valence proton-neutron interaction
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Valence Proton-Neutron Interaction. Development of configuration mixing, collectivity and deformation – competition with pairing Changes in single particle energies and magic numbers - PowerPoint PPT PresentationTRANSCRIPT
Valence Proton-Neutron Interaction
Development of configuration mixing, collectivity and deformation – competition
with pairing
Changes in single particle energies and magic numbers
Partial history: Goldhaber and de Shalit (1953); Talmi (1962); Federman and Pittel ( late 1970’s); Casten et al (1981); Heyde et al (1980’s); Nazarewicz, Dobacewski et al (1980’s); Otsuka et al( 2000’s); Cakirli et al (2000’s); and many others.
Measurements of p-n Interaction Strengths
Vpn
Average p-n interaction between last protons and last neutrons
Double Difference of Binding Energies
Vpn (Z,N) = ¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ]
Ref: J.-y. Zhang and J. D. Garrett
Vpn (Z,N) =
¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ]
p n p n p n p n
Int. of last two n with Z protons, N-2 neutrons and with each other
Int. of last two n with Z-2 protons, N-2 neutrons and with each other
Empirical average interaction of last two neutrons with last two protons
-- -
-
Valence p-n interaction: Can we measure it?
Empirical interactions of the last proton with the last neutron
Vpn (Z, N) = ¼{[B(Z, N ) – B(Z, N - 2)]
- [B(Z - 2, N) – B(Z - 2, N -2)]}
Proton-neutron Interaction Strengths
<p|V|n> = 1/ [ n + l + 1 ]
Orbit dependence of p-n interactions – simple ideas
82
50 82
126
High j, low n
Low j, high n
82
50 82
126
Z 82 , N < 126
11
Z 82 , N < 126
1 2
Z > 82 , N < 126
2
3
3
Z > 82 , N > 126
208Hg208Hg
Can we extend these ideas beyond magic regions?
Away from closed shells, these simple arguments are too crude. But some general predictions can be made
p-n interaction is short range similar orbits give largest p-n interaction
HIGH j, LOW n
LOW j, HIGH n
50
82
82
126
Largest p-n interactions if proton and neutron shells have similar fractional filling
Empirical p-n interaction strengths indeed strongest along diagonal.
82
50 82
126
High j, low n
Low j, high n
New mass data on Xe isotopes at ISOLTRAP – ISOLDE CERNNeidherr et al, preliminary
Empirical p-n interaction strengths stronger in like regions than unlike regions.
Direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths
p-n interactions and the evolution
of structure
W. Nazarewicz, M. Stoitsov, W. Satula
Microscopic Density Functional Calculations with Skyrme forces and
different treatments of pairing
Realistic Calculations
Agreement is remarkable. Especially so since these DFT calculations reproduce known masses
only to ~ 1 MeV – yet the double difference embodied in Vpn allows one to focus on
sensitive aspects of the wave functions that reflect specific correlations
Sensitivity of the mass filter:DFT with different pairing interactions
The new Xe mass measurements at ISOLDE give a new test of the DFT
84
92
96
102
108
116
126
54
58
62
66
70
74
78
82
Z
N 84 96 114 126
54
58
62
66
70
74
78
82
Z
N
Experiment DFT
250 < Vpn < 300
r 350
b250
300 350 bVpn <
hp
hh
pp
SKPDMIX
Vpn (DFT – Two interactions)
84
92
96
102
108
116
126
54
58
62
66
70
74
78
82
Z
N 84 96 114 126
54
58
62
66
70
74
78
82
Z
N
Experiment DFT
250 < Vpn < 300
r 350
b250
300 350 bVpn <
hp
hh
pp
SLY4MIX
55 60 65 70 75 80
200
300
400
500
600
55 60 65 70 75 80
200
300
400
500
600
Vp
n /
keV
55 60 65 70 75 80
Pd
CdSn
TeXe
55 60 65 70 75 80
200
300
400
500
600
Neutron Number N
55 60 65 70 75 80
Vpn
Exp
Vpn
DFT
Vpn
Overlap
Martin’s model (Sn, Te, Cd, Xe, Pd)
Vpn =
A 1/ [ nik + lik + 1 ]} Vi2 Vk
2
Sum is over all pairs of the i-th proton and k-th neutron orbits near the Fermi surface, as seen in
the neighboring odd N and Z nuclei (in practice, levels up to
~ 300 keV). Parameters are A, B and the pairing gap parameter -
chosen once for all the nuclei. Interestingly, the p-n interaction
strength also depends on pairing! Might expect this simple, no-
collective-correlations, model to work for Sn but not much else
(because R4/2 >2 for Te, Cd, Xe, Pd).
Two regions of parabolic
anomalies.
Two regions of octupole
correlations.
Possible signature?
One last thing
Neidherr et al, preliminary
Summary of p-n interactions and Vpn
• p-n interactions and the evolution of collectivity
• Mass filters – Vpn – available far from stability
– Crossing pattern around doubly magic nuclei -- orbit overlaps– Maximum for similar shell filling – along diagonal in N-Z plot– Maximum in like quadrants (p-p or h-h vs p-h or h-p)– Correlation with growth rates of collectivity– Comparison with DFT calculations– Possible signature of octupole correlations
How to know what the nuclei are telling us
(using simple observables)
• R4/2, but you all know that already, right?
• Note: nuclear shapes have TWO variables so generally need TWO observables.
• How shapes change with N and Z: P-factor• Bubbles and shell structure (already discussed)• How shapes change with N and Z: AHV model• How shapes change with spin – E-GOS• IBA (Its really simple, don’t be scared)
How does R4/2 vary and why do we care
We care because it is the only observable whose value
immediately tells us something (not everything !!!)
about the structure.
We care because it is easy to measure, especially far
from stability
Other observables, like E(2+) and masses, are
measurable even further from stability. They can give
valuable information in the context of regional behavior.
0+
2+
6+. . .
8+. . .
Vibrator (H.O.)
E(I) = n ( 0 )
R4/2= 2.0
n = 0
n = 1
n = 2
Rotor
E(I) ( ħ2/2I )I(I+1)
R4/2= 3.33
Doubly magic plus 2 nucleons
R4/2< 2.0
Guide to R42 values in Collective Nuclei (R42 < 2 near closed shells)
Sph.
Def.
R42~2.2
R42~2.9
Axial Sym.
Axi
al A
sym
.
E(02 ) > E(2
)
E(02) < E(2)
Broad perspective on structural evolution
Quantum phase transitions as a function of proton and neutron number
86 88 90 92 94 96 98 100
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
Nd Sm Gd Dy
R4/
2
N
Vibrator
RotorTransitional
Competition between pairing and the p-n interactions
A simple microscopic guide to the evolution of structure
(The next slides allow you to estimate the structure of any nucleus by multiplying and dividing two numbers
each less than 30)(or, if you prefer, you can get the same result from 10 hours of
supercomputer time)
NpNn p – n
PNp + Nn pairing
p-n / pairing
P ~ 5
Pairing int. ~ 1 MeV, p-n ~ 200 keV
P~5
p-n interactions perpairing interaction
Hence takes ~ 5 p-n int. to compete with one pairing int.
Comparing with the data
Comparison with the data
Simple ways to see underlying shell structure
Onset of deformation Onset of deformation as a phase transition
mediated by a change in shell structure
Mid-sh.
magic
“Crossing” and “Bubble” plots as indicators of phase transitional regions mediated by sub-shell changes
A~100
52 54 56 58 60 62 64 66
1,6
1,8
2,0
2,2
2,4
2,6
2,8
3,0
3,2
Z=36 Z=38 Z=40 Z=42 Z=44 Z=46
R4/
2
Neutron Number
36 38 40 42 44 46
1,6
1,8
2,0
2,2
2,4
2,6
2,8
3,0
3,2
N=52 N=54 N=56 N=58 N=60 N=62 N=64 N=66
R4/
2
Proton Number
Often, esp. in exotic nuclei, R4/2 is not available.
E(21+) [as 1/ E(21
+) ], is easier to measure, works as well !!!
Two very useful techniques:
Anharmonic Vibrator model (AHV)(works even for nuclei very far from vibrator)
E-GOS Plots(Paddy Regan, inventor)
Anharmonic Vibrator
Very general tool, BOTH for yrast states and structural evolution
R4/2 not exactly 2.0 in vibrational nuclei. Typ. ~2.2–ish. Write: E(4) = 2 E(2) + where is a phonon - phonon interaction.
How many interactions in the 3 – phonon 6+ state? Answer: 3How many in the n – phonon J = 2n state? Answer: n(n-1)/2
Hence we can write, in general, for the any “band”:
E(J) = n E(2) + [n(n – 1)/2] n = J/2
General. E.g., pure rotor, E(4) = 3.33E(2), hence = 1.33 E(2)
/E(2) varies from 0 for harmonic vibrator to 1.33 for rotor !!!
E-GOS PlotsEGamma Over Spin
Plot gamma ray energies going up a “band” divided by the spin of the initial state
E-Gos plots are, for spin, what AHV plots are for N or Z
Evolution of structure with J, N, or Z
E-GOS plots in limiting cases
Vibrator: E(I) = h
Hence E (I) = h
Rotor: E(I) = h2/2J [I (I + 1)]
Hence E(I) = h2/2J [4I - 2]
AHV (R4/2 = 3.0): E(4) =
2E(2) Hence E(I) = 2(I + 2)
R = 2 = constant
R4/2 = 3.0 (AHV)
Comparisons of AHV and E-GOS with data
J = 2 4 6 8 10 12 14
98-Ru 652 1397 2222 3126 4001 4912 5819
Egamma 652 745 825 904 875 911 907
AHV 652 1397 2235 3166 4190 5307 6517 ( /E(2) = 0.14
E-GOS 326 189 137 113 87.5 76 64.8
----------------------------------------------------------------------------------------------------------------166-Dy 77 253 527 892 1341 1868 2467
Egamma 77 176 274 365 449 527 599
AHV 77 253 528 902 1375 1947 2618( /E(2) = 1.29
E-GOS 38 44 46 46 45 44 42.8
102 Ru
R4/2 = 3.0 (AHV)
Vibr rotor
0+2+4+
6+
8+
Rotational states
Vibrational excitations
Rotational states built on (superposed on)
vibrational modes
Why do the energies of rotational states differ from the rotor model??
[You can see this by comparing with the rotor formula E ~ J ( J + 1). You can also see it deviations of E-GOS plots from the rotor
expression, or with the AHV.]
Look at the rotor formula:
E(J) = [h2 /2I] x [ J ( J = 1)] where I is the moment of inertia.
I is linear in BETA, the quadrupole deformation. There is a centrifugal force. If the potential in beta is SOFT, the nucleus can stretch, and beta will increase, so I will increase, so the energies will Decrease relative to the rotor expression with fixed I.
Phase transitional critical point
nucleus
What to do with very sparse data???
Example: 190 W. How do we figure out its structure?
W-Isotopes
N = 116
IBA Model, flexible 2- parameter model for a wide variety of structures. Three benchmark symmetries:
Symmetry triangle (see lectures last year).
= 2.9R4/2
R42 not enough. Need another observable: the second 2+
Backups
Neutron rich Kr isotopes (Z = 36)
E N 2 4 R42 P factor Structure 86 50 1565 2250 1.44 0 Magic88 52 775 1644 2.12 1.33 Vib ???90 54 707 ---- ----92 56 769 1804 2.34 2.40 Vib ???94 58 665 1519?? 2.28 2.67 Vib ???
Sr (Z = 38)
88 50 1836 --- Doubly Magic?90 52 831 1655 1.9992 54 814 1673 2.0694 56 836 2146 2.57 !!! Why 4+ higher?96 58 814 1792 2.2098 60 144 434 3.01 !!! Deformed
BEWARE OF FALSE BEWARE OF FALSE CORRELATIONS!CORRELATIONS!
BEWARE OF FALSE BEWARE OF FALSE CORRELATIONS!CORRELATIONS!
Proton-neutron Interaction Strengths
Simple estimate of the p-n matrix element
<p|V|n> = 1/ [ n + l + 1 ]
The IBA – a flexible, parameter-efficient, collective model
• Enormous truncation of the Shell Model: valence nucleons in pairs coupled to L = 0 (s bosons) and L = 2 (d bosons), simple interactions • Three dynamical symmetries, intermediate structures, symmetry triangle
• Two parameters (except scale)
)2(
)2()0(
1
22
E
EE)2(
)4(
1
1
E
EVibrator Rotor
γ - soft
Mapping Structure with Simple Observables – Technique of Orthogonal Crossing Contours
Burcu Cakirli et al.Beta decay exp. + IBA calcs.
SU(3)U(5)
O(6)
3.3
3.1
2.92.7
2.5
2.2
-3.0
-1.0-2.0
-0.1
+0.1
+1.0
+2.0
+2.9
U(5) SU(3)
O(6)
R4/2
)2(
)2()0(
1
2
E
EE
= 2.3 = 0.0
156Er
119 123 127 131 135 139100
200
300
400
500
84Po
90U
86Rn
88Ra
Vpn
(Zev
en,N
odd)
(keV
)
75 77 79 81 83 85 87 89 91
200
300
400
500
54Xe
52Te
56Ba
58Ce
Vpn
(Zev
en,N
odd)(
keV
)
74 76 78 80 82 84 86 88 90 92200
300
400
500
Neutron Number
54Xe
52Te
56Ba
58Ce
Vpn
(Zev
en,N
even
)(ke
V)
What happens as the numbers of valence neutrons and protons increase?
Case of few valence nucleons:Lowering of energies, development of multiplets.
R4/2 ~2-2.4
This evolution is the emergence
of collective behavior
Lots of valence nucleons of both types:emergence of deformation and therefore rotation (nuclei
live in the world, not in their own solipsistic enclaves)
R4/2 ~3.33
0+2+4+
6+
8+
Rotor
E(I) ( ħ2/2I )I(I+1)
R4/2= 3.33
Deformed nuclei – rotational spectra
Think about the striking regularities in these data.
Take a nucleus with A ~100-200. The summed volume of all the nucleons is ~ 60 % the volume of the nucleus, and they
orbit the nucleus ~ 1021 times per second!
Instead of utter chaos, the result is very regular behavior, reflecting ordered, coherent, motions of these nucleons.
This should astonish you.
How can this happen??!!!!
Much of understanding nuclei is understanding the relation between nucleonic motions and collective behavior
6+ 690
4+ 330
0+ 0
2+ 100
J E (keV)
?Without
rotor
paradigm
Paradigm
Benchmark
700
333
100
0
Rotor J(J + 1)
Amplifies structural
differences
Centrifugal stretching
Deviations
Identify additional
degrees of freedom
Value of paradigms
Two-neutron separation energies
Sn
Ba
Sm Hf
Pb
5
7
9
11
13
15
17
19
21
23
25
52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132
S(2
n)
MeV
Neutron Number
Normal behavior: ~ linear segments with drops after closed shellsDiscontinuities at first order phase transitionsS2n = A + BN + S2n (Coll.) Curvature in isotopic chains – collective effects: deviations from linearity are a few hundred keV
Use any collective model to calculate the collective contributions to S2n.
Binding Energies