Nuclear Size and Density - Physics and Astronomyphysics.unm.edu/Courses/Bassalleck/PHYS450/notes/Nuclear_Models.… · Nuclear Size and Density ... • Assume a Velcro ball model,

Nuclear Size and Density• How does the limited range of the nuclear force affect the

size and density of the nuclei?• Assume a Velcro ball model, each having radius r, volume V

= 4/3π r3. Then the volume of the entire nucleus is just A·V, or the radius of the nucleus is just rnucl = const ·A1/3.

• This also implies that the (core) density of nuclei is independent of the size (A).

• Charge density found from scattering experiments to be: with a=1.07A1/3F, b=0.55F.

• Mass density found to be ~ 1018kg/m3!bare

r /)(1)0()( −+

=ρρ

Thursday, January 23, 2003 30

Nuclear Size and Density


Modeling the Binding Energy

• What we know:– Total binding energy scales with A.– Interior densities are independent of A.– General tendency to have Z = N.– With increasing A, more likely that N>Z.– Many more stable nuclei if N and Z are even

(166), than if N or Z are odd (110), than if both are odd (8).


Modeling the Binding Energy• Liquid Drop Model (Weizsacker Formula, Semi-

Empirical Mass Formula)– Five parameter fit to Atomic Masses:


21

2

31

232

4)(),(

AAZNa

AZaAaAaZMZMNMAZM acsvepn

δ+

−+++−++=

av = 15.67MeV/c2

as = 17.23MeV/c2

ac = 0.714MeV/c2

aa = 93.15MeV/c2

δ = ⎪⎩

⎪⎨

⎧

+

−

N and Zoddfor /2.11A oddfor /0

N and even Zfor /2.11

2

2

2

cMeVcMeV

cMeV

Modeling the Binding Energy• Semi-Empirical Mass Formula

21

2

31

232

4)(),(

AAZNa

AZaAaAaZMZMNMAZM acsvepn

δ+

−+++−++=

First three terms are just the masses of the constituents

Fourth term is volume term. This accounts for most of the binding energy.

Fifth term is the surface term. Reduces the binding energy (fewer nearest neighbors on the surface).

Sixth term is Coulomb term. Repulsion between protons reduces binding.

Seventh term is asymmetry term. Reduces binding as you go away from N=Z.

Last term is pairing term. This accounts for the increased binding in even Z or N nuclei.


What have we learned?• Nuclear force is short ranged (saturation of binding

energy).• Surface effects are important (like in liquid-drop

model).• Coulomb effects are important.

– Why is there a tendency towards N>Z for larger A?• Density ~ constant and very high.• Radius ~ A1/3.• Tendency for N ~ Z.• N > Z for heavier nuclei.

Tuesday, January 28, 2003 37

Mystery of the Magic Numbers

At certain values of N (and Z), there is an abrupt change in how well the Semi-empirical Mass Formula predicts the binding energy. At these “magic numbers”, the nuclei are particularly stable. This is reminiscent of the stability of the noble gases when there are closed shells. However, this also implies that the nucleons are basically independent of one another moving in the nuclear potential. Considering the density of nuclear matter, how can this be?


Fermi Gas Model• Nucleons behave in a similar fashion to free electrons in a

metal. • Each nucleon moves in an attractive net nuclear potential

(three-dimensional finite square well).• In the ground state, the nucleons (fermions) occupy the lowest

possible energy levels without violating the exclusion principle.

• Then, since all available energy levels are filled, nucleon–nucleon scattering is suppressed (except for exchange scattering which still leaves the nucleus unchanged).

• So, even though the density is extremely high, the nucleons move as if they were independent particles trapped in a well.


Fermi Energy

Each state can contain two protons and two neutrons

For Z=N=A/2


Results from Fermi Gas Model• In order to be stable, the Fermi

energy level for neutrons and protons must be the same.

• Since protons feel an additional Coulomb repulsion, the potential depth is shallower, hence there must be more neutrons: neutron excess for higher A nuclei.

• The Fermi Gas model also gives a firm foundation for the idea that the nucleons move independently of each other.

• One can now look into the idea that nucleons fill discreet states (as in the atomic case).


Fermi Gas Model and Neutron Stars

When the core of a star “burns out” and collapses under gravitational pressure, the Fermi energy of the electrons increase until the reaction:

dominates the inverse reaction:

since the electron will be Pauli blocked. Eventually, all of the protons will be converted to neutrons, the coulomb repulsion between nuclei disappears and you are left with “neutron matter”.

enep ν+→+ −

eepn ν++→ −


Radius of Neutron Stars

km12≈Tuesday, January 28, 2003 43

Hyperon Spectroscopy• In the Fermi model, all energy levels of a nucleus are filled up to the

Fermi energy and interactions between nucleons are suppressed, other than (unobservable) swaps between indistinguishable nucleons.

• How then to “mark” nucleons in a particular energy level?• Replace a nucleon with a hyperon (hadron with strange quark content),

e.g. a Lambda Λ.

energy recoil)( 2 +⋅−+−+=

+Λ→+

+→+

ΛΛ

−−

−Λ

−

cMMEEBBnK

AAK

nKn π

π

π


Hyperon Spectroscopy

• The Lambda can go into any of the energy levels, since it is not affected by the Pauli exclusion of the other nucleons.

• The pion spectra then reflects the binding energy of the Lambda.

• For heavier nuclei, the potential depth stays approximately constant, but the radius of the well increases as A1/3, and hence the energy of a particular state decreases approximately linearly with the radius squared.


Shell Model(Back to Magic Number Problem)


Shell Model• With the Fermi Gas Model, we can now understand the nucleus as a

collection of independent nucleons, each of which are moving in a spherical potential well whose depth is determined by the net potential from the rest of the nucleons and whose extent is given by the nuclear radius (proportional to A1/3).

• Next step is to solve Schrödinger's equation for this problem. The wave functions can be separated into two parts, radial and spherical:

• Energy is independent of m (=integer between +/-ℓ)• Since nucleons have spin ½, the nℓ levels are 2·(2ℓ+1) degenerate.

⎩⎨⎧

=+=

momentumangular orbital ,,,,,1nodes ofnumber ,4,3,2,1

:

),(),(

Ll

L

ll

gfdpsn

with

YrR mn ϕθ


Shell Model• First, must assume some form of the potential. For all but the lightest

nuclei, start with Woods-Saxon Potential to fit the density distribution:

• With solutions to Schrödinger's equation, can construct the energy level diagram based on the degeneracy of each level:

• Remember, that magic numbers were: 2,8,20,28,50,82,126• One Can see that for higher magic numbers, this doesn’t work.• Many different forms for the potential were tried with no success.• Something missing?!

aRrcentral eVrV /)(

0

1)( −+

−=


Shell Model with S·L• Remember from atomic physics, that LS coupling splits

degenerate levels into “fine structure”.– ~10-4 effect on the energy (order α2).– higher j (=ℓ±½) has higher energy (less bound)– Magnetic in origin (electron’s spin couples with magnetic field created

by its orbital motion)• Spin-Orbit coupling in nuclei is different in two important

respects:– It is strong (on the same order as ∆Enℓ)– It is inverted from the atomic case. (lower j has higher energy).– Both are indications that this coupling is not magnetic in origin.


Shell Model with S·L

2)()()(h

ll

srVrVrV scentral +=• New term in potential:

• Then,

)(2

12

21for 2)1(

21for 2)1()1()1(

2)1()1()1(

2

2

222

2222

rVE

j

jssjj

s

sssjj

sssj

ss ll

l

ll

llll

h

l

lhllhh

lll

⋅+

=∆

⎪⎩

⎪⎨⎧

−=+−+=

=+−+−+=

⇒++++=+

++=+=



2

8

20

40

58

70

92

2

8

2028

50

82

126

Predictions


• Total angular momentum of one-particle and one-hole states– For nuclei with one too many

nucleons to fill a shell (or one too few), the total nuclear spin is determined by the spin of the extra nucleon (or hole).

– Examples: • 7N15 is doubly magic except for a

proton hole in the 1pl/2 subshell. So it should have a spin i equal to the value j = 1/2 for that subshell. This prediction agrees with measurement.

• 8O17 is doubly magic except for an extra neutron in the 1d5/2 subshell. So it should have i = j = 5/2, in agreement with measurement.

• 19K39 is predicted to be doubly magic except for a proton hole in the ld3/2subshell, so it should have i = j = 3/2. It does.

• 83Bi2O9 is doubly magic except for an extra proton in the 1h9/2 subshell. So its spin should be i = j = 9/2. This agrees with measurement.


Predictions• Total angular momentum

of one-particle and one-hole states

• 82Pb207 is doubly magic except for a neutron hole in the 1i13/2subshell. So the exclusion principle predicts that it should have a spin i = j = 13/2. However, the measured spin is i = 1/2.

• Need correction for pairing term, and also small coulomb corrections for the proton spectra. This drives the 1i13/2 level below the 3p1/2 level.

Summary of Pairing

• Each nucleon couples L with S to form total angular momentum, J.

• Nucleons interact to form pairs with total angular momentum = 0.

• Nucleons want to be close (spatially) because the nucleon-nucleon interaction is strong and attractive.

• Odd-A nuclei have total spin of the odd (unpaired) nucleon.

Tuesday, February 4, 2003 62

Parity• Parity = 1 (nuclear eigenfunction is even

function of space variables)• Parity = -1 (nuclear eigenfunction is odd

function of space variables)• Since the nucleons move basically

independently, nuclear eigenfunctions are just the product of the nucleon eigenfunctions →

• Total parity of the nuclei = parity of the eigenfunction of the odd nucleon = (-1)ℓ


Predictions• Magnetic moments

– Can be determined from the properties of the extra (or missing) nucleon in one-particle and one-hole states.

• E.g., 16O is doubly magic and therefore each nucleon is paired with another to produce zero total angular momentum. It’s nuclear magnetic moment is then zero.

– This works for lighter nuclei, but in heavier nuclei, breaks down because of nucleon-nucleus interactions. Again, complicated systems!



Magnetic Moments

Collective Model• A “blend” between the shell model and the liquid

drop model.• Nucleons in unfilled subshells move in a net nuclear

potential produced by the core of filled subshells.• That potential is not the spherically symmetric

potential of the shell model, but can undergo deformations in shape similar to what one would expect from the liquid drop model.

• Even in the “ground state” the core of nucleons are affected by the nucleons in the unfilled shell. Their (non spherically symmetric) presence can distort the core and create a “tidal” effect.


Collective Model• This tidal effect can create a

net charge current with its own angular momentum, adding to the net magnetic moment of the unpaired nucleon from the shell model.

• This corrects the shell model predictions and accounts for the magnetic moments of the nuclei.

• This can also account for another effect that the shell model does not correctly predict: electric quadrupole moments.

Extra proton Proton hole


Negative electric quadrupole moment

Positive electric quadrupole moment

Deformations and Pairing Force• The pairing force also tends

to keep nucleons in close spatial proximity = neighboring values of |m|.

• This effect adds to the collective distortions from spherical symmetry and can create large quadrupole moments in half-filled shells of large nuclei.


Collective Model• The collective model

does an excellent job of predicting the electric quadrupole moments of most nuclei.

• Notice the change in sign of the moments at the magic numbers.

• The collective model can also be used to understand other collective deformations of nuclei, including dipole and quadrupole resonances and fission.


Models SummaryName Assumptions Theory Used

Properties Predicted

Liquid Drop ModelSimilar mass densities,

binding energies proportional to masses

Classical (asymmetry and pairing terms

introduced with no justification)

Accurate average masses and binding energies (SEMF)

Fermi Gas ModelNucleons move

independently in a net nuclear potential

Quantum statistics of Fermi gas of nucleons

Depth of net nuclear potential, asymmetry

term

Shell Model

Nucleons move in net nuclear potential with

strong and inverted spin-orbit term

Schrödinger equation solved for potential

Magic numbers, nuclear spins, nuclear parities, pairing term

Collective ModelNet nuclear potential

undergoes deformations

Schrödinger equation solved for non-

spherical potential

Magnetic dipole moments, electric

quadrupole moments


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Nuclear Size and Density - Physics and Astronomyphysics.unm.edu/Courses/Bassalleck/PHYS450/notes/Nuclear_Models.… · Nuclear Size and Density ... • Assume a Velcro ball model,