Effects of diabaticity on fusion of heavy nuclei in the dinuclear model ...

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Chapter 2 The study <strong>of</strong> nucleus-nucleus collisions at energies <strong>of</strong> some few MeV/u above the interaction barrier stimulated the development <strong>of</strong> theoretical concepts for large-amplitude collective nuclear motion [44]. We note in particular the introduction <strong>of</strong> transport theories like the random- matrix approach [45], the one-body dissipation model [46, 47] and the linear response theory [48]. These theories assume (either explicitly or implicitly) a statistical equilibrium within the intrinsic degrees <strong>of</strong> freedom throughout the collision described by a few collective variables. Such a local equilibrium, however, is not expected to be realistic during the initial stage <strong>of</strong> a nucleus-nucleus collision, a process which starts from the approach <strong>of</strong> two nuclei in their ground states. Because <strong>of</strong> the long mean free path, the motion <strong>of</strong> the individual nucleons during the approach phase is governed by their self-consistent mean potential which evolves in time. Therefore, this stage should be well described by the time-dependent Hartree-Fock (TDHF) theory [49]. <strong>Effects</strong> from residual two-body collisions and the corresponding transition to local equilibrium were formulated in extensions <strong>of</strong> TDHF [50]. An alternative model was introduced in [51] by the concept <strong>of</strong> dissipative diabatic dynamics (DDD) as a time-dependent shell-model approach. This model takes into account simultaneously the coherent and incoherent forms <strong>of</strong> motion <strong>of</strong> the nucleons in nucleus-nucleus collisions. For sufficiently large collective velocities (typically the collective kinetic energy per nucleon needs to be larger than 0.03 MeV) the nucleons no longer adjust their wave functions adiabatically but diabatically, thereby preserving the nodal structure (character) <strong>of</strong> the wave functions. Adiabatic basis states are not always useful [52]. It was realized by Landau, Stueckelberg and Zener [53] that it is advantageous at 17
higher collective velocities to replace the adiabatic basis states at pseudo-crossings by diabatic states [54]. Diabatic states belong to crossing levels. This situation is illustrated in Fig. 2-1. 2-1: Diabatic and adiabatic levels at the Landau-Zener-Stückelberg (quasi-)crossing. In Figure diabatic motion the nucleon keeps the nodal structure (character) <strong>of</strong> the wave function. the Only for very small collective velocities, the nucleons are able to adjust their wave functions adiabatically and follow the adiabatic levels. At higher collective velocities, a nucleon occupying the lower level before the crossing with an unoccupied level will stay (while preserving the character <strong>of</strong> its wave function) on the diabatic level and finds itself on the upper level after the crossing. Quantitatively, this jump probability, which measures the degree <strong>of</strong> <strong>diabaticity</strong>, is given by [53] where = exp(− J 1 ), (2.1) ∆ | ∂(ɛ1 − ɛ2)/∂q |·| ∆=¯h · /2π | H q| ′ 12 |2 . (2.2) The quantity ∆ is proportional to the difference in the slopes <strong>of</strong> the diabatic levels and to 18
Page 1 and 2: Effects of
Page 3 and 4: 1 Introduction 6 1.1 The DNS-concep
Page 5 and 6: Chapter 1 The elements existing in
Page 7 and 8: observed the rapid fall-of<
Page 9 and 10: Figure 1-1 illustrates the compound
Page 11 and 12: smaller than the extra-extra push e
Page 13 and 14: fusion and quasi-fission processes
Page 15: In the present chapter we have pres
Page 19 and 20: 2.1 General considerations We want
Page 21 and 22: couplings. Diabatic basis states ψ
Page 23 and 24: Chapter 3 The synthesis of<
Page 25 and 26: is calculated within the TCSM using
Page 27 and 28: E adiab (MeV) n E diab (MeV) n 30 2
Page 29 and 30: E diab (MeV) n 60 55 50 45 40 100Mo
Page 31 and 32: 3-5: Diabatic potentials for the sy
Page 33 and 34: 3-7: The same as in Fig. 3-6 for th
Page 35 and 36: The estimated collective velocities
Page 37 and 38: 3-10: Diabatic contributions as a f
Page 39 and 40: effects give a justification for th
Page 41 and 42: The similarity of
Page 43 and 44: 4.1.3 Alternative microscopical met
Page 45 and 46: to the DNS concept [18, 34], cross
Page 47 and 48: U (MeV) 28 27 26 25 24 23 22 21 20
Page 49 and 50: U (MeV) 35 30 25 20 15 10 90Zr + 12
Page 51 and 52: 4-1: Fusion barriers B Table TCSM
Page 53 and 54: U (MeV) 40 30 20 10 20 10 0 a) b) 9
Page 55 and 56: experimental separation energy allo
Page 57 and 58: means a neglect of
Page 59 and 60: the case of an ind
Page 61 and 62: single particle density matrix exte
Page 63 and 64: following ratio in the limit <stron
Page 65 and 66: 5-1: Dependence of
Page 67 and 68:
the microscopical mass parameter <s
Page 69 and 70:
5-3: Mass parameter Mεε as a func
Page 71 and 72:
5-4: Mass parameter Mεε as a func
Page 73 and 74:
the approximate expressions (Eqs. (
Page 75 and 76:
(10 3 m fm 2) M , M λλ εε 25 20
Page 77 and 78:
(10 3 m fm 2) M εε M (10 λλ 4 m
Page 79 and 80:
~ f k (MeV -1) 0,20 0,15 0,10 0,05
Page 81 and 82:
V (MeV) 30 25 20 15 10 5 0 -5 -10 1
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to the dependence of</stron
Page 85 and 86:
where the lifetime t 0 of</
Page 87 and 88:
6-2: a) Time-dependent dynamical po
Page 89 and 90:
Table 6-1: Quasi-fission and inner
Page 91 and 92:
6-3: Time-dependent average width <
Page 93 and 94:
Chapter 7 In the quasi-fission proc
Page 95 and 96:
The value of t0 is
Page 97 and 98:
Y Z Y A 0,10 0,08 0,06 0,04 0,02 0,
Page 99 and 100:
estimated from the difference betwe
Page 101 and 102:
elements from Z=104 to Z=112 in the
Page 103 and 104:
fission process is an important fac
Page 105 and 106:
the neck coordinate ε and
Page 107 and 108:
• The time-dependent transition b
Page 109 and 110:
Appendix A For describing nucleus-n
Page 111 and 112:
where The volume is V0 = 1 2 m0ϖ 2
Page 113 and 114:
ϕ nz(z) = ⎧ ⎪⎨ ⎪⎩ C−1
Page 115 and 116:
Appendix B The collective response
Page 117 and 118:
[1] S.G.Nilsson et al., Phys. Lett.
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[27] N.V.Antonenko et al., in 1 st
Page 121 and 122:
[63] Yu.F.Smirnov and Yu.M.Tchuvil
Page 123 and 124:
[97] H. Hofmann et
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