Lectures 4 & 5

Oct 2006, Lectures 4&5 1

Lectures 4 & 5

The end of the SEMF and the nuclear shell model


4.1 Overview

4.2 Shortcomings of the SEMF magic numbers for N and Z spin & parity of nuclei

unexplained magnetic moments of nuclei value of nuclear density values of the SEMF coefficients

4.3 The nuclear shell model choosing a potential L*S coupling Nuclear “Spin” and Parity Shortfalls of the shell model


4.2 Shortcomings of the SEMF


4.2 Shortcomings of the SEMF(magic numbers in Ebind/A)

SEMF does not apply for A<20

(2,2)

2*(2,2)= Be(4,4)

E94keV

(6,6)

(8,8)

(10,10)(N,Z)

There are systematic deviations from SEMF for A>20


4.2 Shortcomings of the SEMF

(magic numbers in numbers of stable isotopes and isotones)

ProtonMagic

Numbers

N

Z

• Magic Proton Numbers (stable isotopes)• Magic Neutron Numbers (stable isotones)

Neutron MagicNumbers


4.2 Shortcomings of the SEMF(magic numbers in separation energies)

Neutron separation energies

saw tooth from pairing term

step down when N goes across magic number at 82

Ba Neutron separation energy in MeV


N=

50

Z=

50

N=

82

Z=

82

N=

126

iron mountain

4.2 Shortcomings of the SEMF(abundances of elements in the solar system)

Complex plot due to dynamics of creation, see lecture on nucleosynthesis

no A=5 or 8


4.2 Shortcomings of the SEMF(other evidence for magic numbers, Isomers)

Nuclei with N=magic have abnormally small n-capture cross sections (they don’t like n’s)

Close to magic numbers nuclei can have “long lived” excited states (>O(10-6 s) called “isomers”. One speaks of “islands of isomerism” [Don’t make holydays there!]

They show up as nuclei with very large energies for their first excited state (a nucleon has to jump across a shell closure)

208Pb

First excitation energy


4.2 Shortcomings of the SEMF(others)

spin & parity of nuclei do not fit into a drop model

magnetic moments of nuclei are incompatible with drops

actual value of nuclear density is unpredicted

values of the SEMF coefficients except Coulomb and Asymmetry are completely empirical


4.2 Towards a nuclear shell model

How to get to a quantum mechanical model of the nucleus?

Can’t just solve the n-body problem because: we don’t know if a two body model makes sense (it

does not make much sense for a normal liquid drop)

if it did make sense we don’t know the two body potentials (yet!)

and if we did, we could not even solve a three body problem

But we can solve a two body problem! Need simplifying assumptions


4.3 The nuclear shell model

This section follows Williams, Chapters 8.1 to 8.4


4.3 Making a shell model(Assumptions)

Assumptions: Each nucleon moves in an averaged potential

neutrons see average of all nucleon-nucleon nuclear interactions protons see same as neutrons plus proton-proton electric repulsion the two potentials for n and p are wells of some form (nucleons are

bound) Each nucleon moves in single particle orbit corresponding to its

state in the potential We are making a single particle shell model Q: why does this make sense if nucleus full of nucleons and typical

mean free paths of nuclear scattering projectiles = O(2fm) A: Because nucleons are fermions and stack up. They can not loose

energy in collisions since there is no state to drop into after collision Use Schroedinger Equation to compute Energies (i.e. non-

relativistic), justified by simple infinite square well energy estimates

Aim to get the correct magic numbers (shell closures) and be content


4.3 Making a shell model (without thinking, just compute)

Try some potentials; motto: “Eat what you know”

Coulomb infin. square harmonic

desiredmagic

numbers

2

8

20

28

50

82

126


4.3 Making a shell model (with thinking)

We know how potential should look like! It must be of finite depth and … If we have short range nucleon-nucleon potential then … … the average potential must look like the density

flat in the middle (you don’t know where the middle is if you are surrounded by nucleons)

steep at the edge (due to short range nucleon-nucleon potential)

R ≈ Nuclear Radiusd ≈ width of the edge

15

4.3 Making a shell model (what to expect when rounding off a potential well)

Higher L solutions get larger “angular momentum barrier” Higher L wave functions are “localised” at larger r and thus closer to “edge”

Radial Wavefunction U(r)=R(r)*r for the finite square well Rounding the edge

affects high L states most because they are closer to the edge then low L ones.

High L states drop in energy because

can now spill out across the “edge”

this reduces their curvature

which reduces their energy

So high L states drop rounding the well!!


4.3 Making a shell model(with thinking)

The “well improvement program”

Harmonic is bad Even realistic well

does not match magic numbers

Need more splitting of high L states

Include spin-orbit coupling a’la atomic

magnetic coupling much too weak and wrong sign

Two-nucleon potential has nuclear spin orbit term

deep in nucleus it averages away

at the edge it has biggest effect

the higher L the bigger the split


4.3 Making a shell model (spin orbit terms)

Q: how does the spin orbit term look like? Spin S and orbital angular momentum L in our model are that of

single nucleon in the assumed average potential In the middle the two-nucleon interactions average to a flat potential

and the two-nucleon spin-orbit terms average to zero Reasonable to assume that the average spin-orbit term is strongest

in the non symmetric environment near the edge 1 ( )( )

dV rW r

r dr:

2

LS LS

1`3

0 0

( ) ( ) ( )

1 ( )where: ( )

1and V V ( ) and ( ) (Woods-Saxon)

1 exp

with and 1.2 and 0.75 "thickness of edge"

LS

nucleon

V r V r W r L S

dV rW r V

m c r dr

E V rr ad

a R A R fm d fm

g

hDimension: Length2

compensate 1/r * d/dr

LS analogy to atomic physics:

and and ( )

and and and

and using spherical symmetry gives:

1 ( )

E B B v E E V r

gS p mv L r p

dV rE L S

mr dr

r r r r rrrg

r rr rr rr

rrg


4.3 Making a shell model (spin orbit terms)

Good quantum numbers without LS term : l, lz & s=½ , sz from operators L2, Lz, S2, Sz with

Eigenvalues of l(l+1)ħ2, s(s+1)ħ2, lzħ, szħ With LS term need operators commuting with new

H J=L+S & Jz=Lz+Sz with quantum numbers j, jz, l, s

Since s=½ one gets j=l+½ or j=l-½ (l≠0) Giving eigenvalues of LS [ LS=(L+S)2-L2-S2 ]

½[j(j+1)-l(l+1)-s(s+1)]ħ2 So potential becomes:

V(r) + ½l ħ2 W(r) for j=l+½ V(r) - ½(l+1) ħ2 W(r) for j=l -½ we can see this asymmetric splitting on slide 16


4.3 Making a shell model (fine print)

There are of course two wells with different potentials for n and p

We currently assume one well for all nuclei but … The shape of the well depends on the size of the

nucleus and this will shift energy levels as one adds more nucleons

Using a different well for each nucleus is too long winded for us though perfectly doable

So lets not use this model to precisely predict exact energy levels but to make magic numbers and …


4.3 Predictions from the shell model

(total nuclear “spin” in ground states) Total nuclear angular momentum is called

nuclear spin = Jtot

Just a few empirical rules on how to add up all nucleon J’s to give Jtot of the whole nucleus

Two identical nucleons occupying same level (same n,j,l) couple their J’s to give J(pair)=0Jtot(even-even ground states) = 0

Jtot(odd-A; i.e. one unpaired nucleon) = J(unpaired nucleon) Carefull: Need to know which level nucleon occupies. I.e. more or less accurate shell model wanted!

|Junpaired-n-Junpaired-p|<Jtot(odd-odd)< Junpaired-n+Junpaired-p

there is no rule on how to combine the two unpaired J’s


4.3 Predictions from the shell model

(nuclear parity in groundstates)

Parity of a compound system (nucleus):

1, 1,

1,

( -nucleon system) ( ) ( )

where ( 1) and ( ) 1

( -nucleon system) ( 1)

i

i

intrinsic i i ii A i A

li intrinsic i

l

i A

P A P nucleon P nucleon

P P nucleon

P A

P(even-even groundstates) = +1 because all

levels occupied by two nucleons P(odd-A groundstates) = P(unpaired nucleon) No prediction for parity of odd-odd nuclei


4.3 Shortcomings of the shell model

The fact that we can not predict spin or parity for odd-odd nuclei tells us that we do not have a very good model for the LS interactions

A consequence of the above is that the shell model predictions for nuclear magnetic moments are very imprecise

We can not predict accurate energy levels because: we only use one “well” to suit all nuclei we ignore the fact that n and p should have separate wells of

different shape As a consequence of the above we can not reliably

predict much (configuration, excitation energy) about excited states other then an educated guess of the configuration of the lowest excitation

Documents

Lectures 4 & 5