Consistent chiral three-nucleon interactions in ... - Theory Center

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1 Introduction of at least E3max = 14. The standard energy limit before was E3max 9 according to fig 1. As already mentioned above, solving the Schrödinger equation numerically is not straight forward. The complex nuclear interaction induces strong correlations between the nucleons that result in the need of very large model spaces in order to incorporate these correlations and to obtain converged results. Therefore, the bare interaction is not suitable to perform calculations for heavier nuclei, since the results in tractable model spaces are still not converged with respect to model space sizes. One possible way to overcome this obstacle is the use of unitarily transformed interactions. Here, we will apply the similarity renormalization group (SRG) transformation. This can be done at the level of the origi- nal fully antisymmetric three-particle Jacobi harmonic oscillator matrix elements. Then the transformation into the coupled scheme can be performed as mentioned above. Afterwards, also heavier nuclei are accessible in a reasonable way within tractable model spaces. Furthermore, we are able to distinguish between the ef- fects of induced three-particle interactions due to the SRG transformation and the impact of the genuine three-particle interaction from χEFT. Finally, many applications and studies of the impact of the three-body matrix elements, especially matrix elements in the framework of χEFT, are accessible. We show results of calculations in the importance-truncated no-core shell model (IT-NCSM), which allows for ab-initio nuclear structure studies beyond the domain of the full no-core shell model NCSM [6]. The principal idea is to reduce the model space in which the matrix eigenvalue problem has to be solved using an a priori importance measure that determines whether a basis state is included in the model space or not. The IT-NCSM yields eigenstates of the Hamilton op- erator which are the ingredient for calculations of various observables. We show results including the bare N2LO three-particle interaction for 4 He as well as the SRG-transformed three-body interaction for 4 He and 6 Li. Thereby, we can distinguish between the induced and genuine three-body force influences. Moreover, we present first results of Hartree-Fock calculations with the SRG- transformed chiral interactions including the N2LO three-body force. With this method the whole nuclear chart is accessible. As example we investigate 4 He, 16 O, 40 Ca, 48 Ca and 90 Zr. This thesis is organized as follows: In section 2 we give a short introduction into chiral effective field theory. For the rest of the thesis matrix elements of the chiral interactions are our input. We transform these matrix elements given in three-particle relative states into a form depending on Cartesian coordinates that is suitable for many-body calculations in section 4. However, to carry out this 4
transformation we need some mathematical background and tricks, presented in section 3. After the transformation of the three-body interaction matrix elements we discuss the unitary transformation of the interaction via the similarity renormalization group in section 5. An important issue in this section will be the proper separation of the three-body part of the interaction form the two-body part. In section 6 we present the results of IT-NCSM and Hartree-Fock calculations and discuss the impact of the chiral three-body forces. Finally, we give a résumé of this thesis including an outlook in section 7. In the appendices the interested reader can find some technical remarks on our implementation of the transformation presented in section 4 and the SRG transformation in appendix A. Moreover, in appendix B one can find the formula of the two-body Talmi transformation, which is the analog transformation as in section 4 but on two-body level. J. Langhammer 5
Page 1: Consistent chiral three-nucleon int
Page 4 and 5: Contents 5.2.1 Evolution in three-b
Page 6 and 7: 1 Introduction state of 10 B which
Page 11 and 12: SECTION 2 Chiral effective field th
Page 13 and 14: comparable to the high-precision Ar
Page 15 and 16: Figure 3 - Trajectories in the cD-c
Page 17 and 18: SECTION 3 Mathematical basics In th
Page 19 and 20: 3.1 Angular momentum coupling are f
Page 21 and 22: • |[j1, (j2, j3)J23]J ′ M ′
Page 23 and 24: and r2, respectively, these are def
Page 25 and 26: 3.3 Harmonic oscillator brackets (H
Page 27 and 28: in coordinate basis representation
Page 29 and 30: matrix. Moreover, we can use the re
Page 31 and 32: write down the definition 〈〈n1l
Page 33 and 34: 3.3 Harmonic oscillator brackets (H
Page 35 and 36: 3.4 Antisymmetrizer in basis repres
Page 37 and 38: 3.4 Antisymmetrizer in basis repres
Page 39 and 40: Thus the isospin matrix element is
Page 41 and 42: to arrive at the final result 〈[(
Page 43 and 44: SECTION 4 Three-body Jacobi matrix
Page 45 and 46: 4.1 Matrix elements of the three-nu
Page 51 and 52: mations. 4.2 Calculation of the T -
Page 53 and 54: ◮ (nala, nblb)Lab −→ (N12L12(
Page 55 and 56: 4.2 Calculation of the T -coefficie
Page 57 and 58: 4.2 Calculation of the T -coefficie
Page 59 and 60:
4.2 Calculation of the T -coefficie
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Page 63 and 64:
Page 65 and 66:
4.4 J , T -coupling of the m-scheme
Page 67 and 68:
plug eq. (4.72) into eq. (4.70) and
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Page 71:
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5 Similarity renormalization group
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Page 78 and 79:
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. . . 5 Similarity renormalization
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Page 86 and 87:
Page 89 and 90:
SECTION 6 Chiral three-body force i
Page 91 and 92:
6.1 Studies of the N2LO three-body
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energy [MeV] . -18 -20 -22 -24 -26
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energy [MeV] . energy [MeV] . -22 -
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6.2 Three-body interactions and the
Page 103 and 104:
E−Eexp A [MeV] . 5 2.5 0 -2.5 -5
Page 105:
6.2 Three-body interactions and the
Page 108 and 109:
7 Summary and outlook flow paramete
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APPENDICES
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A Implementation dition (−1) (l12
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A Implementation We want to briefly
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A Implementation CGC ,SixJ ,NineJ I
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A Implementation • L12 Lower Bou
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A Implementation which can be found
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B Two-body Talmi Moshinsky transfor
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Page 128 and 129:
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References [13] P. Navratil, G. P.
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References [41] N. Imai, H. J. Ong,
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ERKLÄRUNG ZUR EIGENSTÄNDIGKEIT Hi
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