The Deuteron
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Transcript of 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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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…?
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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The Deuteron
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Nuclear Force
Read Secs. 4.4 and 4.5 in Krane.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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 …..
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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, …
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model
3) Natural abundances.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model5) Binding energy of the last neutron
(Separation Energy). (The measured values are plotted relative to the calculations without ).
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model
Pb (even-A) isotopes.
6) Excited states.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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 ….. ?
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
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2
1rmV
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell modelcentrifugal potential
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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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
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model
2(2x3 + 1) = 14
2j+1
1f7/2
First time
l = 3
j
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).
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Shell model
2)]1()1()1([2
1. sslljjSL
HW 20HW 20
0,)12(2
1 2 llgap