-orbit potential DOES . . .
The spin-orbit potential splits the l+1/2 configuration
from the l-1/2 configuration!
By adjusting the strength of the
Spin-Orbit perturbation
Mayer and Jensen were able to fit the
Magic Numbers
Many-Body Nuclear Wave Functions
A particular configuration can be represented by a Slater determinant of occupied single-particle orbitals.
Note that the creation operators for different orbitals anti-commute, ensuring overall anti-symmetry.
choose “model space” of single-particle orbitals
“vacuum” =
closed core,
e.g., 40Ca
Can represent this at machine-level in a computer as a string of ones (occupied) and zeros (unoccupied)
for a specified order of orbitals. In this case operators are like “masks.”
We can then get the total many-body wave function by forming a coherent sum
of Slater determinants (configurations). The complex amplitudes Aare determined
by diagonalizing a residual nucleon-nucleon Hamiltonian and coupling to
good energy, angular momentum, and isospin:
Solving for nuclear energy levels,
wave functions . . .
Hit many-body trial wave function with Hamiltonian
many times. Use Lanczos to iterate and get successively
the ground state and excited states, each coupled to
good total angular momentum and isospin.
Compare to experiment . . .
Adjust ingredients (two-body Hamiltonian, single particle
energies, model space) appropriately to get agreement
Hoyle level
in 12C
Stars “burn” hydrogen to helium,
and then helium to
carbon and oxygen. The latter
is tricky as there are no stable
nuclei at mass 5 or 8.
Helium burning in red giants:
T~ 10 keV, density ~ 105 g cm-3
Build up equilibrium concentration of
8Be via
Then
through an s-wave resonance
The Weak Interaction
changes neutrons to protons and vice versa
strength of the Weak Interaction:
typically some 20 orders of magnitude weaker than
electricity (e.g., Thompson cross section)
Nuclear weak interactions:
beta decay, positron decay, electron capture, etc.
How could you predict nuclear
(ground state) masses?
Semi-impirical mass formula (liquid drop model)
bulk
surface
a1 = 15.75 MeV
a2 = 17.8 MeV
Coulomb
a3 = 0.710 MeV
symmetry
a4 = 23.7 MeV
pairing
a5 = 34 MeV ( = +1 odd-odd, -1 even-even, 0 even-odd)
so-called valley of beta stability
RIA will produce nuclei of interest
in the r-Process
Beam Parameters:
400 kW (238U 2.4x1013)
400 MeV/u
p process
RIA intensities (nuc/s)
> 1012 102
1010
10-2
10-6
106
protons
neutrons
• Low energy beams for (p,g) and
(d,p) to determine (n,g)
• Mass measurements
• High energy beams for studying
Gamow-Teller strength
Independent Particle, Collective Potential Model for the Nucleus
(ignore Coulomb potential for protons)
Nucleon “quasi-particles” behave like non-interacting particles moving in a collective
potential well (e.g., spherical square well or harmonic oscillator ) - they have quantum numbers
similar to those of bare (in vacuum) nucleons.
Ground State - like zero temperature Fermi gases for neutrons/protons
eF
~10 MeV
eF
V0~50 MeV
mp=eF-V0
mn=eF-V0
PROTONS
NEUTRONS
Now turn on the “residual” interaction between nucleons:
Particle/hole pairs are excited by residual interaction and the actual ground state in this model,
now with “configuration mixing,” might look like this . . .
Ground
Zero
State
Order
withGround
residualState
interaction
real ground state with somewhat “smeared” Fermi surfaces
eF
~10 MeV
eF
V0~50 MeV
mp=eF-V0
mn=eF-V0
PROTONS
NEUTRONS
Schematic “Nucleus” in Thermal Bath
(ignore Coulomb potential for protons)
Finite Temperature,
Zero Temperature
i.e., excited states
Excited States: excitation of particles above the Fermi surface, leaving holes below
eF
~10 MeV
eF
V0~50 MeV
mp=eF-V0
mn=eF-V0
PROTONS
NEUTRONS
Nuclear Level Density
Bethe formula:
The level density for most all systems is exponential with excitation
energy E above the ground state. Nuclei are no exception. A fit to experimental
nuclear level data gives . . .
where
and where the back-shifting parameter is 
and the level density parameter is
nuclear mass number
Number of nucleons excited above the Fermi surface
N nucleons ~ a T
where the level density parmeter is

A
a  MeV -1
8
Each nucleon so excited has an excitation
~T
so that the mean excitation energy of the nucleus is
E ~ a T2

For example, at a temperature T  2 MeV,
a nucleus with mass number A ~ 200,
which is typical during the late satges of infall/collapse,
will have mean excitation energy
200
2
E ~ a T2 
 2 MeV   100 MeV
8 MeV