Post on 21-Jan-2016
Quantum Phase Transitions (QPT)in Finite Nuclei
R. F. Casten
June 21, 2010, CERN/ISOLDE
Themes and challenges 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
Degrees of freedom: nucleon coordinatesDescription: nucleon orbits, interactions
Degrees of freedom: nuclear shape variables, , bgDescription: shapes, symmetries, quantum numbers of the many-body system as a whole
Quantum (equilibrium) phase transitions in the shapes of strongly interacting finite nuclei as a function of neutron
and proton number ord
er
para
mete
r
control parametercritical point
Broad perspective on structural evolution
56 58 60 62 64 66 68 701.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
N=84 N=86 N=88 N=90 N=92 N=94 N=96R
4/2
Z84 86 88 90 92 94 96
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
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 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
Microscopic mechanism of first order phase transition (Federman-Pittel, Heyde)
Monopole shift of proton s.p.e. as function of neutron number
Gap obliteration
2-space 1-space
(N ~ 90 )
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 shapes of nuclei with N, Z
For nuclear shape phase transitions the control parameter is nucleon number
Potential as function of the ellipsoidal deformation of the nucleus
E
β
1 2
3
4
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
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(2n
) MeV
E
β
1 2
3
4
Nuclear Shape Evolution b - nuclear ellipsoidal deformation ( =0b is spherical)
Vibrational Region Transitional Region Rotational Region
)(V
)(V
)(V
nEn )1(~ JJEJCritical Point
Few valence nucleons Many valence Nucleons
R4/2= 3.33R4/2= ~2.0
Nuclear Shape Evolution b - nuclear ellipsoidal deformation ( =0b 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 SymmetriesFirst 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
2 3 5
Comparison of relative energies with X(5)
Based on idea of Mark Caprio
Li et al, 2009
Flat potentials in b validated by microscopic calculations
Minimum in energy of first excited 0+ state
Li et al, 2009
E E
β
1 2
3
4
Other signatures
Isotope shifts
Li et al, 2009
Charlwood et al, 2009
Treating QPT with the IBA
H = c [
ζ ( 1 – ζ ) nd
4NB
Qχ ·Qχ - ]
ζ
χ
U(5)0+
2+ 0+
2+
4+
0
2.01
ζ = 0
O(6)
0+
2+
0+
2+
4+
0
2.51
ζ = 1, χ = 0
SU(3)
2γ+
0+
2+
4+ 3.33
10+ 0
ζ = 1, χ = -1.32
E(5)
X(5)
1st order
2nd order
Axially symmetric
Axi
ally
asy
mm
etri
c
Sph.
Def.
Order parameters
Li et al., 2009 Bonatsos et al
Q-invariants: model independent shape determinations
Good to better than 95% for k =3
~
Cartoon interpretation of “crossing” of b values across a transitional region
Werner et al, 2009
Li et al, 2009
E0 transitions in the IBA and phase Transitional Regions
• For 35 years, it was thought that E0 transitions to the ground state should peak in shape/phase transition regions because of the radius change (E0s had long been associated with changes in radii). It was thought that they should be small elsewhere.
• The IBA predicts something completely different: E0 transitions should be small for spherical nuclei, should grow rapidly in the shape/phase transition region, and REMAIN large throughout deformed nuclei.
• This is a robust prediction of the IBA so it is crucial to test it with measurements of E0 transitions to the g.s. in deformed nuclei.
• Very difficult experimentally.
E0 transitions in IBA.
Brentano et al, 2004
Li et al, 2009
Delaroche et al, 2009
Traditional model, for 35
years
Crit. Pt.
E0
0+i 0+
0
E0 transitions: 0+i 0+
0 Data and IBA calculations
0+
0+
2+
2+
If inertial properties of ground and excited sequences are very
similar, as is likely, it is very difficult to isolate the 0+ 0+
transition.
Wimmer et al, Priv. Comm., 2009
Enhanced density of 0+ states at the critical point
Meyer et al, 2006
E
β
1 2
3
4
Where else? Look at other N=90 nulei
= NpNn p – n
PNp + Nn pairing
What is the locus of candidates for X(5)
p-n / pairing
P ~ 5
Pairing int. ~ 1 MeV, p-n ~ 200 keV
p-n interactions perpairing interaction
Hence takes ~ 5 p-n int. to compete with one pairing int.
Comparison with the data
Summary
• QPT• Geometrical and IBA treatments• Microscopic calculations
• Principle collaborators:•
Victor Zamfir, E. A. McCutchan, Deseree Meyer, Jan Jolie, Peter von Brentano, Mark Caprio, Dennis Bonatsos, Volker Werner