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Lecture 4 Quantum Phase Transitions and the microscopic drivers of structural evolution. Quantum phase transitions and structural evolution in nuclei. Increasing valence nucleon number. E. 1. 2. 3. 4. β. Quantum phase transitions in equilibrium shapes of nuclei with N , Z.

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Presentation Transcript
slide4

Increasing valence nucleon number

E

1

2

3

4

β

Quantum phase transitions in equilibrium shapes of nuclei with N, Z

Potential as function of the ellipsoidal deformation of the nucleus

Transitional

Rotor

Vibrator

For nuclear shape phase transitions the control parameter is nucleon number

nuclear shape evolution b nuclear ellipsoidal deformation b 0 is spherical
Nuclear Shape Evolutionb - nuclear ellipsoidal deformation (b=0 is spherical)

Vibrational RegionTransitional RegionRotational Region

Critical Point

New analytical solutions, E(5) and X(5)

R4/2= ~2.0

R4/2= 3.33

Few valence nucleonsMany valence Nucleons

slide7

X(5)

E

E

1

2

3

4

β

Critical Point Symmetries

First Order Phase Transition – Phase Coexistence

Energy surface changes with valence nucleon number

Bessel equation

Iachello

potential energy surfaces of 136 134 132 ba

Shimizu et al

More neutron holes

Potential energy surfaces of 136,134,132Ba

100keV

134Ba

132Ba

136Ba

<H>

×

×

×

minimum

<HPJ=0>

×

×

×

(Nn,Np)= (-2,6) (-4,6) (-6,6)

slide15

Isotope shifts

Charlwood et al, 2009

Li et al, 2009

where else

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
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

Proton-neutron interactions

A crucial key to structural evolution

valence proton neutron interaction

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.

slide21

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

slide24

Seeing structural evolution

Different perspectives can yield different insights

Mid-sh.

magic

Onset of deformation

as a phase transition

mediated by a change in shell structure

Onset of deformation

“Crossing” and “Bubble” plots as indicators of phase transitional regions mediated by sub-shell changes

slide25

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 and Nucleonic Interactions

Masses:

  • 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

Macro

Micro

measurements of p n interaction strengths
Measurements of p-n Interaction Strengths

dVpn

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

slide29

Valence p-n interaction: Can we measure it?

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

orbit dependence of p n interactions

82

50

82

126

Orbit dependence of p-n interactions

Low j, high n

High j, low n

slide31

Z  82 , N < 126

Z  82 , N < 126

3

1

1

2

3

2

1

126

82

82

50

Z > 82 , N < 126

Z > 82 , N > 126

Low j, high n

High j, low n

slide36

126

82

LOW j, HIGH n

HIGH j, LOW n

50

82

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

Largest p-n interactions if proton and neutron shells are filling similar orbits

slide37

82

50

82

126

Empirical p-n interaction strengths indeed strongest along diagonal.

Empirical p-n interaction strengths stronger in like regions than unlike regions.

Low j, high n

High j, low n

New mass data on Xe isotopes at ISOLTRAP – ISOLDE CERN

Neidherr et al., PR C, 2009

slide38

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
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
slide40

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 xNn

general p n strengths

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.

slide42

What is the locus of candidates for X(5)

NpNn

p – n

=

P

Np + Nn pairing

p-n / pairing

p-n interactions per

pairing interaction

Pairing int. ~ 1.5 MeV, p-n ~ 300 keV

Hence takes ~ 5 p-n int. to compete with one pairing int.

P ~ 5

w nazarewicz m stoitsov w satula
W. Nazarewicz, M. Stoitsov, W. Satula

Realistic Calculations

Microscopic Density Functional Calculations with Skyrme forces and different treatments of pairing

slide45

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

slide48

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.

special thanks to
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
slide54

One more intriguing thing

Two regions of parabolic anomalies.

Two regions of octupole correlations

Possible signature?

slide55

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

2.7

2.9

2.5

3.1

3.3

2.2

R4/2

Contours of constant

NB = 10

slide58

Axially asymmetric

2nd order

E(5)

Def.

1st order

Sph.

X(5)

Axially symmetric