some (more) nuclear structure paddy regan department of physics univesity of surrey guildford, uk...
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Some (more) Nuclear Structure
Paddy ReganDepartment of Physics
Univesity of SurreyGuildford, UK
Lecture 2Low-energy Collective Modes and Electromagnetic
Decays in Nuclei
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Outline of Lectures 1& 2
• 1) Overview of nuclear structure ‘limits’– Some experimental observables, evidence for
shell structure– Independent particle (shell) model– Single particle excitations and 2 particle
interactions.
• 2) Low Energy Collective Modes and EM Decays in Nuclei.– Low-energy quadrupole vibrations in nuclei– Rotations in even-even nuclei– Vibrator-rotor transitions, E-GOS curves
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What about 2 nucleons outside a closed shell ?
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Residual Interactions?
• We need to include any addition changes to the energy which arise from the interactions between valence nucleons.
• This is in addition the mean-field (average) potential which the valence proton/neutron feels.
• Hamiltonian now becomes H = H0 + Hresidual
• 2-nucleon system can be thought of as an inert, doubly magic core plus 2 interacting nucleons.
• Residual interactions between these two ‘valence’ nucleons will determine the energy sequence of the allowed spins / parities.
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What spins can you make?• If two particles are in identical orbits (j2), then what spins are allowed?
Two possible cases:• Same particle, e.g., 2 protons or 2 neutrons = even-even nuclei like
42Ca, 2 neutrons in f7/2 = (f7/2)2 We can couple the two neutrons to make states with spin/parity J=0+,
2+, 4+ and 6+ These all have T=1 in isospin formalism, intrinsic spins are anti-aligned with respect to each other.
• Proton-neutron configurations (odd-odd)e.g., 42Sc, 1 proton and 1 neutron in f7/2
We can couple these two make states with spin / parity 0+, 1+, 2+, 3+, 4+, 5+, 6+ and 7+.
Even spins have T=1 (S=0, intrinsic spins anti-aligned); Odd spins have T=0 (S=1, intrinsic spins aligned)
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m – scheme showing which Jtot values are allowed for (f7/2)2 coupling of two identical particles (2 protons or 2 neutrons).
Note, that only even spin states are allowed.
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Schematic for (f7/2)2 configuration. 4 degenerate states if there are no residual interactions.
Residual interactions between two valence nucleons give additional binding, lowering the (mass) energy of the state.
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Geometric Interpretation of the Residual Interaction for a j2 Configuration Coupled to Spin
J
1
121cos for
cos11111
cos2
121
22112211
2122
21
2
jj
jjJJjjj
jjjjjjjjJJ
therefore
jjjjJ
111 jj 1JJ
Use the cosine rule and recall that the magnitude of the spin vector of spin j = [ j (j+1) ]-1/2
122 jj
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interaction gives nice simple geometric rationale for Seniority Isomers from
E ~ -VoFr tan (/2)
for T=1, even J
0
2
4
6
8
180
E(j2J)
90 0
2
468
e.g. J= (h9/2)2 coupled to
0+, 2+, 4+, 6+ and 8+.
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interaction gives nice simple geometric rationale
for Seniority Isomers from E ~ -VoFr tan (/2)
for T=1, even J
02
4
68
See e.g., Nuclear structure from a simple perspective, R.F. Casten Chap 4.)
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A. Jungclaus et al.,
Note, 2 neutron or 2 proton holes in doubly magic nuclei show spectra like 2 proton or neutron particles.
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Basic EM Selection Rules?
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1,, the highest energy transitions for the lowest are (generally) favoured. This results in the preferential population of yrast and near-yrast states.
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1,, the highest energy transitions for the lowest are (generally) favoured. This results in the preferential population of yrast and near-yrast states.
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1,, the highest energy transitions for the lowest are (generally) favoured. This results in the preferential population of yrast and near-yrast states.
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1,, the highest energy transitions for the lowest are (generally) favoured. This results in the preferential population of yrast and near-yrast states.
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1,, the highest energy transitions for the lowest are (generally) favoured. This results in the preferential population of yrast and near-yrast states.
= gamma-ray between yrast states
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1, (for E2 decays E5)
Thus, the highest energy transitions for the lowest are usually favoured. Non-yrast states decay to yrast ones (unless very different , K-isomers
= ray from non-yrast state.
= ray between yrast states
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1, (for E2 decays E5)
Thus, the highest energy transitions for the lowest are usually favoured. Non-yrast states decay to yrast ones (unless very different , K-isomers
= ray from non-yrast state.
= ray between yrast states
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'Near-Yrast' decays
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10
Spin of decaying state, I
Ex
cit
ati
on
en
erg
y
The EM transition rate depends on E2+1, (for E2 decays E5)
Thus, the highest energy transitions for the lowest are usually favoured. Non-yrast states decay to yrast ones (unless very different , K-isomers
= ray from non-yrast state.
= ray between yrast states
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Schematic for (f7/2)2 configuration. 4 degenerate states if there are no residual interactions.
Residual interactions between two valence nucleons give additional binding, lowering the (mass) energy of the state.
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Excitation energy (keV)
Ground stateConfiguration.Spin/parity I=0+ ;Ex = 0 keV
2+
0+
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4+/2+ energy ratio:mirrors 2+ systematics.
Excitation energy (keV)
Ground stateConfiguration.Spin/parity I=0+ ;Ex = 0 keV
2+
0+
4+
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B(E2; 2+ 0+ )
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What about both valence neutrons and protons?
In cases of a few valence nucleons there is a lowering of energies, development of
multiplets. R4/2 ~2-2.4
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Quadrupole Vibrations in Nuclei ?
• Low-energy quadrupole vibrations in nuclei ?– Evidence?– Signatures?– Coupling schemes ?
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2
V
2
En
n=0
n=1
n=2
n=3
http://npl.kyy.nitech.ac.jp/~arita/vib
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We can use the m-scheme to see what states we can make when we coupletogether 2 quadrupole phonon excitations of order J=2ħ. (Note phonons are bosons, so we can couple identical ‘particles’ together).
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From,Nuclear StructureFrom a SimplePerspective, byR.F. Casten,Oxford UniversityPress.
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For an idealised quantum quadrupole vibrator, the(quadrupole) phonon (=‘d-boson’) selection rule is n=1 , where n=phonon number.
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For an idealised quantum quadrupole vibrator, thephonon (=‘d-boson’) selection rule is n=1
4+ →2+ E2 from n=3 →n=1 is ‘forbidden’ in an idealised quadrupole vibrator by phonon selection rule.
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For an idealised quantum quadrupole vibrator, thephonon (=‘d-boson’) selection rule is np=1
4+ →2+ E2 from n=3 →n=1 is ‘forbidden’ in an idealised quadrupole vibrator by phonon selection rule.
Similarly, E2 from 2+→0+ from n=3 →n=0 not allowed.
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Collective (Quadrupole) Nuclear Rotations and Vibrations
• What are the (idealised) excitation energy signatures for quadrupole collective motion (in even-even nuclei) ?– (extreme) theoretical limits
2 (4 ) 4(5) 20( 1), 3.33
2 (2 ) 2(
(4 ) 2 = 2.00
3)
( 1
6
2 )N
J
EE N
EE J J
E
E
Perfect, quadrupole (ellipsoidal), axially symmetric quantum rotor with a constant moment of inertia (I) has rotational energies given by (from Eclass(rotor) = L2/2I)
Perfect, quadrupole vibrator has energies given by the solutionto the harmonic oscilator potential (Eclassical=1/2kx2 + p2/2m ).
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Collective (Quadrupole) Nuclear Rotations and Vibrations
• What are the (idealised) excitation energy signatures for quadrupole collective motion (in even-even nuclei) ?– (extreme) theoretical limits
2 (4 ) 4(5) 20( 1), 3.33
2 (2 ) 2(
(4 ) 2 = 2.00
3)
( 1
6
2 )N
J
EE N
EE J J
E
E
Perfect, quadrupole (ellipsoidal), axially symmetric quantum rotor with a constant moment of inertia (I) has rotational energies given by (from Eclass(rotor) = ½ L2/2I)
Perfect, quadrupole vibrator has energies given by the solutionto the harmonic oscilator potential (Eclassical=1/2kx2 +
p2/2m ).
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Other Signatures of (perfect) vibrators and rotors
Decay lifetimes give B(E2) values. Also selection rules important (eg. n=1).
For (‘real’) examples, see J. Kern et al., Nucl. Phys. A593 (1995) 21
E=ħE(J→J-2)=0 Ex=(ħ2/2I)J(J+1)i.e., E(J→J-
2)= (ħ2/2I)[J(J+1) – (J-2)(J-3)] = (ħ2/2I)
(6J-6); E=(ħ2/2I)*12=const.
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Other Signatures of (perfect) vibrators and rotors
Decay lifetimes give B(E2) values. Also selection rules important (eg. n=1).
Ex=(ħ2/2I)J(J+1)i.e., E(J→J-2)=(ħ2/2I)[J(J+1) – (J-2)(J-3)] = (ħ2/2I)
(6J-6); E=(ħ2/2I)*12=const.
Ex=(ħ2/2I)J(J+1)
++
+
+
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Other Signatures of (perfect) vibrators and rotors
Decay lifetimes give B(E2) values. Also selection rules important (eg. n=1).
For (‘real’) examples, see J. Kern et al., Nucl. Phys. A593 (1995) 21
E=ħE(J→J-2)=0 Ex=(ħ2/2I)J(J+1)i.e., E(J→J-
2)= (ħ2/2I)[J(J+1) – (J-2)(J-3)] = (ħ2/2I)
(6J-6); E=(ħ2/2I)*12=const.
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So, what about ‘real’ nuclei ?
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Many nuclei with R(4/2)~2.0 also show I=4+,2+,0+ triplet states at ~2E(2+).
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Note on ‘near-yrast feeding’ for vibrational states in nuclei.
If ‘vibrational’ states are populated in very high-spin reactions (such as heavyion induced fusion evaporation reactions), only the decays betweenthe (near)-YRAST states are likely to be observed.
The effect is to (only?) see the ‘stretched’ E2 cascade from Jmax →Jmax-2 for each phonon multiplet.
= the ‘yrast’ stretched E2 cascade.
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Note on ‘near-yrast feeding’ for vibrational states in nuclei.
If ‘vibrational’ states are populated in very high-spin reactions (such as heavyion induced fusion evaporation reactions), only the decays betweenthe (near)-YRAST states are likely to be observed.
The effect is to (only?) see the ‘stretched’ E2 cascade from Jmax →Jmax-2 for each phonon multiplet.
= the ‘yrast’ stretched E2 cascade.
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Nuclear Rotations and Static Quadrupole Deformation
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B(E2: 0+1 2+
1) 2+1 E20+
12
2+
0+T (E2) = transition probability = 1/ (secs); E = transition energy in MeV
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B(E2: 0+1 2+
1) 2+1 E20+
12
2+
0+
Rotational model, B(E2: I→I-2) gives:
Qo = INTRINSIC (TRANSITION) ELECTRIC QUADRUPOLE MOMENT.
This is intimately linked to the electrical charge (i.e. proton) distribution within the nucleus.
Non-zero Qo means some deviation from spherical symmetry and thus somequadrupole ‘deformation’.
T (E2) = transition probability = 1/ (secs); E = transition energy in MeV
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Bohr and Mottelson, Phys. Rev. 90, 717 (1953)
Isomer spin in 180Hf, I>11 shown later to be I=K=8- by Korner et al. Phys. Rev. Letts. 27, 1593 (1971)).
K-value very important in understanding isomers.
Ex = (ħ2/2I)*J(J+1)
I = moment of inertia. This depends on nucleardeformation and I~ kMR2
Thus, I ~ kA5/3
(since rnuc=1.2A1/3fm )
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Therefore, plotting the moment of inertia, divided by A5/3 should give
a comparison of nuclear deformations across chains of nuclei and mass regions….
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Nuclear static moment of inertia for E(2+) states divided by A5/3 trivial mass dependence. Should show regions of quadrupole deformation.
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Lots of valence nucleons of both types:emergence of deformation and therefore
rotation
R4/2 ~3.33 = [4(4+1)] / [2(2+1)]
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Perfect rotor limit R(4/2) = 3.33 = 4(4+1) / 2(2+1)
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Best nuclear ‘rotors’ have largest values of N.N This is the product of the number of ‘valence’ protons, N X the number of valence neutrons N
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Alignments and rotational motion in ‘vibrational’ 106Cd (Z=48, N=58),
PHR et al. Nucl. Phys. A586 (1995) p351
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Some useful nuclear rotational,‘pseudo-observables’…
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Some useful nuclear rotational,‘pseudo-observables’…
Rotational ‘frequency’, given by,
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2qp states, Ex~2
4qp states, Ex~4
6qp states, Ex~6
8qp states, Ex~8
C.S.Purry et al., Nucl. Phys. A632 (1998) p229
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Transitions from Vibrator to Rotor?
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PHR, Beausang, Zamfir, Casten, Zhang et al., Phys. Rev. Lett. 90 (2003) 152502
24
24
2 :Rotor
0 : Vibrator
)2(
242
),1(2
:Rotor
,2
:Vibrator
22
22
J
J
J
n
JR
JR
J
JJER
JEJJE
EJ
nE
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PHR, Beausang, Zamfir, Casten, Zhang et al., Phys. Rev. Lett. 90 (2003) 152502
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PHR, Beausang, Zamfir, Casten, Zhang et al., Phys. Rev. Lett. 90 (2003) 152502
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Vibrator-Rotator phase change is a feature of near stable (green) A~100.
‘Rotational alignment’ can be a crossing between quasi-vibrational GSB & deformed rotational sequence.(stiffening of potential by population of high-j, equatorial (h11/2) orbitals).
PHR, Beausang, Zamfir, Casten, Zhang et al., Phys. Rev. Lett. 90 (2003) 152502