EVEN-ODD EFFECT IN THE YIELDS EVEN-ODD EFFECT IN THE YIELDS

OF NUCLEAR-REACTION PRODUCTSOF NUCLEAR-REACTION PRODUCTS

M. Valentina Ricciardi

GSI, Darmstadt, Germany

EVEN-ODD EFFECT IN THE YIELDS EVEN-ODD EFFECT IN THE YIELDS

OF NUCLEAR-REACTION PRODUCTSOF NUCLEAR-REACTION PRODUCTS

OUTLINE

Experimental evidence of even-odd staggering in the yields: common properties and differences

GSI results: a powerful overview

Understanding the even-odd staggering

Implications for the study of properties of hot nuclear matter

(Main consequences of our simple idea)

What is pairing?What is pairing?

The pairing interaction can be defined as an attractive (thus binding) residual interaction acting between two nucleons.

By pairing nucleons, the nuclear binding energy is increased compared to its value predicted by the independent-particle model.

Even-even nuclei have all nucleons paired and gain binding energy. Odd nuclei have always one unpaired nucleon.

The ground state (cold nucleus) corresponds to the most possible coherent motion of nucleons.

If I want to bring two paired nucleons in an excited state, I have to pay energy to break the pair.

The excited states correspond to configurations of the (hot) nucleus where a less coherent motion is achieved.

Il Gattopardo. Il ballo

Disco dancing

Consequences of pairingConsequences of pairing

The most evident consequence of pairing can be found in the nuclear binding energy.

Another rather evident (but not obvious) consequence of pairing is the even-odd effect in the yields of nuclear reactions....

Even-odd effect in the yields: What we are speaking Even-odd effect in the yields: What we are speaking aboutabout

We make a nuclear reaction and we observe the final productseffect structure staggering zigzagWithout pairing With pairing

Even-odd

Experimental evidence of even-odd staggering in the Experimental evidence of even-odd staggering in the yieldsyields

C. N. Knott et al., Phys. Rev. C 53 (1996) 347

40Ca + p

40Ar + target 44 A MeV

Ch. O. Bacri et al., Nucl. Phys. A 555 (1998) 477

Steinhäuser et al., Nuc. Phys.A 634 (1998) 89

low-energy fission

Sl. Cavallaro et al., Phys. Rev. C 57 (1998) 731

35Cl + 24Mg 8 A MeV

What do yields have in common in all these different reactions? What is different?

In common: an even-odd staggering is observedIndependent of the reaction mechanism PAIRING is responsible for the even-odd effect

Different: the characteristics of the staggering can be differentthe PHASE through which the nucleus is passing

superfluid

liquid

coexistence

gas

E*/MeV

5

7010 300

A25

0.5

T/M

eV

SUPERFLUIDThe nucleus warms up a bit and cools down without fully exiting the superfluid phase

LIQUIDThe nucleus warms up, enters the liquid phase, then it cools down by evaporation back into the superfluid phase

COEXISTENCE LIQUID-GASThe nucleus warms up, experiences a co-existence phase, then the liquid pieces cool down by evaporation back into the superfluid phase

EVEN-ODD STRUCTURE IN LOW-ENERGY FISSION

Results from e.m.-induced fission of 70 different secondary projectiles (Steinhäuser et al., Nuc. Phys.A 634 (1998) 89 K. H. Schmidt et al., Nucl. Phys. A 665 (2000) 221 )

Conclusion: Structural properties survive at low energy

(F. Rejmund et al., Nucl. Phys. A 678 (2000) 215-234)

SUPERFLUIDITY

Zfiss=90 Zfiss=89

de-excitation by fission / evaporation

cold fragment superfluid

compound nucleus

collision

E.g.: transfer, abrasion

LIQUID PHASE

we describe successfully some aspects of the nucleus as a liquid drop

Keep in mind!

Final nuclei observed in a great variety of nuclear reactions have experienced a de-excitation process (evaporation)!

Nuclei pass from the liquid to the superfluid state.

Nuclei pass from being "structureless" to "structurefull".

Evaporation residues: Which Evaporation residues: Which are the characteristics of this type of final yields?are the characteristics of this type of final yields?

C. N. Knott et al., Phys. Rev. C 53 (1996) 347

40Ca + p

Ch. O. Bacri et al., Nucl. Phys. A 555 (1998) 477

Yields of nuclei with even Z are enhanced

Yields of nuclides with even N=Z number are enhanced

Nowadays we can tell much more...

x2, x4 Bt2, t4 velocity

flight path

x2, t2

x4, t4

beam monitor

target

scintillator

scintillator

ionisationchamber

ionisationchamber

beam

2ZΔE

A/Z from time and position:

Z from IC:

cβBρ

m

e

Z

A

0

Experimental set-up at the FRagment Separator (FRS), GSI

A powerful overview: GSI dataA powerful overview: GSI data

Use of high-resolution magnetic spectrometers to measure production yields:

- full Z, A identification - entire production range - very precise measurement

Sequence:N-Z=constant

A powerful overview: GSI dataA powerful overview: GSI data

Fragmentation data:

1 A GeV 238U on Ti

● N=Z ■ N=Z+2 ▲N=Z+4 N=Z+6 Data reveal complex structural effects!

M. V. Ricciardi et al., Nucl. Phys. A 733 (2003) 299

● N=Z+1 ■ N=Z+3 ▲N=Z+5

Experimental results for Experimental results for 238238U fragmentationU fragmentation

C. Villagrasa-Canton et al., Phys. Rev. C 75 (2007)

044603

P. Napolitani et al., Phys. Rev. C 70 (2004)

054607

Same complex behavior observed in a large bulk

of new data.

Can we explain this complex behavior?Can we explain this complex behavior?

evaporation

cold fragment with structure

compound nucleus

collision

E.g.: transfer abrasion

without structure

We have this picture in mind:

The compound nucleus is hot and structureless

In each evaporation step the mass and excitation energy are reduced. The new compound nucleus is still structureless.

Only below Ecritical (~10 MeV) structural effect can exist

Below 10 MeV of excitation energy particle evaporation normally stops

we test the idea that pairing is restored in the last evaporation step, i.e. the even-odd effect in the yields is determined in the last evaporation step

How does particle evaporation proceeds?How does particle evaporation proceeds?

The dominant particle-decays in the last

evaporation step are n and p emission

o.o. o.e. e.e. e.o.

Svmother

Svmother

ground state

ground state

ground state

ground state

daughter mother daughter mother

Svdaughter Sv

daughter

Sv

2

o.o. o.e. e.e. e.o.

Svmother

Svmother

ground state

ground state

ground state

ground state

daughter mother daughter mother

Svdaughter Sv

daughter

Sv

2

o.o. o.e. e.e. e.o.

Svmother

Svmother

ground state

ground state

ground state

ground state

daughter mother daughter mother

Svdaughter Sv

daughter

Sv

2

o.o. o.e. e.e. e.o.

Svmother

Svmother

ground state

ground state

ground state

ground state

daughter mother daughter mother

Svdaughter Sv

daughter

Sv

2

o.o. o.e. o.e. /e.o. o.o. /e.e e.e. e.o. o.o. o.e. e.e. e.o.

Svmother

Svmother

ground state

ground state

ground state

ground state

daughter mother daughter mother

Svdaughter Sv

daughter

Sv

2

o.o. o.e. e.e. e.o.

Svmother

Svmother

ground state

ground state

ground state

ground state

daughter mother daughter mother

Svdaughter Sv

daughter

Sv

2

Understanding the staggering in the yieldsUnderstanding the staggering in the yields

Last step in the evaporation cascade (assuming only n and p evaporation)

The lowest particle separation energy is the key quantity!

The key role of the separation energyThe key role of the separation energy

"Energy range" = "Particle threshold" = min(Sn, Sp)

data from Audi-Wapstra

keV

The key role of the separation energyThe key role of the separation energy

The complex features of the even-odd staggering are reproduced in this fishbone pattern!

data from Audi-Wapstra

"Energy range" = "Particle threshold" = min(Sn, Sp)

Conclusions concerning the yields from nuclei Conclusions concerning the yields from nuclei in thein the LIQUID PHASE

The even-odd staggering of final nuclei which have experienced a de-excitation process (evaporation) is determined in the last evaporation step

in other words:

Structural properties do not survive in excited nuclei, but the pairing interaction is restored once the nucleus falls below the critical energy

The characteristics of the staggering correlate strongly with the lowest n p particle separation energy of the final experimentally observed nuclei.

Coexistence liquid-gas

Multifragmentation establishing the caloric curve

Heat bath at temperature T

Excitation energy: measured from the masses of observed final

fragments

Temperature: measured from the ratio of the yields of observed final

fragments

Temperature: a very important variableTemperature: a very important variable

Yield ~ e-Ei/kT

Heat bath at temperature T

light fragments

investigated

The temperature is determined by the Boltzmann factor.

The Boltzmann factor is a weighting factor that determines the relative probability of a state i in a multi-state system in thermodynamic equilibrium at temperature T.

The ratio of the probabilities of two states is given by the ratio of their Boltzmann factors.

kT

E

kT

EE

kT

E

kT

E

1

2

1

2 ee

e

e

Y

Y

p

p 12

2

1

Under the assumption that only (mostly) the ground state is populated, T can be deduced from the yields and the binding energies of two final fragments (ISOTOPE THERMOMETER).

Using very light final fragments for the thermometer, one hopes that close heavier fragments are also cold and do not evaporate(no SIDE FEEDING).

the ground state has highest probability to be populated

Even-odd effect in the yields of multifragmentation Even-odd effect in the yields of multifragmentation productsproducts

56Fe on Ti at 1000 A MeV P. Napolitani et al., Phys. Rev. C 70 (2004) 054607

We focus now on the very light nuclei, undoubtly attributed to multifragmentation.

Let us consider these 2 chains:

N=Z even-odd effect N=Z+1 odd-even effect

Following the footprints of the data...Following the footprints of the data...

Light multifragmentation products: Yield ~ e-E/T

Let's assume that evaporation does not play any role the staggering in the yields should be correlated to that in binding energies

N=Z

Staggering in binding energy (MeV) (BEexp from Audi Wapstra – BEcalc from pure LDM Myers, Swiatecky)

N=Z+1 ?

Production cross sections (mb) 56Fe on Ti at 1 A GeV

cross sections

binding energies binding energies

cross sections

Overview on the staggering in the binding energy

0 ½ 0 ½ 0 ½ 0 ½ 0 ½ ½ 1 ½ 1 ½ 1 ½ 2 ½ 10 ½ 0 ½ 0 ½ 0 ½ 0 ½ ½ 1 ½ 1 ½ 2 ½ 1 ½ 10 ½ 0 ½ 0 ½ 0 ½ 0 ½ ½ 1 ½ 2 ½ 1 ½ 1 ½ 10 ½ 0 ½ 0 ½ 0 ½ 0 ½ ½ 2 ½ 1 ½ 1 ½ 1 ½ 10 ½ 0 ½ 0 ½ 0 ½ 0 ½

oddZNfor2

oddoddfor1

evenoddfor2

1evenevenfor0

withAA

ZN

2

3

Extra binding energy associated with the presence of congruent pairs:

most bound

less bound

staggering in the ground-state energies

(Myers Swiatecki NPA 601, 1996, 141)

N=Z N=Z+1

e o e o e o e o e o

e

o

e

o

e

o

e

o

It is not the binding energy responsible for the staggering in the

cross sections

Expected even-odd staggering in evaporation residuesExpected even-odd staggering in evaporation residues

FISHBONE PATTERN

"Energy range" = min(Sn, Sp)

data from Audi-Wapstra

Staggering in yields vs. Staggering in yields vs. min(Sn,Sp)min(Sn,Sp)

Production cross sections (mb)Staggering in binding energy (MeV)Particle threshold = lowest particle separation energy (MeV)

The lowest particle separation energy reproduces perfectly the staggering

The even-odd staggering in the yields of evaporation residues does not correlate with the binding energies but with the lowest particle threshold!

the sequential de-excitation process plays a dominant role!

N=Z N=Z+1

cross sections cross sectionsparticle threshold

particle threshold

binding energiesbinding energies

Conclusions concerning the yields from nuclei Conclusions concerning the yields from nuclei in thein the COEXISTENCE PHASE

The even-odd staggering of final nuclei does not correlate with the binding energy

It is not the binding energy (pure Boltzmann approach) that is responsible for the staggering in the yields of multifragmentation process but the lowest particle separation energy

Even the yields of the lightest multifragmentation products (e.g. Li) are governed by evaporation (side feeding is relevant!)

Warning to all methods based on Boltzmann statistics when determining directly the properties of hot nuclear matter

SUMMARISING THE SIMPLE IDEA OF "PARTICLE THRESHOLD"SUMMARISING THE SIMPLE IDEA OF "PARTICLE THRESHOLD"

• All residual products (yields) from any reaction which passed through at least one evaporation step show an even-odd staggering

• The even-odd effect is complex even qualitatively

• The complex behavior of the even-odd staggering can be reproduced qualitatively by the lowest separation energy (threshold energy) but not by the binding energy

J. Hüfner, C. Sander and G. Wolschin, Phys. Let. 73 B (1978) 289.X. Campi and J. Hüfner, Phys. Rev. C 24 (1981) 2199.

Consequences of this simple idea...

1. consequence: a statistical evaporation code with correct ingredients (masses, level densities, competition of all possible channels –gamma emission-) should be able to reproduce the even-odd staggering

2. consequence: all other even-odd observables (e.g. Z distributions) do not contain any additional information and can be explained by the "fishbone pattern"

A complete statistical evaporation code: ABRABLA07A complete statistical evaporation code: ABRABLA07

A complete statistical evaporation code: ABRABLA07A complete statistical evaporation code: ABRABLA07

CONCLUSIONSCONCLUSIONS

The measurements of cross-sections of all nuclides (A,Z) in the entire production range made at FRS, GSI, offered the possibility for the 1st time to observe and systematically investigate the complexity of the even-odd effect in the yields

(Structural properties survive at low energies )

In nuclei which are the final products of an evaporation process, the even-odd structure is restored in the last evaporation step

It is not the binding energy that is responsible for the staggering in the yields; the characteristics of the staggering correlate strongly with the lowest n p particle separation energy of the final experimentally observed nuclei.

Even the yields of the lightest multifragmentation products (e.g. Li) are governed by evaporation (model independent!). Warning to all methods based on a pure Boltzmann approach when determining directly (neglecting evaporation) the properties of hot nuclear matter

A statistical model can reproduce (in principle) the complexity of the even-odd effect.

More complex effects (eg. anomaly) are in reality just a consequence of the observed characteristics of the even-odd effect in the individual yields.