3/11/2009 1

Intro to Nuclear and Particle Intro to Nuclear and Particle Physics (5110)Physics (5110)

March 11, 2009Mostly:

Nuclear Shell Model

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Nuclear Energy LevelsNuclear Energy LevelsDaughter nucleus in α or β decay is sometimes in an excited state that can be unstable or a relatively long-lived “metastable” nuclear isomer state. Saw one example already:

214 214 21083 84 82

β α→ →Bi Po Pb

(dominant)

214 214 21083 84 82

β α→ →*Bi Po Pb

(rare)

Nuclear isomers (meta-stable excited states of a nuclide) can also be produced in nuclear scattering:

a A b B+ → + a is the incident particle, A a target nucleus, b the lighter and B the heavier product.( ),A a b B

14 177 8pα + → +N O ( )14 17

7 8, pαN OExample:

Sometimes scattering experiments find a set of closely related processes:

( ),A a b B a A b B Q+ → + +*1 1b B Q→ + +*2 2b B Q→ + +

M

Precise measurements of Q values (b KE) give the energy levels: nuclear spectroscopy

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A AZ Z γ∗ → +X X

Nuclear energy levels are revealed in scattering and in α-decay by energy spectra with closely separated α energies. All but the highest energy αare accompanied by γ

228 22488→90Th Ra

γγ--DecayDecay

• Typical spacing of nuclear energy levels: tens of keV (hard X-rays) to a few MeV (gammas)

• Electromagnetic transitions dominate, with typical lifetimes of 10-9 s to 10-15 s (Γ < ~1 eV)– Angular momentum conservation affects rates – big changes in l require

“higher multipole” radiation, suppressing rate– Longer-lived states (low-energy, high-multipole transitions) are called

“isomers,” e.g. 110Agm has t1/2 = 235 d• For excitation energies above B/A, an excited state may decay by

nucleon emission – strong interaction

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Back to The Model of the Nucleus:Back to The Model of the Nucleus:• How well does the Liquid Drop Model Work?

– For A > ~20, the computed binding energy is good to about 1.5%– Good enough for quite reliable predictions of nuclear stability

alpha and beta decays– As always, departures are powerful clues to something deeper

Magic Numbers

Specific values of Z and N that have larger than expected Binding Energy

(Odd-A Nuclei)

Z or N = 2, 8, 20, 28, 50, 82, (126)

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Nuclear Shell ModelNuclear Shell Model Single Particle Shell Model (SPSM)• Single nucleon moves in a potential that represents the average effect

of all of the other nucleons.• Guess at “reasonable” potential, plug into Schrodinger’s Equation (SE),

solve for wave function, demonstrate magic #’s, etc. as consequences.

( )( )

0:

: 0

r R V r V

r R V r

< = −

> =

Square Well

( ) 2 212

V r M rω=

Harmonic Oscillator

( ) 0

1 exp

VV rr R

d

−=

− +

Woods -Saxon

• Aside: How can nucleons occupy atom-like orbitals in nuclei without interacting?– Fermions! Pauli exclusion principle

prohibits orbit-changing collisions because there is no unoccupied level for a nucleon to move to.

• Candidate potentials:– Square well - solvable– Harmonic oscillator - solvable– Woods-Saxon – reasonable nuclear charge

distribution, but…

• Solve for discrete eigen-functions by “fitting” waves into effective potential (Variational Principle) (Neutron Potentials)

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( ) ( )2

2 ( )2

V r r E rm

ψ ψ − ∇ + =

h r r

Back to NR Quantum Mechanics (Review)Back to NR Quantum Mechanics (Review)H atom → Electron in Multi-Electron Atoms → Nucleon in Nucleus

⇒Central potential energy eigenstates are angular momentum eigenstates. 2 2 2

2 2 2

1 1r Lr r r r

∂ ∂∇ = −

∂ ∂

r

h Spherical coordinates :

2Lr

Apply usual separation of variables. EF's of are spherical harmonics.

( ) ( )( ) ,l l

nlnlm lm

u rr Y

rψ θ φ=

r n = 1, 2, … Principle Q. No.=s+l+1 l = 0, 1, 2, … Orbital Q. No.ml = -l+1,-l,…,l-1,l Magnetic Q. No.

( ) ( ) ( )22

2 2 2

12 02nl nl

l ld m E V r u rdr mr

+ + − − =

h

h

Note: 1D SE w. centrifugal barrier; unl(r) must be regular at r=0 and 0 for r→∞. n and l can have any integer values.

Substitute into SE and get:

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• Boundary condition (infinite barrier) ⇒ unl(r) vanishes at r = R, giving energy quantization:

( ) ( ) 00,1,2,... 1, 2,...

nl l nlu R j k Rl n l

= =

= = and for any

th th 0nl

nl

k n lE

∴ ↔ of the spherical B.F.,which give the quantized energies

Infinite Square WellInfinite Square Well ( ) 0

r RV r

∞ ≥=

for otherwise

( ) ( )22

2 2 2

12 02nl nl

l ld m E u rdr mr

+ + − =

h

hInside:

( ) ( ) nl 2

2 k nlnl l nl

mEu r j k r= =h

Solutions are spherical Bessel Functions

2 4 6 8 10 12 14-0.2

0.2

0.4

0.6

0.8

1

***Note from this point on we are using n as the “radial” quantum number

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Count the available statesCount the available states Closed Shells

Observed Magic Numbers: Z or N = 2, 8, 20, 28, 50,

82, (126)

2n = 1 l = 0 → 2

l = 1 → 6

l = 2 → 10

l = 3 → 14

l = 4 → 18

8

18• All zeroes of the jl’s are distinct ⇒ no energy degeneracy in n and l.

• For given n and l, all ml’s are degenerate in energy: (2l + 1)

• For given n, l and ml, there are ↑ and ↓ spin states for n and p.

• Total degeneracy: 2(2l +1) slots to fill for every n and l.

32

50

Good News: predicts existence of nuclear magic numbers.

Bad News: Only some of these predicted magic numbers match observed (2, 8, 20, 28, 50,…).

Larger n’s bring in other closed shells, but still not the right set.

What do we try next? More/different/better potentials to start.

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( ) 2 212

V r M rω=Harmonic OscillatorHarmonic Oscillator

122nlE n l ω = + −

hEnergy Levels :

Has some degeneracy, reduced set of magic #’s.

Still not the right set!

Coulomb Infinite Square Well

Harmonic Oscillator

Observed Magic Numbers: Z or N = 2, 8, 20, 28, 50,

82, (126)

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• Tweak the wells: Get rid of features like infinite barriers/ sharp edges– “Reasonable” nuclear potentials (solve

numerically): finite square wells, Woods-Saxon, etc. → adjust energy levels, break degeneracies, give new magic numbers

– Still the wrong ones!• Bottom line: No central potential

can produce all of the observed magic numbers and only the observed magic numbers

• Need more complicated strong interaction for nucleon in nucleus– Break rotational symmetry

Keep trying…Keep trying…

Observed Magic Numbers: Z or N = 2, 8, 20, 28, 50,

82, (126)

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Maria Maria GoeppertGoeppert MayerMayer

Nobel Prize in Physics 1963

Maria Goeppert Mayer (1906 - 1972) was an accomplished physicist from the beginning of her career until the end and she made numerous contributions to the field of physics.

For an account of her life and work, see Robert G. Sachs, Biographical Memoirs of the National Academy of Sciences, Volume 50. See also “Nobel Lectures, Physics 1963-1970”, Elsevier Publishing Company, Amsterdam.

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( ) ( )TotV V r f r L S= − ⋅rr

Atomic Physics

Nuclear Physics

Spin of e → magnetic dipole moment, interacts with EM field (magnetic field component in e rest frame).

SpinSpin--Orbit InteractionOrbit Interaction

Nucleon-nucleon strong interactionpostulated to have spin-orbit term. Nucleon deep in nucleus feels no effect (symmetry), but one near surface feels net interaction with all others.

Differs from atomic:f(r) – to be chosenSign – chosen to match observed level splitting

Next step is to apply what we know about angular momentum and deduce the form of the resulting level splitting.

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( ) ( )TotV V r f r L S= − ⋅rr

Compute the energy shifts (perturbatively):

J L S= +rr r

2 2 2 2J L S L S= + + ⋅r rr r r

2 2 2, , , zL S J L S J⋅r rr r r commutes with

( ) ( )

( ) ( ) ( )

23

23

12 2

112 2

nl nl

nl nl

E j l d r r f r

lE j l d r r f r

ψ

ψ

∆ = + = −

+ ∆ = − =

∫

∫

h r

h r

2

2

, , , jl s j m⇒ good q. nos. are

( ) ( ) ( )1 1 12

L S j j l l s s ⋅ + − + − + rr h2

=

Total splitting:

( ) ( ) 31 12 4

L S j j l l ⋅ + − + −

rr h2

=

( ) ( )2312 nll d r r f rψ ∆ = +

∫r

h2

( )

1 2 2

11 2 2

l j lL S

l j l

= −⋅

− + = +

hrr

h

2

2=

Key observation: Splitting grows with l⇒ can reorder the

levels.

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2

2028

50

8

• The SPSM has been coaxed to do what we need: provide a physics rationale for observed level ordering in the nucleus.

• Deduce consequences!

• Even-Even Nuclei JP = 0+

• Odd-A Spin-Parity Assignments– p and n levels fill independently

paired in every level ↑↓⇒Last unpaired nucleon’s j

determines nuclear spin and its ldetermines nuclear parity (-1)l.

136Example : C

6 protons, all paired ⇒no effect

7 neutrons: 2 4 1

1 3 12 2 2

1 1 1S P P

j = 1/2

l = 1 ⇒ odd parity

12

PJ−

=

Other “successes” – magnetic moments

Omission – odd-odd nuclei

And it does…And it does…

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Nuclear PhysicsNuclear Physics ?• “A pastiche of slightly related ideas and techniques”

– Assessment of theoretical nuclear physics by W.S.C. Williams, an experimental nuclear physicist

• Models do not represent a coherent, fundamental framework, because…– Underlying interaction (strong force) is not well understood.– Many-body effects are very difficult to handle, central to

“collective models” of “nuclear matter,” but…• Still, it works quite well on an empirical level

– Choose your functions intelligently and give them enough, but not too many, free parameters and you get a phenomenology with many impressive applications

– Pragmatism: Use the models and empirical tools to analyze behavior of nuclei, especially radioactive decay, but also the energy-producing processes of fission and fusion