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Wright Nuclear Structure Laboratory, Yale
Quantum Phase Transitions in Nuclear Physics
R. F. Casten, WNSL, Yale
The study of nuclei links phenomena ranging over 42 orders of magnitude in distance scale—from sub-nucleonic (<10–15 m) to the cosmos (1027 m).
Nuclei are the interface between QCD, the nanoscale physics of atomic phenomena, and the macroscopic world.
Concepts we will discuss
• Evolution of structure in nuclei
• Signatures of structural evolution• Excitation energies• Masses, separation energies
• Quantum Phase Transitions (QPT)
• Critical Point Symmetries (CPS)
• Femtoscopic basis for structural evolution• Competition between pairing and the p-n interaction• Signatures of phase transitions mediated by sub-shell changes
Themes and challenges of Modern Science
•Complexity out of simplicity
How the world, with all its apparent complexity and diversity can be constructed out of a few elementary building blocks and their interactions
• How do the forces between protons and neutrons lead to the nuclei we
observe? The WHY
•Simplicity out of complexity
How the world of complex systems can display such remarkable regularity and simplicity
• What are the simple patterns and symmetries that nuclei exhibit?
The WHAT
r = |ri - rj|
Vij
r
Ui
Shell structure
Clusters of levels shell structure
Pauli Principle (≤ 2j+1 nucleons in orbit with angular momentum j) magic numbers, inert cores
Concept of valence nucleons – key to structure. Many-body few-body: each body counts. Addition of 2 neutrons in a nucleus with 150 can drastically alter structure
= nl , E = EnlH.O. E = ħ (2n+l) E (n,l) = E (n-1, l+2) E (2s) = E (1d)
– Pairing – coupling of two identical nucleons to angular momentum zero. No preferred direction in space, therefore drives nucleus towards spherical shapes
– p-n interactions – drives towards deformation
What determines how nuclear structure evolves?
Nucleons orbit in a potential but that would never produce correlations or collective phenomena. However, there are crucial extra (residual) interactions (beyond mean field) among valence nucleons (those outside closed shells)
These interactions dominate the evolution of structure
1000 4+
2+
0
400
0+
E (keV) Jπ
Si
m pl e O bs er v a bl
es
- E ve n- E ve n
N u cl ei
. .
)2(
)4(
1
12/4
E
ER
Masses1400 2+
T1/2(ps)
R4/2= 3.33 DeformedR4/2= 2.0 Spherical
Astonishing regularities that nuclei exhibit
Why is this amazing? What is the origin of ordered motion of complex nuclei?
Complex systems often display astonishing simplicities. How is it
that a heavy nucleus, with hundreds of nucleons, occupying 60 % of the volume of the nucleus,
and executing 1021 orbits/sec without colliding, can exhibit such
simple collective motions.
Symmetric Rotor
E(I) ( ħ2/2I )I(I+1)
R4/2= 3.33
0+
2+
6+. . .
8+. . .
Vibrator (H.O.)
E(I) = n ( 0 )
R4/2= 2.0
Spherical vibrator
Multi-phonon states
n = 0,1,2,3,4,5 !!
n = 0
n = 1
n = 2
n = 3
B(E2; 2+ 0+ )
Emergence of collectivity with valence nucleon number
Broad perspective on structural evolution
Note sharp increase
R4/2= 2.0 Spherical R4/2= 3.33 Deformed
From Cakirli
Quantum Phase Transitions in Finite Atomic Nuclei
order parameter
control parametercritical point
Nuclei: Changes in equilibrium shape (spherical to deformed) as a function of neutron and proton number
Vibrator Soft Rotor
Deformation
Spherical
Ene
rgy Transitiona
lDeformed
Order parameter: Nucleon number
Control parameter:
Deformation(note: not an observable)
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
Phase Coexistence
Critical Point Symmetries
E E
β
1 2
3
4
Energy surface changes with valence nucleon numberX(5)
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
Modeling phase transitional
behavior in the A ~ 150 region
Parameter- free except for scale
Casten and Zamfir
E-GOS Plots (aka Paddy Plots)
Yrast
Classifying Structure -- The Symmetry TriangleClassifying Structure -- The Symmetry Triangle with its 3 traditional paradigmswith its 3 traditional paradigms
Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Mapping the triangle.
Sph.
Deform
ed
Unique signature of phase
transitional line?
X(5)
E(5)
= 2.9R4/2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.0
0.5
1.0
1.5
2nd
order
1st order = -1.32
= -0.75 = 0.00
N = 100
E(6
+ 1)
/ E(0
+ 2)
Energy ratio between 6+ of ground state and first excited 0+
)0(
)6(
2
1
E
E { 1.5 U(5)
→ 0 SU(3)
Empirical signature of 1st and 2nd order
~1 at Ph. Tr ~ X(5)
Vibrator
Rotor
w/Bonatsos and McCutchan
First order
Second order
Degeneracies point to underlying symmetries
w/Bonatsos and McCutchan
Special properties of flat-bottomed potentials
Remarkable generalization to any flat potential
E(0+n) / E(0+
2) = A n [ n + (D + 1)/2 ]
depends ONLY on D, the number of dimensions!
n( n + 3)
5 4
3 2
1 0
V(
)
842
E ~ n ( n + x)
• Competition of p-n and pairing
P = NpNn/ (Np+Nn)
Numerator ~ number of p-n interactions;
Denominator ~ number of pairing interactions.
Ratio reflects competition between spherical-driving pairing forces and ellipsoidally-driving p-n forces.
Understanding the evolution of structure without complex models or super computers
The P-factor(calculated from the numbers of valence nucleons only)
Critical value of P
• Pairing interaction has a strength of ~ 1 MeV
• p-n interaction has a strength of ~ 200 keV
• Therefore, it takes ~ 5 p-n interactions to compete against 1 pairing interaction
• Pcrit > 5 defines onset of deformed nuclei
NpNn p – n
PNp + Nn pairing
Contours define locus of possible X(5) nuclei and enclose regions of deformation
p-n / pairing
P ~ 5
168W130Ce178Os
Study of symmetry phases
deformation
-decay -decay
But .. Spectroscopy is not enough
Sn
Ba
SmHf
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
Neutron Number
S(2
n)
MeV
Energy required to remove two neutrons from nuclei
(2-neutron binding energies = 2-neutron “separation” energies)
N = 82
N = 84
N = 126
Ba Ce Nd
Sm
Gd
Dy
Er
Yb
11
12
13
14
15
16
17
84 86 88 90 92 94 96
Neutron Number
S (
2n)
MeV
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
Neutron Number
S(2
n)
MeV
Two nucleon separation energies as test of candidates for critical point nuclei
Sn
Ba
SmHf
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
Neutron Number
S(2
n)
Me
V
178Os168W130Ce
Masses are an essential complement to level scheme data. In 178Os, for example, the level scheme suggests X(5) character, while
masses show that there is no first order phase transtion in this nucleus
Femtoscopy – Why, How, what are the underlying mechanisms
Different perspectives can yield different insights
Onset of deformation Onset of deformation as a phase transition
mediated by a change in shell structure
mid-sh.
magic
Note change in curves from concave to convex
Subshell changes as a microscopic driving mechanism
for phase transitions
“Crossing” and “Bubble” plots
(Plus, seeing beyond the integer nucleon number problem)
This has recently been generalized
w/ Cakirli
A~150
Crossing Bubble Visually eliminating the transitional nucleiConcave and convex curves
Identifying the “subshell “ kind of nucleon
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
A~120
52 54 56 58 60
2,0
2,2
2,4
2,6
2,8
3,0
3,2
N=58 N=60 N=62 N=64 N=66 N=68 N=70
R4/
2
Proton Number
58 60 62 64 66 68 70
2,0
2,2
2,4
2,6
2,8
3,0
3,2
Z=52 Z=54 Z=56 Z=58 Z=60
R4/
2
Neutron Number
A~190
80 78 76 74 72 702,2
2,4
2,6
2,8
3,0
3,2
N=102 N=104 N=106 N=108 N=110 N=112 N=114
R4/
2
Proton Number
114 112 110 108 106 104 1022,2
2,4
2,6
2,8
3,0
3,2
Z=70 Z=72 Z=74 Z=76 Z=78 Z=80
R4/
2
Neutron Number
A~150
An alternate, simpler, observable, useful far from stability
1--------E(2+
1 )
A~100
THEORY
84 86 88 90 92 940,000
0,004
0,008
0,012
Z=56 Z=58 Z=60 Z=62 Z=64 Z=66 Z=68
1/E(
21+
)
Neutron Number
56 58 60 62 64 66 68
N=84 N=86 N=88 N=90 N=92 N=94
Proton Number
DATA
84 86 88 90 92 940,000
0,004
0,008
0,012A~150
Z=56 Z=58 Z=60 Z=62 Z=64 Z=66 Z=68
1/E(
21+
)
Neutron Number
56 58 60 62 64 66 68
N=84 N=86 N=88 N=90 N=92 N=94
Proton Number
Comparison with Femto-theory – Gogny force –Bertsch et al
Valence Proton-Neutron InteractionDevelopment of configuration mixing, collectivity and deformation
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) and many others.
Empirical average (last) p-n interaction
Double difference of binding energies (Garrett and Zhang)
Vpn (Z,N) = ¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ]
w/Cakirli
p n
82
50 82
126
low j, high n
high j, low n
Hence, if the protons and neutrons are filling similarly (similar fractional filling), the p-n interaction should be largest.
Generic sequencing of
shell model orbits
First extensive tests of specific interactions in heavy nuclei with
Density Functional Theory
Stoitsov, Cakirli et alFirst direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths
Cakirli et al
Summary
• R. Burcu Cakirli• Witek Nazarewicz• Mario Stoitsov• Libby McCutchan• Dennis Bonatsos• Victor Zamfir
Refs: PRL, 85,3584(2000) 87,52503(2001)
94, 092501(2005) 96, 132501(2006)
98, 132502(2007) 100, 142501(2008)
10X, in press
•Structural evolution in nuclei
•Quantum Phase Transitions and Critical Point Symmetries
•Empirical signatures
•The femtoscopic origins
• sub-shell changes
• p-n interactions
Backup slide
A~190-II