Nucleons & NucleiNucleons & Nuclei

a quick guide to the real essentials in the subjectwhich particle and nuclear physicists won’t tell you

The ParadoxThe Paradox

Do electron scattering on nuclei and deep inelastic scattering and find that bare nucleons havea radius ~ 1 fm. Therefore, in a big nucleus like 208Pb the nucleons should be overlapping andyou would think that the structure and dynamics of the system would depend on quark and gluon degreesof 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, nearlynon-interacting nucleons moving in a collective potential well

Nucleon - (www.jlab.org)3 basic quarks plus a sea ofgluons and quark-antiquark pairs

~ 1 fm

Nucleon-Nucleon PotentialNucleon-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 fmto 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 nucleonsat 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(Infinite) Nuclear MatterConsider 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 functions

Spherically symmetric, central potentialso 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 configurationfrom the l-1/2 configuration!

By adjusting the strength of theSpin-Orbit perturbationMayer and Jensen were able to fit theMagic Numbers

Many-Body Nuclear Wave FunctionsMany-Body Nuclear Wave FunctionsA 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.

“vacuum” =closed core,e.g., 40Ca

choose “model space” of single-particle orbitals

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 sumof Slater determinants (configurations). The complex amplitudes Aare determinedby diagonalizing a residual nucleon-nucleon Hamiltonian and coupling togood energy, angular momentum, and isospin:

Solving for nuclear energy levels,wave functions . . .

Hit many-body trial wave function with Hamiltonianmany times. Use Lanczos to iterate and get successivelythe ground state and excited states, each coupled togood total angular momentum and isospin.

Compare to experiment . . .

Adjust ingredients (two-body Hamiltonian, single particleenergies, model space) appropriately to get agreement

Hoyle level in 12C

Stars “burn” hydrogen to helium,and then helium tocarbon and oxygen. The latteris tricky as there are no stablenuclei 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 Interactionchanges neutrons to protons and vice versa

strength of the Weak Interaction:

typically some 20 orders of magnitude weaker thanelectricity (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 a1 = 15.75 MeV

surface a2 = 17.8 MeV

symmetry a4 = 23.7 MeV

Coulomb a3 = 0.710 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 interestin the r-Process

rp p

roce

ss

r process

neutrons

protons

p process

RIA intensities (nuc/s)> 1012

1010

106

102

10-2

10-6

Beam Parameters:400 kW (238U 2.4x1013)400 MeV/u

• Low energy beams for (p,) and (d,p) to determine (n,)

• Mass measurements

• High energy beams for studying Gamow-Teller strength

V0~50 MeV

PROTONS NEUTRONS

~10 MeV

Independent Particle, Collective Potential Model for the NucleusIndependent Particle, Collective Potential Model for the Nucleus(ignore Coulomb potential for protons)

F

p=F-V0

n=F-V0

F

Nucleon “quasi-particles” behave like non-interacting particles moving in a collectivepotential well (e.g., spherical square well or harmonic oscillator ) - they have quantum numberssimilar to those of bare (in vacuum) nucleons.

Ground State - like zero temperature Fermi gases for neutrons/protons

V0~50 MeV

PROTONS NEUTRONS

~10 MeV

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 . . .

F

F

p=F-V0

n=F-V0

Zero Order Ground StateGround State with residual interaction

real ground state with somewhat “smeared” Fermi surfaces

V0~50 MeV

PROTONS NEUTRONS

~10 MeV

Schematic “Nucleus” in Thermal BathSchematic “Nucleus” in Thermal Bath(ignore Coulomb potential for protons)

F

F

p=F-V0

n=F-V0

Zero TemperatureFinite Temperature, i.e., excited states

Excited States: excitation of particles above the Fermi surface, leaving holes below

Nuclear Level DensityNuclear Level DensityBethe formula:

The level density for most all systems is exponential with excitationenergy E above the ground state. Nuclei are no exception. A fit to experimentalnuclear 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

Nnucleons ~ a T

where the level density parmeter is a ≈A

8 MeV -1

€

Each nucleon so excited has an excitation ~ T

so that the mean excitation energy of the nucleus is

E ~ a T 2

€

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

E ~ a T 2 ≈200

8 MeV⋅ 2 MeV( )

2≈100 MeV