R. F. Casten WNSL, Yale KERNZ08, Dec. 2, 2008
description
Transcript of R. F. Casten WNSL, Yale KERNZ08, Dec. 2, 2008
(An outgrowth of our studies of shape/phase transitions and empirical signatures for them)
A) An enhanced link between nuclear masses and structure
B) 2-nucleon transfer reactions: a measure of structural change
R. F. Casten
WNSL, Yale
KERNZ08, Dec. 2, 2008
Collectivity, Masses, and Transfer Reactions
Two-neutron separation energies
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
S(2
n)
MeV
Neutron Number
Normal behavior: ~ linear segments with drops after closed shellsDiscontinuities at first order phase transitionsS2n = A + BN + S2n (Coll.) Curvature in isotopic chains – collecitve effects: deviations from linearity are a few hundred keV
Use a collective model to calculate the collective contributions to S2n. We will use the IBA.
Binding Energies
The IBA – a flexible, parameter-efficient phenomenological collective model
• Enormous truncation of the Shell Model: valence nucleons in
pairs coupled to L = 0 (s bosons) and L = 2 (d bosons), simple interactions
• Three dynamical symmetries, intermediate structures
• Two parameters (except scale)
• Symmetry triangle
Sph.
Def.
Shape/phase trans.
The IBA: convenient model that spans the entire The IBA: convenient model that spans the entire triangle of colllective structurestriangle of colllective structures
H = ε nd - Q Q Parameters: , (within Q)/ε
Sph. Driving Def. Driving
H = c [
ζ ( 1 – ζ ) nd 4NB Qχ ·Qχ - ]
Competition: : 0 to infinity /ε
Span triangle with and
Parameters already known for many nuclei
c is an overall scale factor giving the overall energy scale. Normally, it is fit to the first 2+ state.
Use the IBA to calculate the collective component of the binding energy
• The same interactions in the IBA that give excitation spectra and E2 transition rates also give the collective component of the binding energy – that is, those interactions depress the ground state due to s-d mixing which gives added binding compared to the vibrator [U(5)] limit.
• We will use the IBA in exactly the same way binding energies have been calculated with it numerous times since its inception (see, e.g., Scholten et al., 1978) except we will focus on its sensitivity to the parameters and the structure.
Which 0+ level is collective and which is a 2-quasi-particle state?
So, can do alternate collective model fits, assuming one or the other state is the collective one. Look at implications for masses.
How much will the calculated collective components of the binding energies change for two fits – to the 0+ states at 1222 and 1422 keV?
Evolution of level energies in rare earth nuclei:Fit levels, B(E2) values, then calculate BE’sBut note:
McCutchan et al
H = c [ ζ ( 1 – ζ ) nd -
4NB
Qχ ·Qχ ]
Now, lets look at the calculated
collective components of
binding energies (S2n(Coll.) values)
with these different sets of
parameters chosen to fit
different excited states.
McCutchan et al
!!!
Note: These two levels differ by 200 keV
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
S(2
n)
MeV
Neutron Number
These collective binding energies added to A + BN cannot both be
consistent with empirical S2n trends
Valence nucleonnumber
Effects are largest for large numbers of
valence nucleons and for well-deformed nuclei. Previous studies (e.g., Scholten et al., others) were in regimes where the effects were 5-10
times smaller and hence did not stand out.
Conclusions These results show a link between masses and structure
that is much more sensitive than heretofore realized.
Effect is strongest in well-deformed nuclei near mid-shell
Two-Nucleon Transfer Reactions to 0+ States
Recent explosion of ultra high resolution (Munich) data, leading to discovery of three to four TIMES the number of known 0+ states in
about 20 nuclei.
Need to understand cross sections and implications for structure
A new interpretation
Empirical survey of (p,t) reaction strengths to 0 + states
(p, t)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
82 86 90 94 98 102 106 110 114 118 122 126
Neutron Number
Str
on
ge
st
0+
/GS
Pt
Os
W
Hf
Yb
Er
Dy
Gd
Sm
XO
Nearly always: cross sections to excited 0+
states are a small percentage of the
ground state cross section.
In the spherical – deformed transition
region at N = 90, excited state cross sections are comparable to those of
the ground state.
The “standard interpretation” (since ca. 1960s) of 2-nucleon transfer reactions to excited 0 + states in collective nuclei
• Most nuclei: Cross sections are small because the collective components add coherently for the ground state but cancel for the orthogonal excited states.
• Phase transition region: Spherical and deformed states coexist and mix. Hence a reaction such as (p,t) on a deformed 156 Gd target populates both the “quasi-deformed” ground and “quasi-spherical” excited states of 154 Gd. Well-known signature of phase transitions.
Are these interpretations correct? Use IBA model to calculate two-nucleon transfer cross sections (rel. to g.s.)
IBA well-suited to this: embodies wide range of collective structures and, being based on s and d bosons, naturally
contains an appropriate transfer operator for L=0 -- s-boson
• Parameters for initial, final nuclei known so calculations are parameter-free
162Hf
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10
162Hf
164Hf
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10
164Hf
166Hf
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10
166Hf
168Hf
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10
168Hf
170Hf
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 2 3 4 5 6 7 8 9 10
170Hf
172Hf
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 2 3 4 5 6 7 8 9 10
172Hf
Look at Hf isotopes as an example: Exp – all excited state cross sections are small
Gd Isotopes: Undergo rapid shape transition at N=90. Excited state cross sections are comparable to g.s.
Sph.
Def.
Shape/phase trans. line
~ 105 calculations
Big
Small
So, the model works well and can be used to look at predictions for 2-nucleon transfer strengths
Expect:
Let’s see what we get !
Huh !!???R4/2
R4/2
Cross section ratios across triangle Ratios as function of R4/2
Monotonically grow
The QQ term mixes the s and d boson basis states, spreading the strength. The further “apart” the two nuclei are, the greater the difference in the distributions
of s, d amplitudes, hence the greater the spreading of cross section.
Example: U(5) target: ground state has (ns, nd) = (N,0).
Therefore, only amplitude that contributes to cross section, is
(ns, nd) = (N-1,0).
H = ε nd - Q Q WHY?
Thus, we find a completely unexpected result that leads to a new interpretation of
these cross sections
• The cross sections are large in the transitional region but they are far larger in other cases. There is nothing special about the phase transition region.
• Rather, the cross sections depend only on the change in structure between initial and final nuclei ! This change can be “measured” by R4/2
• Why the earlier interpretation? Look at R4/2 values.
• Can we find a case that does NOT involve a phase
transitional region. Not easy but one case exists.
Conclusions, Implications
• A single framework now accounts for both the (usual) small cross sections (since most adjacent nuclei have small R4/2 values), and for the large cross sections in regions of rapid change.
• The cross section distribution is a mixing effect but not of collective modes. Rather it is mixing at the shell model level (nucleon pairs coupled to spin 0) and therefore is general.
• Test in new nuclei by searching for large R4/2 values and doing 2-nucleon transfer in inverse kinematics.
Two nucleon transfer cross sections and structural change in nuclei
Collaborators
• Sensitivity of binding energies to structure: Burcu Cakirli (Istanbul), Ryan Winkler (Yale),
Klaus Blaum (Heidelberg), Magda Kowalska (CERN-ISOLDE)
• Two – nucleon transfer cross sections: Rod Clark (LBNL), Linus Bettermann (Koeln),
Ryan Winkler (Yale)
This work could not have been done without the prior mapping of nuclei into the triangle by McCutchan et al, Phys. Rev C 69, 064306 (2004) and
subsequent mapping papers