中重核領域における 原子核殻模型計算 N. Shimizu Univ. of Tokyo T. Otsuka Univ. of Tokyo/CNS/RIKEN T. Mizusaki Senshu Univ. M. Honma Aizu Univ. Y. Utsuno JAEA Thanks the chairman for the introduction. We also thank organizers for giving me an opportunity to have a talk on such a wonderful conference. Title of my talk is Monte-Carlo Shell Model calculations of triaxially deformed nuclei around Ba134. I would like to discuss the quadrupole collective states around Ba134, which is known to show some features of the critical point symmetry, E(5). In this region, the shapes of the nuclei show the phase transition from spherical vibrator to triaxially deformed shape.

扱うべきハミルトニアン行列の次元の大きさ 原子核構造計算の目的 原子核 ... 有限個の陽子(Z)と中性子(N)の集合 フェルミオン多体系 有限量子多体系の記述の難しさ 励起スペクトル 遷移確率 モーメント ... I would remind you some features of the quadrupole collective motion. This triangle is called Casten's triangle including three vertexes of which correspond to the dynamical symmetries of the Hamiltonian in the IBM has. These symmetries correspond the three typical collective motions, respectively. U(5) corresponds to the spherical vibrator, SU(3) corresponds to the axially symmetric rotor, and O(6) corresponds to the triaxial deformation. We focus mainly on the O(6) symmetry, which describes the triaxial deformation, and the transitional region between these vertices. In the study of the quadrupole collective states using the intrinsic framework, as you know, we often write the potential energy surface, parameterized by beta, the degree of freedom of axially symmetric deformation, and gamma, the degree of freedom of the triaxial deformation. We can classify the nuclear shapes according to the location of the energy minimum. The minimum of U(5) nuclei is expected to be located around the place where the beta equals zero, The minimum of SU(3) nuclei is located around the place where the gamma equals zero, which is called prolate deformation. The O(6) nuclei is expected to have its minimum around the region where the gamma is thirty degree. If the minimum of the potential is stable, its shape shows rigid triaxial rotor, and if the energy surface shows rather flat in the direction of gamma, this feature is called gamma-unstable. 原子核殻模型においては 扱うべきハミルトニアン行列の次元の大きさ

原子核構造計算の趨勢 50 100 質量数 A 少数系の精密計算 Green Function Monte Carlo 50 100 質量数 A 少数系の精密計算 Green Function Monte Carlo No-core Shell Model, etc. ガウス展開法 (AMD等) 原子核殻模型 集団運動模型 相互作用するボゾン模型 I would remind you some features of the quadrupole collective motion. This triangle is called Casten's triangle including three vertexes of which correspond to the dynamical symmetries of the Hamiltonian in the IBM has. These symmetries correspond the three typical collective motions, respectively. U(5) corresponds to the spherical vibrator, SU(3) corresponds to the axially symmetric rotor, and O(6) corresponds to the triaxial deformation. We focus mainly on the O(6) symmetry, which describes the triaxial deformation, and the transitional region between these vertices. In the study of the quadrupole collective states using the intrinsic framework, as you know, we often write the potential energy surface, parameterized by beta, the degree of freedom of axially symmetric deformation, and gamma, the degree of freedom of the triaxial deformation. We can classify the nuclear shapes according to the location of the energy minimum. The minimum of U(5) nuclei is expected to be located around the place where the beta equals zero, The minimum of SU(3) nuclei is located around the place where the gamma equals zero, which is called prolate deformation. The O(6) nuclei is expected to have its minimum around the region where the gamma is thirty degree. If the minimum of the potential is stable, its shape shows rigid triaxial rotor, and if the energy surface shows rather flat in the direction of gamma, this feature is called gamma-unstable. Mean-Field calc. (Skyrme, Gogny Hartree-Fock) Relativistic Mean Field calc. 　　+ Generator Coordinate Method, etc.

原子核殻模型における計算の困難 56Niは、40Caを閉殻として、1f7/2,2p3/2,1f5/2,2p1/2の４本の軌道、２０個の一粒子軌道に８粒子が存在する。すべての配位の配位混合を考慮して固有状態を求めたい。 例　56Niの殻模型計算における配位 × ... 粒子　　○... ホール 単純に考えて の配位の配位混合がありうる。 空間の対称性を考慮に 入れても1.1x109次元の行列の 対角化が必要となる。 I would remind you some features of the quadrupole collective motion. This triangle is called Casten's triangle including three vertexes of which correspond to the dynamical symmetries of the Hamiltonian in the IBM has. These symmetries correspond the three typical collective motions, respectively. U(5) corresponds to the spherical vibrator, SU(3) corresponds to the axially symmetric rotor, and O(6) corresponds to the triaxial deformation. We focus mainly on the O(6) symmetry, which describes the triaxial deformation, and the transitional region between these vertices. In the study of the quadrupole collective states using the intrinsic framework, as you know, we often write the potential energy surface, parameterized by beta, the degree of freedom of axially symmetric deformation, and gamma, the degree of freedom of the triaxial deformation. We can classify the nuclear shapes according to the location of the energy minimum. The minimum of U(5) nuclei is expected to be located around the place where the beta equals zero, The minimum of SU(3) nuclei is located around the place where the gamma equals zero, which is called prolate deformation. The O(6) nuclei is expected to have its minimum around the region where the gamma is thirty degree. If the minimum of the potential is stable, its shape shows rigid triaxial rotor, and if the energy surface shows rather flat in the direction of gamma, this feature is called gamma-unstable.

原子核殻模型による構造計算 魔法数 最近の展開 伝統的な模型空間 82-126 中重核領域 50-82 pfg (本間さん） 2d3/2,3s1/2, h11/2 50-82 1d5/2,2g7/2 pfg (本間さん） pf-shell I would remind you some features of the quadrupole collective motion. This triangle is called Casten's triangle including three vertexes of which correspond to the dynamical symmetries of the Hamiltonian in the IBM has. These symmetries correspond the three typical collective motions, respectively. U(5) corresponds to the spherical vibrator, SU(3) corresponds to the axially symmetric rotor, and O(6) corresponds to the triaxial deformation. We focus mainly on the O(6) symmetry, which describes the triaxial deformation, and the transitional region between these vertices. In the study of the quadrupole collective states using the intrinsic framework, as you know, we often write the potential energy surface, parameterized by beta, the degree of freedom of axially symmetric deformation, and gamma, the degree of freedom of the triaxial deformation. We can classify the nuclear shapes according to the location of the energy minimum. The minimum of U(5) nuclei is expected to be located around the place where the beta equals zero, The minimum of SU(3) nuclei is located around the place where the gamma equals zero, which is called prolate deformation. The O(6) nuclei is expected to have its minimum around the region where the gamma is thirty degree. If the minimum of the potential is stable, its shape shows rigid triaxial rotor, and if the energy surface shows rather flat in the direction of gamma, this feature is called gamma-unstable. sd+pf （宇都野さん) sd-shell Ref. M.G.Meyer and J.H.D.Jensen, Elementary Theory of Nuclear Shell Structure p.58(1955)

中重核領域への原子核殻模型への適用 方法論の発展と 計算機資源の増強 56Baアイソトープにおける次元数 模型空間 ... Z=50-82, N=82-126 旧来の対角化法では、このような巨大次元の行列の対角化は困難 方法論の発展と 計算機資源の増強

殻模型計算におけるさまざまなアプローチ 量子モンテカルロ法 Lanczos法による厳密対角化 conventional t-particle t-hole truncation scheme VAMPIRE (Variation After Mean-Field Projection In Realistic model space) 密度行列繰り込み群法 Shell Model Monte Carlo 　　　　　　(量子モンテカルロ法） モンテカルロ殻模型(MCSM) GCM, TDA, RPA, .... 量子モンテカルロ法 ハミルトニアンHは２体演算子 虚時間発展 Hubbard-Stratonovich 変換 ハミルトニアンh（s） 1体演算子+補助場 波動関数を得られない 不符号問題 補助場sはモンテカルロ積分される

大次元殻模型計算におけるモンテカルロ的解法 （モンテカルロ殻模型,MCSM) 発生させた乱数のセット これらのMCSM基底によって作られた部分空間で 波動関数を記述する MCSM次元（~40) I show the formulation of the MCSM. I told you that the MCSM is a stochastic importance truncation, which means the whole Hilbert space is truncated into around 40 bases selected stochastically for lowering the eigenvalue of the Hamiltonian matrix in terms of the variational principle. The selected bases are called QMCD bases, which is named after the selection technique, Quantum Monte-Carlo Diagonalization method. These bases are generated using the auxiliary field technique. The auxiliary field are generated randomly. これらのMCSM基底は、乱数に基づいて生成された多数の基底から 固有値を下げるように選ばれた少数の(~40)基底である。

Monte Carlo Shell Model Stochastic “importance” truncation 計算量のほとんどは浮動小数点演算 並列計算可能 (厳密対角化は並列化が困難）

MCSM基底 カノニカル多体基底 Slater 行列式 粒子数射影 HFB 波動関数 対称性の回復 Projection method ... 調和振動子基底の生成演算子 Slater 行列式 粒子数射影 HFB 波動関数 対相関が取り入れられる 一体ハミルトニアンの演算によって 基底の表現を変えない （Baker-Haussdorf theorem) 対称性の回復 Projection method

MCSMの収束性 56Ni MCSM results (FPD6 interaction) 8 protons and 8 neutrons in pf-shell 1.1 x 109 M-scheme dimension Slater determinant MCSM basis MCSM basis dimension Ref. T.Otsuka, M.Honma,T.Mizusaki, N.Shimizu, and Y.Utsuno Prog. Part. Nucl. Phys. 46 319(2001)

MCSMと従来の対角化計算との比較 56Ni in pf-shell t-particle t-hole truncation v.s. MCSM calculation Ref. T.Otsuka, M.Honma,T.Mizusaki, N.Shimizu, and Y.Utsuno Prog. Part. Nucl. Phys. 46 319(2001)

MCSMにおける並列計算 対称性の回復(課運動量射影) MCSMの並列計算効率

Alphleet-1 since 1999 Compaq DS-20 (Alpha 2cpu) x 73 Myrinet network RIKEN

Alphleet-2 since 2002 HP ES-45(Alpha 4cpu) x 28 HP GS-1280(Alpha 32cpu) x 2 Myrinet network Dept. of Physics, Univ. of Tokyo CNS, Univ. of Tokyo RIKEN 112+64=176

x86 Linux サーバ Dell Intel Xeon (2cpu) x 20 Gigabit Ethernet Since 2003 HP AMD Opteron (2cpu) x 8 Gigabit Ethernet Since 2005

殻模型計算におけるさまざまなアプローチ Exact calculation for 56Ni (?) MCSM for 56Ni Ref. T.Otsuka, M.Honma,T.Mizusaki, N.Shimizu, and Y.Utsuno Prog. Part. Nucl. Phys. 46 319(2001)

四重極集団運動状態のB(E2)遷移確率 Experimental value 軸対称変形、球形、 非軸対称変形状態 の間の遷移領域を Semi-magic nuclei (spherical) axial deformation triaxial deformation axially symmetric deformation Experimental value O(6) U(5) SU(3) 軸対称変形、球形、 非軸対称変形状態 の間の遷移領域を 殻模型で微視的に 記述する。 This figure shows the E2 transition probability from the ground 0+ state to the first excited 2+ state of tin, tellurium, xenon, Barium even-even isotopes, against the neutron numbers. These values are obtained experimentally. At the semi-magic nuclei, N=82, the structure shows some features of spherical vibrator, including the small B(E2) values. if we add neutron pairs, the shape becomes axially symmetric rotor, and B(E2) increases smoothly. On the other hand, if we add neutron hole pairs, the shape becomes triaxially deformed. We focus on this transitional region, between triaxially deformed rotor and spherical vibrator. Beyond mean-field calculation is essential for describing the structure of these transitional nuclei. Then, we did a microscopic study using the nuclear shell model.

B(E2;0+→2+) MCSMによる計算結果 単一の取り扱いのもと、 ３つの典型的な 集団運動状態と その間の遷移状態の 記述に成功した。 Semi-magic nuclei (spherical) axial deformation triaxial deformation axially symmetric deformation exp. MCSM 単一の取り扱いのもと、 ３つの典型的な 集団運動状態と その間の遷移状態の 記述に成功した。 We will discuss the three physical observables of xeon isotopes in the following screens. First, We return to the issue of the B(E2). This figure shows B(E2) values against neutron number. These blue symbols denote the experimental values, which are the same as the previous figure. The red ones show the MCSM results. In the MCSM calculation, the model space of the nuclear shell model is different from each other. However, the same proton-proton interaction is used as common. The MCSM calculation reproduce the experimental values of B(E2) in the whole region, including the triaxially deformed nuclei, semi-magic nuclei, and axially symmetric deformed nuclei. These quadrupole collective states are described in a single microscopic framework consistently. on the other hand (side) ( )

Experiment : G. Jakob et al. Magnetic moment of Xe isotopes (g factors) Experiment : G. Jakob et al. Phys. Rev. C65, 024316 (2002) truncated shell model calc. (N=64 and Z=64 subshells assumed) IBM-2 Z/A MCSM 2+1 g factor 4+1 g factor Thirdly, this figure shows the Magnetic moments of Xe isotopes. The red square symbols show the experimental values of g-factors of 2+ states measured recently by Jakob and his collaborators using the Coulomb excitation technique. The black line shows the prediction of the IBM-2, which is rather rough estimation, And the green stars show the results of the shell model calculation, where N=Z=64 sub-shell closure is assumed. This is consistently close to that of the IBM-2. These values are twice as large as the experimental values, except for 136Xe, which is a semi-magic nucleus. The collective picture provides the g=Z/A, shown by the orange line. These values are rather close to the experimental values, but the tendency is opposite. The g-factors are smoothly decreased by increasing the neutron number. The open symbols with black line show old experimental values. The green symbols show the results of the shell model calculation assuming the sub shell closure in this paper. In the truncated shell model calculation, the collectivity is not sufficient for these nuclei. --- interplay of collective picture and single-particle picture. --- at the same time Interestingly, The g-factor of 130Xe is agreed well with the collective picture, Z/A and the results of IBM-2, truncated shell model, the MCSM results, and the experimental value. Intuitive explanation N Large model space solves the problem naturally. Spin quenching 0.7

Summary 計算方法論の発展と計算機資源の増強によって、原子核殻模型計算を中重核領域に適用可能とした。 中重核領域における有限多体量子系に特有な緩やかな遷移を記述した。

モンテカルロ殻模型 Monte-Carlo Shell Model The nuclear shell model is very useful model to study nuclear structure microscopically. However, the Hilbert space of the Hamiltonian becomes too huge in medium-heavy nuclei. Monte-Carlo Shell Model Stochastic “importance” truncation of the full Hilbert space to the subspace spanned by about 40 highly selected bases. The nuclear shell model is one of the most useful model to study the nuclear structure in the transitional region microscopically. However, huge dimension of the Hilbert space of the shell-model Hamiltonian often prevents us to diagonalize the Hamiltonian matrix especially of the medium-heavy nuclei, because we require the larger model space about the medium-heavy nuclei than light nuclei. For example, 10 to 14 dimension is required for describing the structure of barium 148. In order to overcome such difficulty, we use the Monte-Carlo shell model technique, which is called the MCSM. The MCSM is the stochastic importance truncation to the full shell model calculations. We verified the usefulness of the MCSM by applying it to the Ba isotopes in this reference. The transition from spherical to deformed shapes in the Ba isotopes was studied using the MCSM. Ref. N.Shimizu, T.Otsuka, T.Mizusaki, and M.Honma Phys. Rev. Lett. 86 1171 (2001)

MCSM計算の進展 56Ni in pf-shell Exact calculation for 56Ni (?) t-particle t-hole truncation v.s. MCSM calculation Development of the conventional shell-model diagonalization method Ref. T.Otsuka, M.Honma,T.Mizusaki, N.Shimizu, and Y.Utsuno Prog. Part. Nucl. Phys. 46 319(2001)

Structure of 136Te Energy levels of 134Te, 136Te, and 134Sn Quadrupole collectivity is not strong enough. valence particles 2p 2p2n 2n