the deuteron

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

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The Deuteron

• Deuterium (atom).•The only bound state of two nucleons simplest bound state• Neither di-proton nor di-neutron are stable. Why?

• Experimentally 2.224 MeV (Recoil..!).• Also inverse (,n) reaction using Bremsstrahlung (Recoil…!). mc2 = 2.224…??…MeV Very weakly bound.• Compare n-p to n-n and p-p Charge independence of nuclear force.• Only ground state. (There is an additional virtual state).

HHn 21


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The DeuteronV(r) = -V0 r < R = 0 r > R• Oversimplified.HW 17HW 17 Show that VShow that V00 35 MeV. 35 MeV.

(Follow Krane Ch.4 and (Follow Krane Ch.4 and Problem 4.6), or Problem 4.6), or similarly any other similarly any other reference.reference.• Really weakly bound.• What if the force were a bit weaker…?


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The Deuteron


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The Deuteron• Experiment deuteron is in triplet state = 1.• Experiment even parity.• = l + sn + sp parity = (-1)l

• Adding spins of proton and neutron gives: s = 0 (antiparallel) or s = 1 (parallel).• For = 1

parallel s-state evenparallel p-state oddantiparallel p-state oddparallel d-state even


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• Experiment = 0.8574376 N spins are aligned…..But.?• Direct addition 0.8798038 N.

• Direct addition of spin components assumes s-state (no orbital component).• Discrepancy d-state admixture.

= a00 + a22

= a020 + a2

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HW 18HW 18 In solving HW 17 you assumed an s-state. How good was that assumption?

The Deuteron


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The Deuteron

• S-state No quadrupole moment.• Experiment +0.00288 b.HW 18HW 18Discuss this discrepancy.

• From and Q, is it really admixture?• What about other effects?• Important to know the d-state wavefunction.


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Nuclear Force

Read Secs. 4.4 and 4.5 in Krane.


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Nuclear Models

• Nuclear force is not yet fully understood.• No absolutely satisfying model, but models.• Specific experimental data specific model.• Model success in a certain range.• Some are:

Individual particle model. (No interaction, E. states, static properties, …).

Liquid drop model. (Strong force, B.E., Fission, …).

Collective model. -particle model. Optical model. others …..


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Shell model

• Electron configuration…. 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 ….

• AtomicAtomic magic numbers: 2, 10, 18, 36, 54, … Common center of “external” attraction. Well understood Coulomb force. One kind of particles. Clear meaning for electron orbits. …

• NuclearNuclear magic numbers: 2, 8, 20, 28, 50, 82,126, …


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Shell model

Evidence:Evidence:1) End of radioactive series:

thorium series 208Pb

uranium series 206Pb

actinium series 207Pb

neptunium series 209Bi

2) At Z and N mn’s there are relatively large numbers of isotopes and isotones.


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Shell model


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Shell model


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Shell model

3) Natural abundances.


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Shell model

NEUTRON NUMBER

NE

UT

RO

N C

AP

TU

RE

C

RO

SS

SE

CT

ION

4) Neutron capture cross section.


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Shell model5) Binding energy of the last neutron

(Separation Energy). (The measured values are plotted relative to the calculations without ).


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Shell model

Pb (even-A) isotopes.

6) Excited states.


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Shell model

HW 19HW 19

Work out more examples for the above evidences. For example, take part of a plot and work on a group of relevant nuclides.

7) Quadrupole moments ….. ?


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Shell model

• Nucleons are in definite states of energy and angular momentum.• Nucleon orbit ?? Continuous scattering expected ..!!• No vacancy for scattering at low energy levels.• Potential of all other nucleons.• Infinite square well:

• Harmonic oscillator:

Rr

RrV

0

22

2

1rmV


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Shell model

MeVVe

VrV

aRr57~

1)( 0/

0

• More realistic:• Finite square well potential:

• Rounded well potential:

• Correction for asymmetry (n-p has more possibilities than n-n or p-p) and Coulomb repulsion.

Rr

RrVV

00


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Shell model• Separation of variables:

• For a given spherically symmetric potential V(r), the bound-state energy levels can be calculated from radial wave equation for a particular orbital angular momentum l. • Notice the important centrifugal potential.

1s 1p 1d 2s 1f 2p 1g 2d 3s

2(2l +1) 2 6 10 2 14 6 18 10 2

Total 2 8 18 20 34 40 58 68 70

ml

ms

),()()()()(),,( mlYrRrRr


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Shell modelcentrifugal potential


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Shell model


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Shell model

Infinite spherical well(R=8F)

Harmonic oscillator

???

2(2l + 1)accounts correctly for the number of nucleons in each level.But what about magic numbers?

)( 23E

)2( 21 lnEnl


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Shell model

• So far, 2(2l + 1) accounts correctly for the number of nucleons in each level, since we already considered both orbital angular momentum, and spin, but still not for closed shells.

sl ms

mlsl Ymsml ,,,

Spherical Harmonics,

Eigenfunctions of L2 and Lz.

smsmS

sssS

smss

msz

ms

ms

ss

ss

21)1( 22


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Shell model

• 2, 8, 20 ok.• What about other magic numbers?• Situation does not improve with other potentials.• Something very fundamental about the single-particle interaction picture is missing in the description…..!!!!!• Spin-orbit coupling.


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Shell model

Spin-Orbit CouplingSpin-Orbit Coupling• M. G. Mayer and independently Haxel, Jensen, and Suess.• Spin-Orbit term added to the Hamiltonian:

Central, attractive

No longer

Spherically symmetric

Orientation

LSrVrVm

pH SO .)()(

2

2


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Shell model

LLantiparallel

ULparallel

L SJ

LSJLSJLS 2/)(. 222

21,)1(

,....2,1,0,)1(

,

,)1(

22

22

22

slsjmsslsjmS

llsjmlllsjmL

jmjlsjmmlsjmJ

sljsllsjmjjlsjmJ

jj

jj

jjjjz

jj


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Shell model

2(2x3 + 1) = 14

2j+1

1f7/2

First time

l = 3

j


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Shell model

2)]1()1()1([2

1. sslljjSL

HW 20HW 20

0,)12(2

1 2 llgap

the deuteron

Documents

shell model nucleons

shell model sepa

model success

collective model

optical model

shell model binding

individual particle

shell modelhw