Reduction methods for fusion and total reaction cross sections

Transcription

1 Reduction methods for fusion and total reaction cross sections L.F Canto 1,2, D. Mendes Jr 2, P.R.S. Gomes 2 and J. Lubian 2 1) Universidade Federal do Rio de Janeiro 2) Universidade Federal Fluminense

2 The fusion and reaction cross sections B l V(r) fusion direct reactions Coulex E r σ F = absorption inside the barrier σ R = σ F + all non-elastic channels

3 Some considerations about σ F 1. Depends on Z P, Z T, A P, A T through the barrier of the bare potential (more later) 2. Depends on channel coupling effects «Modifies effective barriers (enhancement of suppression) «Attenuates incident current but there is fusion through non-elastic channels «Net effects: Ø Enhancement at sub-barrier fusion (may be very strong) Ø Small effects above the barrier (few exceptions like CF of weakly bound)

4 Some considerations about σ R It is a sum of short-range (fusion), mid range (nuclear DR) and long range (coulex) In a potential scattering view, with V bare + V pol, there is absorption inside the barrier (W bare ) and outside (W pol ) Thus, σ R is not a barrier transmission problem and it is unlikely to scale as σ F

5 Reduction of σ F and σ R To study the effects of some particular nuclear property, e.g. low breakup threshold or deformation: 1. Reduce the data. That is, make transformaons on E and σ to eliminate the influence of Z P, A P, Z T,A T. 2. Compare cross secons for many different systems 3. Look for a signature of the property under invesgaon

6 Reduction methods 1. The traditional method Inspired by the classical expression for the fusion cross secon. One starts from the paral- wave expansion, (E) = k 2 X (2l + 1) Tl (E) and replacse the transmission coefficient by its classical approximaon, T l (E) = 1 for E B l = 0 for E<B l

7 Ond gets the analycal expression, apple = R 2 V B B 1 E, with V B = B l=0 (barrier for l = 0) In the tradional method, one performs the transformaons: E! " (1) = E V B and! (1) = R 2 B If the classical expression holds, the reduced cross secon becomes the universal funcon (1) (" (1) )=1 1 " (1) Therefore, the reduc-on procedure would eliminate any dependence on the poten-al (and thus on Z P, Z T, A P, A T )

8 2. The simplified traditional method Based on the tradional method and on the approximate relaons, R B = a [A 1/3 P + A 1/3 T ] and V B = b apple Z P Z T A 1/3 P + A 1/3 T Gomes et al* proposed the reducon method, " (2) = E apple A 1/3 P + A 1/3 T ; (2) = Z P Z P A 1/3 P + A 1/3 2. T If a and b are system independent, the two methods are equivalent (just changes of scales) * PRC 1, 0101 (2005)

9 3. The fusion function method Inspired by Wong s quantum mechanical fusion cross secon: apple = R 2 ~! B 2E ln 1 + exp 2 E V B. ~! The fusion funcon transformaons the are* " (3) = E V apple B 2E and (3) = ~! ~! R 2 B If Wong s expression is valid, the reduced x- secon becomes the Universal Fusion Func-on (system independent) (3) (" (3) )=1+ln[2 " (3) ] Universal (system independent) funcon, above and below V B *NPA 821, 51 (2009)

10 Checking performances: Potential scattering with standard U opt Real part: Akyüz-Winther Imaginary part: - Fusion: short-range absorption Woods-Saxon with r 0 = 1 fm, a = 0.2 fm - Reaction: absorption with longer range W(r) = 0.8 V(r) Eight systems in different mass ranges: Li + 2 Al, Li + 4 Zn, Li Bi, O Sm, O Sm, O Sm, 58 Ni + 4 Ni, 40 Ca Pb

11 Fusion (single channel calculation) Traditional method (b) (a) Li + 2 Al Li + 4 Zn Li Bi O Sm O Sm 40 Ca Pb Class. limit Good at high enough energies (ε (1) > 1.05) converge to classical universal function heavier systems faster Strong system dependence below the barrier (ε (1) < 1) Higher sub-barrier cross sections for lighter systems, why? (explanation later)

12 Simplified traditional method (b) (a) Li + 2 Al Li + 4 Zn Li Bi O Sm O Sm 40 Ca Pb Good for very similar systems ( O + 144,154 Sm on top of each other) Very poor otherwise, below and above the barrier Cross sections for lighter systems remain higher, as in the previous figure

13 Questions: 1. why methods based on the same ideas have so different performances? They would be equivalent if the normalizaon factors were proporonal. That is, if α and β are system independent: Z P Z T α and β are system dependent, mainly for light systems Then, the two methods can have very different performances

14 1. Why sub-barrier cross sections are larger for lighter systems? Wong s formula explains. Take its low energy limit: (E) = ~! 2E R2 B exp apple 2 E V B ~!, Then reduce it by the tradional method: (E)! (1) (" (1) )=C exp[ V B ] /E C = ~!/2 and = 2 ~! [1 "(1) ] Since ~! has a weak dependence on the system, For a fixed reduced energy, the cross secon decreases exponenally with V B, as observed

15 The fusion function method 10 8 (a) (b) Li + 2 Al Li + 4 Zn Li Bi O Sm O Sm 40 Ca Pb UFF Very good, above and below the barrier Slightly worse for light systems, where Wong s formula is not very accurate

16 Total reaction (single channel calculation) Traditional method Li + 2 Al Li + 4 Zn Li Bi O Sm O Sm 40 Ca Pb Class. limit Very poor, except for similar systems (the curves for O+ 144 Sm and O+ 154 Sm are nearly the same) At high energies, the curves are above the universal function and the difference is the contribution from direct processes to σ R

17 Simplified traditional method Li + 2 Al Li + 4 Zn Li Bi O Sm O Sm 40 Ca Pb Slightly better. It works well very similar systems Disadvantage: there is neither a reference value of the reduced energy (barrier), nor of the cross section (geometrical), and no reference curve.

18 Fusion (reaction) function method Li + 2 Al Li + 4 Zn Li Bi O Sm O Sm 40 Ca Pb UFF As poor as the other two methods Difference to the universal curve is the contribution from direct processes. 40 Ca+ 208 Pb is so large that is out of the plot

19 Conclusions about this theoretical study: Fusion: Traditional method: satisfactory at ε (1) >1.05, but spurious system dependence below the barrier. Not surprising since it is inspired by a classical expression Simplified traditional: much poorer, except for very similar systems Fusion function: works fine. Eliminate Z P,Z T,A P,A T dependence above and below the barrier. Total reaction: None of the method succeed in eliminating Z P,Z T,A P,A T dependence on a broad mass range They all work reasonably in comparisons of very similar systems The failure results from the fact that total reaction cannot be treated as a barrier transmission problem. Absorption at the surface and beyond plays a very important role

20 Application to data: Fusion Traditional method (a) e (1) (b) [10] O Sm [10] O Sm [11] 4 He Bi [12] He Bi [13] 9 Be + 2 Al [14] 1 F Pb [15] Li Bi (CF) [15] Li Bi (CF) [] 9 Be Pb (CF) [1] Li + 90 Zr (CF) [18] 9 Be Sm [19] 58 Ni + 4 Ni Spurious trends below V B for example: strongly enhanced σ F for 58 Ni + 4 Ni becomes one of the lowest curves (owing to factor exp (- γ V B )) Good above V B (as in single-channel study). CF of weakly bound below the universal curve (suppression)

21 Simplified traditional method (a) (b) Same problems below V B. Again, 58 Ni + 4 Ni is one of the lowest curves Also poor above V B, showing spurious trends. The highest curves are for Li Sm and 9 Be+ 144 Sm!

22 Fusion function method fusion function (a) Much better above and below V B. Weakly bound system show suppression fusion function (b) x Reasonable below V B : the strongly enhanced 58 Ni + 4 Ni and O Sm are on top. Difference among systems are channel coupling

23 Check of CC effects: Renormalize reduced cross section, multiplying by the factor σ opt /σ CC, with all inelastic channels renormalized fusion function (a) Below V B : All curves are close to the universal function. The exception is 58 Ni+ 4 Ni, because σ CC neglects transfer channels renormalized fusion function (b) x Above V B : fine! Only CF of weakly bound systems are suppressed

24 Application to data: Total reaction Traditional method O + 58 Ni_Joao (2) O Bi (3) O Pb (4) O Sm (5) O Ba () O Sn (8) O + 58 Ni (9) O + 58 Ni_Joao (10) O + 40 Ca (11) O + 2 Al (12) 12 C Bi (13) 12 C Pb (14) 11 Be + 4 Zn (1) 11 B Bi (18) 10 Be + 4 Zn (19) 9 Be Pb (20) 9 Be Sm (21) 9 Be + 4 Zn (22) 9 Be + 2 Al (23) 8 B + 58 Ni (24) Li Ba (25) Li + 2 Al M(2) Be + 58 Ni (2) Li Bi (28) Li Ba (29) Li + 58 Ni (30) Li + 2 Al (31) He Bi (32) He Pb (33) He Sn (34) He + 2 Al (35) 4 He Bi (3) 4 He Pb (3) 4 He + 19 Au (38) 4 He Sm (39) 4 He Sn (40) 4 He + 58 Sn (41) Universal Curve Hard to find signatures of the nonelastic channels. Different processes seem to give contributions of same order. Universal curve is a lower bound to σ R

25 Simplified traditional method (a) One can not extract useful information from this plot. Systems with very different properties are mixed together Reaction function method (b) The situation is also bad. The only interesting point is that the universal curve is a lower bound for σ R. The difference between the data and this curve is the contribution from direct reactions.

26 No signature for specific processes (weakly bound syst. are not special) However. Strong Coulex clearly emerge Li Pb Cubero et al., PRL 109, 2201 (2012) 18 O W Thorn et al., PRL 38, 384 (19)

27 Summary We have investigated the main reduction methods, applying them to theoretical single-channel cross sections and to a large set of experimental fusion and total reaction data We concluded that: Fusion: Ø The fusion function method works very well, above and below V B Ø The traditional method is reasonable above V B, but it is fails completely below V B Ø The simplified traditional method can only be used for very similar systems Total reaction: Ø None of the methods works Ø They should not be used to compare reaction data of very different weakly bound system