Lecture 4 Quantum Phase Transitions and the microscopic drivers of structural evolution.
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Transcript of Lecture 4 Quantum Phase Transitions and the microscopic drivers of structural evolution.
Lecture 4
Quantum Phase Transitions and the microscopic drivers of structural
evolution
Quantum phase transitions and structural evolution in nuclei
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
RotorTransitionalE
β
1 2
3
4
Quantum phase transitions in equilibrium Quantum phase transitions in equilibrium shapes of nuclei with shapes of nuclei with NN, , ZZ
For nuclear shape phase transitions the control parameter is nucleon number
Potential as function of the ellipsoidal deformation of the nucleus
Nuclear Shape Evolution - nuclear ellipsoidal deformation ( is spherical)
Vibrational Region Transitional Region Rotational Region
)(V
)(V
)(V
nEn )1(~ JJEJCritical Point
Few valence nucleons Many valence Nucleons
New analytical solutions, E(5) and
X(5)
R4/2= 3.33R4/2= ~2.0
2
21 0;
v
z z
Bessel equation
0. w
1/21 9
3 4
L Lv
Critical Point Symmetries
First Order Phase Transition – Phase Coexistence
E E
β
1 2
3
4
Energy surface changes with valence nucleon number
Iachello
X(5)
Casten and Zamfir
Comparison of relative energies with X(5)
Based on idea of Mark Caprio
Li et al, 2009
Flat potentials in validated by microscopic calculations
Potential energy surfaces of 136,134,132Ba
<H>
136Ba 134Ba 132Ba
<HPJ=0>
×minimum
× ×
× × ×
100keV
(N,N)= (-2,6) (-4,6) (-6,6)
More neutron holes
Shimizu et al
Isotope shifts
Li et al, 2009
Charlwood et al, 2009
Look at other N=90 nulei
Where else?
In a few minutes I will show some slides that will 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)
Where we stand on QPTs
• Muted phase transitional behavior seems established from a number of observables.
• Critical point solutions (CPSs) provide extremely simple, parameter-free (except for scales) descriptions that are surprisingly good given their simplicity.
• Extensive work exists on refinements to these CPSs.• Microscopic theories have made great strides, and
validate the basic idea of flat potentials in at the critical point. They can also now provide specific predictions for key observables.
Proton-neutron interactions
A crucial key to structural evolution
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) and many others.
Microscopic perspective
Sn – Magic: no valence p-n interactions
Both valence protons and
neutrons
Two effects
Configuration mixing, collectivity
Changes in single particle energies and shell structure
Concept of monopole interaction changing shell structure and inducing collectivity
A simple signature of phase transitions
MEDIATED
by sub-shell changes
Bubbles and Crossing patterns
Seeing structural evolution 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
“Crossing” and “Bubble” plots as indicators of phase transitional regions mediated by sub-shell changes
Often, esp. in exotic nuclei, R4/2 is not available. A easier-to-obtain observable, E(21
+), in the form of 1/ E(21
+), can substitute equally well
• Shell structure: ~ 1 MeV• Quantum phase transitions: ~ 100s keV• Collective effects ~ 100 keV• Interaction filters ~ 10-15 keV
Total mass/binding energy: Sum of all interactions
Mass differences: Separation energies shell structure, phase transitions
Double differences of masses: Interaction filters
Masses:
Macro
Micro
Masses and Nucleonic Interactions
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
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?
Orbit dependence of p-n interactions
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
High j, low n
Low j, high n
208Hg
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 are filling similar orbits
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., PR C, 2009
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
Exploiting the p-n interaction
• Estimating the structure of any nucleus in a trivial way (example: finding candidat6e for phase transitional behavior)
• Testing microscopic calculations
The NpNn Scheme and the P-factor
If the p-n interaction is so important it should be possible to use it to simplify our understanding of how structure evolves. Instead of plotting
observables against N or Z or A, plot them against a measure of the p-n interaction. Assume all p-n interactions are equal. How many are there:
Answer: Np x Nn
A simple microscopic guide to the evolution of structure
General p – n strengths
For heavy nuclei can approximate them as all constant.
Total number of p – n interactions is NpNn
Compeition between the p-n interaction and pairing: the P-factor
Pairing: each nucleon interacts with ONLY one other – the nucleon of the same type in the same orbit but orbiting in the opposite direction. So, the total number of pairing interactions scales as the number of valence nucleonss.
NpNn p – n
PNp + Nn pairing
What is the locus of candidates for X(5)
p-n / pairing
P ~ 5
Pairing int. ~ 1.5 MeV, p-n ~ 300 keV
p-n interactions perpairing interaction
Hence takes ~ 5 p-n int. to compete with one pairing int.
Comparison with the data
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
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
So, now what?
Go out and measure all 4000 unknown nuclei? No way!!! Choose those that tell us some physics, use simple paradigms
to get started, use more sophisticated ones to probe more deeply, and study the new physics that emerges. Overall, we
understand these beasts (nuclei) only very superficially.
Why do this?
Ultimately, the goal is to take this quantal, many-body system interacting with at least two forces, consuming 99.9% of visible matter, and understand its structure and symmetries, and its
microscopic underpinnings from a fundamental coherent framework.
We are progressing. It is your generation that will get us there.
The End
Thanks for listening
Special Thanks to:
• Iachello and Arima
• Dave Warner, Victor Zamfir, Burcu Cakirli, Stuart Pittel, Kris Heyde and others i9 didn’t have time to type just before the lecture
Backups
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
Two regions of parabolic
anomalies.
Two regions of octupole
correlations
Possible signature?
One more intriguing thing
Agreement is remarkable (within 10’s of keV). Yet these DFT calculations reproduce known masses only to ~ 1 MeV. How is this possible? Vpn focuses on sensitive
aspects of the wave functions that reflect specific correlations. It is designed to be insensitive to others.
Contours of constant
Vibrator Rotor
- soft
U(5) SU(3)
O(6)
3.3
3.1
2.9
2.7
2.5
2.2
R4/2
NB = 10
E(5)
X(5)
1st order
2nd order
Axially symmetric
Axi
ally
asy
mm
etri
c
Sph.
Def.