The M4 transitions of isomeric states

of Isomeric States

Master’s thesis, 27.9.2015

Author:

Lotta Jokiniemi

Instructor:

Jouni Suhonen

ABSTRACT

Jokiniemi, Lotta

The M4 transitions of isomeric states Master’s thesis

Department of Physics, University of Jyväskylä, 2015, 57 pages.

This master’s thesis concerns the reduced matrix elements of magnetic hexadecapole (M4) γ transitions of isomeric states. The transitions examined in the thesis are stretched M4 transitions in the mass regions of double beta decay, A=85-115 and A=135-143. The aim of the thesis is to compare the experimental nuclear matrix elements M_exp(M4) of the example nuclei to the quasiparticle NMEs M_qp obtained via BCS theory and to the ones obtained via MQPM theory, MM QP M. The experimental NMEs are calculated from the experimental data and the quasiparticle and MQPM NMEs are obtained using the available computer programs.

The ratio of experimental and quasiparticle NMEs is found to be roughly 0.29, while the ratio between experimental and MQPM NMEs is roughly 0.33. The MQPM NMEs are closer to the experimental ones but are not reduced as much as anticipated. In addition, the effect of changes in the gyromagnetic ratios g_l andg_s on the MQPM NMEs is investigated. It seems that the changes in the spin g factors g_s affect the NMEs quite strongly, while the orbital g factors g_l have little effect on them.

Keywords: Electromagnetic transitions, M4 transition, nuclear matrix elements, BCS theory, QRPA theory, MQPM theory

TIIVISTELMÄ

Jokiniemi, Lotta

Isomeeristen tilojen M4-siirtymät Pro gradu -tutkielma

Fysiikan laitos, Jyväskylän yliopisto, 2015, 57 sivua

Tässä pro gradu -tutkielmassa tutkitaan isomeeristen tilojen magneettisten M4-gammasiirtymien redusoituja matriisielementtejä. Tutkittavat siirtymät ovat venyneitä M4-siirtymiä kak-

soisbeetahajoamisten massa-alueilla A=85-115 ja A=135-143. Tutkielman tarkoituksena on verrata kokeellisia ydinmatriisielementtejäM_expkvasihiukkasmatriisielementteihinM_qp ja MQPM-teorian avulla laskettuihin matriisielementteihin MM QP M. Kokeelliset ma- triisielementit lasketaan kokeellisesti määritettyjen arvojen avulla ja kvasihiukkas- sekä MQPM-matriisielementit määritetään tietokoneohjelmien avulla.

Kokeellisten ja kvasihiukkasmatriisilementtien välinen suhde osoittautui olevan noin 0.29 ja kokeellisten ja MQPM-matriisielementtien välinen suhde noin 0.33. MQPM-matriisielementit ovat lähempänä kokeellisia tuloksia, mutta eivät poikkea kvasihiukkasarvoista niin paljon kuin olisi voinut olettaa. Lisäksi gyromagneettisten tekijöiden g_l ja g_s muutosten vaiku- tusta MQPM-matriisielementtien arvoihin tutkittiin. Osoittautui, että spin-g-tekijöiden muuttaminen vaikutti melko voimakkaasti matriisielementtien arvoihin, kun taas rata-g- tekijöiden muuttamisella ei ollut suurta vaikutusta.

Avainsanat: Sähkömagneettiset siirtymät, M4-siirtymä, ydinmatriisielementit, BCS- teoria, QRPA-teoria, MQPM-teoria

PREFACE

At first, I would like to thank my supervisor, Prof. Jouni Suhonen, for excellent guidance.

I would also like to thank Prof. Hiro Ejiri for introducing the interesting topic and also for the valuable discussion with him. Also, I would like to thank Ph.D. Emanuel Ydrefors for providing the computer codes and also for great advice. I would also like to thank M.Sc. Pekka Pirinen for his great advice regarding the computer programs.

A great deal of thanks belongs also to my dear fellow physicists, who have helped me a lot with my summer practise and have also been wonderful company, Especially I would like to thank Joakim Linja, Marko Suomalainen, Juuso Manninen and Joni Lämsä for helping me a lot with my computer problems.

LIST OF ABBREVIATIONS

BCS Bardeen Cooper Scrieffer -theory

BR Biedenharn-Rose phase convention

DBD Double beta decay

EM Electromagnetic

GT Gamow-Teller excitation

MQPM Microscopic quasiparticle-phonon model

NME Nuclear matrix element

pn-QRPA Proton-neutron QRPA

RPA Random phase approximation

SD Spin-dipole excitation

SP Single-particle

QP Quasiparticle

QRPA Quasiparticle RPA

CONTENTS

1 Introduction 1

2 Electromagnetic Transitions Theory 2

2.1 Coupling of Angular Momenta . . . 2

2.2 Wigner 3j-symbols . . . 4

2.3 Radial Integrals . . . 6

2.4 EM transitions . . . 7

2.4.1 Selection rules . . . 7

2.4.2 Transition Probabilities . . . 8

2.4.3 Single particle matrix elements . . . 9

3 Nuclear Models 11 3.1 Woods-Saxon Potential . . . 11

3.2 BCS Theory . . . 14

3.3 QRPA theory . . . 17

3.4 MQPM theory . . . 20

4 Results 23 4.1 Computing the Experimental NMEs . . . 23

4.2 BCS calculations . . . 25

4.2.1 The single-particle NME for the transition 0g_9/2 ←→1p_1/2 . . . 26

4.2.2 The single-particle NME for the transition 0h_11/2 ←→1d_3/2 . . . 28

4.2.3 Determining the BCS occupation and vacancy amplitudes u and v . 29 4.3 QRPA calculations . . . 35

4.4 MQPM calculations . . . 36

4.4.1 MQPM nuclear matrix elements . . . 36

4.4.2 Four different parts of the MQPM NMEs . . . 45

5 Conclusions 52

References 56

1 INTRODUCTION

In the thesis, the magnetic hexadecapole (M4) γ transitions of isomeric states of nuclei in the mass regions of A=85-115 and A=135-143 are examined. The nuclei of inter- est are either proton-odd or neutron-odd. The transitions analysed in the thesis are mainly stretched M4 γ transitions of nuclei in the mass regions of double beta decay.

The stretched transitions mean the transitions for which the change in the total angular momentum is maximal, i.e. for stretched M4 transitions ∆J = 4.

The aim of the work is to compare the experimental nuclear matrix elementsM_exp(M4) of the M4 transitions with the single-quasiparticle nuclear matrix elements (NMEs)M_qp(M4) and MQPM (microscopic quasiparticle model) NMEs M_{M QP M}(M4). The investigation aims to solve how much the experimental NMEs are reduced compared to the theoret- ical ones. The investigation may help to study the higher multipole NMEs associated with 0νββ decays and higher energy components of astro neutrinos. Since there is not much experimental data concerning the rare third forbidden M4β transitions, instead the more common M4 γ transitions are examined. So for here on, only the electromagnetic transitions are discussed in this thesis.

There are earlier works [1, 2] concerning Gamow-Teller GT(1⁺) and spin-dipole SD(2⁻) NMEs, which show that the allowed and first forbidden β transitions are reduced much in comparison with the quasiparticle and proton-neutron QRPA NMEs. That is why the higher multipole (4⁻) NMEs are examined to see how they are reduced by the nucleonic and non-nucleonic correlations.

The experimental NMEs are calculated using the experimental data obtained from the NNDC(National Nuclear Data Center) database [3]. Then, the quasiparticle NMEs are calculated by solving the requisite single-particle NMEs by hand and then solving the BCS occupation and vacancy amplitudes by using available computer programs. Finally, the MQPM NMEs are solved numerically with MQPM computer programs.

In the second section of the thesis, some mathematical tools are first introduced in order to build up the theory of electromagnetic transitions. Then, some basic information of EM transitions is discoursed. In the third section, the used nuclear models are introduced, including the Woods-Saxon potential, BCS theory, QRPA theory and MQPM theory. In the fourth section, the actual calculations are performed and the results are analysed and commented. Finally, in the fifth chapter, there are drawn some conclusions of the gained results.

2 ELECTROMAGNETIC TRANSITIONS THEORY

In this section, the general theory of electromagnetic transitions is discussed. To build the theory, some mathematical tools, like angular momenta coupling, Wigner 3j-symbols and radial integrals need to be introduced. It is done in the first subsections.

2.1 Coupling of Angular Momenta

The quantity J is said to be an angular momentum vector, if its components J_i satisfy the following relations [4, 5]

J_k^† =J_k ,k=1,2,3, (2.1)

i.e. the components are hermitian, and

[J_i, J_j] =i~^X

k

_ijkJ_k, (2.2)

where _ijk is the completely antisymmetric three-dimensional Levi-Civita tensor, which can take three different values according to

ijk=











0, if two of the indices are equal

+1, if (i, j, k) is an even permutation of (1,2,3)

−1, if (i, j, k) is an odd permutation of (1,2,3).

(2.3)

Usually the three different components of the angular momentum are labelled as (x, y, z) instead of (1,2,3). This is a natural choice in the cartesian coordinates and the notation used in this work, too.

Often, the quantities J² and J_z commute with the Hamiltonian H of the system. This is the case, when the Hamiltonian is invariant under rotations. Consequently, it is beneficial to determine the common eigenfunctions of J² and J_z.

The total angular momentum operator J is usually a sum of either the total angular momentum operators of distinct systems, or sets of dynamical variables of the system, such as the orbital angular momentum L and the spin angular momentum S. That is why it is essential to know how to couple angular momenta. [6]

The operator J fulfils the basic eigenvalue equations

J²|j mi=j(j+ 1)~²|j mi (2.4a) Jz|j mi=m~|j mi , (2.4b) where the states are orthonormal, i.e.

hj m|j⁰ m⁰i=δjj⁰δmm⁰ (2.5) and the projection quantum numbers mcan take one of the 2j+ 1 values −j,−j+ 1, ..., j as usual. The quantum number j corresponds to the length of the angular momentum vector and m the length of the z-projection of the angular momentum vector. [5]

Now, when the basic properties of the angular momenta have been introduced, one can finally start to add two commuting angular momenta J₁ and J₂. The commutation relations read

[J₁,J2] = 0, , i.e. [J_1k, J2l] = 0 , for all k= 1,2,3. (2.6) Both of the states satisfy the eigenvalue equations

J²_k|j_k m_ki=j_k(j_k+ 1)~²|j_k m_ki , (2.7a) J_kz|j_k m_ki=m_k~|j_k m_ki , k= 1,2. (2.7b) It is easily seen that the sum vectorJ =J₁+J₂ is also an angular momentum vector since it satisfies the commutation relations (2.2) and is hermitian as a sum of two hermitian angular momentum vectors. The eigenstates of this kind of system can be written as the products of the eigenstates of the angular momentum vectors J₁ and J₂ [4]

|j1 m1 j2 m2i=|j1 m1i |j2 m2i , (2.8) which are the eigenstates of the complete operator set {J²₁, J_1z,J²₂, J_2z}. The eigenvalue equations of the set are [5]

J_k|j₁ m₁ j₂ m₂i=j_k(j_k+ 1)~²|j₁ m₁ j₂ m₂i (2.9a) Jkz|j1 m1 j2 m2i=mk~|j1 m1 j2 m2i , k = 1,2. (2.9b) It follows from (2.5) and the definition of the product states (2.8) that the product states are orthonormal if the members of the product form an orthonormal set. The orthonor- mality condition can be written [5]

hj1 m1 j2 m2|j₁⁰ m⁰₁ j₂⁰ m⁰₂i=hj1 m1|j₁⁰ m⁰₁i hj2 m2|j₂⁰ m⁰₂i

=δ_j1j⁰

1δ_m1m⁰

1δ_j2j⁰

2δ_m2m⁰

2 . (2.10)

The orthonormal set {|j1 m1 j2 m2i} is called the uncoupled basis. The coupled basis {|j₁ j₂ j mi} is then formed by coupling the two angular momenta. The coupled angular momentum belongs to the set {J²₁,J²₂,J², J_z} of mutually commuting angular momenta.

The set has a complete set of common eigenvectors.

The eigenvalue equations of the coupled basis {|j₁ j₂ j mi}are

J²_k|j₁ j₂ j mi=j_k(j_k+ 1)~²|j₁ j₂ j mi , k = 1,2, (2.11a) J²|j₁ j₂ j mi=j(j+ 1)~²|j₁ j₂ j mi , (2.11b) Jz|j₁ j₂ j mi=m~|j₁ j₂ j mi . (2.11c) The coupled basis is orthonormal, so that

hj₁ j₂ j m|j₁ j₂ j⁰ m⁰i=δ_jj⁰δ_mm⁰ . (2.12) By using the orthonormality of the uncoupled basis one can write the identity operator as

1= ^X

m1 m2

|j1 m1 j2 m2i hj1 m1 j2 m2| . (2.13)

With the help of the identity operator (2.13) the coupled states can be written in the form

|j₁ j₂ j mi= ^X

m1 m2

|j₁ m₁ j₂ m₂i hj₁ m₁ j₂ m₂|j₁ j₂ j mi

≡ ^X

m1 m2

(j₁ m₁ j₂ m₂|j m)|j₁ m₁ j₂ m₂i . (2.14) The quantity (j₁ m₁ j₂ m₂|j m) is called the Clebsch-Gordan coefficient. The Clebsch- Gordan coefficients have a few basic properties [5]:

• The addition law of the projection quantum numbers m:

(j₁ m1 j2 m2|j m) = 0 unless m1+m2 =m . (2.15)

• The triangular condition of the coupled angular momenta j:

|j₁−j₂| ≤j ≤j₁+j₂ (2.16)

• Integer rule of the total angular momenta j:

j ∈ {|j₁−j₂|,|j₁−j₂|+ 1, . . . , j₁+j₂−1, j₁+j₂}, (2.17) where the total angular momenta quantum numbers j₁ and j₂ have to be either integers or half integers.

• The Clebsch-Gordan coefficients are real and

(j₁ j1 j2 j2|j1+j2 j1+j2) = +1 and (j₁ m1j2 −j2|j m)≥0. (2.18)

2.2 Wigner 3j-symbols

The Wigner 3j-symbols provide a more symmetric way to couple angular momenta of two different systems than the Clebsch-Gordan coefficients. They are denoted by

j₁ j₂ j₃ m₁ m₂ m₃

!

, (2.19)

where all of the parameters are either integers or half-integers. The 3j-symbols can also be defined with the help of the Clebsch-Gordan coefficients as [5]

j₁ j₂ j₃ m₁ m₂ m₃

!

≡(−1)^{j1−j2−m3b}j₃⁻¹(j₁m₁j₂m₂|j₃–m₃), (2.20) where (j₁m₁j₂m₂|j₃–m₃) is the corresponding Clebsch-Gordan coefficient. They satisfy the following selection rules [7]:

1. −|j₁| ≤m₁ ≤ |j₁|,

−|j₂| ≤m₂ ≤ |j₂| and

−|j₃| ≤m₃ ≤ |j₃| 2. m₁+m₂ =−m₃

3. Triangular inequality: |j₁−j₂| ≤j₃ ≤j₁+j₂

4. Integer perimeter rule: j1+j2+j3 ∈Z. If the above conditions are not satisfied, then

j₁ j₂ j₃ m1 m2 m3

!

= 0.

Some basic properties of the 3j-symbols are explained briefly in the next paragraphs. The exchange of two angular momenta leads to the symmetric phase factor, namely

j₂ j₁ j₃ m2 m1 m3

!

= (−1)^j1+j2+j3 j₁ j₂ j₃ m1 m2 m3

!

, (2.21)

and similarly the change of the signs of the projection quantum numbers, j₁ j₂ j₃

−m₁ −m₂ −m₃

!

= (−1)^j1+j2+j3 j₁ j₂ j₃ m₁ m₂ m₃

!

. (2.22)

From (2.20) it follows that the 3j-symbols obey j₁ j₂ 0

m1 m2 0

!

= (−1)^j1−m1j^b₁⁻¹δ_j1j2δ_m1,−m2 , (2.23) j₁ j₂ j₃

0 0 0

!

= 0 unless j₁+j₂+j₃ = even. (2.24) There is a particularly useful relation that simplifies the calculations of the 3j-symbols of the form that is present in the matrix elements of the EM-transitions [8]:

j₁ j₂ j₃

1

2 −¹₂ 0

!

=



















−

r(−j1+j2+j3)(j1−j2+j3+1) (2j1+1)(2j2+1)





j₁+ ¹₂ j₂− ¹₂ j₃

0 0 0



 if J is even

r(j1+j2−j₃)(J+1) (2j1+1)(2j2+1)





j₁− ¹₂ j₂− ¹₂ j₃

0 0 0



 if J is odd

(2.25)

whereJ =j₁+j₂+j₃. The 3j-symbol on the right-hand side can be calculated according to the formula [8]

j₁− ¹₂ j₂− ¹₂ j₃

0 0 0

!

=(−1)^J2

v u u t

(J−2(j₁− ¹₂))!(J−2(j₂− ¹₂))!(J−2j₃)!

(J+ 1)!

× (¹₂J)!

J−2(j₁−1 2) 2

!

! ^J−2(j2−

1 2) 2

!

!^J−2j₂ ³! ,

(2.26)

where J =j1− ¹₂ +j2− ¹₂ +j3 =j1+j2+j3−1.

2.3 Radial Integrals

The so called radial integrals are encountered when one computes the single-particle matrix elements. There are some beneficial properties of the radial integrals, which are discussed next. The discussion is based on Ref. [5].

The radial integrals can be written in the form R^(λ)=_ab b^λ

Z ∞ 0

dx˜g_nala(x)x^λ+2g˜_nblb(x)≡b^λR˜^(λ)_ab , (2.27) where ˜g_nl are defined as

˜

g_nl(x)≡b⁻²³g_nl(r, b), (2.28) where g_nl are the harmonic oscillator wave functions and x≡ ^r_b, where r is the distance from the center and b the harmonic oscillator length. The wave functions can be written as

g_nl(r) =

v u u t

2n!

b³Γ(n+l+ ³₂)

r b

l

e^−r2/2b2L^(l+

1 2)

n (r²/b²), (2.29) where L^(l+

1 2)

n is the associated Laguerre polynomial.

One can derive some useful formulas for the radial integrals. The most important ones are

Z ∞ 0

g_0l(r)g_0l⁰(r)r²dr=

2 π

p/2 (l+l⁰+ 1)!!

q(2l+ 1)!!(2l⁰+ 1)!!

, (2.30)

Z ∞ 0

g_1l(r)g_1l⁰(r)r²dr=

2 π

p/2 (l+l⁰+ 1)!![3 +l+l⁰− ¹₂(l−l⁰)²]

q(2l+ 3)!!(2l⁰+ 3)!!

(2.31) and

Z ∞ 0

g_0l(r)g_1l⁰(r)r²dr=

2 π

p/2 (l⁰−l)(l+l⁰ + 1)!!

q2(2l+ 1)!!(2l⁰+ 3)!!

, (2.32)

where

p=







0, l+l⁰ = even 1, l+l⁰ = odd .

Using the recursion relation for the associated Laguerre polynomials one can derive the following relation for the eigenfunctions:

rg_nl(r) =b^qn+l+ ³₂g_n,l+1(r)−b√

ngn−1,l+1(r). (2.33)

Substituting (2.33) into (2.27) yields the recursion relation R˜^(λ)_n

alanblb =^qn_b+l_b +³₂R˜^(λ−1)_n

alanb,lb+1−√

n_bR˜_{nala,nb−1,lb+1} , (2.34) which enables one to calculate the higher λ integrals.

2.4 EM transitions

The excited states of nuclei can decay to the ground states by emitting one or more gamma rays. This process is called electromagnetic transition, or γ decay. Gamma rays are typically in the energy range of 0.1-10 MeV, which corresponds to the energy differences between nuclear states [9]. The transitions can be of magnetic or electric type.

In this thesis, especially the magnetic M4 transitions are of interest.

2.4.1 Selection rules

Electromagnetic radiation is generated either by an oscillating charge or by a varying current or magnetic moment. The former causes an oscillation in the external electric field and the radiation caused by the process is called electric (E) radiation. The latter, in turn, raises a varying magnetic field, which causes magnetic (M) radiation. The classification of the different electromagnetic processes is based on the conservation of angular momentum and parity. [9, 10]

A photon carries an angular momentumL~, whereLis a positive integer≥1. Lcannot be 0, because photon always carries some angular momentum. The multipolarity of photons is said to be L. For example, L= 1 radiation is called dipole, L = 2 quadrupole, L = 3 octapole radiation and so on. The angular momentum conservation law states that the angular momenta of the initial and final states, I_i and I_f and the L have to fulfil the equation

I_i =I_f +L. (2.35)

With the triangle inequality, (2.35) leads to the condition

|I_i−I_f| ≤L≤ |I_i+I_f|. (2.36) From (2.36) one can see that the transition 0^π ←→0^π, whereπ is the parity + or -, is not allowed, sinceLmust be greater than zero. Even though such electromagnetic transitions are forbidden, the decay is possible via internal conversion, which is not discussed further in this thesis. [10] Transitions with the maximum angular momentum change are called stretched transitions. Especially the stretched transitions are studied in this thesis.

Parity is conserved in electromagnetic transitions. The radiation field resulted from an EM transition can have either even or odd parity, depending on the L and whether the transition is electric or magnetic. The parity conservation rules are [9]

π(M L) = (−1)^L+1 (2.37)

π(EL) = (−1)^L (2.38)

The angular momentum and parity conservation rules for a few lowest EM transitions are summarized in the Table 1.

Table 1: Angular momentum and parity selection rules for EM transitions.

L π_iπ_f Multipolarity Radiation type

1 -1 Dipole E1

1 +1 Dipole M1

2 -1 Quadrupole M2

2 +1 Quadrupole E2

3 -1 Octupole E3

3 +1 Octupole M3

4 -1 Hexadecapole M4 4 +1 Hexadecapole E4 ...

2.4.2 Transition Probabilities

The transition probability T_{f i} of an electromagnetic transition from an initial state i to a final state f tells the probability of the transition per unit time. The half-life of the transition can be written with the help of the transition probability as [5]

t_1/2 = ln 2

T_{f i} . (2.39)

The reduced transition probability of an electromagnetic transition is defined as B(σλ;ξ_iJ_i →ξ_fJ_f) = 1

2J_i+ 1|(ξ_fJ_fkM_σλkξ_iJ_i)|² , (2.40) where J_i/f are the total angular momenta of the final/initial state andξ_i/f stands for the rest of the quantum numbers needed to describe the states. In this thesis, the notation M_{M λ} =M_λ for the magnetic operator will be used in order to simplify the notation.

The magnetic tensor operator is of the form M_λµ = µ_N

~cζ^{(M λ)}

A

X

j=1

2

λ+ 1g_l^(j)l(j) +g_s^(j)s(j)

·∇_j^hr^λ_jY_λµ(Ω_j)ⁱ , (2.41) where j refers to a nucleon and l(j) and s(j) are the orbital and spin angular momenta, correspondingly. The gyromagnetic ratios g_s and the orbital g factors g_l are

g_s =

( 5.586 ^µ_c^N for protons

−3.826 ^µ_c^N for neutrons (2.42)

g_l =

( 1 ^µ_c^N for protons

0 ^µ_c^N for neutrons (2.43)

and the nuclear magneton µ_N is

µ_N = e~

2m_p , (2.44)

where mp is the proton mass.

The phase factors ζ^{(M λ)} are in this thesis expressed in the Biedenharn-Rose phase con- vention as

ζ^{(M λ)} =i^λ−1 . (2.45)

The magnetic transition probability can be expressed in the following useful numerical form:

T_{f i}^{M λ}= 6.080×10²⁰f(λ) Eλ[MeV]

197.33

!2λ+1

B(M λ)[(µ_N/c)²fm^2λ−2]1/s, (2.46) where

f(λ)≡ λ+ 1

λ[(2λ+ 1)!!]² . (2.47)

2.4.3 Single particle matrix elements

A general one-body spherical tensor operator T_λµ can be written in the form [5]

T_λµ =^X

αβ

hα|T_λµ|βic^†_αc_β =λ^b−1X

ab

(akT_λkb)^hc^†_a˜c_bⁱ

λµ , (2.48)

where

˜

c_α≡(−1)^ja+mac−α, c−α =ca,−mα (2.49) is an annihilation operator and

λb ≡√

2λ+ 1 . (2.50)

Since in the electromagnetic transitions a nucleon is annihilated from the initial state and created in the final state, the electromagnetic transition matrix elements are of the type

hξ_fJ_fM_f|T_λµ|ξ_iJ_iM_ii=^X

αβ

hα|T_λµ|βi hξ_fJ_fM_f|c^†_αc_β|ξ_iJ_iM_ii , (2.51) where ξ includes the quantum numbers besides J and M needed to specify the state.

When one applies the Wigner-Eckart theorem to (2.51), one gets the reduced matrix element

(ξ_fJ_fkT_λkξ_iJ_i) = λ^b−1X

ab

(akT_λkb)(ξ_fJ_fk[c^†_a˜c_b]_λkξ_iJ_i), (2.52) which is called the transition amplitude.

For the electromagnetic operator M_σλ, the equation (2.52) gives (ξ_fJ_fkM_σλkξ_iJ_i) = λ^b−1X

ab

(akM_σλkb)(ξ_fJ_fk[c^†_a˜c_b]_λkξ_iJ_i). (2.53)

Using (2.41) and the information that the reduced matrix element for the spherical tensor Y_λ can be obtained from

(l¹₂jkY_λkl⁰¹₂j⁰) = 1

√4π(−1)^j0−12+λ1 + (−1)^l+l0+λ

2 ^bjj^b0λ^b j j⁰ λ

1

2 −¹₂ 0

!

(2.54)

the reduced matrix element (2.53) can be cast to the form (akM_λkb) =ζ_ab^{(M λ)}µ_N/c

√4π(−1)^jb+λ−121−(−1)^la+lb+λ

2 λ^bj^b_aj^b_b j_a j_b λ

1

2 −¹₂ 0

!

×(λ−κ)

g_l

1 + κ λ+ 1

− ¹₂g_s

R^(λ−1)_ab ,

(2.55)

where

κ= (−1)^la+ja+12(j_a+¹₂) + (−1)^lb+jb+12(j_b+ ¹₂) (2.56) and

ζ_ab^(mλ) = (−1)^{12(lb−la+λ+1)} (2.57) in the chosen BR convention. The hat factors are defined as in (2.50).

3 NUCLEAR MODELS

In this section, the nuclear models used in the thesis are introduced. We will start from the Woods-Saxon potential, which is used to describe the nuclear potential caused by the interactions between nucleons. The Woods-Saxon potential is used to form the single- particle basis for protons and neutrons needed in the further calculations. The BCS (Bardeen Cooper Scrieffer) theory is then applied to the single-particle valence basis to produce the single-quasiparticle excitations of the nuclei of interest. The QRPA (quasi- particle random phase approximation) theory is used to describe the excited states of the nuclei. Finally, the MQPM (microscopic quasiparticle model) theory couples the BCS quasiparticles and QRPA phonons together.

3.1 Woods-Saxon Potential

The nuclear mean-field potential V is generated by the mutual interactions of nucleons.

Therefore, the form of the potential is similar to that of the density of the nuclear matter in the nucleus. Such potential can be approximated with the Woods-Saxon potential.

The Woods-Saxon potential can be parametrized in the form [11]:

V_{W S}(r) = −V₀

1 + exp[(r−R)/a] , (3.1)

where V₀ is a constant (the depth of the potential), R the radius, where the potential is half its central value and a describes the diffuseness of the nucleus.

In this work, the usual parametrization is used [5]:

R =r₀A^1/3 = 1.27A^1/3 fm, (3.2)

a= 0.67 fm, (3.3)

V₀ =

51±33N −Z A

MeV, (3.4)

where the negative sign is for neutrons and positive sign for protons.

The central Woods-Saxon potential can be modified to reproduce the observed single- particle energies by adding a spin-orbit interaction term to it. It splits the mean-field potential states of the same angular momentum quantum number l into two distinguish- able states with the total single-particle angular momenta quantum numbers asj =l+¹₂ and j =l−¹₂. The energy splitting caused by the spin-orbit interaction is on the million electronvolts scale. The spin-orbit potential V_LS can be written is the form [12]

V_LS =V_LS⁽⁰⁾

r_o

~

2"

d dr

1

1 + exp{(r−R)/a}

#

. (3.5)

Due to the derivative, the spin-orbit interaction is at its strongest at the nuclear surface region, as it should be since the nuclear potential prevents the nucleons from escaping

from the nucleus [13]. The Woods-Saxon parameters of Ref. [11] are used and the V_LS⁽⁰⁾ is taken to be

V_LS⁽⁰⁾ = 0.44V₀ . (3.6)

Furthermore, as charged particles, the protons also experience the Coulomb interaction, which has to be taken into account in the mean field potential. The Coulomb interaction makes protons repel each other, which causes the potential well for protons to be shallower than that for neutrons. The Coulomb potential can be written in the form [5]

V_C = Ze² 4π₀







3−(r/R)²

2R r≤R,

1

r r > R, (3.7)

where the charged radiusR is given in (3.2). For neutrons the Coulomb potential is taken to be zero. The total Woods-Saxon potential, where the spin-orbit and Coulomb terms are added, is showed schematically for protons and neutrons in Figure 1.

The wave functions that fulfil the Schrödinger equation with the Woods-Saxon potential, Coulomb potential and spin-orbit coupling are called Woods-Saxon wave functions. They can be constructed either in a harmonic oscillator wave function basis or by solving the corresponding differential equation using a dedicated solver. The complete Hamiltonian for the interaction is

h=− ~²

2m_N +V(r) +VLS(r)L·S

=− ~²

2m_N ∇²_r− L²/~² r²

!

+V_{W S}(r) +V_C(r) +V_LS(r)L·S ,

(3.8)

where the radial Laplacian operator ∇²_r in polar coordinates is

∇²_r ≡ 1 r²

d dr r² d

dr

!

. (3.9)

If one assumes that the states l ¹₂ j m^Eare the eigenstates of the operators L² and L·S present in the Hamiltonian (3.8), we get, with the help of (2.11a–2.11c), the eigenvalue equations

L²l ¹₂ j m^E=l(l+ 1)~²

l ¹₂ j m^E , (3.10a)

S²l ¹₂ j m^E= ³₄~²

l ¹₂ j m^E , (3.10b)

J²l ¹₂ j m^E=j(j + 1)~²

l ¹₂ j m^E , (3.10c)

Jz

l ¹₂ j m^E=m~

l ¹₂ j m^E (3.10d)

The total angular momentum operator can be written as a sum of the orbital and spin angular momentum operators as J =L+S. Therefore, one can write J² = (L+S)² = L²+ 2L·S+S². Combining this with Equations (3.10a–3.10d) one gets

j(j + 1)~²

l ¹₂ j m^E=^hl(l+ 1)~²+ 2L·S +³₄~²

i

l ¹₂ j m^E , (3.11) and by arranging the terms

L·Sl ¹₂ j m^E= ¹₂^hj(j+ 1)−l(l+ 1)− ³₄ⁱ l ¹₂ j m^E . (3.12)

Finally, one can solve the eigenstates of the Hamiltonian (3.8) in the n l ¹₂ j m^E basis, where the quantum number n is the principal quantum number of the state. Combining (3.8) with (3.12), the time-independent Schrödinger equation for the ansatz becomes

(

− ~² 2m_N

"

∇²_r− l(l+ 1) r²

#

+V_{W S}(r) +V_C(r) + ¹₂[j(j + 1)−l(l+ 1)− ³₄]~²V_LS(r)

)

n l ¹₂ j m^E

≡hlj(r)n l ¹₂ j m^E

=_nljn l ¹₂ j m^E

.

(3.13) The energy eigenvalues do not depend on the projection quantum number m, since the Hamiltonian is spherically symmetric. For that reason, the Schrödinger equation (3.13) is actually radial, and therefore the wave functions n l ¹₂ j m^E can be expressed in the form n l ¹₂ j m^E=f_nlj(r)l ¹₂ j m^E, where the radial functions f_nlj(r) satisfy

h_lj(r)f_nlj(r) =_nljf_nlj(r). (3.14) This differential equation can be solved for the eigenenergies _nlj and eigenstates f_nlj(r).

The states f_nlj(r) are chosen so that they form a real, orthogonal and normalized basis.

So it stands _{Z ∞}

0

r²f_nlj(r)f_n⁰_lj(r)dr =δ_nn⁰ . (3.15)

r r

v Neutrons

2 8

0p -space 20

1s-0d -space 28

50

1p-0f -space Protons

0s1/2

2

0p3/2 0p1/2 0d5/2

8

1s1/2

0d3/2

20

0f7/2

28

1p3/2 1p1/2 0f5/2

0g9/2

50

Figure 1: Schematic picture of the Woods-Saxon potential for protons and neutrons.

The magic numbers are indicated with circled numbers.

3.2 BCS Theory

The BCS theory can be used to approximate the quasiparticle energies of nuclei. The BCS, or Bardeen-Cooper-Scrieffer, theory was proposed by John Bardeen, Leon Cooper and John Robert Scrieffer in 1957. They developed a microscopic theory to describe the superconductivity of metals, which arises from the interactions between electrons and lattice vibrations. The electrons form correlated pairs with a zero total spin. At low temperatures, electrons cannot cross the energy gap formed by the bosonic pairs condensing to the ground state. Instead, they stay in the ground state without losing energy in the collisions with the lattice. This causes an electric current flow without resistance, i.e. superconductivity [14].

In 1958-1959 the theory was adapted to nuclear physics by Bohr, Mottelson and Belyaev [15, 16]. They proposed that a similar collective condensate would be present in nuclei.

The valence nucleons of nuclei feel a strong attractive force, which can be explained with

a BCS-like theory. Nowadays this theory is one of the most important theories used to describe the nature of nuclei.

The BCS ground state |BCSi can be defined with the ansatz [5]

|BCSi= ^Y

α>0

(u_a−v_ac^†_α˜c^†_α)|COREi , (3.16) where u_a and v_a are variational parameters, c^†_α the particle creation operator and ˜c^†_α its time-reversal conjugate, |COREithe ground state of the even-even reference nucleus, and α= (a, m_α), where a= (n_a, l_a, j_a) (3.17) and n, l and j are the usual principal, orbital and total angular momentum quantum numbers, respectively.

In order to normalize the state (3.16) one has to require for all a

|u_a|²+|v_a|² = 1, (3.18)

and because the amplitudes u_a and v_a are chosen to be real, equivalently

u²_a+v_a² = 1. (3.19)

The BCS ground state (3.16) can be expanded as a Taylor series as [5]

|BCSi= ^Y

β>0

u_b^X

n

1

n! −^X

α>0

v_a u_aA^†_α

!n

|COREi . (3.20) The BCS ground state can be written as a sum of eigenstates |Ni:

|BCSi= ^Y

β>0

u_b ^X

N=even

1

(N/2)! |Ni , (3.21)

where the nucleon number eigenstates |Ni are written as

|Ni ≡ −^X

α>0

v_a ua

A^†_α

!N/2

|COREi . (3.22)

The Bogoliubov-Valatin transformation to quasiparticles , introduced for example in [17], are used to diagonalize the BCS Hamiltonian. They are

a^†_α =u_ac^†_α+v_a˜c_α , (3.23a)

˜

a_α =u_a˜c_α−v_ac^†_α , (3.23b) where ˜a^†_α =a^†_−α(−1)^j+m and ˜c^†_α =c^†_−α(−1)^j+m. The Hermitian conjugates of (3.23) are

a_α =u_ac_α+v_a˜c^†_α , (3.24a)

˜

a^†_α =u_a˜c^†_α−v_ac_α . (3.24b)