1. EVEN-ODD EFFECT IN THE YIELDS OF NUCLEAR-REACTION PRODUCTS
M. Valentina Ricciardi
GSI, Darmstadt, Germany

2. EVEN-ODD EFFECT IN THE YIELDS OF 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)

3. 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.
Il Gattopardo. Il ballo
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.
Disco dancing

4. Consequences 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....

5. Even-odd effect in the yields: What we are speaking about
We make a nuclear reaction and we observe the final products
effect structure staggering zigzag
Even-odd
Without pairing With pairing

6. Experimental evidence of even-odd staggering in the yields
Steinhäuser et al., Nuc. Phys.A 634 (1998) 89
Sl. Cavallaro et al., Phys. Rev. C 57 (1998) 731
low-energy fission
35Cl + 24Mg 8 A MeV
Ch. O. Bacri et al., Nucl. Phys. A 555 (1998) 477
C. N. Knott et al., Phys. Rev. C 53 (1996) 347
40Ar + target 44 A MeV
40Ca + p

7. What do yields have in common in all these different reactions
What do yields have in common in all these different reactions? What is different?
In common: an even-odd staggering is observed
Independent of the reaction mechanism PAIRING is responsible for the even-odd effect
Different: the characteristics of the staggering can be different
the PHASE through which the nucleus is passing
SUPERFLUID
The nucleus warms up a bit and cools down without fully exiting the superfluid phase
LIQUID
The nucleus warms up, enters the liquid phase, then it cools down by evaporation back into the superfluid phase
COEXISTENCE LIQUID-GAS
The nucleus warms up, experiences a co-existence phase, then the liquid pieces cool down by evaporation back into the superfluid phase
superfluid
liquid
coexistence
gas
E*/MeV 5 70 10
300
A25
0.5
T/MeV

8. SUPERFLUIDITY Conclusion: Structural properties survive at low energy
EVEN-ODD STRUCTURE IN LOW-ENERGY FISSION
Zfiss=90
Zfiss=89
Results from e.m.-induced fission of 70 different secondary projectiles (Steinhäuser et al., Nuc. Phys.A 634 (1998) 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) )

9. LIQUID PHASE Keep in mind!
de-excitation by fission / evaporation
cold fragment superfluid
compound nucleus
collision
E.g.: transfer, abrasion
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".

10. Nowadays we can tell much more...
Evaporation residues: Which are the characteristics of this type of final yields?
C. N. Knott et al., Phys. Rev. C 53 (1996) 347
Ch. O. Bacri et al., Nucl. Phys. A 555 (1998) 477
40Ca + p
Yields of nuclei with even Z are enhanced
Yields of nuclides with even N=Z number are enhanced
Nowadays we can tell much more...

11. A powerful overview: GSI data
Use of high-resolution magnetic spectrometers to measure production yields:
- full Z, A identification entire production range very precise measurement
Experimental set-up at the FRagment Separator (FRS), GSI
Z from IC:
A/Z from time and position:

12. A powerful overview: GSI data
Fragmentation data:
1 A GeV 238U on Ti
Sequence:
N-Z=constant

13. Experimental results for 238U fragmentation
● N=Z ■ N=Z+2 ▲N=Z+4 N=Z+6
● N=Z+1 ■ N=Z+3 ▲N=Z+5
Data reveal complex structural effects!
M. V. Ricciardi et al., Nucl. Phys. A 733 (2003) 299

14. Same complex behavior observed in a large bulk of new data.
C. Villagrasa-Canton et al., Phys. Rev. C 75 (2007)
P. Napolitani et al., Phys. Rev. C 70 (2004)
Same complex behavior observed in a large bulk of new data.

15. Can we explain this complex behavior?
We have this picture in mind:
evaporation
cold fragment with structure
compound nucleus
collision
E.g.: transfer abrasion
without structure
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

16. How does particle evaporation proceeds?
The dominant particle-decays in the last evaporation step are n and p emission

17. Understanding the staggering in the yields
Last step in the evaporation cascade (assuming only n and p evaporation)
o.o. o.e o.e. /e.o. o.o. /e.e e.e e.o.
The lowest particle separation energy is the key quantity!

18. The key role of the separation energy
"Energy range" = "Particle threshold" = min(Sn, Sp)
keV
data from Audi-Wapstra

19. The key role of the separation energy
"Energy range" = "Particle threshold" = min(Sn, Sp)
data from Audi-Wapstra
The complex features of the even-odd staggering are reproduced in this fishbone pattern!

20. Conclusions concerning the yields from nuclei in 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.

21. 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

22. Yield ~ e-Ei/kT Temperature: a very important variable
Heat bath at temperature T
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 ground state has highest probability to be populated
Yield ~ e-Ei/kT
The ratio of the probabilities of two states is given by the ratio of their Boltzmann factors.
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).
light fragments investigated

23. Even-odd effect in the yields of multifragmentation products
56Fe on Ti at 1000 A MeV P. Napolitani et al., Phys. Rev. C 70 (2004)
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

24. 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
cross sections
cross sections
binding energies
binding energies
N=Z
N=Z+1 ?
Production cross sections (mb) 56Fe on Ti at 1 A GeV
Staggering in binding energy (MeV) (BEexp from Audi Wapstra – BEcalc from pure LDM Myers, Swiatecky)

25. Overview on the staggering in the binding energy
Extra binding energy associated with the presence of congruent pairs:
most bound
less bound
e o e o e o e o
0 ½ 0 ½ 0 ½ 0 ½ 0 ½
½ 1 ½ 1 ½ 1 ½ 2 ½ 1
½ 1 ½ 1 ½ 2 ½ 1 ½ 1
½ 1 ½ 2 ½ 1 ½ 1 ½ 1
½ 2 ½ 1 ½ 1 ½ 1 ½ 1
N=Z N=Z+1
staggering in the ground-state energies
(Myers Swiatecki NPA 601, 1996, 141)
It is not the binding energy responsible for the staggering in the cross sections
e o e o e o e o e o

26. Expected even-odd staggering in evaporation residues
FISHBONE PATTERN
"Energy range" = min(Sn, Sp)
data from Audi-Wapstra

27. Staggering in yields vs. min(Sn,Sp)
N=Z
N=Z+1
cross sections
cross sections
particle threshold
particle threshold
binding energies
binding energies
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!

28. Conclusions concerning the yields from nuclei in 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

29. 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"

30. A complete statistical evaporation code: ABRABLA07

31. A complete statistical evaporation code: ABRABLA07

32. CONCLUSIONS
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.