Lecture 4
Quantum Phase Transitions and
the microscopic drivers of structural
evolution
Quantum phase transitions and
structural evolution in nuclei
Quantum phase transitions in equilibrium
shapes of nuclei with N, Z
Potential as function of the ellipsoidal
deformation of the nucleus
3.4
3.2
Transitional
Rotor
E
3.0
1
2
R4/2
2.8
3
2.6
Vibrator
2.4
2.2
2.0
86
Nd
Sm
Gd
Dy
88 nuclear
90
92 shape
94
96 phase
98 100
For
transitions the control
parameter
is nucleon number
N
4
β
Nuclear Shape Evolution
b - nuclear ellipsoidal deformation (b=0 is spherical)
Vibrational Region
V (b )
Transitional Region
V (b )
b
En = n
R4/2= ~2.0
Rotational Region
V (b )
b
b
Critical Point
New analytical solutions,
E(5) and X(5)
Few valence nucleons
EJ ~ J ( J  1)
R4/2= 3.33
Many valence Nucleons
Critical Point Symmetries
First Order Phase Transition – Phase Coexistence
E
Energy surface
changes with
valence nucleon
number
1
X(5)
E
2
3
4
βb
b
Bessel equation
2

v 
     1  2   = 0;
z 
z 

Iachello
  b w  = 0.
 L  L  1 9  1/ 2
v=


3
4


Casten and Zamfir
Comparison of relative energies with X(5)
Based on idea of Mark Caprio
Flat potentials in b validated by microscopic calculations
Li et al, 2009
Potential energy surfaces of
136,134,132Ba
Shimizu et al
More neutron holes
134Ba
136Ba
100keV
132Ba
<H>
×
×
×
minimum
<HPJ=0>
(Nn,Np)=
×
(-2,6)
×
×
(-4,6)
(-6,6)
Isotope shifts
Charlwood et al, 2009
Li 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 b at the
critical point. They can also now provide specific
predictions for key observables.
Proton-neutron interactions
A crucial key to structural
evolution
Microscopic perspective
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.
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
3.4
3.4
3.0
R4/2
2.8
2.6
2.4
2.2
3.0
2.6
2.4
1.8
1.8
1.6
1.6
88
90
92
94
N
Onset of deformation
96
magic
2.2
2.0
86
Mid-sh.
2.8
2.0
84
N=84
N=86
N=88
N=90
N=92
N=94
N=96
3.2
R4/2
Ba
Ce
Nd
Sm
Gd
Dy
Er
Yb
3.2
56
58
60
62
64
66
68
70
Z
Onset of deformation
as a phase transition
mediated by a change in shell structure
“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
Masses and Nucleonic Interactions
Masses:
Total mass/binding energy: Sum of all interactions
Mass differences: Separation energies
shell structure, phase transitions
Double differences of masses: Interaction filters
Macro
Micro
•
•
•
•
Shell structure:
Quantum phase transitions:
Collective effects
Interaction filters
~ 1 MeV
~ 100s keV
~ 100 keV
~ 10-15 keV
25
23
21
S(2n) MeV
19
17
15
13
Sm
11
9
Hf
Ba
Pb
7
Sn
5
52
56
60
64
68
72
76
80
84
88
92
96
Neutron Number
100
104
108
112
116
120
124
128
132
Measurements of p-n Interaction Strengths
dVpn
Average p-n interaction between
last protons and last neutrons
Double Difference of Binding Energies
dVpn (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
Valence p-n interaction: Can we measure it?
dVpn (Z,N) =
¼ [ {B(Z,N)
-
B(Z, N-2)}
p
n
-
{B(Z-2, N)
p
n
Int. of last two n with Z protons,
N-2 neutrons and with each other
p
-
- B(Z-2, N-2)} ]
n
p
n
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
Orbit dependence of p-n
interactions
126
82
Low j, high n
High j, low n
50
82
p
n
Z  82 , N < 126
3
82
1
3
2Low j, high n
1
126
High j, low n
2
50
82
p
n
Z > 82 , N > 126
Z > 82 , N < 126
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
126
82
LOW j, HIGH n
HIGH j, LOW n
50
82
Largest p-n interactions if proton and neutron
shells are filling similar orbits
Empirical p-n interaction strengths
indeed strongest along diagonal.
82
126
Empirical p-n interaction strengths
stronger in like regions than unlike
regions.
High j, low n
50
82
p
n
New mass data on Xe isotopes at
ISOLTRAP – ISOLDE CERN
Neidherr et al., PR C, 2009
p-n interactions
and the evolution
of structure
Direct correlation of observed
growth rates of collectivity with
empirical p-n interaction strengths
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
A simple microscopic guide to the
evolution of structure
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
Compeition between the p-n interaction and
pairing: the P-factor
General p – n strengths
For heavy nuclei can approximate them as all
constant.
Total number of p – n interactions is NpNn
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.
What is the locus of candidates for X(5)
p-n / pairing
P=
NpNn
Np + Nn
p–n

pairing
Pairing int. ~ 1.5 MeV,
p-n interactions per
pairing interaction
p-n ~ 300 keV
Hence takes ~ 5 p-n int. to compete with one pairing int.
P~5
Comparison with the data
Realistic Calculations
Microscopic Density Functional
Calculations with Skyrme forces and
different treatments of pairing
W. Nazarewicz, M. Stoitsov, W. Satula
Agreement is remarkable. Especially so since
these DFT calculations reproduce known masses
only to ~ 1 MeV – yet the double difference
embodied in dVpn 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
dVpn (DFT – Two interactions)
Experiment
DFT
DFT
82 82
82
78 78
78
hp
74 74
h
70 70
102
62 62
126
96
58 58
54 54
84 84
hh
108
Z Z66 66
350
00
74
116
92
pp
96
Z
r350
300 bdVpn < 350
250 < dVpn < 300
b250
NN
114
SLY4MIX
126126
70
66
62
58
54
84
96
N
114
SKPDMIX
126
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
Z=36
Z=38
Z=40
Z=42
Z=44
Z=46
3,2
3,0
2,8
2,4
3,0
2,8
2,6
R4/2
R4/2
2,6
2,4
2,2
2,2
2,0
2,0
1,8
1,8
1,6
1,6
52
54
56
58
60
Neutron Number
62
64
66
N=52
N=54
N=56
N=58
N=60
N=62
N=64
N=66
3,2
36
38
40
42
Proton Number
44
46
One more intriguing thing
Two regions of
parabolic
anomalies.
Two regions of
octupole
correlations
Possible
signature?
Agreement is remarkable (within 10’s of keV). Yet these
DFT calculations reproduce known masses only to ~ 1
MeV. How is this possible? dVpn focuses on sensitive
aspects of the wave functions that reflect specific
correlations. It is designed to be insensitive to others.
Contours of constant R4/2
O(6)
 - soft
2.7
2.9
2.5
2.2
U(5)
Vibrator
NB = 10
3.1
3.3
SU(3)
Rotor
2nd order
E(5)
Def.
1st order
Sph.
X(5)
Axially symmetric