Giant Resonances in Atomic Nuclei

Charles University in Prague Faculty of Mathematics and Physics

MASTER THESIS

Anton Repko

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 functional 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 interaction 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 resonances, 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):

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

{ψj(x)}, 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 {ψj(x)}j=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 ˆ ˆ symmetrical, V (x1, x2) = V (x2, x1)): ˆ X ˆ X ˆ H = T (xi) + V (xj, xk) (2.2) i j

n n ˆ X ˆ X h ˆ E0 = hΨ|H|Ψi = hψi|T |ψii + hψj(x1)ψk(x2)|V (x1, x2)|ψj(x1)ψk(x2)i i=1 j

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

ψj in the expression for E0 (2.3) leads to:

n ˆ X h ˆ 0 = hδψ|T |ψji + hδψ(x1)ψk(x2)|V (x1, x2)|ψj(x1)ψk(x2)i k=1 ˆ i −hδψ(x1)ψk(x2)|V (x1, x2)|ψk(x1)ψj(x2)i − εjhδψ|ψji (2.5)

Since the δψ is arbitrary (up to phase factor), the above expression can be refor- mulated as (almost) the eigenvalue problem:

h n Z i ˆ ˆ X ∗ ˆ h ψj(x1) = T ψj(x1) + dx2 ψk(x2)V (x1, x2)ψk(x2) ψj(x1) k=1 n h Z i X ∗ ˆ − dx2 ψk(x2)V (x1, x2)ψj(x2) ψk(x1) k=1 = εjψj(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 {φj}j=1,...N , where N n and wavefunctions ψj are found by iteration (i.e. self-consistently). We obtain not only ψj≤n (occupied, or hole states), but also ψj>n (particle states). Finally, we can write single-particle form of Hamiltonian ( + and are cre- aˆj aˆj ation and annihilation operators for ψj):

ˆ X + (2.7) h = εjaˆj aˆj j

However, this cannot be completely adapted for full Hamiltonian Hˆ (we cannot build the whole many-particle state from the ground up, because mean-eld is changed by this process). But it can be used describe 1ph excitations (in average ˆ ˆ eld HAV, not counting residual interaction Vres):

ˆ ˆ ˆ ˆ X + (2.8) H = HAV + Vres HAV = E0 + εj :ˆaj aˆj : j where normal ordering is understood in the sense of particle ( + , ) or hole aˆj aˆj j > n ( +, ) excitations. −aˆjaˆj j ≤ n

2.2 Density functional theory

Many-body problems can be alternatively formulated in terms of energy functional depending on the one-body density Jρ:

n ˆ(n) ˆ(n) X (2.9) E = H[Jρ(x)] = H[hΨ|Jρ (x)|Ψi] Jρ (x) = δ(xj − x) j=1

4 Index ρ means that we are dealing with ordinary density, index (n) means that it is an operator on n-particle Hilbert space. One-body density means that the operator probes density at single point x and therefore is not able to distinguish spatial correlations in many-body wavefunction. For Slater determinant |Ψi = |1, 2, . . . ni (composed of {ψj(x)}j=1,...n) its expectation value is

n ˆ(n) X ˆ (2.10) Jρ(x0) = hΨ|Jρ (x0)|Ψi = hψj|δ(x − x0)|ψji = Tr(Jρ(x0)ˆρ) j=0 ˆ where we dened density operator Jρ on one-particle Hilbert space (in X representation) and density matrix ρˆ:

n n ˆ X X Jρ(x0) = δ(x − x0)ρ ˆ = ρˆj = |ψjihψj| (2.11) j=1 j=1 This formulation is useful for the application of variational principle to the energy functional. We dene variation δj :ρ ˆj 7→ ρˆj + δρˆ and impose constraint Trˆρ = n. This leads to Z δH ˆ 0 = δj(H[Jρ] − εjTrˆρ) = dx0 Tr(Jρ(x0)δρˆ) − εjTr(δρˆ) (2.12) δJρ(x0) This result can be generalised to functionals depending on various other densities (involving also space derivatives). Then index α instead of ρ will enumerate particular densities (ordinary ρ, kinetic-energy τ, spin-orbital J etc.), which will be dened later. δρˆ can be expressed by means of wavefunction ψj and its variation δψ:

2 δρˆ = |ψj + δψihψj + δψ| − |ψjihψj| = |δψihψj| + |ψjihδψ| + O(δψ ) (2.13)

Since δψ is arbitrary, the result can be formulated as the eigenvalue problem:

Z X δH hˆ ψ (x) = dx Jˆ (x ) ψ (x) = ε ψ (x) (2.14) j 0 δJ (x ) α 0 j j j α α 0 This problem must be solved iteratively, because, in general, H has a non-linear dependence on the densities. In realistic calculations, one has to deal with non-local Coulomb interaction. This wouldn't constitute problem for the classical formulation of energy functional. But on quantum level, non-local interactions give rise to an exchange term (see equation (2.6)) which involves two-body density. However, it can be approximated by the following expression derived by Slater [3] (LDA stands for local density approximation):

3 3 1/3 e2 Z LDA 4/3 (2.15) Hxc = − dx Jρ (x) 4 π 4π0 2.2.1 Skyrme functional Calculations performed by the program Skyax are based on Skyrme energy functional, which was derived from phenomenological interaction proposed by Skyrme

5 [4], now used in the form [5] (with parameters t0, t1, t2, t3, x0, x1, x2, x3, t4, α): ˆ ˆ V (~r1, ~r2) = t0(1 + x0Pσ)δ(~r1 − ~r2) 1 −→ −→ ←− ←− − t (1 + x Pˆ )[δ(~r − ~r )(∇ − ∇ )2 + (∇ − ∇ )2δ(~r − ~r )] 8 1 1 σ 1 2 1 2 1 2 1 2 1 ←− ←− −→ −→ + t (1 + x Pˆ )(∇ − ∇ ) · δ(~r − ~r )(∇ − ∇ ) 4 2 2 σ 1 2 1 2 1 2 1 ~r + ~r + t (1 + x Pˆ )δ(~r − ~r )ρα 1 2 6 3 3 σ 1 2 2 i ←− ←− −→ −→ + t (~σ + ~σ ) · [(∇ − ∇ ) × δ(~r − ~r )(∇ − ∇ )] (2.16) 4 4 1 2 1 2 1 2 1 2

Arrows over derivatives indicate action on bra- or ket-vector. Term at t3 was originally formulated as a three body interaction, but now it is usually rewritten ˆ as a density dependent two-body interaction. Pσ is the spin-exchange operator: 1 1 Pˆ = (1 + ~σ · ~σ ) = (1 + σ σ ) + σ σ + σ σ (2.17) σ 2 1 2 2 1z 2z 1+ 2− 1− 2+ 0 1 0 −i 1 0 0 1 0 0 σ = σ = σ = σ = σ = x 1 0 y i 0 z 0 −1 + 0 0 − 1 0 Skyrme assumed contact interaction with dependence on relative momenta of the interacting particles. Local behaviour of the interaction permits particulary eective numerical implementation and allows formulation by density functional (including exchange term). In the following, we will label densities by α instead ˆ of Jα to keep expressions clear (labeling of the operators remains Jα) Skyrme force can be rewritten into a density functional by procedure described in [6]. Besides ordinary density ρ, it is neccesary to introduce also kinetic-energy ~ density τ, spin-orbital tensor density Jij, its vector component J, current density ~j, spin density ~σ and kinetic energy-spin density T~ (last three are time-odd; for time reversal symmetry see appendix A) and corresponding operators:

n X 2 ρ(~r) = |ψk(~r)| k=1 n X ~ 2 τ(~r) = |∇ψk(~r)| k=1 n i X n o J (~r) = [∇ ψ∗(~r)]σ ψ (~r) − ψ∗(~r)σ [∇ ψ (~r)] ij 2 i k j k k j i k k=1 n i X n o ~J(~r) = − ψ∗(~r)[(∇~ × ~σ)ψ (~r)] − [(∇~ × ~σ)ψ∗(~r)]ψ (~r) = J 2 k k k k ijk ij k=1 n i X n o ~j(~r) = [∇~ ψ∗(~r)]ψ (~r) − ψ∗(~r)[∇~ ψ (~r)] 2 k k k k k=1 n X ∗ ~σ(~r) = ψk(~r)~σψk(~r) k=1 n ~ X ~ ∗ ~ (2.18) Tj(~r) = [∇ψk(~r)] · σj[∇ψk(~r)] k=1 6 ˆ Jρ(~r0) = δ(~r − ~r0) ˆ ←− −→ Jτ (~r0) = ∇ · δ(~r − ~r0)∇ i ←− −→ [Jˆ (~r )] = [∇ δ(~r − ~r )σ − δ(~r − ~r )σ ∇ ] J 0 ij 2 i 0 j 0 j i ˆ i −→ ←− J~ (~r ) = − [δ(~r − ~r )(∇ × ~σˆ) − (∇ × ~σˆ)δ(~r − ~r )] J 0 2 0 0 ˆ i ←− −→ J~ (~r ) = [∇δ(~r − ~r ) − δ(~r − ~r )∇] j 0 2 0 0 ˆ~ J σ(~r0) = δ(~r − ~r0)~σ ˆ~ ←− −→ (2.19) [J T (~r0)]j = ∇ · σjδ(~r − ~r0)∇ Skyrme functional then reads [2, 5] (s labels protons/neutrons, or their sum for no label):

Z 0 nb0 b X X H = d3~r ρ2 − 0 ρ2 + b (ρτ − ~j2) − b0 (ρ τ − ~j2) Sk 2 2 s 1 1 s s s s=p,n s=p,n 0 0 b2 b X b3 b X − ρ∆ρ + 2 ρ ∆ρ + ρα+2 − 3 ρα ρ2 2 2 s s 3 3 s s=p,n s=p,n ~ ~ ~ ~ 0 X ~ ~ ~ ~ −b4[ρ∇ · J + ~σ · (∇ × j)] − b4 [ρs(∇ · Js) + ~σs · (∇ × js)] s=p,n o ˜ ~ 2 ˜0 X ~ 2 (2.20) +b4(~σ · T − J ) + b4 (~σs · Ts − Js) s=p,n with 2 P and newly introduced parameters: J = ij JijJij 1 1 1 1 1 1 b = t 1 + x b0 = t + x b = t 1 + x b0 = t + x 0 0 2 0 0 0 2 0 3 4 3 2 3 3 4 3 2 3 1h 1 1 i 1h 1 1 i b = t 1 + x + t 1 + x b0 = t + x − t + x 1 4 1 2 1 2 2 2 1 4 1 2 1 2 2 2 1h 1 1 i 1h 1 1 i b = 3t 1 + x − t 1 + x b0 = 3t + x + t + x 2 8 1 2 1 2 2 2 2 8 1 2 1 2 2 2 1 t x + t x t − t b = b0 = t ˜b = 1 1 2 2 ˜b0 = 2 1 (2.21) 4 4 2 4 4 8 4 8 The above expression includes also terms containing time-odd densities ~j, ~σ and T~ which are omitted from ground state-calculation for even-even nuclei, but their inclusion in SRPA calculation leads to better agreement with experiments [7]. There are also additional terms which violate Galilean and gauge symmetry [8,9] but they have important contribution for M1 resonances [10]: Z ht0 X t3 X Hextra = d3~r x ~σ2 − ~σ2 + ρα x ~σ2 − ~σ2 Sk 4 0 s 24 3 s s=p,n s=p,n 3t1x1 − t2x2 3t1 + t2 X i − ~σ · ∆~σ + ~σ · ∆~σ (2.22) 32 32 s s s=p,n

7 Overall density functional contains also kinetic energy term, and direct and exchange Coulomb term.

H = Hkin + HSk + HCoul (2.23) 2 Z H = ~ d3~rτ(~r) kin 2m e2 h Z ρ (~r )ρ (~r ) 3 3 1/3 Z i 3 3 p 1 p 2 3 4/3 (2.24) HCoul = d ~r1d ~r2 − d ~rρp (~r) 8π0 |~r1 − ~r2| 2 π There is also a pairing contribution to total energy, but it needs to be evaluated separately (see chapter about BCS). Next, using variational principle described in the previous section (equation (2.14)) we get Hartree-Fock equation (after integrating out delta-functions from ˆ Jα(~r0)) [11]: ˆ ←− −→ ~ −→ hsψj(~r) = [Us(~r) + ∇ · Bs(~r)∇ − iWs(~r) · (∇ × ~σ)]ψj(~r) = εjψj(~r) (2.25) Left derivative should be understood in the sense that the expression will be multiplied by bra-vector in order to solve the eigenvalue problem for matrix hi|hˆ|ji. Densities used in HF equation are:

δH 0 0 0 α + 2 α+1 Us(~r) = = b0ρ − b0ρs + b1τ − b1τs − b2∆ρ + b2∆ρs + b3 ρ δρs(~r) 3 0 b3 α−1 X 2 α 0 − [αρ ρ 0 + 2ρ ρ ] − b ∇~ · ~J − b ∇~ · ~J 3 s s 4 4 s s0 2 Z 1/3 e h 3 ρp(~r1) 3ρp i +δs,p d ~r1 − (2.26) 4π0 |~r − ~r1| π δH 2 ~ 0 (2.27) Bs(~r) = = + b1ρ − b1ρs δτs(~r) 2m δH W~ (~r) = = b ∇~ ρ + b0 ∇~ ρ − ˜b ~J − ˜b ~J (2.28) s ~ 4 4 s 4 4 δJs(~r)

Time-odd part of HF Hamiltonian (not including extra) used for derivation of HSk the responses is:

i ←− −→ hˆodd(~r) = (∇ · A~ (~r) − A~ (~r) · ∇) + S~ (~r) · ~σ (2.29) s 2 s s s with densities δH ~ ~ 0 ~ ~ 0 ~ (2.30) As(~r) = = −2b1j + 2b1js − b4(∇ × ~σ) − b4(∇ × ~σs) δ~js(~r) δH ~ ~ ~ 0 ~ ~ ˜ ~ ˜0 ~ (2.31) Ss(~r) = = −b4(∇ × j) − b4(∇ × js) + b4T + b4Ts δ~σs(~r)

Calculation of M1 resonances involves also terms from extra (they are not listed HSk here).

8 2.3 Simple example of collective phenomena

In the following sections we will deal with pairing and collective vibrations. Al- though they are somewhat understandable from the particle point of view, quantum mechanics deals with wavefunctions and we need to interpret them also from the wave point of view (where they have no analogy in wave dynamics). Here we will demonstrate them qualitatively on an example of two interacting particles in a 1D box. Energy eigenfunctions of the particle in a 1D box of length L are:

r 2 jπx ψ (x) = sin (2.32) j L L We will consider two fermions in a singlet spin state. We can construct their wavefunctions from ψj: 1 hx1, x2|jji = √ ψj(x1)ψj(x2)(↑↓ − ↓↑) (2.33) 2 1 hx , x |jki = [ψ (x )ψ (x ) + ψ (x )ψ (x )](↑↓ − ↓↑) (2.34) 1 2 2 j 1 k 2 k 1 j 2 State |jji is in a form of a Slater determinant, while |jki is already partially correlated (by spins), but we will not consider it as a collective state yet. Several of these wavefunctions are depicted on gure 2.1. These wavefunctions can be used as a basis for solution of a system with (attractive) interaction. Attractive interaction should lead to formation of a

1 1

0.5 0.5

2 0 2 0 x x

-0.5 -0.5

a) -1 b) -1 x1 x1 1 1

0.5 0.5

2 0 2 0 x x

-0.5 -0.5

c) -1 d) -1 x1 x1

Figure 2.1: Non-collective states a) |11i, b) |12i, c) |13i, d) |22i

9 1 1

0.5 0.5

2 0 2 0 x x

-0.5 -0.5

a) -1 b) -1 x1 x1 1 1

0.5 0.5

2 0 2 0 x x

-0.5 -0.5

c) -1 d) -1 x1 x1

Figure 2.2: Collective states a) |11i + 0.6|22i + 0.2|33i, b) |12i + 0.4|23i, c) |13i + 0.5|24i + 0.5|35i, d) |11i + 0.8|12i + 0.32|22i bound state, which is manifested by concentration of wavefunction along the diagonal x1 = x2. It is demonstrated on gure 2.2. As can be seen from gures, to achieve more tightly bound state, we need to add higher wavefunctions. This must be compensated by a stronger interaction term in Hamiltonian. The bound state can itself oscilate in the box (gures 2.2b, c). It was argued in previous section that one-body density cannot distinguish collective states from simple Slater determinants. This can be directly seen on the gures 2.1d and 2.2b, which lead to nearly the same density (during the calculation one coordinate is integrated out). To establish a link between the density functional theory and collective states, it is neccesary to use coherent states [1] (p. 412). They are created by linear combination of energy eigenfunctions and so they evolve in time. This gives them clear classical interpretation of oscillating wave-packets. Our example on gure 2.2d was constructed by the following procedure: We dene a bosonic operator

1 Cˆ+ = √ (ˆa+ aˆ +a ˆ+ aˆ ) (2.35) 2 2↑ 1↑ 2↓ 1↓ and let it act on a state |11i

Cˆ+|11i = |12i Cˆ+|12i = |22i (2.36) 1 1 1 √ ψ1ψ1(↑↓ − ↓↑) → (ψ1ψ2 + ψ2ψ1)(↑↓ − ↓↑) → √ ψ2ψ2(↑↓ − ↓↑) 2 2 2

10 The coherent state is created by action of the exponential of the operator Cˆ+:

2 + a eaCˆ = 1 + aCˆ+ + Cˆ+Cˆ+ + ... (2.37) 2 In this case a = 0.8 was chosen

e0.8Cˆ+ |11i = |11i + 0.8|12i + 0.32|22i (2.38)

Wave-packet can be clearly distinguished from the ground state by the density functional (because it can be described by one Slater determinant by Thouless theorem [1], p. 615) and will be used instead of pure bosonic excitation in our derivation of SRPA later.

2.4 BCS: Pairing description

Description of nuclear pairing is similar to the description of superconductivity developed by Bardeen, Cooper and Schrieer [12] (they assumed attractive interaction between electrons of opposite momenta). This description was later adapted also for nuclear physics, where it is in agreement with spin 0 ground state and large energy gap between the ground state and the rst excited state in even-even nuclei, and also with variation of binding energy between even and odd nuclei. In the following derivation we rst consider even-even nuclei. The short-range nature of nucleon-nucleon force suggests that correlated motion of nucleons is energetically favorable. This requires that nucleons in the correlated pair have wavefunctions with high positive overlap (we mean pp overlap h0|δ(x1 − x2)|ψ1ψ2i, not ph hψ1|δ(x1 − x2)|ψ2i which is always zero for orthogonal wavefunctions). It is best achieved for given state and its time reversal (ψj and ψ¯j, see appendix A). Correlation itself is accomplished by superposition with excited pairs (see gure 2.2a). This is possible only for pairs near Fermi level and in the presence of low-lying unoccupied states. Particles in the pair then feel higher than average density and this eect must be included into Hamiltonian separately. This can be eectively achieved by the following term:

ˆ X + + (2.39) Hpair = −G aˆj aˆ¯j aˆk¯aˆk j,k>0

ˆ Exact solution involving Hpair would require diagonalization of the many-body wavefunction in the set of Slater determinants created by 2ph (and higher) excitations of the ground state. This is numerically dicult, so following approximation is taken: We require that BCS state is in the form of one Slater determinant created on the basis of Thouless theorem [1] (p. 229, 615):

v Y k + + Y + + (2.40) |BCSi = uk exp aˆk aˆk¯ |0i = (uk + vkaˆk aˆk¯ )|0i uk k>0 k>0 Normalization condition then requires

Y ∗ ∗ + + Y 2 2 ¯ 1 = hBCS|BCSi = h0| [(uk + vkaˆkaˆk)(uk + vkaˆk¯ aˆk )]|0i = (|uk| + |vk| ) k>0 k>0

11 2 2 |uk| + |vk| = 1 (2.41)

For realistic potentials, real and positive uk and vk lead to the lowest energy [1] (p. 228). It is clear that the BCS state doesn't conserve the number of particles, but we can at least impose a correct mean value:

ˆ X + + X 2 (2.42) N = hBCS|N|BCSi = hBCS| (ˆak aˆk +a ˆk¯ aˆk¯)|BCSi = 2vk k>0 k>0 To obtain BCS ground state, we begin with the Hartree-Fock mean-eld Hamil- tonian (2.8) combined with the pairing term (2.39):

X + + X + + ˆ ¯ (2.43) H = εk(ˆak aˆk +a ˆk¯ aˆk¯) − G aˆj aˆ¯j aˆkaˆk k>0 j,k>0 In the above expression the ground-state energy was omitted for simplicity. For correct implementation of the variational principle, we need to x average particle number. This leads to a chemical potential µ (Fermi energy) as a Lagrange multiplier. 0 = δhBCS|Hˆ 0|BCSi = δhBCS|Hˆ − µNˆ|BCSi (2.44) ˆ ˆ X + + X 2 hHAV − µNi = hBCS| (εk − µ)(ˆak aˆk +a ˆk¯ aˆk¯ )|BCSi = 2vk(εk − µ) k>0 k>0 2 X + + X 2 X X X 4 hBCS| aˆj aˆ¯j aˆk¯aˆk|BCSi = vk + ujvjukvk = ukvk + vk j,k>0 k>0 j6=k k>0 k>0

The variation then means a dierentiation with respect to vk: ∂ ∂ h X X q 2 X i hBCS|Hˆ 0|BCSi = 2v2(ε − µ) − G v 1 − v2 − G v4 ∂v ∂v j j j j j k k j>0 j>0 j>0

X v2 = 4v (ε − µ − Gv2) − 2G u v u − k = 0 (2.45) k k k j j k u j>0 k

It is convenient to introduce new quantities: energies ε˜k and a gap ∆ (the meaning of the gap will become clear in the next section). 2 X (2.46) ε˜k = εk − µ − Gvk ∆ = G ukvk k>0 Then we will obtain expression for 2: vk ∆ 1 1 u v ε˜ = (u2 − v2) = ∆ − v2 → (1 − v2)v2ε˜2 = ∆2 − v2 + v4 k k k 2 k k 2 k k k k 4 k k 2 2 2 2 2 2 ∆ 4 2 ∆ (˜εk + ∆ )(1 − vk)vk = → vk − vk + 2 2 = 0 4 4(˜εk + ∆ ) s 1 1 ∆2 1 ε˜ 1 ε˜ v2 = − − = 1 − k u2 = 1 + k k 2 4 4(˜ε2 + ∆2) 2 p 2 2 k 2 p 2 2 k ε˜k + ∆ ε˜k + ∆ (2.47)

12 And nally the gap equation:

X X ∆ 1 G X ∆ ∆ = G u v = G − v2 = k k k p 2 ε˜k 2 2 2 k>0 k>0 k>0 ε˜k + ∆

G X 1 1 = (2.48) 2 p 2 2 k>0 ε˜k + ∆ For the weak pairing force G, there is a possibility that the gap equation has no solution. Then we are left with ∆ = 0 and the ground state is unchanged (i.e. uk = 0, vk = 1 for εk < µ and uk = 1, vk = 0 for εk > µ). Equations (2.47) and (2.48) can be solved iteratively ( 2 εk, µ → ∆ → vk → ). Term 2 in is usually neglected and then the equations of the BCS N... Gvk ε˜k problem are solved in one step.

G X 1 1 εk − µ 1 = v2 = 1 − (2.49) 2 p 2 2 k 2 p 2 2 k>0 (εk − µ) + ∆ (εk − µ) + ∆ 2.4.1 Quasiparticle formalism For treatment of excited states and odd nuclei, it is convenient to dene quasiparticle creation and annihilation operators (i.e. Bogoljubov transformation):

+ + + + ¯ αˆk = ukaˆk − vkaˆk αˆk = ukaˆk − vkaˆk¯ {αˆj, αˆk } = δjk + + + (2.50) ¯ ¯ αˆk¯ = ukaˆk¯ + vkaˆk αˆk = ukaˆk + vkaˆk {αˆj, αˆk} = 0 BCS ground state is then a vacuum for and : αˆk αˆk¯

+ Y + + αˆk|BCSi = (ukaˆk − vkaˆk¯ ) (uj + vjaˆj aˆ¯j )|0i j Y + + + + (2.51) = (uj + vjaˆj aˆ¯j )(ukvkaˆk¯ − vkukaˆk¯ )|0i = 0 j6=k

Excited states are formed by action of + and +. αˆk αˆk¯

When we choose αˆk with εk µ (vk ≈ 1), it acts as an annihilation operator for ¯ (there is no ¯ in ), but when , it acts as +. Near Fermi level k k |ki εk µ aˆk µ, it cannot be so precisely dened, but its action removes states k and k¯ from correlated wavefunction (so called blocking efect [1], p. 237). The average number of particles in the state |ki is:

ˆ X + + X 2 2 (2.53) hk|N|ki = hk| (ˆaj aˆj +a ˆ¯j aˆ¯j )|ki = 1 + 2vj = N + 1 − 2vk j j6=k

13 The expectation value of Hˆ 0 is: 2 ˆ ˆ X 2 X X 4 (2.54) hk|H − µN|ki = εk − µ + 2vj (εj − µ) − G ujvj − G vj j6=k j6=k j6=k We would also like to evaluate particle and hole excitation energies. In that case it is necessary to correct wrong particle number [1] (p. 237):

dE Ek − EBCS = hk|Hˆ |ki − hBCS|Hˆ |BCSi + (hBCS|Nˆ|BCSi + 1 − hk|Nˆ|ki) N+1 N dN ˆ ˆ ˆ ˆ 2 = hk|H − µN|ki − hBCS|H − µN|BCSi + µ = (εk − µ)(1 − 2vk) 2 2 2 4 ∆ 2 2 +2∆ukvk − Gukvk + Gvk + µ = εk − µ + − Gvk (1 − 2vk) + µ ε˜k ε˜2 + ∆2 q k 2 2 2 (2.55) = (1 − 2vk) + µ = ε˜k + ∆ + µ ε˜k q k BCS 2 2 (2.56) EN−1 − EN = ε˜k + ∆ − µ

Similar procedure can be applied to excitations ( + + ), and then it 1ph αˆp αˆh |BCSi becomes clear why ∆ is called the gap (i.e. the rst excited state is at least 2∆ above the ground state). q q ph BCS 2 2 2 2 2 2 2 2 (2.57) EN − EN = ε˜p + ∆ + ε˜h + ∆ − G(upvh + uhvp) & 2∆

A term proportional to 4 can be neglected and we can express the mean-eld Gvk Hamiltonian as (considering E0 = 0): q ˆ X 2 2 + + (2.58) HAV = ε˜k + ∆ (ˆαk αˆk +α ˆk¯ αˆk¯ ) k>0 2.5 RPA: Collective vibrations

Excited states of the even-even nucleus can be compared to the excitations of harmonic oscillator and described by phonon creation and annihilation (bosonic) operators [11]

1 X (ν−) (ν+) 1 X (ν+)∗ (ν−)∗ Cˆ+ = (c aˆ+aˆ − c aˆ+aˆ ) Cˆ = (−c aˆ+aˆ + c aˆ+aˆ ) ν 2 ph p h ph h p ν 2 ph p h ph h p p,h p,h (2.59) The nucleus can be excited to various modes, which are labeled by index ν. By the operators ˆ+ we can describe simple excitations as well as collective Cν 1ph states with contribution of many 1ph congurations. Ground state with respect to these excitations will be labeled by |RPAi (RPA stands for Random Phase Approximation) [1] (p. 302)

ˆ ˆ+ (2.60) Cν|RPAi = 0 Cν |RPAi = |νi

Energy of the state ν will be denoted Eν ˆ ˆ H|νi = Eν|νi H|RPAi = E0|RPAi

14 ˆ ˆ+ ˆ+ (2.61) [H, Cν ]|RPAi = (Eν − E0)Cν |RPAi The solution of RPA problem then consists of the solution of operator equation

ˆ ˆ+ ˆ+ (2.62) [H, Cν ] = ~ωνCν ~ων = Eν − E0 It can be solved by comparison of coecients at + and + (this will be aˆp aˆh aˆh aˆp done in SRPA approach later), or by evaluation of 1ph matrix elements followed by a matrix eigenvalue problem [1] (p. 303) this will be briey described here. We multiply equation (2.61) from left by hRPA|δCˆ, where δCˆ is a shorthand for + or + . aˆp aˆh aˆh aˆp

ˆ ˆ ˆ+ ˆ ˆ+ (2.65) hRPA|[δC, [H, Cν ]]|RPAi = ~ωνhRPA|[δC, Cν ]|RPAi In the following, the RPA ground state will be replaced by the HF ground state. By this, ground-state correlations are ignored, that is, we assume that (ν+) 2ph cph in (2.59) are small. ˆ is replaced rst by + , and then by + : δC aˆh aˆp aˆp aˆh

[A, [B,C]]† = (ABC − ACB − BCA + CBA)† = C†B†A† − B†C†A† − A†C†B† + A†B†C† = [A†, [B†,C†]] (2.68)

We will use also (2.78)

+ + + + + + (2.69) [âp aˆh , aˆp0 aˆh0 ] = 0 [âp aˆh , aˆh0 aˆp0 ] = δhh0 aˆp aˆp0 − δpp0 aˆh0 aˆh 1 1 hHF|[â+aˆ , Cˆ+]|HFi = c(ν−) hHF|[â+aˆ , Cˆ+]|HFi = c(ν+) (2.70) h p ν 2 ph p h ν 2 ph By substitution of (2.59) into (2.66) and using (2.67), (2.70) we obtain matrix equation AB c(ν−) 1 0 c(ν−) = ω (2.71) B∗ A∗ c(ν+) ~ ν 0 −1 c(ν+)

15 As can be seen, equation (2.71) involves matrices of dimension proportional to the dimension of 1ph space. This eigenvalue problem requires large numerical eort and will be referred to as a full RPA. Finally, a normalization of (ν±) can be xed by imposing the correct commu- cph tation relations on ˆ ˆ+ using (2.70) Cν , Cν

1 X (ν−) (ν+) 1 = hHF|[Cˆ , Cˆ+]|HFi = (|c |2 − |c |2) (2.72) ν ν 4 ph ph ph

2.5.1 Evaluation of matrix elements of 1ph operators In the following, we will frequently represent one-particle operators by creation and annihilation operators.

ˆ X ˆ + (2.73) A = (j|A|k)ˆaj aˆk j,k

Round brackets are used to denote matrix elements of Aˆ in one-particle Hilbert space Z ˆ ∗ ˆ (2.74) (j|A|k) = dx ψj (x)Aψk(x) Following notation will be used for the Hartree-Fock ground state and the excited states (particle and hole states):

n Y + + (2.75) |i = |HFi = aˆj |0i |pi =a ˆp |HFi |hi =a ˆh |HFi j=1

The 1ph excitation is then understood as

+ (2.76) |phi =a ˆp aˆh |HFi and its matrix element with the HF ground state: Z ˆ ∗ ˆ ˆ (2.77) h|A|phi = dx ψhAψp = (h|A|p)

This can be seen by expansion of Slater determinants (ψp replaces ψh; we consider orthogonal set {ψk}). Next, we will evaluate expectation value of a commutator hHF|[A,ˆ Bˆ]|HFi:

+ + + + + + [âk aˆl , aˆmaˆn] =a ˆk [âl , aˆmaˆn] + [âk , aˆmaˆn]âl + + + + + + + + =a ˆk {aˆl , aˆm}aˆn − aˆk aˆm{aˆl , aˆn} + {aˆk , aˆm}aˆnaˆl − aˆm{aˆk , aˆn}aˆl + + (2.78) =a ˆk aˆnδlm − aˆmaˆl δkn

X hHF|[A,ˆ Bˆ]|HFi = (h|Aˆ|k)(k|Bˆ|h) − (k|Aˆ|h)(h|Bˆ|k) h,k X = (h|Aˆ|p)(p|Bˆ|h) − (p|Aˆ|h)(h|Bˆ|p) (2.79) p,h

16 (p is used for particle states, h for hole states and k, l, m, n for arbitrary states). We will restrict ourselves to hermitian operators Aˆ, Bˆ and use (A.15): ˆ A ¯ ˆ ¯ ˆ ¯ A ˆ ¯ ¯ ˆ A ¯ ˆ (j|A|k) = γT (k|A|j)(j|A|k) = −γT (k|A|j)(j|A|k) = −γT (k|A|j) Considering that p and h wavefunctions come in pairs with time reversed coun- terparts in the HF ground state, we can further simplify (2.79):

ˆ ˆ A B X ˆ ˆ A B X ˆ ˆ hHF|[A, B]|HFi = (1 − γT γT ) (h|A|p)(p|B|h) = (1 − γT γT ) h|A|phihph|B|i p,h p,h (2.80) We see that only the commutator of time-even and time-odd operator has a non-zero expectation value (for two time-even or time-odd operators we get zero): X X hHF|[A,ˆ Bˆ]|HFi = 2 (h|Aˆ|p)(p|Bˆ|h) = 2 h|Aˆ|phih|Bˆ|phi∗ (2.81) p,h p,h For the treatment of BCS ground and excited states, it is necessary to evaluate commutators of quasiparticle creation and annihilation operators. Because |BCSi is a ground state with respect to all quasiparticle annihilation operators, we have the following correspondence between |HFi and |BCSi: + + + + ¯ (2.82) aˆp 7→ αˆp aˆp 7→ αˆp aˆh 7→ αˆh¯ aˆh 7→ αˆh so there is no longer distinction between particle and hole states. Corresponding commutators are:

+ + + + + + [ˆαk αˆl , αˆmαˆn] = δlmαˆk αˆn − δlnαˆk αˆm − δkmαˆl αˆn + δknαˆl αˆm + δkmδln − δknδlm + [ˆαk αˆl , αˆmαˆn] = (δkmαˆn − δknαˆm)ˆαl [ˆαkαˆl, αˆmαˆn] = 0 + + + + + + + + + + (2.83) [ˆαk αˆl , αˆmαˆn ] =α ˆk (δlmαˆn − δlnαˆm) [ˆαk αˆl , αˆmαˆn ] = 0 It can be seen that only the commutator [ˆα+αˆ+, αˆαˆ] leads to a non-zero expectation value in the BCS state. 1ph operators can be rewritten into the quasiparticle representation by the inverse Bogoljubov transformation (see (2.50)):

+ + + ¯ aˆk = ukαˆk + vkαˆk aˆk = ukαˆk + vkαˆk¯ + + + (2.84) ¯ ¯ aˆk¯ = ukαˆk¯ − vkαˆk aˆk = ukαˆk − vkαˆk For a hermitian operator ˆ with time parity A we have (for even-even nucleus): A γT X + A + + + ˆ ˆ ˆ ¯ ¯ ¯ ˆ A = [(j|A|k)(âj aˆk + γT aˆk¯ aˆ¯j) + (j|A|k)âj aˆk + (j|A|k)â¯j aˆk ] j,k>0 X A + + A A ˆ ¯ = {(j|A|k)[(ujvk + γT vjuk)(ˆαj αˆk¯ + γT αˆjαˆk) + (1 + γT )vjvkδjk j,k>0 A + A + +(ujuk − γT vjvk)(ˆαj αˆk + γT αˆk¯ αˆ¯j)] ˆ ¯ + + + + +(j|A|k)(−ujvkαˆj αˆk + vjukαˆ¯jαˆk¯ + ujukαˆj αˆk¯ + vjvkαˆk αˆ¯j) ¯ ˆ + + + + (2.85) +(j|A|k)(ujvkαˆ¯j αˆk¯ − vjukαˆjαˆk + ujukαˆ¯j αˆk + vjvkαˆk¯ αˆj)} Commutator of time-even Aˆ and time-odd Bˆ is then: ˆ ˆ X 2 2 ˆ ˆ ˆ ¯ ¯ ˆ (2.86) hBCS|[A, B]|BCSi = 2(vj − vk)[(j|A|k)(k|B|j) + (j|A|k)(k|B|j)] j,k>0 Commutator of two time-even or time-odd operators is again zero.

17 2.5.2 SRPA in wavefunction formalism

Hartree-Fock method is used to nd a set {ψj}j≤n which minimizes energy. This can be compared to nding a value ~x0 where a function f(~x) has its minimum ~ (∇f(~x0) = 0). 1 f(~x) = f(~x ) + ∇~ f(~x ) · (~x − ~x ) + [∂ ∂ f(~x )](~x − ~x ) (~x − ~x ) + ... (2.87) 0 0 0 2 i j 0 0 i 0 j Evaluation of excited states (by full RPA) then corresponds to the diagonalization of matrix ∂i∂jf(~x0) and treating its eigenvalues as squares of vibrational frequencies of harmonic oscillator. In practice, at the beginning of solution of the HF problem, we need to evaluate matrix elements of the Hamiltonian Hˆ = Tˆ + Vˆ (where Tˆ is one-body and ˆ two-body operator) in a trial basis , i.e. φ R ∗ ˆ ( 2 el- V {φj}j=1...N Tij = dx φi T φj N ements) and φ R ∗ ∗ ˆ ( 4 elements). The wavefunctions Vijkl = dx φi φj V φkφl N {ψj} obtained in iterations are linear combinations of the trial wavefunctions {φj}, and matrix elements of ˆ and ˆ in the basis can be computed from φ T V {ψj} Tij and φ by a simple matrix multiplication (change of basis ). ψ is then Vijkl φ → ψ hij obtained by corresponding antisymmetrization and traces. The diagonalization is then performed on the matrix ψ . However, computation of a full RPA N × N hij is equivalent to the diagonalization of a matrix of full Hamiltonian ψ (anti- Hijkl symmetrized), with the simplication that only elements of ψ are taken 2ph Hijkl 2 2 into account in matrices Aph,p0h0 and Bph,p0h0 . These large matrices (with O(n N ) elements) are neccesary if we want to use whole information contained in ψ . Vijkl It is therefore desirable to somehow extract signicant information from ψ Hijkl into a matrix of smaller dimension. We will assume that the residual interaction ˆ Vres from the full Hamiltonian (2.8) ˆ ˆ ˆ ˆ X + + (2.88) H = HAV + Vres HAV = εj(ˆaj aˆj +a ˆ¯j aˆ¯j) j>0 can be to a large accuracy written in a separable form [11]

sep 1 X Vˆ (x , x ) = − [κ 0 Xˆ (x )Xˆ 0 (x ) + η 0 Yˆ (x )Yˆ 0 (x )] (2.89) res i j 2 kk k i k j kk k i k j k,k0 with the ad-hoc property that we exclude the exchange term in evaluation of its ˆ ˆ matrix elements. We also distinguish time-even operators Xk and time-odd Yk. ˆ We will apply Hartree-Fock variational procedure to our Hamiltonian HSRPA = ˆ ˆ sep to obtain single-particle Hamiltonian ˆ in analogy with (2.6) HAV + Vres h ˆ ˆ X ˆ ˆ ˆ ˆ hSRPA = h0 − (κkk0 XkhΨ|Xk0 |Ψi + ηkk0 YkhΨ|Yk0 |Ψi) (2.90) kk0 ˆ where we omited the exchange term. h0 stands for a single-particle version of ˆ the mean-eld Hamiltonian HAV. The ground-state wavefunction was denoted by |Ψi, in which we will do certain variations, which we believe to span possible vibrational excitations.

K ˆ ˆ Y −iqkPk −ipkQk |Ψ(qk, pk)i = e e |HFi (2.91) k=1 18 This ansatz is inspired by TDHF (time dependent Hartree-Fock) scaling transformation, where qk and pk are harmonic functions dependent on time [11]. Op- ˆ ˆ erators Qk are time-even and Pk are time-odd (we usually use simple multipole l operators, e.g. r (Yl,m + Yl,−m)). In practice, we always choose one of these operators and the other is derived by

ˆ ˆ ˆ ˆ ˆ ˆ Pk = i[H, Qk] or Qk = i[H, Pk] (2.92) for example (certain tricks used here will be explained later):

ˆ ˆ X ˆ ˆ ˆ (p|Pk|h) = iεph(p|Qk|h) − iηk0k00 (p|Yk0 |h)hHF|[Yk00 , Qk]|HFi k0,k00 ˆ ˆ = iεph(p|Qk|h) − (p|Yk|h) (2.93)

The wavefunction |Ψi perturbed in parameter qk or pk then reads

ˆ 2 ˆ 2 (2.94) |Ψ(qk)i = (1−iqkPk +O(qk))|HFi |Ψ(pk)i = (1−ipkQk +O(pk))|HFi

Single-particle Hamiltonian (2.90) in perturbed mean eld (qk, pk 6= 0) is then

ˆ(q,p) ˆ ˆ ˆ ˆ ˆ ˆ ˆ (2.95) hSRPA = h0 +iqkκk0k00 Xk0 hHF|[Xk00 , Pk]|HFi+ipkηk0k00 Yk0 hHF|[Yk00 , Qk]|HFi

ˆ ˆ ˆ 0 ˆ 0 where we assumed hHF|Xk|HFi = hHF|Yk|HFi = 0 (i.e. (h|Xk|h ) = (h|Yk|h ) = ˆ 0; mean eld of the ground state is already described by h0). We also used implicit summation over repeated indices k, k0, k00 this will be used from here on. Up to ˆ ˆ now we have not xed normalization nor chosen basis of sets {Xk} and {Yk}. We can do it now by imposing

ˆ ˆ ˆ ˆ Xk = iκk0k00 Xk0 hHF|[Xk00 , Pk]|HFi (2.96) ˆ ˆ ˆ ˆ Yk = iηk0k00 Yk0 hHF|[Yk00 , Qk]|HFi Then the single-particle Hamiltonian is

ˆ(q,p) ˆ ˆ ˆ (2.97) hSRPA = h0 + qkXk + pkYk and inverses of the strength matrices κkk0 and ηkk0 are

−1 ˆ ˆ X ˆ ˆ κkk0 = ihHF|[Xk, Pk0 ]|HFi = 2i (h|Xk|p)(p|Pk0 |h) p,h −1 ˆ ˆ X ˆ ˆ (2.98) ηkk0 = ihHF|[Yk, Qk0 ]|HFi = 2i (h|Yk|p)(p|Qk0 |h) p,h where commutators were evaluated using (2.81). So far, only our special form of ˆ sep was considered. It is time to derive Vres ˆ ˆ response operators Xk and Yk directly from the original Hamiltonian. This can be done by using (2.94) in (2.6) (again omitting exchange term)

19 and comparing it to (2.97):

ˆ ˆ ˆ Xk(x0) = −ihHF|[V (x0, x), Pk(x)]|HFi (2.100) ˆ ˆ ˆ Yk(x0) = −ihHF|[V (x0, x), Qk(x)]|HFi Or as explicit matrix elements: Z Z ˆ X h ˆ ˆ i (i|Xk|j) = −2i dx1dx2 ψi(x1)ψh(x2)V ψj(x1)ψp(x2) dx ψp(x)Pkψh(x) p,h Z Z ˆ X h ˆ ˆ i (i|Yk|j) = −2i dx1dx2 ψi(x1)ψh(x2)V ψj(x1)ψp(x2) dx ψp(x)Qkψh(x) p,h (2.101) Only the time-even part of Vˆ contributes to the rst equation and only the time- odd part of Vˆ contributes to the second. Now, when we obtained separable residual interaction with corresponding matrix elements, it is time to solve the RPA equations (2.62) [2]

ˆ ˆ+ ˆ+ (2.102) [H, Cν ] = ~ωνCν Let us remind the prescription for the Hamiltonian (2.88), (2.89) and ˆ+ (2.59): Cν

X + 1 Hˆ = ε aˆ aˆ − (κ 0 Xˆ Xˆ 0 + η 0 Yˆ Yˆ 0 ) SRPA j j j 2 kk k k kk k k j

1 X (ν−) (ν+) Cˆ+ = (c aˆ+aˆ − c aˆ+aˆ ) ν 2 ph p h ph h p p,h After substitution of these equations into (2.102) and using (2.78), we are interested in 1ph part of the obtained operator equation. However, we will encounter also terms like ˆ ˆ ˆ+ which are obviously . We need to extract eective Xk[Xk0 , Cν ] 2ph ˆ (mean-eld) 1ph part of this term. Since hHF|Xk|HFi = 0 and we omit exchange terms, the only possibility is ˆ ˆ ˆ+ (it will be denoted as XkhHF|[Xk0 , Cν ]|HFi ˆ ˆ ˆ+ for simplicity). We will also use . Resulting equation Xkh[Xk0 , Cν ]i εph = εp − εh is:

1 X (ν−) + (ν+) + + + (ε c aˆ aˆ + ε c aˆ aˆ ) − κ 0 Xˆ h[Xˆ 0 , Cˆ ]i − η 0 Yˆ h[Yˆ 0 , Cˆ ]i 2 ph ph p h ph ph h p kk k k ν kk k k ν p,h

1 X (ν−) (ν+) = ω (c aˆ+aˆ − c aˆ+aˆ ) (2.103) 2~ ν ph p h ph h p p,h

By comparison of coecients at + and + we obtain: aˆp aˆh aˆh aˆp

1 (ν−) + + 1 (ν−) ε c − κ 0 (p|Xˆ |h)h[Xˆ 0 , Cˆ ]i − η 0 (p|Yˆ |h)h[Yˆ 0 , Cˆ ]i = ω c 2 ph ph kk k k ν kk k k ν 2~ ν ph 1 (ν+) + + 1 (ν+) ε c − κ 0 (h|Xˆ |p)h[Xˆ 0 , Cˆ ]i − η 0 (h|Yˆ |p)h[Yˆ 0 , Cˆ ]i = − ω c 2 ph ph kk k k ν kk k k ν 2~ ν ph (2.104) We will dene , (notation is inspired by correspondence ˆ ˆ ˆ+) q¯k p¯k iqkPk, ipkQk ↔ Cν ˆ ˆ+ ˆ ˆ+ (2.105) q¯k = κkk0 h[Xk0 , Cν ]i p¯k = ηkk0 h[Yk0 , Cν ]i

20 and evaluate values of the commutators by (2.79) (we cannot use (2.81) because ˆ+ is not hermitian and has no denite time parity) Cν

−1 ˆ ˆ+ 1 X ˆ (ν−) ˆ (ν+) κ 0 q¯ 0 = h[X , C ]i = [(h|X |p)c + (p|X |h)c ] kk k k ν 2 k ph k ph p,h −1 ˆ ˆ+ 1 X ˆ (ν−) ˆ (ν+) η 0 p¯ 0 = h[Y , C ]i = [(h|Y |p)c + (p|Y |h)c ] (2.106) kk k k ν 2 k ph k ph p,h Coecients (ν±) can be obtained from (2.104): cph ˆ ˆ ˆ ˆ (ν−) q¯k0 (p|Xk0 |h) +p ¯k0 (p|Yk0 |h) (ν+) q¯k0 (h|Xk0 |p) +p ¯k0 (h|Yk0 |p) cph = 2 cph = 2 εph − ~ων εph + ~ων (2.107) This can be substituted into (2.106) to get

−1 (XX) (XY ) κ 0 q¯ 0 = F 0 q¯ 0 + F 0 p¯ 0 kk k kk k kk k (2.108) −1 (YX) (YY ) ηkk0 p¯k0 = Fkk0 q¯k0 + Fkk0 p¯k0 where we dened (using time reversal symmetry (A.15) and considering the even- even nucleus)

ˆ ˆ ˆ ˆ (XX) X 2εph(h|Xk|p)(p|Xk0 |h) (XY ) X 2~ων(h|Xk|p)(p|Yk0 |h) Fkk0 = 2 2 Fkk0 = 2 2 ε − ( ων) ε − ( ων) p,h ph ~ p,h ph ~ ˆ ˆ ˆ ˆ (YX) X 2~ων(h|Yk|p)(p|Xk0 |h) (YY ) X 2εph(h|Yk|p)(p|Yk0 |h) Fkk0 = 2 2 Fkk0 = 2 2 ε − ( ων) ε − ( ων) p,h ph ~ p,h ph ~ (2.109) Equations (2.108) can be rewritten into a compact matrix form:

F (XX) − κ−1 F (XY ) q¯ 0 DR~ = = (2.110) F (YX) F (YY ) − η−1 p¯ 0 where we dened hermitian matrix D (D† = D) and vector R~ = (¯q, p¯)>. We can nd RPA solutions ων by imposing det D = 0. Since F matrices (2.109) depend on ων in a non-linear way, it is possible, in principle, to obtain much more roots than a dimension of . If we are interested also in structure coeecients (ν±), it D cph ~ is neccesary for given solution ων to nd eigenvector R of D (2.110), substitute it into (2.107) and normalize according to (2.72)

1 X (ν−) (ν+) 1 = (|c |2 − |c |2) 4 ph ph ph or we can normalize directly R~ (this will be useful for calculation of a strength function):

∗ ˆ ∗ ˆ ˆ ˆ (ν−) 2 [¯qk(h|Xk|p) +p ¯k(h|Yk|p)][(p|Xk0 |h)¯qk0 + (p|Yk0 |h)¯pk0 ] |cph | = 4 2 (εph − ~ων) ∗ ˆ ∗ ˆ ˆ ˆ (ν+) 2 [¯qk(p|Xk|h) +p ¯k(p|Yk|h)][(h|Xk0 |p)¯qk0 + (h|Yk0 |p)¯pk0 ] |cph | = 4 2 (εph + ~ων) 21 ˆ ˆ ∗ ˆ ˆ ∗ X n4εph~ων[(h|Xk|p)(p|Xk0 |h)¯qkq¯k0 + (h|Yk|p)(p|Yk0 |h)¯pkp¯k0 ] 1 = 2 2 2 [ε − ( ων) ] p,h ph ~ 2 2 ˆ ˆ ∗ ˆ ˆ ∗ 2[εph + (~ων) ][(h|Xk|p)(p|Yk0 |h)¯qkp¯k0 + (h|Yk|p)(p|Xk0 |h)¯pkq¯k0 ]o + 2 2 2 [εph − (~ων) ] ∂D 0 ∗ kk (2.111) = Rk Rk0 ∂(~ων) 2.5.3 SRPA in density formalism Formulation of SRPA residual interaction in the density formalism is somewhat less cumbersome because we don't need to take care of quantum eects (the exchange term). We can write separable residual interaction directly as a density functional: 1 V = − [κ 0 Tr(Xˆ ρˆ)Tr(Xˆ 0 ρˆ) + η 0 Tr(Yˆ ρˆ)Tr(Yˆ 0 ρˆ)] (2.112) res 2 kk k k kk k k

We proceed to eective single-particle operator vˆres by variation of δψ similar to (2.14) ˆ ˆ ˆ ˆ vˆres = −[κkk0 XkTr(Xk0 ρˆ) + ηkk0 YkTr(Yk0 ρˆ)] (2.113) Instead of the ground-state density ρˆ = |HFihHF| we use a perturbed density with the wavefunction Ψ (2.94) to obtain in analogy with (2.97)

(q,p) ˆ ˆ (2.114) vˆres = qkXk + pkYk Procedure similar to (2.99) applied to the DFT eective Hamiltonian (2.14)

Z X δH hˆ = dx Jˆ (x ) 0 δJ (x ) α 0 α α 0 leads to

Z 2 (q,p) X δ H ˆ ˆ ˆ ˆ ˆ vˆres = dx0dx1 Jα(x0){−iqkh[Jα0 (x1), Pk]i−ipkh[Jα0 (x1), Qk]i} δJα(x0)δJα0 (x1) α,α0 (2.115) from which we can obtain [11,13]

Z 2 ˆ X δ H ˆ ˆ ˆ Xk = −i dx0dx1 Jα(x0)h[Jα0 (x1), Pk]i δJα(x0)δJα0 (x1) α,α0 Z 2 ˆ X δ H ˆ ˆ ˆ Yk = −i dx0dx1 Jα(x0)h[Jα0 (x1), Qk]i (2.116) δJα(x0)δJα0 (x1) α,α0 and the inverse strengths matrices (2.98) these matrices are symmetric in k, k0:

Z 2 −1 X δ H ˆ ˆ ˆ ˆ κkk0 = dx0dx1 h[Jα(x0), Pk0 ]ih[Jα0 (x1), Pk]i δJα(x0)δJα0 (x1) α,α0 Z δ2H −1 X ˆ ˆ ˆ ˆ (2.117) ηkk0 = dx0dx1 h[Jα(x0), Qk0 ]ih[Jα0 (x1), Qk]i δJα(x0)δJα0 (x1) α,α0

22 Then, we can continue with solution of the RPA equation as in previous section. Now we can briey mention the treatment of SRPA over the BCS ground state. Densities are calculated with 2 weighting factors. Matrix elements of the vk ˆ ˆ ˆ ˆ operators Qk, Pk, Xk, Yk are taken as before, but quasiparticle representation is according to (2.85):

X + + + + ˆ ˆ ¯ ¯ ¯ ¯ ˆ A = [(j|A|k)(vjukαˆjαˆk − ujvkαˆj αˆk ) + (j|A|k)(ujvkαˆ¯j αˆk¯ − vjukαˆjαˆk) j,k>0 A + + A ˆ ¯ (2.118) +(j|A|k)(ujvk + γT vjuk)(ˆαj αˆk¯ + γT αˆjαˆk)] + Terms αˆ α,ˆ δjk were discarded, because they don't contribute to the commuta- ˆ ˆ tors. This is also required by the condition hBCS|Xk|BCSi = hBCS|Yk|BCSi = 0 as in (2.95). The operator ˆ+ can be expressed similarly. Solution of the RPA Cν equation is analogous to the previous case and won't be treated here. ˆ ˆ So far we haven't distinguished protons and neutrons. The operators Qsk, Psk will carry one index s ∈ p, n to indicate whether the proton or neutron mean-eld is perturbed. The inverse strength matrices will carry two indices −1 −1 . ηsk,s0k0 , κsk,s0k0 Matrix elements of response operators will carry two indices one to indicate perturbed mean-eld, and the other for identication of 1ph state in which the matrix element is calculated ( ), e.g. ˆ s . The phonon matrix ele- ph ∈ s (h|Xs0k0 |p) ments (ν±) will carry one index and matrices two indices. Dimension of cph,s Fsk,s0k0 ˆ ˆ matrix D is therefore 4× number of initial operators (Qk or Pk). More details can be found in [11].

2.5.4 Transition probabilities and strength function Reduced transition probability is calculated as a square of the matrix element of ˆ transition operator fλµ (it contains es, eective charge, or g-factor, whose value depends on a chosen ph ∈ s either proton or neutron state): ˆ ˆ ˆ ˆ ˆ hν|fλµ|RPAi = hRPA|Cνfλµ|RPAi = hRPA|[Cν, fλµ]|RPAi X 1 (ν−)∗ (ν+)∗ ≈ [(p|fˆ |h)c + (h|fˆ |p)c ] 2 λµ ph λµ ph p,h X ∗ (X) ∗ (Y ) X ∗ (2.119) = (¯qkAk +p ¯kAk ) = RkAk k k where we used (2.59), (2.79), (2.107)

1 X (ν+)∗ (ν−)∗ X Cˆ = (−c aˆ+aˆ + c aˆ+aˆ ) fˆ = (i|fˆ |j)ˆa+aˆ ν 2 ph p h ph h p λµ λµ i j p,h i,j

∗ ˆ ∗ ˆ ∗ ˆ ∗ ˆ (ν−)∗ X q¯k(h|Xk|p) +p ¯k(h|Yk|p) (ν+)∗ X q¯k(p|Xk|h) +p ¯k(p|Yk|h) cph = 2 cph = 2 εph − ων εph + ων k ~ k ~ and dened vector A~ = (A(X),A(Y ))> for an electric transition Eλµ (time-even ˆ fλµ): ˆ ˆ ˆ ˆ (X) X 2εph(p|fλµ|h)(h|Xk|p) (Y ) X 2~ων(p|fλµ|h)(h|Yk|p) (2.120) Ak = 2 2 Ak = 2 2 ε − ( ων) ε − ( ων) p,h ph ~ p,h ph ~

23 ˆ or for a magnetic transition Mλµ (time-odd fλµ): ˆ ˆ ˆ ˆ (X) X 2~ων(p|fλµ|h)(h|Xk|p) (Y ) X 2εph(p|fλµ|h)(h|Yk|p) (2.121) Ak = 2 2 Ak = 2 2 ε − ( ων) ε − ( ων) p,h ph ~ p,h ph ~

For calculation of ˆ 2 P ∗ ∗ we need to evaluate ∗ . |hν|fλµ|RPAi| = i,j Ri Rj Ai Aj Ri Rj P ∗ ∂Dkk0 We will employ the normalization condition 1 = 0 R Rk0 (2.111). Before kk k ∂(~ων ) that, we need to investigate some algebraic properties of the matrix D and its determinant [14]. Determinant can be expanded by j-th row:

X j+k (jk) X 0 = det D = (−1) Djk det D = Djkdjk (2.122) k k D(jk) is a submatrix created from the matrix D by omission of j-th row and k-th j+k (jk) column. We dene an algebraic supplement djk = (−1) det D . We see that fullls condition required of : P , but only for -th row. To see djk Rk k DjkRk = 0 j that it actually fullls it for any other row j0, we make use of another property of determinants: we can add j0-th row (j0 6= j) to the j-th row without changing the value of determinant. By this, algebraic supplements djk remain unchanged, because they don't depend on j-th row:

X j+k (jk) X X 0 = det D = (−1) (Djk+Dj0k) det D = (Djk+Dj0k)djk = 0+ Dj0kdjk k k k (2.123) djk is therefore proportional to Rk with a normalization factor depending on j. This can be expressed by (j is arbitrary): R d k = jk (2.124) Rk0 djk0

We will also need the following property of djk (of hermitian D): ∗ j+k (jk) ∗ j+k (jk)† j+k (kj) djk = (−1) (det D ) = (−1) det D = (−1) det D = dkj (2.125) Derivative of a determinant can be expressed by chain rule the determinant is a function of its own matrix elements:

∂ det D X ∂ det D ∂Dij X ∂Dij = = d (2.126) ∂( ω ) ∂D ∂( ω ) ij ∂( ω ) ~ ν i,j ij ~ ν i,j ~ ν Now we can rewrite the normalization condition (2.111) into a more usable form: ∗ ∗ X ∗ ∂Dkk0 ∗ X Rk ∂Dkk0 Rk0 ∗ X djk ∂Dkk0 dkk0 1 = Rk Rk0 = Ri Rj ∗ = Ri Rj ∗ ∂( ων) R ∂( ων) Rj d ∂( ων) dkj kk0 ~ kk0 i ~ kk0 ji ~ ∗ ∗ Ri Rj X ∂Dkk0 Ri Rj ∂ det D = dkk0 = (2.127) dij ∂( ων) dij ∂( ων) kk0 ~ ~ Reduced transition probability [1] (p. 589) (from ground state I = 0) is then: A d A∗ ˆ 2 X ∗ ∗ X i ij j (2.128) B(Eλµ/Mλµ; ν) = |hν|fλµ|RPAi| = Ri Rj Ai Aj = ∂ det D i,j i,j ∂(~ων )

24 Strength function of L-th order is dened as

X L (2.129) SL(Eλµ/Mλµ; ~ω) = (~ω) B(Eλµ/Mλµ; ν)δ(~ω − ~ων) ν To take into account experimental resolution, nite lifetime of excited states and other decay modes (emission of p, n, complex congurations), we use Lorentz smoothing function instead of δ function:

1 ∆ (2.130) δ∆(~ω − ~ων) = 2 2 2π (~ω − ~ων) + (∆/2) It is possible to evaluate strength function from matrix D without actually nding RPA solutions ων [11, 14]. To do that, we will need some properties of functions of complex variable. Analytic function f(z) with only rst-order poles which vanishes for z → ∞ can be expanded into a series: X 1 lim f(z) = 0 ⇒ f(x) = Res f(z) (2.131) z→∞ x − z z=zj j j

We can assume real and shift it by imaginary number ∆ : x i 2

∆ X 1 X x − zj − i∆/2 f x + i = Res f(z) = 2 2 Res f(z) 2 x − z + i∆/2 z=zj (x − z ) + (∆/2) z=zj j j j j (2.132) If we assume that f(z) has poles only on the real axis and their residues are real, the Lorentz function (2.130) can be recognized in the imaginary part of (2.132):

1 h ∆i X − = f x + i = δ∆(x − zj) Res f(z) (2.133) π 2 z=zj j Now we will investigate properties of the following function (derived from (2.128)) where we assume replacement (complex conjugation of is ~ων 7→ z Aj applied before substitution ): ~ων 7→ z ∗ X Ai (z)dij(z)Aj (z) f(z) = (2.134) det D(z) i,j This function has poles in (because ) and in . z = ±~ων det D(~ων) = 0 z = ±εph To see this, let us investigate its behavior in the limit z → εph (for given ph pair). From each matrix element Dij (2.109) survives only a term proportional to 2 2 −1 −1 −1. Denominator is then −N (εph − z ) = (εph − z) (εph + z) O((εph − z) ) −N−1 (N is a dimension of the matrix D) and numerator O((εph − z) ) (2.120). The function f(z) has therefore rst-order poles in ±εph. We can apply equation (2.133) to (2.134):

∗ 1 h ∆i X X Ai (z)dij(z)Aj (z) − = f x + i = δ∆(x − εph) Res + π 2 z=εph det D(z) ph i,j A d A∗ X X i ij j X (2.135) δ∆(x − ~ων) + ... ∂ det D ν i,j ∂(~ων ) ~ων −εph,−ων

25 The second term on the right-hand side can be identied with the strength function S0 (2.129) smoothed by δ∆, terms denoted by (...) will be neglected (they are small for x > 0). We will briey describe evaluation of the residue in the rst term. Matrix elements of D (2.109) and A (2.120) will be e.g.:

2 ˆ ˆ ˆ ˆ 0 2 (X) (Y )∗ 4εph(p|fλµ|h)(h|fλµ|p)(h|Xk|p)(p|Yk |h) lim (z − εph) Ak (z)Ak0 (z) = 2 z→εph (εph + εph) ˆ ˆ (XY ) 2εph(h|Xk|p)(p|Yk0 |h) (2.136) lim (z − εph)Fk,k0 (z) = − z→εph εph + εph The residue is then A (z)d (z)A∗(z) X i ij j ˆ ˆ X dij(εph)Dij(εph) Res = −(p|fλµ|h)(h|fλµ|p) z=εph det D(z) det D(ε ) i,j i,j ph ˆ ˆ = −(p|fλµ|h)(h|fλµ|p) (2.137) Overall result can be generalized to L = 0, 1, 2:

X L ˆ ˆ SL(Eλµ/Mλµ; ~ω) = (~ω) (p|fλµ|h)(h|fλµ|p)δ∆(~ω − εph) p,h ∗ 1 X Ai (z)dij(z)Aj (z) − = zL (2.138) π det D(z) ∆ i,j z=~ω+i 2 A numerator of the second term can be conveniently rewritten into a determinant:

∗ X ∗ 0 Aj Dij Ai (2.139) Ai (z)dij(z)Aj (z) = − det = − det ∗ Ai Dij A 0 i,j j We can interpret structure of (2.138) as a consisting of two parts: unperturbed ph excitation spectrum (rst term) and a contribution of the residual interaction (second term). We can express photon absorption cross-section using the strength function [10] (p. 2830):

X S2λ−1(Eλµ; E) + S2λ−1(Mλµ; E) λ + 1 σ(E) = 8π3α (2.140) ( c)2λ−2 λ[(2λ + 1)!!]2 λ,µ ~ ˆ This is valid for the following choice of operators fλµ:

ˆ(E) λ ∗ fλµ = esr [Yλµ(ϑ, ϕ) + Yλµ(ϑ, ϕ)] µ n 2 o fˆ(M) = N pλ(2λ + 1)rλ−1 qg [ˆs ⊗ Y ] + g [ˆl ⊗ Y ] (2.141) λµ ec s λ−1 λµ l λ + 1 λ−1 λµ

−19 where e = 1.602176 · 10 C is elementary charge, es is eective charge (in units 2 e e~ of e), α = = 1/137.036 is the ne structure constant, µN = is Bohr 4π0~c 2mp magneton (µN /ec = 0.105154 fm), gs,p = 5.5857 is the proton spin g-factor, gs,n = −3.8261 is the neutron spin g-factor; the orbital gyromagnetic factors are gl,p = 1, gl,n = 0. Quenching factor q = 0.7 is a correction for eective spin g-factors in a nucleus.

26 3. Numerical calculations

3.1 Skyrme parametrizations

In our calculations we used 6 Skyrme parametrizations: SkM∗, SGII, SkT6, SkI3, SLy6 and SV-bas. Because Skyrme parameters are underdetermined by static properties of the nuclei (ground state energy, radius), various other nuclear phenomena are employed in particular parametrizations. Here we describe their motivation and specic properties. SkM∗ is based on an older SkM [15] (1980) which was adjusted to the experimental monopole (E0) and dipole (E1) giant resonances of 208Pb and other magic nuclei using TDHF-based macroscopic uid-dynamical calculations and their comparison to RPA sum rules. SkM∗ [16] (1982) corrected low surface ten- 240 sion of SkM (modifying t1 and t2) to better reproduce the ssion barrier of Pu and also to yield to more accurate ground-state energies based on microscopic

HF+BCS. Both parametrizations have x1 = x2 = x3 = 0. SGII [17] (1981) is based on SkM with inclusion of spin-orbit term t4 and nonzero x1, x2, x3. Its aim was to reproduce spin-orbit splitting of 1p3/2 and 1p1/2 in 16O, pairing matrix elements and Gamow-Teller resonances of 48Ca, 50Zr and 208Pb by HF+TDA (Tamm-Danco approximation). SkT6 [18] (1984) was one of 11 various parametrizations which were investi- gated by authors to properly explore parameter space of Skyrme interaction. Fit was done mainly to well reproduce groud-state energies of 16O, 56Fe, 90Zr, 118Sn, 132Sn, 138Ba, 208Pb and charge radius of 208Pb. Moreover, SkT6 also reproduces neutron-skin thickness of heavy nuclei and keeps constant eective mass m∗ = m. Spectroscopic properties and compressibility are partially taken into account. SkI3 [19] (1995) was meant to correct too high spin-orbit splitting of SkM∗ (due to the incorrect isospin dependence) observed in a chain of Pb isotopes on a basis of charge radius (isotope shifts). A correction inspired by Relativistic Mean Field theory was to release 0 constraint (2.21) and set 0 . Fitting b4 = b4 b4 = 0 procedure was based on the ground-state energies, difraction radii and surface thicknesses of spherical light, medium and heavy nuclei. SLy6 [20] (1998) was proposed to better describe unsymmetric nuclear matter, in the rst place to agree with the assumed properties of neutron stars [21], but also compressibility, symmetric matter properties and nite nuclei (double magic 16O, 40,48Ca, 56Ni, 132Sn, 208Pb) were taken into account. Among various proposed SLy parametrizations, the tting of SLy6 was done with the center-of- mass correction included in the energy functional. SV-bas [22] (2009) is based on a large selection of data: either ground-state (energy, charge radius, surface thickness, odd-even staggering) as well as giant monopole, dipole and quadrupole resonances of 208Pb. The nuclei (mainly spherical) were chosen with the aim of low ground-state correlations. SV-bas employs density dependent pairing force.

27 3.2 Description of the codes

3.2.1 Skyax (HF and BCS calculation) Program Skyax was developed by prof. P.-G. Reinhard at University of Erlangen. It calculates wavefunctions and energies of axially deformed nuclei by Skyrme HF + BCS with a constaint on quadrupole deformation β. Its conguration le contains the input nucleus (Z, N), parameters of the used parametrization ( 0 ), pairing force strength ( or other), mesh spacing t0, x0, . . . t3, x3, b4, b4, α Gp,Gn in r and z directions (0.7 fm was used) and the upper bound of energy of conguration space (35 MeV was used). The deformation is specied by the interval

(βmin, βmax) and step βstep. In the case βmin < βmax the program will calculate energy of given nucleus for a series of deformation parameters β. From these we can choose a deformation with minimum energy and continue calculations with this deformation. Binary output of this program contains the wavefunctions as spinors evaluated on a mesh in cylindrical coordinates r and z (ϕ is treated analytically), their projection of angular momentum on z: Kj, one-particle energy εj, gap ∆j and occupation factor vj. Time-inverted states (K¯j < 0) are not stored (they can be easily calculated).

3.2.2 Skyax_me (calculation of matrix elements) ˆ ˆ ˆ ˆ ˆ Program Skyax_me calculates matrix elements of operators Qk, Pk, Xk, Yk, fλµ. The axial symetry enables to take advantage of selection rules for given µ (projection of angular momentum of the transition operator), so only the matrix elements of selected ph pairs are actually calculated. ˆ We choose either operators Qk √ ˆ 2 2 (l+p)/2 ∗ ˆ 2 2 ∗ (3.1) Qk = (r + z ) [Ylµ + Ylµ] or Qk = Jλ(qzλ r + z /rdiﬀ )[Yλµ + Yλµ] ˆ or Pk (in pairs for spin and orbital contribution) √ Pˆ(s) = pλ(2λ + 1)( r2 + z2)λ−1+p[ˆs ⊗ Y ] k √ λ−1 λµ (l) p 2 2 λ−1+p ˆ (3.2) Pk = λ(2λ + 1)( r + z ) [l ⊗ Yλ−1]λµ where zλ is the rst zero point of the Bessel function Jλ(zλ) = 0. We also choose operators ˆ with higher multipole ( for ) and for Bessel functions Qk l = 3 E1 √ q = 0.6, 0.9, 1.2. We can also choose higher powers of r2 + z2 by p = 0, 2, 4. ˆ ˆ ˆ ˆ For the every input operator Qk (or Pk), corresponding operator Pk (or Qk) with opposite time parity is derived by (2.92). ˆ ˆ ˆ ˆ In the case of Qk input operator, calculation proceeds by Qk → Yk → Pk → ˆ . In the case of ˆ input operator, calculation proceeds by ˆ(s,l) ˆ (s,l) Xk Pk Pk → Xk → ˆ(s,l) ˆ (s,l). For given input operators we obtain ˆ operators and Qk → Yk N 2N X 2N Yˆ operators (protons and neutrons are perturbed separately). Dimension of κ−1 and η−1 matrices is thus 2N × 2N and dimension of D matrix (calculated in Sky_srpa) is 4N × 4N. Skyax_me calculates also matrix elements of the transitions operators (2.141).

Magnetic transition operator is implemented without µN /ec factor. M1 strength function (calculated by Sky_srpa) is then in the nuclear magneton units.

28 3.2.3 Sky_srpa (calculation of strength function and RPA states) Program Sky_srpa takes the matrix elements calculated by the previous program, builds matrices κ−1, η−1,D,A and calculates the strength function (for given ∆). Optionally, it can nd individual RPA solutions and print their structure (ν±). cph Several weighting factors can be set to tune contribution of spin and orbital parts ˆ ˆ of the operators Xk, Yk (in magnetic case). Also eective charges are set to choose either isovector (Zep + Nen = 0, ep − en = 1) or isoscalar modes (Zep + Nen = Z, ep −en = 0), or their combination (ep = 1, en = 0). In the dipolar E1 case there is only isovector mode which corresponds to the vibration of protons and neutrons out of phase. In-phase vibration (isoscalar) corresponds to motion of the nucleus as a whole, which violates conservation of momentum, and in practice gives rise to the spurious modes (low-lying excitations). We can cancel spurious modes by removing monopole part of the nuclear electric charge (imposing Zep +Nen = 0), keeping only dipolar part (ep − en = 1). This leads to ep = N/A and en = −Z/A.

3.3 E1 resonances

Calculation of the E1 resonances was done for nuclei 40Ca, 42Ca, 44Ca, 48Ca, 50Ti, 52Cr, 54Fe, 56Fe. These nuclei were considered as spherical, so we used deformation parameter β = 0.001 to avoid singularities from the degeneracy of energy levels. Previous calculations with Skyax code were done mainly for deformed nuclei in lanthanide region [10,23]. ˆ 3 As input operators we used 5 time-even Qk operators rY1µ, r Y3µ, J1(0.6r)Y1µ, J1(0.9r)Y1µ, J1(1.2r)Y1µ. Strength function (with ∆ = 0.1 MeV)) was calculated for transition operators ˆ(E) and ˆ(E) and the results were summed to obtain full f10 f11 E1 strength function (these contributions are represented by thinner lines on g- ures 3.13.4). We expressed results in terms of energy dependent photoabsorption cross section by (2.140)

8π3α(λ + 1) σ (E) = S (E1; E) = 0.4022 S (E1; E) (3.3) E1 λ[(2λ + 1)!!]2 1 1 Due to the increased decay rate of higher-exited states (mainly by the emission of nucleons), smoothing width ∆ should increase with increasing energy. We implemented this by double-folding procedure: Z (3.4) σ∆(E) = dE1σ(E1) δ∆0+max(0,a(E1−E0))(E1 − E) where we chose ∆0 = 0.1 MeV, a = 0.2, E0 = 10 MeV. Calculated cross sections were compared with experimental data (gures 3.1 3.4), which were obtained from program Janis 3.2 (database Exfor; available at http://www.oecd-nea.org/janis/) or directly from articles (for 42Ca, 44Ca). E1 strength constitutes a major part of the experimental photoabsorption cross section (2.140) to which it can be directly compared, with remaining multipo- larities (M1, E2) being negligible (due to multiplication by 2 or −2). Total µN (~c) photoabsorption (γ, any) was measured only for 40Ca [24], natural Ca [25] and

29 12 12 Ca(nat) exp 42Ca exp (nx+p) 8 8 4 4 0 0 12 40 Ca exp 12 8 42Ca SV-bas 4 8 0 4 12 40Ca SV-bas 0 8 12 4 42Ca SLy6 ] 8 2 0 4 12 40 Ca SLy6 ] 8 2 0 4 12 42 (E1) [fm 0 8 Ca SkI3 σ 12 40

(E1) [fm 4 8 Ca SkI3 σ 0 4 12 0 42 8 Ca SkT6 12 40 8 Ca SkT6 4 4 0 0 12 42 12 Ca SGII 40Ca SGII 8 8 4 4 0 0 12 ∗ 12 40 ∗ 42Ca SkM 8 Ca SkM 8 4 4 0 0 5 10 15 20 25 30 5 10 15 20 25 30 E [MeV] E [MeV]

Figure 3.1: E1 cross section of 40Ca and 42Ca. Natural calcium contains 96.9% 40Ca. Thin lines represent µ = 0 and µ = 1 contribution. natural Fe [26]. These measurements are connected with large uncertainities, because cross section is dominated by electron photoabsorption and this has to be subtracted. Remaining experimental data were evaluated as a sum of neutron and proton emission cross sections (γ, nx) + (γ, p): 42Ca [27], 44Ca [28], 48Ca [29] ((γ, nx) stands for (γ, n) + (γ, 2n) + (γ, np)). Other experimental data are for (γ, n) + (γ, 2n): 50Ti [30], or (γ, nx): 52Cr [31], or (γ, n) + (γ, p): 54Fe [32]. The main contribution comes from (γ, n), but at higher energies (> 20 MeV) also other reactions take place and their omission leads to underestimation of energy and intensity of the giant resonance (mainly for 50Ti and 54Fe). It can be seen that calculations are in relatively good agreement with experimental data, mainly for SV-bas parametrization. Previous calculations for deformed nuclei [10,23] showed the best agreement for SLy6 force. In the present work, SLy6 showed too low fragmentation and shift to lower energies, with the exception of 54Fe, where it led to better results than SV-bas. This is probably caused by optimization of SLy6 force for neutron-rich nuclei, which is not our case. In the case of 44Ca we imposed stronger smoothing (a = 0.4), because all calculations yielded too narrow resonances. Experimental data for 40Ca and 54Fe have large uncertainities but they suggest presence of pygmy modes in the region of ∼ 12 MeV. Since these modes should contain signicant contribution of 2ph excitations, they are only weakly visible in our calculations. Whole calculated range (045 MeV) is shown for 48Ca on gure 3.5. Calcu- lated strength functions are compared with pure 1ph strength (with no residual interaction). Residual interaction thus plays crucial role for description of excited states to be in agreement with experiments.

30 9 44Ca exp (nx+p) 15 48 6 10 Ca exp (nx+p) 3 5 0 0 9 15 44Ca SV-bas 48 6 10 Ca SV-bas 3 5 0 0 9 44Ca SLy6 15 48 6 10 Ca SLy6 3 5 ] ]

2 0 2 0 9 15 44Ca SkI3 48 6 10 Ca SkI3

(E1) [fm 3 (E1) [fm 5

σ 0 σ 0 9 15 44Ca SkT6 48 6 10 Ca SkT6 3 5 0 0 9 15 44Ca SGII 48 6 10 Ca SGII 3 5 0 0 9 ∗ 15 44Ca SkM 48 ∗ 6 10 Ca SkM 3 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 E [MeV] E [MeV]

Figure 3.2: E1 cross section of 44Ca and 48Ca. For 44Ca we used a = 0.4 smoothing parameter.

15 50 15 52 10 Ti exp (n+2n) 10 Cr exp (nx) 5 5 0 0 15 50 15 52 10 Ti SV-bas 10 Cr SV-bas 5 5 0 0 15 50 15 52 10 Ti SLy6 10 Cr SLy6 5 5 ] ]

2 0 2 0 15 50 15 52 10 Ti SkI3 10 Cr SkI3

(E1) [fm 5 (E1) [fm 5

σ 0 σ 0 15 50 15 52 10 Ti SkT6 10 Cr SkT6 5 5 0 0 15 50 15 52 10 Ti SGII 10 Cr SGII 5 5 0 0 15 50 ∗ 15 52 ∗ 10 Ti SkM 10 Cr SkM 5 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 E [MeV] E [MeV]

Figure 3.3: E1 cross section of 50Ti and 52Cr.

31 15 54 18 Fe(nat) exp 10 Fe exp (n+p) 12 5 6 0 0 15 54 18 56 10 Fe SV-bas 12 Fe SV-bas 5 6 0 0 15 18 54 56Fe SLy6 10 Fe SLy6 12 5 6 ] ]

2 0 2 0 15 54 18 56 10 Fe SkI3 12 Fe SkI3

(E1) [fm 5 (E1) [fm 6

σ 0 σ 0 15 54 18 56 10 Fe SkT6 12 Fe SkT6 5 6 0 0 15 54 18 56 10 Fe SGII 12 Fe SGII 5 6 0 0 15 54 ∗ 18 56 ∗ 10 Fe SkM 12 Fe SkM 5 6 0 0 5 10 15 20 25 30 5 10 15 20 25 30 E [MeV] E [MeV]

Figure 3.4: E1 cross section of 54Fe and 56Fe. Natural iron contains 5.84% 54Fe and 91.75% 56Fe.

SRPA 1ph 15 exp ] 2 48Ca SV-bas (E1) 10 (E1) [fm σ 5

0 SRPA ]

1 1ph

− 8

MeV 6 · 2 N µ 4 48Ca SkI3 (M1)

(M1) [ 2 S

0 0 5 10 15 20 25 30 35 40 45 E [MeV]

Figure 3.5: E1 and M1 resonances of 48Ca with comparison of unperturbed (1ph) and perturbed (SRPA) strength.

32 3.4 M1 resonances

We used only the spin part of input operators (3.2), because the orbital part shouldn't contribute for spherical nuclei (it corresponds to scissor shape vibra- 2 tions of protons and neutrons). Full set of input operators was then sˆ1µ, r sˆ1µ, 4 2 4 r sˆ1µ, r Y2µ, r Y2µ (last two time-even operators were included to enable cou- pling with E2 states). The results are expressed as a strength function S0(M1; E) −1 [µN MeV ] with smoothing ∆ = 0.1 MeV. Due to the dominance of E1 transitions in the photoabsorption spectra, experimental M1 strength function must be obtained indirectly, e.g. by inelastic electron scattering. All data used in this section are taken from article [33]. The experimental data for 50Ti, 52Cr, 54Fe are given as reduced transition probabilities B(M1) [µN ] (µN is understood in SI units as µN /ec = 0.105154 fm). To compare experimental B(M1) and calculated S(M1; E), we multiplied B(M1) by a factor which corresponds to the smoothing ∆ = 0.1 MeV: 2 S (M1, ω) = δ (0)B(M1; ω) = B(M1; ω) (3.5) 0 ~ ∆ ~ π∆ ~ and also performed this smoothing (gures 3.63.9). Experimental data for 40Ca, 42Ca, 48Ca (there were observed no signicant transitions for 44Ca) are given only as positions of the transitions and are indicated by arrows. The used parametrizations lead to quite dierent results, the best agreement can be seen for SkI3, except for 50Ti and 52Cr where SLy6 and SV-bas (for 50Ti) or SGII (for 52Cr) perform a bit better. Particular disagreement with experimental data was observed for the doubly magic 40Ca, where only SV-bas showed signicant resonance structure. Also the calculated fragmentation was lower as expected, but this can be explained by the omission of 2ph contribution (its inclusion leads to a larger fragmentation in a simple shell-model calculation [33]). Previous calculations [10,34] showed similar non-uniform behavior of Skyrme parametrizations, where SkI4 usualy performed quite well, except for deformed 238U (where SGII was much better). Relatively good agreement of SkI3 and SkI4 forces with experiment is apparently based on their tting procedure which took into account also spin-orbit splitting. If we like to obtain the M1 contribution to the photoabsorption cross section (dominantly given by E1) we need to multiply by 2 and use (2.140): S1(M1) µN 8π3α(λ + 1) µ 2 σ (E) = N S (M1; E) = 4.448 · 10−3S (M1; E) (3.6) M1 λ[(2λ + 1)!!]2 ec 1 1

33 0.4 4 40Ca SV-bas 42Ca SV-bas 0.2 2 0 0 0.4 4 40Ca SLy6 42Ca SLy6

] 0.2 ] 2 1 1 − 0 − 0 MeV MeV

· 0.4 · 4

2 40 2 42 N Ca SkI3 N Ca SkI3 µ 0.2 µ 2 0 0 (M1) [ (M1) [

S 0.4 S 4 40Ca SkT6 42Ca SkT6 0.2 2 0 0 0.4 4 40Ca SGII 42Ca SGII 0.2 2 0 0 0.4 4 40Ca SkM∗ 42Ca SkM∗ 0.2 2 0 0 4 6 8 10 12 14 16 18 4 6 8 10 12 14 16 18 E [MeV] E [MeV]

Figure 3.6: M1 strength function of 40Ca and 42Ca. The experimental data (indicated by arrows) are from [33].

15 8 48Ca SV-bas 44Ca SV-bas 10 4 5 0 0 15 8 48Ca SLy6 44Ca SLy6 10

] 4 ] 1 1 5 − 0 − 0

MeV MeV 15 · 8 · 48Ca SkI3 2 44 2 N Ca SkI3 N 10 µ µ 4 5 0 0 (M1) [ (M1) [

S S 15 8 48Ca SkT6 44Ca SkT6 10 4 5 0 0 15 8 48Ca SGII 44Ca SGII 10 4 5 0 0 8 15 ∗ ∗ 48Ca SkM 44Ca SkM 10 4 5 0 0 4 6 8 10 12 14 16 18 4 6 8 10 12 14 16 18 E [MeV] E [MeV]

Figure 3.7: M1 strength function of 44Ca and 48Ca.

34 8 8 6 50 6 52 4 Ti exp (e,e) 4 Cr exp (e,e) 2 2 0 0 15 50 30 52 10 Ti SV-bas 20 Cr SV-bas 5 10 0 0 15 30 50Ti SLy6 52Cr SLy6

] 10 ] 20 1 1

− 5 − 10 0 0

MeV 15 MeV 30 · 50 · 52Cr SkI3 2 Ti SkI3 2 N 10 N 20 µ 5 µ 10 0 0

(M1) [ 15 50 (M1) [ 30 52 S 10 Ti SkT6 S 20 Cr SkT6 5 10 0 0 15 50 30 52 10 Ti SGII 20 Cr SGII 5 10 0 0 15 50 ∗ 30 52 ∗ 10 Ti SkM 20 Cr SkM 5 10 0 0 4 6 8 10 12 14 16 18 4 6 8 10 12 14 16 18 E [MeV] E [MeV]

Figure 3.8: M1 strength function of 50Ti and 52Cr. Experimental transition probabilities B(M1) are from [33].

8 8 56 6 54 6 Fe exp (e,e) 4 Fe exp (e,e) 4 2 2 0 0 40 20 54Fe SV-bas 56Fe SV-bas 20 10 0 0

40 54 20 56

] Fe SLy6 ] Fe SLy6 1 20 1 10 − − 0 0 MeV MeV

· 40 · 20

2 54 2 56 N Fe SkI3 N Fe SkI3 µ 20 µ 10 0 0

(M1) [ 40 (M1) [ 20 S 54Fe SkT6 S 56Fe SkT6 20 10 0 0 40 20 54Fe SGII 56Fe SGII 20 10 0 0 40 20 54Fe SkM∗ 56Fe SkM∗ 20 10 0 0 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 E [MeV] E [MeV]

Figure 3.9: M1 strength function of 54Fe and 56Fe. Experimental data of 56Fe (up to 9 MeV) are from [35] (see section 3.5).

35 3.5 Deformed 56Fe

In the all previous calculations we assumed spherical nuclei. However, we should check this assumption by performing calculation of the ground-state energy in some range of deformations β and indentify a minimum. The calculation con- rmed sphericity for 40Ca54Fe, but 56Fe showed the minumum of energy at relatively large deformation around β = 0.23 for 5 forces (only SV-bas suggests β = 0). Corresponding energy surfaces are depicted on gure 3.10 (left). It is necessary to look for the experimental data which would conrm non- zero quadrupole deformation of 56Fe. Since the spin of 56Fe nucleus is 0+, the deformation can be obtained only indirectly. Analysis of alpha scattering within the optical model [36] leads to β = 0.241 (and even triaxiality with γ = 19◦). Another study [37] implements the soft-rotator model for the description of level scheme up to 5.5 MeV (where rst two excited states, 2+ at 0.847 MeV and 4+ at 2.085 MeV, are treated as a part of ground-state rotational band), and subsequent analysis of (n, n0) and (p, p0) inelastic scattering to obtain β = 0.231 (among other results). However, widely cited table of theoretical deformations [38] gives β = 0 for 56Fe (but β = 0.189 for 57Fe and β = 0.199 for 58Fe). During the calculation of E1 and M1 strength functions we used β = 0.235 (as an average of 0.245 from SkI3 and 0.225 from SLy6, see gure 3.10). E1 cross section was calculated as before (see section 3.3). For the calculation of µ = 0 component of M1 strength function (where scissor mode is expected) we included also orbital part into the Pˆ input operators in the same ratio as it is in ˆ(M) (2.141). Components and of are shown separately f10 µ = 0 µ = 1 S(M1) on gure 3.11. Smoothing ∆ = 1 MeV was used due to a larger fragmentation of underlying strength. Experimental data on M1 resonances up to 9 MeV were obtained by inelastic electron scattering [35]. They contain scissor mode at 3.449 MeV. E1 cross section calculated for the deformed 56Fe (gure 3.10) shows much better agreement with experimental data than σ(E1) calculated for the spherical 56Fe (gure 3.4) due to larger fragmentation. M1 strength function (gure 3.11) is not directly comparable to the experimental data [35] because they end at 9 MeV, while our calculations suggest giant resonance at 910 MeV. At least, we could observe the scissor mode at ∼ 3 MeV and the rst spin-ip transition at ∼ 7 MeV that are best reproduced by SkI3 force.

36 -478 Fe(nat) exp 10 5 -480 0 SkT6 10 56Fe SV-bas -482 5 SkM∗ 0 10 56Fe SLy6 -484 5 ]

2 0 -486 SkI3 10 56Fe SkI3

Fe) [MeV] 5 (E1) [fm 56 ( σ 0 0 -488 E SLy6 10 56Fe SkT6 5 -490 SV-bas 0 10 56Fe SGII exp -492 5 0 10 56 ∗ SGII Fe SkM -494 5 0 -0.2 -0.1 0 0.1 0.2 0.3 0.4 5 10 15 20 25 30 β E [MeV]

Figure 3.10: Ground-state energy surface of 56Fe for deformations β ∈ (−0.2, 0.4) (left) and E1 cross section of deformed 56Fe (right; β = 0.235). The experimental photoabsorption data are from [26].

3 µ = 1 0.75 µ = 1 56Fe exp (e,e) 56 2 µ = 0 0.5 Fe exp (e,e) µ = 0 1 0.25 0 0 3 56 0.75 56 2 Fe SV-bas 0.5 Fe SV-bas 1 0.25 0 0 3 0.75 56Fe SLy6 56Fe SLy6

] 2 ] 0.5 1 1

− 1 − 0.25 0 0

MeV 3 MeV 0.75 · 56Fe SkI3 · 56 2 2 Fe SkI3 N 2 N 0.5 µ 1 µ 0.25 0 0

(M1) [ 3 56 (M1) [ 0.75 56 S 2 Fe SkT6 S 0.5 Fe SkT6 1 0.25 0 0 3 56 0.75 56 2 Fe SGII 0.5 Fe SGII 1 0.25 0 0 3 56 ∗ 0.75 56 ∗ 2 Fe SkM 0.5 Fe SkM 1 0.25 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 E [MeV] E [MeV]

Figure 3.11: M1 strength function (magnied on the right) of the deformed 56Fe with β = 0.235. The experimental data up to 9 MeV are from [35]. The key (µ = 0, 1) belongs to the calculated strength functions.

37 4. Summary

Density functional methods (particularly Skyrme energy functional) are now widely used for the calculation of ground-state and dynamical nuclear properties around the whole periodic table. There exist several parametrizations which are based on dierent nuclear phenomena. Our aim was to compare and evaluate parametrizations SkM∗, SGII, SkT6, SkI3, SLy6 and SV-bas for the treatment of E1 and M1 giant resonances for a chain of light spherical nuclei 40Ca, 42Ca, 44Ca, 48Ca, 50Ti, 52Cr, 54Fe, 56Fe. HF calculations of ground state wavefunctions were performed with program Skyax_hfb. Giant resonance structure was then obtained by Separable Random Phase Approximation, which allows self-consistent calculation of collective excited states (it was performed by the programs Skyax_me and Skyax_srpa). This procedure requires no additional parameters (except for the original Skyrme and pairing parameters) and can reasonably replace full RPA treatment, utilizing much less computational resources. SRPA formalism allowed direct calculation of the strength functions (with chosen Lorentzian smoothing) without interme- diate evaluation of RPA solutions. To take into account increased broadening in higher energy region (connected with the emission of nucleons and complex congurations), we implemented the double-folding procedure on the E1 strength function. The results of computation show that the best agreement with experimental data is obtained for SV-bas force in the case of E1 resonances. For M1 resonances, usually SkI3 was the best, but in some cases it gives quite unsatisfactory results. It is interesting that the relatively old force SkM∗ performed quite well in nearly all cases. Inability of the new Skyrmes parametrizations to signicantly improve description of the M1 resonances probably lies in the fact that the contact nature of Skyrme force is not a sucient approximation to the real nuclear forces. It is also possible that higher correlations (2ph) could improve description of giant resonances (mainly their fragmentation), but their implementation is technically dicult. Calculations of E1 and M1 resonances showed much better agreement with the experimental data for 56Fe if we allowed non-zero ground-state deformation. This fact supports our initial assumption based on calculated energy surfaces and some experimental data.

The contribution of the author in the present work was following:

• He worked out the details of derivations in the theoretical part (mainly BCS and RPA).

• He thought up the simple example of collective phenomena (section 2.3).

• He rewrote the programs Skyrme_me and Skyrme_srpa from Fortran into C language.

• He performed calculations of strength functions and their double folding.

38 A. Time reversal

The single-particle wavefunctions of nuclear states are at least two-fold degenerate due to the time reversal symmetry of nuclear Hamiltonian:

[T,ˆ Hˆ ] = 0 (A.1) It is therefore convenient take advantage of this symmetry and construct wavefunction basis from the pairs and ˆ . We should note here that ψk T ψk = ψk¯ this symetry is broken for Hamiltonian attached to rotating nucleus, which is constructed in aim to simplify description of such nuclei (under assumption of Born-Oppenheimer approximation). However, we are not working with rotating nuclei in this text. Here we will briey describe some properties of the time reversal operator Tˆ (following Schi [39], p. 229232). Tˆ is antiunitary: ˆ ∗ ˆ ∗ ˆ T (aψ1 + bψ2) = a T ψ1 + b T ψ2 (A.2) This can be the most easily shown by its action on Schrödinger equation where it should reverse time sign: t → −t ∂ ∂ −i Tˆ ψ = Tˆ i ψ = TˆHψˆ = Hˆ Tˆ ψ (A.3) ∂t ∂t This implies that time reversed energy eigenfunction should be proportional to eiEt in contrast to ordinary wavefunction (e−iEt). However, we are working in Heisenberg picture, so this strange behavior of wavefunction is avoided. Time reversal symmetry of common operators follows their classical meaning:

Tˆ−1xˆTˆ =x ˆ Tˆ−1pˆTˆ = −pˆ Tˆ−1LˆTˆ = −Lˆ

Tˆ−1SˆTˆ = −Sˆ Tˆ−1Hˆ Tˆ = Hˆ (A.4) Then we speak of operators as being time-even or time-odd. To describe time reversal of wavefunction of the spin 0 particle in X representation, it is sucient to apply complex conjugation (i.e. operator Kˆ ; this follows from coordinate prescription for operators x,ˆ p,ˆ Lˆ):

Tˆ ϕ(x) = Kϕˆ (x) = ϕ∗(x) (A.5)

In the case of spin 1/2 particles (electrons, nucleons), the form of Tˆ is dependent on actual representation of the spin operator. We assume Pauli matrices:

0 1 0 −i 1 0 1 σ = σ = σ = Sˆ = σ (A.6) x 1 0 y i 0 z 0 −1 j 2 j The time reversal operator is then rewritten as a product of some unitary operator Uˆ acting on spin space and a complex conjugation Kˆ .

Tˆ = UˆKˆ Tˆ−1 = Kˆ Uˆ † (A.7) ˆ ˆ ˆ Since Sx and Sz are real and Sy is imaginary, ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ SxU = −USx SyU = USy SzU = −USz (A.8)

39 and Uˆ can be chosen as a rotation through π radians about the y axis:

ˆ hπ 0 −1i 1 0 π 0 −1 π 0 −1 Uˆ = e−iπSy = exp = cos + sin = = −iσ 2 1 0 0 1 2 1 0 2 1 0 y

0 −1 Tˆ = −iσ Kˆ = Kˆ Tˆ2 = −1 (A.9) y 1 0 We will use the following sign convention:

Tˆ|ji = |¯ji Tˆ|¯ji = −|ji (A.10)

We will derive some useful relations for matrix elements of single-particle hermitian operators. Let Qˆ be time-even and Pˆ time-odd operator. Matrix elements corresponding to single-particle Hilbert space will be denoted by round brackets (j|Aˆ|k). The wavefunctions ψ(x) are considered as two-component (describing spin 1/2 particles). Z Z Z ˆ ∗ ˆ ∗ ˆ−1 ˆ ˆ ∗ ˆ ˆ † ˆ (j|Q|k) = dx ψj Qψk = dx ψj T QT ψk = dx ψj KU Qψk¯ h Z i∗ h Z i∗ h Z i∗ ˆ † ˆ ˆ ∗ ∗ ˆ ∗ ˆ = dx ψjU Qψk¯ = dx (Uψj ) Qψk¯ = dx ψ¯j Qψk¯ h Z i∗ Z ˆ ∗ ∗ ˆ ¯ ˆ ¯ (A.11) = dx (Qψ¯j) ψk¯ = dx ψk¯Qψ¯j = (k|Q|j)

We used denition of time reversal (Tˆ = UˆKˆ ), unitary operator (U −1 = U †, hj|Uˆ †ki = hUjˆ |ki) and hermitian operator (hj|Akˆ i = hAjˆ |ki). Other relations are: Z h Z i∗ ˆ ¯ ∗ ˆ ˆ † ˆ ˆ ∗ ∗ ˆ ˆ ¯ (A.12) (j|Q|k) = − dx ψj KU Qψk = − dx (Uψj ) Qψk = −(k|Q|j) Z h Z i∗ ˆ ∗ ˆ−1 ˆ ˆ ˆ ∗ ∗ ˆ ¯ ˆ ¯ (A.13) (j|P |k) = − dx ψj T P T ψk = − dx (Uψj ) P ψk¯ = −(k|P |j) Z h Z i∗ ˆ ¯ ∗ ˆ ˆ † ˆ ˆ ∗ ∗ ˆ ˆ ¯ (A.14) (j|P |k) = dx ψj KU P ψk = dx (Uψj ) P ψk = (k|P |j)

We can dene A as a time parity of operator ˆ and rewrite the above γT = ±1 A expresions in a compact form:

ˆ A ¯ ˆ ¯ ˆ ¯ A ˆ ¯ ¯ ˆ A ¯ ˆ (j|A|k) = γT (k|A|j)(j|A|k) = −γT (k|A|j)(j|A|k) = −γT (k|A|j) (A.15)

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