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 bound in a collective potential well. This “ 208 Pb 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, 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

r

C to all orders in one type of interaction (ladder graphs) . . .

~

0.4 fm 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,

Let

A

tend toward infinity . . . and

equal

numbers of neutrons and protons.

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 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.,

40 Ca 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 12 C

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 ~ 10 5 g cm -3 Build up equilibrium concentration of 8 Be 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 a 1 surface a 2 = 15.75 MeV = 17.8 MeV Coulomb a 3 symmetry a 4 = 0.710 MeV = 23.7 MeV pairing a 5 = 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 ( 238 U 2.4x10

13 ) 400 MeV/u

p process

protons neutrons RIA intensities (nuc/s) > 10 10 10 10 6 12 10 2 10 -2 10 -6 • 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

e

F ~10 MeV

e

F V 0 ~50 MeV

m

p =

e

F -V 0

m

n =

e

F -V 0 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 . . .

real ground state with somewhat “smeared” Fermi surfaces

e

F ~10 MeV

e

F V 0 ~50 MeV

m

p =

e

F -V 0

m

n =

e

F -V 0 PROTONS NEUTRONS

Schematic “Nucleus” in Thermal Bath

(ignore Coulomb potential for protons )

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

e

F ~10 MeV

e

F V 0 ~50 MeV

m

p =

e

F -V 0

m

n =

e

F -V 0 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 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 