The proton-neutron interaction and the emergence of collectivity in atomic nuclei
R. F. CastenYale University
BNL Colloquium, April 15, 2014
The field of play: Trying to understand how the structure of the nucleus evolves with N and Z,
and its drivers.
Themes and challenges of nuclear structure physics –
common to many areas of Modern Science
•Complexity out of simplicity -- Microscopic How the world, with all its apparent complexity and diversity, can be constructed out of a few elementary building blocks and their interactions
•Simplicity out of complexity – Macroscopic How the world of complex systems* can display such remarkable regularity and simplicity
What is the force that binds nuclei?Why do nuclei do what they do?
What are the simple patterns that emerge in nuclei? What do they tell us about what nuclei do?
* Nucleons orbit ~ 10 21 /s , occupy ~ 60% of the nuclear volume; 2 forces
The p-n Interaction
Linking the two perspectives
1000 4+
2+
0
400
0+
E (keV) Jπ
Simple Observables - Even-Even Nuclei
)2()4(
1
12/4
EER Masses
1400 2+J
B(E2) values ~ transition rates
Geometric Potentials for Nuclei: 2 typesFor single particle motion and for the many-body system
r
Ui
b
V(b)
b
0b
r = |ri - rj|
Vij
r
Ui
Reminder: Concept of "mean field“ or average potential
Simple potentials plus spin-orbit force Clusters of levels shell structure
Pauli Principle (≤ 2j+1 nucleons in orbit with angular momentum j)
Energy gaps, closed shells, magic numbers. Independent Particle Model
Inert core of nucleons, Concept of valence nucleons and residual interactions
Many-body few-body: each body counts
Change by, say, 2 neutrons in a nucleus with 150 nucleons can drastically alter structure
= nl , E = EnlH.O. E = ħ (2n+l) E (n,l) = E (n-1, l+2) E (2s) = E (1d)
64
0+
2+
6+. . .
8+. . .
Vibrator (H.O.)E(I) = n ( 0 )
R4/2= 2.0
n = 0
n = 1
n = 2
RotorE(I) ( ħ2/2I )I(I+1)
R4/2= 3.33
Doubly magic plus 2 nucleons
R4/2< 2.0A few valence
nucleonsMany valence
nucleons
Pairing(sort of…)
ll
With many valence nucleonsMultiple configurations with the same
angular momentum. They mix, produce coherent, collective statesR4/2
A measure of collectivity
Nuclear Shape Evolution b - nuclear ellipsoidal deformation (b0 is spherical)
Vibrational Region Transitional Region Rotational Region
b
)(bV
b
)(bV
b
)(bV
nEn )1(~ JJEJCritical Point
Few valence nucleons Many valence Nucleons
New analytical solutions, E(5) and
X(5)R4/2 = 2 R4/2 = 3.33
10 20 30 40 50 60 70 80 90 10011012013014015010
20
30
40
50
60
70
80
90
100
Prot
on N
umbe
r
Neutron Number
80.00
474.7
869.5
1264
1600
E(21+)
10 20 30 40 50 60 70 80 90 10011012013014015010
20
30
40
50
60
70
80
90
100
Prot
on N
umbe
r
Neutron Number
1.400
1.776
2.152
2.529
2.905
3.200R4/2
The beauty of nuclear structural evolution
The remarkable regularity of these patterns is one of the beauties of nuclear structural evolution and one of the challenges to nuclear theory.
Whether they persist far off stability is one of the fascinating questions for the future.Cakirli
Structural evolutionComplexity and simplicity
BEWARE OF FALSE CORRELATIONS!
Rapid structural changes with N/Z – Quantum Phase Transitions
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
Nuclear Shape Evolution b - nuclear ellipsoidal deformation (b0 is spherical)
Vibrational Region Transitional Region Rotational Region
b
)(bV
b
)(bV
b
)(bV
nEn )1(~ JJEJCritical Point
Few valence nucleons Many valence Nucleons
New analytical solutions, E(5) and
X(5)R4/2 = 2 R4/2 = 3.33
2
21 0;
v
z z
Bessel equation
( ) 0. b w
( ) 1/21 93 4
L Lv
Critical Point SymmetriesFirst Order Phase Transition – Phase Coexistence
E E
β
1 2
3
4
bb
Energy surface changes with valence nucleon number
Iachello, 2001
X(5)
2 3 5
Casten and Zamfir
Remarkable agreement. Discrepancies can be understood in terms of sloped potential wall and the location of the critical point
Param-free
Sn – Magic: no valence p-n interactions
Both valence protons and
neutrons
What drives structural evolution?Valence p-n interactions in competition with pairing
56 58 60 62 64 66 68 701.61.82.02.22.42.62.83.03.23.4
N=84 N=86 N=88 N=90 N=92 N=94 N=96R
4/2
Z84 86 88 90 92 94 96
1.61.82.02.22.42.62.83.03.23.4
Ba Ce Nd Sm Gd Dy Er Yb
R4/
2
N
Seeing structural evolution Different perspectives can yield different insights
Onset of deformation Onset of deformation as a
quantum phase transition(convex to concave contours)
mediated by a change in proton shell structure as a function of neutron number, hence
driven by the p-n interaction
mid-sh.
magic
Look at exactly the same data now plotted against Z
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)]}
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 are filling similar orbits
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
High j, low n
Low j, high n
208Hg
Empirical p-n interaction strengths indeed strongest along diagonal.
82
50 82
126
High j, low n
Low j, high nEmpirical 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
My understanding of Density Functional Theory
Want to see it again?
These DFT calculations accurate only to ~ 1 MeV. Vpn allows one to focus on specific correlations.
M. Stoitsov, R. B. Cakirli, R. F. Casten, W. Nazarewicz, and W. Satula PRL 98, 132502 (2007);D. Neidherr et al, Phys. Rev. C 80, 044323 (2009)
Recent measurements at ISOLTRAP/ISOLDE test these DFT calculations
Comparison of empirical p-n interactions with Density Functional Theory(DFT) with Skyrme forces and surface-volume pairing
The competition of pairing and the p-n interaction. A guide to structural evolution.
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 almost as good results from 100 hours of supercomputer time.)
Valence Proton-Neutron Interactions Correlations, collectivity, deformation. Sensitive to magic numbers.
NpNn
Scheme
Highlight deviant nuclei
A refinement to the NpNn Scheme.
P = NpNn/ (Np+Nn)
No. p-n interactions per
pairing interaction
NpNn p – n
PNp + Nn pairing
p-n / pairing
P ~ 5
Pairing int. ~ 1 – 1.5 MeV, p-n ~ 200 - 300 keV
p-n interactions perpairing interaction
Hence takes ~ 5 p-n int. to compete with one pairing int.
Rich searching ground for
regions of rapid shape/phase
transition far off stability in
neutron rich nucleiMcCutchan
and Zamfir
Competition of p-n interaction with pairing: Simple estimate of evolution of structure with N, Z
Comparison with the data
Comparison with R4/2 data: Rare Earth region
Deformed nuclei
R4/2 data
Special phenomenology of p-n interactions:Implications for the evolution of structure and
the onset of deformation in nuclei
New result, not recognized before, that shows the effect in purer form
Shell model spherical nuclei
Nilsson model Deformed nuclei
Seems complex, huh? Not really. Very simple.Deformation
How to understand the Nilsson diagram
Orbit K1 will be lower in energy than orbit K2 where K is the projection of the angular momentum on the symmetry axis
Nilsson orbit labels (quantum numbers): K [N nz L ] nz is the number of nodes in the z-direction (extent of the w.f. in z)
z
This is the essence of the Nilsson model and diagram. Just repeat this idea for EACH j-orbit of the
spherical shell model. Look at the left and you will see
these patterns.
There is only one other ingredient
needed. Note that some of the lines are curved . This comes from 2-state mixing
of the spherical w.f.’s
(not important for present context)
Note: the last filled proton-neutron Nilsson orbitals for the nuclei where dVpn is largest are usually related by:
K[N,nz,Λ]=0[110]
168 Er
7/2[523]
7/2[633]
7/2[523]
7/2[633]
Specific highly overlapping proton-neutron pairs of orbitals, satisfying 0[110], fill almost synchronously in medium mass and
heavy nuclei and correlate with changing collectivity
1/2[431] 1/2[541]
3/2[422] 3/2[532]
168 Er
Synchronized filling of 0[110] proton and neutron orbit
combinations and the onset of deformation
Similarity of Nilsson patterns as deformation changes, and high overlaps of 0[110] orbit
pairs, leads to maximal collectivity near the Nval ~ Zval line.
0[110]
Summary
• Remarkable regularity and simplicity in a complex system
• Structural evolution – evidence for quantum phase transitions
• Drivers of emergent collectiivity -- valence p-n interaction in competition with pairing
• Empirical extraction of valence p-n interactions – sensitivity to orbit overlaps
• Enhanced valence p-n interactions for equal numbers of valence protons and neutrons
• Key to structural evolution in heavy nuclei – synchronized filling of highly overlapping 0[110] orbits
Principal collaborators
R. Burcu CakirliDennis Bonatsos
Klaus BlaumVictor Zamfir
Special thanks to Jackie Mooney for decades of collaboration and
friendship.
Backups
Competition between valence pairing and the p-n interactions
A simple microscopic interpretation of the evolution of structure
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 others.
0 2 4 6 8 10 12 14 16
2.0
2.5
3.0
3.5
X(5)
144-156NdCritical Point
Rotor
Vibrator
E(4
1+) /
E(2
1+)
Valence Neutron Number
V(b)
b
V(b)
b
V(b)
b
Shape/ Phase Transition and Critical Point
Symmetry
Discrepancies for excited 0+ band are one of scale
Scaled energes
Identifying the nature of the discrepancies. Isolating scale
Based on idea of Mark Caprio
A possible explanation
Discrepancies for transition rates from 0+ band are one of scale
Minimum in energy of first excited 0+ state
Li et al, 2009
E E
β
1 2
3
4
bb
Other signatures
Isotope shifts
Li et al, 2009
Charlwood et al, 2009
Enhanced density of 0+ states at the critical point
Meyer et al, 2006
E
β
1 2
3
4
b
Where else? Look at other N=90 nulei
But, more generally, where else? To understand this we need to discuss the drivers of structural evolution in nuclei with N and Z.
This evolution is a titanic struggle between good and good -- the pairing interaction between like nucleons and the proton-
neutron interaction.
The history of nuclear structure past and future( A VAST oversimplification)
50’s60’s
70’s – 00’s
10’s
10’s
1
0
D| N - Z| Z
Degen.
1<2j + 1>
RIBs:2 mass
units is a big deal
Z
N
<collectivity>
A
Accels.
g-ray det. Arrays
Weak, contaminated beams, fewer levelsMass sep., scintillators, “Back to the Future”
