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Giant Resonances in Atomic Nuclei

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Giant Resonances in Atomic Nuclei Charles University in Prague Faculty of Mathematics and Physics MASTER THESIS Anton Repko Giant Resonances in Atomic Nuclei Institute of Particle and Nuclear Physics Supervisor of the master thesis: prof. RNDr. Jan Kvasil, DrSc. Study programme: Physics Specialization: Nuclear and Subnuclear Physics Prague 2011 Univerzita Karlova v Praze Matematicko-fyzikální fakulta DIPLOMOVÁ PRÁCE Anton Repko Gigantické rezonance v atomových jádrech Ústav £ásticové a jaderné fyziky Vedoucí diplomové práce: prof. RNDr. Jan Kvasil, DrSc. Studijní program: Fyzika Studijní obor: Jaderná a subjaderná fyzika Praha 2011 I would like to thank my supervisor prof. Jan Kvasil for his support and patient explanation of the theoretical and practical aspects of nuclear phenomena and their description. Prohla²uji, ºe jsem tuto diplomovou práci vypracoval samostatn¥ a výhradn¥ s pouºitím citovaných pramen·, literatury a dal²ích odborných zdroj·. Beru na v¥domí, ºe se na moji práci vztahují práva a povinnosti vyplývající ze zákona £. 121/2000 Sb., autorského zákona v platném zn¥ní, zejména skute£nost, ºe Univerzita Karlova v Praze má právo na uzav°ení licen£ní smlouvy o uºití této práce jako ²kolního díla podle 60 odst. 1 autorského zákona. V Praze dne .................... Podpis autora Název práce: Gigantické rezonance v atomových jádrech Autor: Anton Repko Katedra: Ústav £ásticové a jaderné fyziky Vedoucí diplomové práce: prof. RNDr. Jan Kvasil, DrSc., ÚJF Abstrakt: Skyrme funkcionál je £asto pouºíván k popisu základních stav· i dy- namických vlastností atomových jader. V této práci byla pro mikroskopický self- konzistentní popis dynamických vlastností pouºita separabilní Random Phase Approximation (SRPA) metoda vycházející ze Skyrme funkcionálu. Tato práce popisuje teorii Skyrme Hartree-Fock a SRPA a p°edkládá numerické výpo£ty E1 a M1 gigantických rezonancí provedené pro sérii sférických jader 40Ca56Fe. Pro jádro 56Fe existují náznaky jeho nenulové deformace v základním stavu, pro- to byly výpo£ty provedeny i pro tento p°ípad. Výsledky získané pro vybrané parametrizace jsou porovnány s experimentálními daty. Klí£ová slova: silová funkce, kolektivní jevy v jád°e, Skyrme hustotní funkcionál Title: Giant Resonances in Atomic Nuclei Author: Anton Repko Department: Institute of Particle and Nuclear Physics Supervisor: prof. RNDr. Jan Kvasil, DrSc., IPNP Abstract: Skyrme functional is commonly used for the description of ground-state and dynamical properties of atomic nuclei. To describe the dynamical properties in the microscopic self-consistent way, we employed Separable Random Phase Approximation (SRPA) based on Skyrme functional. This work describes theory of Skyrme Hartree-Fock and SRPA and presents numerical calculation of E1 and M1 giant resonances in spherical nuclei 40Ca56Fe. There is some evidence for non-zero ground-state deformation of the nucleus 56Fe, so it is treated also with such assumption. The results obtained for various parametrizations are compared to the experimental data. Keywords: strength function, nuclear collective phenomena, Skyrme density func- tional Contents 1 Introduction 2 2 Theoretical part 3 2.1 Hartree-Fock method . 3 2.2 Density functional theory . 4 2.2.1 Skyrme functional . 5 2.3 Simple example of collective phenomena . 9 2.4 BCS: Pairing description . 11 2.4.1 Quasiparticle formalism . 13 2.5 RPA: Collective vibrations . 14 2.5.1 Evaluation of matrix elements of 1ph operators . 16 2.5.2 SRPA in wavefunction formalism . 18 2.5.3 SRPA in density formalism . 22 2.5.4 Transition probabilities and strength function . 23 3 Numerical calculations 27 3.1 Skyrme parametrizations . 27 3.2 Description of the codes . 28 3.2.1 Skyax (HF and BCS calculation) . 28 3.2.2 Skyax_me (calculation of matrix elements) . 28 3.2.3 Sky_srpa (calculation of strength function and RPA states) 29 3.3 E1 resonances . 29 3.4 M1 resonances . 33 3.5 Deformed 56Fe............................. 36 4 Summary 38 A Time reversal 39 1 1. Introduction Quantum mechanical description of the atomic nucleus has been a challenge from the beginning of quantum physics up to now. Main problem lies in the unknown interaction potential of the nucleons and also in the implementation of many- body techniques (mainly Hartree-Fock method). Experimentaly determined in- teraction potential cannot be used directly because it contains strongly repulsive core which leads to singularities in the common treatment. Various renormaliza- tion procedures have been developed to circumvent this problem, but they are often ambiguous and are restricted to light nuclei. [1] Another route is to use eective interaction Hamiltonian or density functional which is suciently simple to be applicable to the wide range of nuclei, and determine its free parameters by tting to experimental data. Common forces used today are Skyrme, Gogny and Relativistic Mean Field. Here we used Skyrme for its simplicity (contact interaction) and relatively good description of ground- state properties. One of important dynamical properties of the nuclei are so called giant reso- nances, which are manifested by increased photoabsorption cross section in the region of 1030 MeV. They can be classied by their multipolarity and parity, e.g. electric dipole E1 (1−) and quadrupole E2 (2+), magnetic dipole M1 (1+) etc. Early treatment of giant resonances involved macroscopic models with tted parameters. However, our aim is to treat them fully microscopically within the framework of Skyrme force. This can be achieved by Random Phase Approxima- tion (RPA), but it requires large numerical eort, namely the diagonalization of matrices with dimension determined by the number of particle-hole (1ph) states. Therefore, Separable Random Phase Approximation (SRPA) was used, which was developed in the collaboration of IPNP MFF Prague, JINR Dubna and Uni- versity of Erlangen [2]. It extracts residual interaction from Skyrme functional in a separable form of a few terms and eectively reduces the dimension of used matrices. Strength function can be then obtained directly without the actual calculation of excited states. This work is organized as follows: Theoretical part thoroughly describes Hartree-Fock and Density Functional methods and their application to Skyrme functional, treatment of pairing (BCS), and nally describes full and separable RPA methods, and calculation of the strength function. Next part describes used Skyrme parametrizations, programs employed in calculations of E1 and M1 resonances for selected nuclei, and obtained results. 2 2. Theoretical part 2.1 Hartree-Fock method Hartree-Fock method is usually the rst approximation for quantum mechanical solution of many-body problems in atomic and nuclear physics. It's aim is to extract single-particle Hamiltonian from many-body Hamiltonian. Wave function of many body system of identical fermions Ψ(x1; x2; : : : xn) is approximated by Slater determinant (which maintains antisymmetry): 1 X jΨi = j1; 2; 3; : : : j; : : : ni = p sign(P )j P (1)(x1)ij P (2)(x2)i ::: j P (n)(xn)i n! P (2.1) Replacement of Ψ(x1; x2; : : : xn) by Slater determinant is analogous to factoriza- tion of the function of two variables, F (x1; x2) = f(x1)g(x2), and is appropriate only in systems with no collective phenomena. From HF calculation, we expect to get a set of single-particle wavefunctions f j(x)g, of which only n are involved in Slater determinant (i.e. occupied states under Fermi level). From the basic properties of determinants, it follows that we are free to use any linear combination of occupied states f j(x)gj=1;:::n in Slater determinant to get the same many-body wavefunction. Therefore, we always assume orthogonal set. Many-body Hamiltonian contains one-body and two-body terms (which are ^ ^ symmetrical, V (x1; x2) = V (x2; x1)): ^ X ^ X ^ H = T (xi) + V (xj; xk) (2.2) i j<k In nuclear physics we sometimes assume also three-body term (the following derivation can be easily generalized to include it). Mean energy of the system described by Slater determinant is: n n ^ X ^ X h ^ E0 = hΨjHjΨi = h ijT j ii + h j(x1) k(x2)jV (x1; x2)j j(x1) k(x2)i i=1 j<k ^ i −h j(x1) k(x2)jV (x1; x2)j k(x1) j(x2)i n n X 1 X = hijT^jii + hjkjV^ jjk − kji (2.3) 2 i=1 j;k=1 This can be derived by expanding Slater determinant (2.1) and using orthogona- lity of f j(x)g. Solution is found as a minimum of energy by variational principle (δE0 = 0), with constraint hΨjΨi = 1 ^ ^ ^ 0 = δj(hΨjHjΨi − "jhΨjΨi) = hδjΨjHjΨi + hΨjHjδjΨi − "jhδjΨjΨi − "jhΨjδjΨi (2.4) where "j is a Lagrange multiplier and δj means the variation of the j-th single- particle wavefunction, j 7! j + δ . We are free to multiply δ by arbitrary 3 phase factor (i.e. eic) so that the sum of terms with variation in bra-vector is real and thus equal to the remaining terms with variation in ket-vector. Variation in j in the expression for E0 (2.3) leads to: n ^ X h ^ 0 = hδ jT j ji + hδ (x1) k(x2)jV (x1; x2)j j(x1) k(x2)i k=1 ^ i −hδ (x1) k(x2)jV (x1; x2)j k(x1) j(x2)i − "jhδ j ji (2.5) Since the δ is arbitrary (up to phase factor), the above expression can be refor- mulated as (almost) the eigenvalue problem: h n Z i ^ ^ X ∗ ^ h j(x1) = T j(x1) + dx2 k(x2)V (x1; x2) k(x2) j(x1) k=1 n h Z i X ∗ ^ − dx2 k(x2)V (x1; x2) j(x2) k(x1) k=1 = "j j(x1) (2.6) It is not possible to solve this equation directly, mainly because of the second integral (non-local exchange term). The equation is solved by evaluation of matrix elements with some pre-dened set of wavefunctions fφjgj=1;:::N , where N n and wavefunctions j are found by iteration (i.e.
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