(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