Application and extension of a fast parallel code for ... · Theories try to explain these features by trac-ing them back to atomic bound-bound tran-sitions of atoms or ions in the

Application and extension of a fast parallelcode for calculating atomic data in neutron star

magnetic fields

Diploma Thesisby

Akin Yildirim

22.10.2009

First Supervisor: Prof. Dr. Gunter WunnerSecond Supervisor: Dr. Jens Harting

1. Institut fur Theoretische PhysikUniversitat Stuttgart

Pfaffenwaldring 57, 70550 Stuttgart

http://itp1.uni-stuttgart.de

http://www.uni-stuttgart.de

Declaration of authenticity

I hereby declare that I wrote this work independently and used no other sources oraids than those indicated.

Stuttgart, 22.10.2009 Akin Yildirim

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Atoms in magnetic fields 52.1 Hamiltonian of an N-particle atom or ion . . . . . . . . . . . . . . . . . . 52.2 Landau quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Expansion into orthogonal basis functions . . . . . . . . . . . . . . . . . 112.4 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Classification of the magnetic field strengths . . . . . . . . . . . . . . . . 14

3 Hartree-Fock method 153.1 Theoretical basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Numerical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Finite-element method and B-splines . . . . . . . . . . . . . . . . 193.3 Adiabatic approximation Hartree-Fock procedure . . . . . . . . . . . . . 20

3.3.1 Hartree-Fock equations . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Parity attribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.3 Hartree-Fock-Finite-Elements equation . . . . . . . . . . . . . . . 233.3.4 Energy equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Hartree-Fock-Roothaan method . . . . . . . . . . . . . . . . . . . . . . . 253.4.1 Expansion into Landau states . . . . . . . . . . . . . . . . . . . . 253.4.2 Hartree-Fock-Roothaan equations . . . . . . . . . . . . . . . . . . 263.4.3 Energy equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 The general Hartree-Fock-Finite-Elements method . . . . . . . . . . . . . 293.5.1 Self-consistence of the general method . . . . . . . . . . . . . . . 293.5.2 Variational calculation . . . . . . . . . . . . . . . . . . . . . . . . 293.5.3 Hartree-Fock-Finite-Elements equation and energy equation . . . 33

4 Oscillator strengths 354.1 Theoretical basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Algorithm and implementation 39

v

Contents

5.1 Initial wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Program flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Calculations and results 456.1 Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1.1 He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.1.2 He+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2 Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2.1 Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2.2 Li+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2.3 Li+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3 Beryllium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.3.1 Be . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.3.2 Be+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3.3 Be+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3.4 Be+3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.4 Boron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.4.1 Boron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.4.2 B+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.4.3 B+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4.4 B+3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.4.5 B+4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.5 Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Conclusion and Outlook 857.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Bibliography 87

Acknowledgment 89

vi

List of Figures

1.1 Illustration of a Pulsar [18]. . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 CHANDRA X-ray observatory [11]. . . . . . . . . . . . . . . . . . . . . . 21.3 Spectra of the isolated neutron star 1E 1207.4-5209, taken by the CHAN-

DRA X-ray observatory [9]. . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Quantized Landau states perpendicular to the external magnetic fielddirection and free classical motion in the magnetic field direction [17]. . . 10

2.2 Energy values of hydrogen as a continuous function of the magnetic fieldparameter β[14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Contour plots of the spatial probability distribution for hydrogen statesfor 4 different magnetic field strengths [14]. . . . . . . . . . . . . . . . . . 13

5.1 Convergence diagram for Li+1, at a magnetic field strength of B = 1×108

Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Convergence diagram for Li+1, at a magnetic field strength of B = 1×109

Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Flow chart of the algorithm (adopted from ENGEL [6]). . . . . . . . . . 42

6.1 Grotrian diagram for relevant transitions of the outer electron in helium,at B = 108 Tesla. Calculated in adiabatic approximation, transition ener-gies in eV (and the respective oscillator strengths) are shown next to thearrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Grotrian diagram for relevant transitions of the inner electron in helium,at B = 108 Tesla. Calculated in adiabatic approximation, transition ener-gies in eV (and the respective oscillator strengths) are shown next to thearrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3 Spectra of He for 6 different magnetic field strengths. . . . . . . . . . . . 496.4 Grotrian diagram for relevant transitions of the single electron in He+1, at

B = 108 Tesla. Calculated in adiabatic approximation, transition energiesin eV (and the respective oscillator strengths) are shown next to the arrows. 51

6.5 Spectra of He+1 for 6 different magnetic field strengths. . . . . . . . . . . 526.6 Spectra of Li for 6 different magnetic field strengths. . . . . . . . . . . . 546.7 Spectra of Li+1 for 6 different magnetic field strengths. . . . . . . . . . . 566.8 Spectra of Li+2 for 6 different magnetic field strengths. . . . . . . . . . . 58

vii

List of Figures

6.9 Spectra of Be for 6 different magnetic field strengths. . . . . . . . . . . . 616.10 Spectra of Be+1 for 6 different magnetic field strengths. . . . . . . . . . . 636.11 Spectra of Be+2 for 6 different magnetic field strengths. . . . . . . . . . . 666.12 Spectra of Be+3 for 6 different magnetic field strengths. . . . . . . . . . . 686.13 Wave functions of boron at a magnetic field strength of 1 x 108 Tesla.

Left: The ground state. Right: A singly excited state of the innermostelectron, with m = 0 and ν = 1 . . . . . . . . . . . . . . . . . . . . . . . 69

6.14 Spectra of B for 6 different magnetic field strengths. . . . . . . . . . . . . 716.15 Spectra of B+1 for 6 different magnetic field strengths. . . . . . . . . . . 746.16 Spectra of B+2 for 6 different magnetic field strengths. . . . . . . . . . . 776.17 Spectra of B+3 for 6 different magnetic field strengths. . . . . . . . . . . 796.18 Spectra of B+4 for 6 different magnetic field strengths. . . . . . . . . . . 826.19 Spectra of C for 6 different magnetic field strengths. . . . . . . . . . . . . 83

viii

List of Tables

6.1 Relevant transitions and their oscillator strengths. He at a magnetic fieldstrength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2 Relevant transitions and their oscillator strengths. He+1 at a magneticfield strength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.3 Relevant transitions and their oscillator strengths. Li at a magnetic fieldstrength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4 Relevant transitions and their oscillator strengths. Li+1 at a magneticfield strength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.5 Relevant transitions and their oscillator strengths. Li+2 at a magneticfield strength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.6 Relevant transitions and their oscillator strengths. Be at a magnetic fieldstrength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.7 Relevant transitions and their oscillator strengths. Be+1 at a magneticfield strength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . 64



6.10 Relevant transitions and their oscillator strengths. B at a magnetic fieldstrength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.11 Relevant transitions and their oscillator strengths. B+1 at a magnetic fieldstrength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73




6.15 Relevant transitions and their oscillator strengths. C at a magnetic fieldstrength of 108 Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

ix

List of Tables

x

1 Introduction

1.1 Motivation

A neutron star represents the final stage in the evolution of a certain class of stars.

As soon as the reservoir of hydrogen is exhausted, the star begins to burn higher elements.This process continues till the production of the element iron. Iron cannot further fuse,because it has the highest binding energy per nucleon. Therefore a further fusion wouldcost energy instead of producing it.

With the end of the fusion the radiation pressure declines, the gravitational forces beginto dominate, and the star collapses. The collapse proceeds until the neutrons build upa degenerate pressure which stops the further collapse of the star.

Figure 1.1: Illustration of a Pulsar [18].

The newly created neutron star distinguishesitself through its extreme characteristics:

The mass of a typical neutron star fluctuatesbetween 1.4Msolar − 2.1Msolar. The collapseleads to a small diameter of about 104m andtherefore to a high density and to extreme grav-itational forces up to 210 times stronger thanthe gravitational forces on earth.

The conservation of the angular momentumleads to a fast rotation of the small star. Ifthe axis of the magnetic field is inclined withrespect to the rotational axis, the neutron staremits electromagnetic radiation over a widerange of frequencies, as illustrated in Figure

1.1. Pulsars emitting predominantly radio waves are classified as radio pulsars. Theycan be observed by radio astronomers, and were for first detected in 1967, by JocelynBell and Anthony Hewish at the Cavendish Laboratory in Cambridge.

Magnetohydrodynamics furthermore postulates a conservation of the magnetic flux forhighly ionized plasma. This leads to high magnetic field strengths at the end of thecollapse, in the order of 108 Tesla. The surface of the neutron star is hot (estimated as 106

1

1 Introduction

K), and emits thermal radiation in the range of X-rays (E ≈ 1 keV). At the beginning ofthe 90’s, for the first time neutron star spectra were observed with high resolution. Theseobservations were made by X-ray observatories, such as the CHANDRA observatory bythe NASA (Figure 1.2) or the XMM-NEWTON observatory by the ESA.

Figure 1.2: CHANDRA X-ray observa-tory [11].

Especially the observation of the isolated neu-tron star 1E 1207.4-5209 delivered intriguingdata. Its spectrum, which can be seen in Fig-ure 1.3, shows two absorption features at 0.7keV and 1.4 keV.

The origin of these two features is unknown.Theories try to explain these features by trac-ing them back to atomic bound-bound tran-sitions of atoms or ions in the thin neutronstar atmosphere. The width of the two fea-tures could be due to the varying magnetic fieldstrength over the surface of the neutron star.To support this theory, by modelling the atmo-spheres and calculating synthetic spectra, extensive atomic data are absolutely neces-sary.

Since the elemental composition of the neutron star is unknown, bound-bound transi-tions of every element up to iron, in every possible ionization state have to be calculated.The presence of the magnetic field prevents a closed analytical solution. For modellingthe neutron star atmosphere, we therefore perform numerical calculations with an ap-proximation method, such as Hartree-Fock methods. The atomic data depend on themagnetic field strength. This forces us to perform the calculations over a wide rangeof magnetic field strengths. The results can serve as input for the modelling of spec-tra by astronomers, and if agreement is found, this would be a big step forward in thecharacterization and understanding of these important compact cosmic objects.

2

1.2 Outline

Figure 1.3: Spectra of the isolated neutron star 1E 1207.4-5209, taken by the CHANDRAX-ray observatory [9].

1.2 Outline

Chapter 2 will cover the theoretical basics. Our attention is focused on atoms in magneticfields. We will introduce the relevant quantum numbers and consider the behavior ofatoms in intense magnetic fields. The Landau quantization is essential for the treatmentof atoms in magnetic fields.The starting point for our investigations is the hydrogen atom, which represents thesimplest atomic system in magnetic fields. The new quantum numbers and the energylevel schemes for many-electron atoms can be deduced from these investigations. We willalso classify magnetic fields into different categories, where different forces predominatethe behavior of the atom.

In Chapter 3 we will discuss the Hartree-Fock method as an approximation method tosolve the problem of many-electron atoms in intense magnetic fields. Furthermore wewill develop a combined method which consists of two steps: The Hartree-Fock-Finite-Elements (HFFE) part and the Hartree-Fock-Roothaan (HFR) part. Our whole methodrests upon an alternate variation of a longitudinal and a transversal part of the wavefunction. A variational calculation is carried out, which leads to a convergence for bothparts and therefore to a convergence of the total energy.

Chapter 4 will shortly review the concept of oscillator strengths. After the calculationof the wave functions and the associated energy levels, we can calculate the transitionprobabilities.

3

1 Introduction

Chapter 5 will give a brief overview about the flow of the program. The proper selectionof initial wave functions is crucial for the convergence of the calculations. We will shortlydiscuss different ansatzes that have been made to date.

The results will then be summarized in Chapter 6. We will present new results for atomsup to carbon, in different ionization states and for 6 different magnetic field strengths.We will draw conclusions from these calculations, as far as possible.

Chapter 7 summarizes the results and gives an outlook for future investigations.

4

2 Atoms in magnetic fields

2.1 Hamiltonian of an N-particle atom or ion

We start with the examination of atomic systems in external magnetic fields. Thenon-relativistic, infinite-nuclear-mass Hamiltonian of an N -electron atom in an externalmagnetic field is given by:

H =N∑i=1

[1

2me

(−i~∇i + eAi)2 +

e~2me

σiB−Ze2

4πε0|ri|

]+

N∑i,j=1i<j

e2

4πε0|ri − rj|(2.1)

The different terms of equation (2.1) represent the following physical quantities:

• Operator of the kinetic momentum

1

2me

(−i~∇i + eAi)2

• Interaction term between spin and magnetic field

e~2me

σiB ,

where each Pauli matrix σi represents an observable describing the spin of a spin12

particle in the three spatial directions.

• Coulomb potential between nucleus and electron

Ze2

4πε0|ri|

• Electron-electron interaction term

e2

4πε0|ri − rj|

5


The spherical symmetry of the Coulomb potential is perturbated by the cylindrical sym-metry of the magnetic field, which is assumed to point in z-direction. The cylindricalsymmetry of the superimposed magnetic field prohibits a separation of the variables,and therefore an analytical solution. This is a fundamental problem in the solution ofthe respective Schrodinger equation to the Hamiltonian given by (2.1), in addition tothe electron-electron interaction.To approach the problem, we carry out a transition into cylindrical coordinates. Conse-quently, the Hamiltonian of (2.1) can be written as:

H =N∑i=1

[− ~2

2me

(∂2

∂ϕ2i

+1

ρi

∂

∂ρi+

1

ρ2i

∂2

∂ϕ2i

+∂2

∂z2i

)− i~ eB

2me

∂

∂ϕi+e2B2ρ2

i

8me

+ ~eB

2me

σzi −Ze2

4πε0|ri|

]+

N∑i,j=1i<j

e2

4πε0|ri − rj|. (2.2)

Next, we carry out a transformation to atomic Rydberg units. The Bohr radius a0 servesas unit of length, and the Rydberg energy E∞ as unit of energy.

ao =4πε0~2

mee2≈ 5.3 · 10−11 m E∞ =

mee4

(4πε0)22~2≈ 13.61 eV (2.3)

In addition we introduce new parameters, which will be helpful in the further discussionof the problem. From now on, we will represent the magnetic field strength by thedimensionless field strength parameter β.

β =B

B0

, where B0 =2m2

ee3

(4πε0)2~3≈ 4.7 · 105 T . (2.4)

B0 is the magnetic field strength where the Larmor radius aL becomes as long as theBohr radius a0. The Larmor radius is the characteristic length which results from thequantization of the associated problem. In atomic units, the Larmor radius can bewritten as:

aLa0

=1

a0

√2~eB

=1√β. (2.5)

6

2.1 Hamiltonian of an N-particle atom or ion

In atomic units and in cylindrical coordinates, the Hamiltonian of an N -electron atomor ion in an external magnetic field can be written in the form:

H =N∑i=1

[−(∂2

∂ϕ2i

+1

ρi

∂

∂ρi+

1

ρ2i

∂2

∂ϕ2i

+∂2

∂z2i

)− 2iβ

∂

∂ϕi+ β2ρ2

i + 2βσzi −2Z

|ri|

]+

N∑i,j=1i<j

2

|ri − rj|. (2.6)

We can assume that all spins of the electrons are aligned antiparallel to the magneticfield direction. This guarantees the minimal energy of the system. A spin-flip of anelectron would cost energies in the range of several keV, in the examined magnetic fieldstrengths, which is larger than the Coulomb binding energies of the electron. Thereforewe can restrict ourselves to electrons in spin-down states and ignore the spin quantumnumber in what follows.

7


2.2 Landau quantization

The Landau quantization describes the quantization of the energies for charged parti-cles in a magnetic field. This section shall give a short overview about the quantumproperties of charged particles in magnetic fields. We will derive the Landau levels andintroduce the Landau wave functions, which we need in the further discussion of ourstudy.

The uniform magnetic field is assumed to point in z-direction:

B = (0, 0, Bz) , (2.7)

and can be described by a vector potential A via:

B = ∇×A . (2.8)

In the case of a uniform magnetic field, we can set:

A = −1

2B× r . (2.9)

The Lagrange function of a charged particle in a magnetic field is given by:

L =1

2mv2 + qvA , (2.10)

and the classical Hamiltonian is:

H =1

2m[p− qA]2 , (2.11)

where p represents the conjugate momentum of the particle postion r. The momentump is not equal to the mechanical momentum of the particle, given by mv. In a magneticfield, the momentum p is given by:

p = mv + qA . (2.12)

The total motion in the classical problem represents an overlap of a gyration in thexy-plane and a free motion in magnetic field direction. Analogously we can split thequantum Hamiltonian corresponding to equation (2.11) into a transversal and a longi-tudinal part. Since the free longitudinal motion is trivial, we restrict our considerationsto the transversal part of the Hamiltonian:

H = H‖ + H⊥ ,

H⊥ =1

2m

[(px − qAx)2 + (py − qAy)2

]. (2.13)

8

2.2 Landau quantization

The operator V is associated with the velocity of the charged particle, and can be writtenas:

V =1

m[p− qA] . (2.14)

This relation allows us to rewrite the Hamiltonian of equation (2.13) as:

H =m

2V

2. (2.15)

Correspondingly, the transversal part of the Hamiltonian can then be written as:

H⊥ =m

2(V 2

x + V 2y ) . (2.16)

The longitudinal and the transversal part of the Hamiltonian represent commuting op-erators. Hence we can search for the eigenvalues of H⊥ and H‖ separately.

E = E‖ + E⊥ . (2.17)

We know, from the classical analysis, that the charged particle performs a gyration inthe xy-plane, where the frequency of this motion is given by the cyclotron frequencyωc = − qB

m.

For the further discussion, we introduce new operators:

Q =

√m

~ωcVy ,

S =

√m

~ωcVx . (2.18)

Thus, the transversal Hamiltonian becomes:

H⊥ =~ωc2

(Q2 + S2) . (2.19)

This corresponds to the well known Hamiltonian of a 1-dimensional harmonic oscillator.The operators Q and S can be regarded as the position and the momentum of theoscillator, respectivelyThe eigenvalues of the 1-dimensional harmonic oscillator, and hence the eigenvalues ofthe transversal Hamiltonian, are therefore given by:

E⊥ =

(n+

1

2

)~ωc , n = 0, 1, 2, ... . (2.20)

As in the case of the 1-dimensional harmonic oscillator, the eigenvalues are quantized.The corresponding energy levels of the Hamiltonian H are called Landau levels, and thequantum number n the Landau quantum number.

9


The total energy E is the sum of the transversal and the longitudinal energy, and isgiven by:

E =

(n+

1

2

)~ωc +

1

2mv2

z , (2.21)

where mv2z represents the eigenvalues of the longitudinal Hamiltonian H‖, of the free

motion in the z-direction. These eigenvalues are not quantized. Hence, the spectrum ofH‖ is continuous.

Figure 2.1: Quantized Landau states perpendicular to the external magnetic field direc-tion and free classical motion in the magnetic field direction [17].

The states to the corresponding Landau levels are the well known Landau states. Theyare given by [16]:

Φnm(ρ, φ) =1

aL√πn!(n−m!)

eimφ(ρ

aL

)me−ρ2

2a2LLmn−m

(ρ2

a2L

). (2.22)

Here, n represents the Landau quantum number and m the magnetic quantum number.Lmn−m are the well known Laguerre polynomials, which are the canonical solution ofLaguerre’s second order linear differential equation.The Landau functions are the wave functions of charged particles in a uniform magneticfield, pointing in the z-direction. They are orthogonal and normalized:∫ 2π

0

dφ

∫ ∞0

ρ dρ φ∗nm φn′m′ = δnn′ δmm′ . (2.23)

10

2.3 Expansion into orthogonal basis functions

2.3 Expansion into orthogonal basis functions

In the case where the Coulomb forces outweigh the Lorentz forces, the obvious procedureis an expansion of the wave function into appropriate orthogonal basis functions. Herewe make use of spherical coordinates. This leads to:

ψ(r,Θ,Φ) =∑n,l

Rnl(r) Ylm(θ, φ) , (2.24)

where Ylm(Θ,Φ) are the spherical harmonics, and Rnl(r) a set of complete radial basisfunctions.

In the regime around B ≈ B0, the Larmor radius becomes of the same order of magni-tude as the Bohr radius. Above this critical field strength, the cylindrical symmetry ofthe magnetic field begins to outweigh. Thus, we can carry out an expansion in terms ofLandau states, in cylindrical coordinates,

ψ(z, ρ, φ) =∑n,m,ν

gnmν(z) Φnm(ρ, φ) . (2.25)

Here, ν is the new quantum number which defines the number of nodes in magnetic fielddirection of the longitudinal expansion functions gnmν(z).

At very intense magnetic field strengths (B → ∞), the expansion of equation (2.25)reduces itself to the lowest Landau state, because a transition to the next higher Landaustate is in the range of keV (e.g. 50 keV for B = 5·108 T). Hence, the expansion becomesa single product of a longitudinal wave function and the lowest Landau level:

ψ(z, ρ, φ)ad = gadmν(z) Φ0m(ρ, φ) . (2.26)

This is also known as the adiabatic approximation [15].Only the longitudinal part of the wave function gmν(z) has then to be calculated numer-ically.

In our calculations we make use of an ansatz which is more general than the adiabaticapproximation (2.26) but more restrictive than the full expansion (2.25). We make useof the fact that previous calculations for hydrogen have shown that, at intermediate fieldstrengths, the longitudinal expansion functions gnmν(z) do have a very similar structurein all Landau states [14].Therefore, our new ansatz is to calculate an averaged wave function gmν(z) and a setof Landau amplitudes tnmν , which account for the contribution of the different Landaulevels:

ψ(z, ρ, φ) = gnmν(z)

NL∑n=0

tnmν Φnm(ρ, φ) . (2.27)

11


2.4 Hydrogen

In this part, as an example, we will review the hydrogen atom as the simplest atomic sys-tem in external magnetic fields [14]. We can characterize the state of the hydrogen atomby the quantum numbers (n,m, ν). The Landau expansion, derived before, reads:

ψ(z, ρ, φ) =∑n

gnmν(z) Φnm(ρ, φ) . (2.28)

In the case of very high magnetic field strengths, we can reduce the sum to one sum-mand, taking into account only the lowest Landau level (n = 0).States with positive magnetic quantum number (m > 0) are lifted to Landau stateswith positive quantum numbers (n > 0). Therefore we only examine those states withnegative magnetic quantum number (m < 0) which stay in the lowest Landau level(n = 0). The level scheme is shown in Figure 2.2. Beginning with β = 0 we can see the

Figure 2.2: Energy values of hydrogen as a continuous function of the magnetic fieldparameter β[14].

commonly known Rydberg structure of the scheme. With increasing β, the linear andquadratic Zeeman effect occurs. The increasing of β leads to a new arrangement of thelevel structure, which can be finally seen above β = 1.In this regime (the intense-field case), the energies of the states with positive magneticquantum number become less and less negative. These states are shifted to the excitedLandau states, with n > 0.States with vanishing or negative magnetic quantum number move over to the lowestLandau level state. Here again, we have to differ between the nodeless (ν = 0) statesand the states with ν > 0.The nodeless states tend (for every m ≤ 0) for ever increasing β to −∞, whereas thestates with nodes converge against Rydberg series 1

ν2 , with ν = ν2

for ν even, and ν = ν+12

12

2.4 Hydrogen

for ν odd. This is the reason why these states are also labelled as the “hydrogen-likestates”. Figure 2.3 shows the spatial probability distribution for the lowest hydrogen

Figure 2.3: Contour plots of the spatial probability distribution for hydrogen states for4 different magnetic field strengths [14].

states in increasing magnetic field strengths. The magnetic field points in the z-direction.We can see how the probability distribution becomes more and more cigar shaped. Thereason for this behavior is that with increasing magnetic field strength the spatial exten-sion of the Landau states, which scales with the Larmor radius aL = a0√

β, shrinks. While

the longitudinal extension, which is only determined by the influence of the Coulombforce, is hardly affected.

13


2.5 Classification of the magnetic field strengths

Depending on the magnetic field strength, and the predominant symmetry of the system,we can subdivide the magnetic field strength into different regimes:

The weak field regime: This is where the spherical symmetry of the Coulomb potentialpredominates, and where the magnetic field can be treated as a small perturbation.In this regime, we can observe the well known Zeeman effect.aL >> a0 ←→ B0 >> B

The intermediate field: Here, the electrons experience electric and magnetic forces ofthe same order.aL ≈ a0 ←→ B0 ≈ B

The intense field regime: In this regime, the cylindrical symmetry of the magnetic fieldpredominates over the spherical symmetry of the Coulomb potential. This is theregime in which our calculations will take place. The longitudinal motion of theelectrons is caused by the Coulomb potential of the nucleus, whereas the gyrationof the electrons perpendicular to the magnetic field direction is only effected bythe magnetic field itself. Introduced by [15], the adiabatic approximation treatsthe total motion of the electrons as a non perturbating overlap of the longitudinaland the perpendicular motion.aL << a0 ←→ B0 << B

The classification of the magnetic field strengths changes dramatically for higher ele-ments. Here, the dimensionless Z-scaled magnetic field parameter βZ = β

Z2 is the criticalquantity, which decides whether the atom is liable to intermediate or intense magneticfields.For βZ >> 1 we can assume intense magnetic fields, where the adiabatic approximationis feasible. The problem with magnetic fields βZ << 1, where the spherical symmetryof the Coulomb potential dominates, is that an expansion into cylindrical coordinatesbecomes worse and worse.

14

3 Hartree-Fock method

This chapter will give an overview about the Hartree-Fock calculations for N -electronatoms in intense magnetic fields. In the presentation we largely follow reference [6], sincethe latter explains the method in a very complete and intelligible way.Considering the ansatz made in equation (2.27), we are obliged to determine the longi-tudinal wave functions g as well as the Landau amplitudes t.Our approach to solve this problem consists of two steps. The first step is a longitudinalvariational calculation for the determination of the longitudinal wave functions g. Forthis we will keep the Landau amplitudes t constant. Thereupon, we will determine theLandau amplitudes t in a transversal variational calculation. This second step is realizedby keeping the longitudinal wave functions g constant.The calculations are reiterated until the total energy of the N -electron system converges,which corresponds to the convergence of the two single steps.The variational calculation leads to a Hartree-Fock equation for the longitudinal part ofthe wave function, and to a Hartree-Fock-Roothaan equation for the transversal part ofthe wave function.

3.1 Theoretical basics

The Hamiltonian of a N -particle system is given by equation (2.1). This Hamiltonianconsists of single-particle and two-particle terms. Thus we can split the Hamiltonian upinto:

H = H1 + H2 =N∑i=1

h(i) +N∑

i,j=1i<j

w(i,j) . (3.1)

If we first disregard the Coulomb interactions, which represent the two-particle term in(3.1), the Hamiltonian reduces itself to the single-particle term:

H =N∑i=1

hi . (3.2)

15


With the ansatz:ψtot(r1, ..., rN) = ψ1(r1)...ψN(rN) , (3.3)

(which is known as the Hartree product) for the total wave function ψtot, we obtain Nindependent single-particle Schrodinger equations:

hiψi(ri) = εiψi(ri) . (3.4)

A fermionic system requires an antisymmetric total wave function of ortho-normalizedsingle-particle wave functions. This requirement is not fullfilled by the Hartree productbut can be guaranteed by the Slater determinant which also satisfies Pauli’s exclusionprinciple, according to which two electrons cannot occupy the same state. The Slaterdeterminant is given by:

ψtot =1√N !

∣∣∣∣∣∣∣∣∣∣ψ1(r1) · · · ψ1(rN). .. .. .

ψN(r1) · · · ψN(rN)

∣∣∣∣∣∣∣∣∣∣. (3.5)

The single-particle wave functions are, as stated before, products of a z-dependent func-tion and a Landau function:

ψi(z, ρ, φ) = gνi,mi(z) Φ0mi(ρ, φ) . (3.6)

Since the Coulomb interactions of the electrons represent a non-negligible perturbation,a separation into N single-particle Schrodinger equations is not possible.The challenge is to find optimal single-particle wave functions. The method to solve thisproblem is the Hartree-Fock method.Since the energy of the system is a functional of the total wave function ψtot, a variation ofψtot delivers the minimum of the energy functional, therefore the best Slater determinant,and thus the proper single-particle wave functions ψi(ri):

δ

[< ψtot, H, ψtot > −

N∑i=1

εi < ψi, ψi >

]= 0 . (3.7)

The Lagrange parameters εi represent the single-particle energies. The Lagrange param-eters include the interaction with each electron. A sum over the Lagrange parameterswould count this interaction term twice. Therefore the sum over all Lagrange parametersdoes not correspond to the total energy. We have to subtract the interaction energy toobtain the total energy of the system

Etot =∑i

εi− < H2 >=< H1 > + < H2 > . (3.8)

16


The total Energy Etot can be written as:

< ψtot, H, ψtot > =N∑i=1

∫ψ∗i hiψi dτ

+1

2

∑i,ji 6=j

∫ ∫ψ∗i (r1)ψ∗j (r2) g12 ψi(r1)ψj(r2) dτ1 dτ2

− 1

2

∑i,ji 6=j

∫ ∫ψ∗i (r1)ψ∗j (r2) g12 ψj(r1)ψi(r2) dτ1 dτ2 , (3.9)

with gij =1

|ri − rj|.

The Hartree-Fock equations in adiabatic approximation are given by [7]:

(−1

2

δ2

δz2+ V eff

i (z)− εi)gi(z) +

N∑j=1j 6=i

∫ ∞−∞

dz′ gj(z′)gj(z

′)V Diij (z, z′)

gi(z)

−N∑j=1j 6=i

gj(z)

∫ ∞−∞

dz′gj(z′)gi(z

′)V Exij (z, z′) = 0 , (3.10)

where V effi represents the effective potential, which includes the kinetic energy, the

Coulomb potential between electron and nucleus, and the interaction term betweencharge and spin of the electron in a magnetic field.V Diij is the Coulomb potential between the electrons.V Exij represents the exchange interaction, due to Pauli’s principle.

By introducing the Fock operator, the Hartree-Fock equations can be rewritten as:

F |i >= εi|i > , (3.11)

where |i > represents the single-particle states, which are a linear combination of or-thonormalized basis states |φn >

|i >=

NL∑n=0

cn|φn > , < φk|φn >= δkn ,

NL∑n=0

|cn|2 = 1 , (3.12)

while the corresponding bra-vectors < i′| are given by:

< i′| =NL∑k=0

d′k < φk| . (3.13)

17


The Fock matrix is an approximating matrix for the single electron energy operator in agiven set of basis vectors, and is defined by the Fock operator. The Fock matrix in thebasis of the states |i > is:

< i′|F |i >=

NL∑n,k=0

d′kcn < φk|F |φn >= εi

NL∑n.k=0

d′kcn < φk|φn >= εi

NL∑n=0

d′ncn . (3.14)

The matrix equation for the Fock matrix F and the coefficient vectors of the basis, isthen given by:

d† · F · c = εid† · c⇐⇒ dd†︸︷︷︸

1

·F · c = εi dd†︸︷︷︸1

·c . (3.15)

The multiplication with the coefficient vector d leads to the Hartree-Fock-Roothaanequations for the coefficients in the orthonormalized basis:

Fc = εic . (3.16)

Similar to the Hartree-Fock equations, the Fock matrix depends on its own solution andthus on the coefficients. It has to be solved iteratively, initiating with a starting coeffi-cient vector. For each electron we obtain a Hartree-Fock-Roothaan equation and thusan eigenvalue problem.

The Hamiltonian of the N -electron atom in an external magnetic field β is given byequation (2.1), or equation (3.1), respectively, and has the general form:

H =N∑i=1

h(i)T + h

(i)S + h

(i)K + h

(i)V︸︷︷︸

H1

+N∑

i,j=1i<j

2

|ri − rj|︸︷︷︸H2

. (3.17)

As discussed in Chapter 2, the different terms in this equation stand for the followingphysical quantities:

• transversal Landau term

hiT = −(∂2

∂ϕ2i

+1

ρi

∂

∂ρi+

1

ρ2i

∂2

∂ϕ2i

)− 2iβ

∂

∂ϕi+ β2ρ2

i ,

• spin term

hiS = 2βσzi ,

• longitudinal kinetic energy

hiK =∂2

∂z2i

,

18

3.2 Numerical techniques

• Coulomb attraction between nucleus and electrons

hiV =2Z

|ri|.

The single-particle orbitals are characterized by the magnetic quantum number m andthe longitudinal quantum number ν. The orbitals are a linear combination of productsof longitudinal single-particle electron states and transversal Landau amplitudes:

|i >= |mν >=

NL∑n=0

tnmν |nmν > , (3.18)

< r|i >=< r|mν >= gmν(z)

NL∑n=0

tnmν Φnm(ρ, ϕ) > . (3.19)

Hartree-Fock alone serves the purpose of the self-consistent calculation of the longitudi-nal and transversal wave function. The method we use is a combination of Hartree-Fockand Hartree-Fock Roothan calculations.The longitudinal and the transversal amplitudes are varied in an alternating way untilthe energy minimum is reached.The process starts with the longitudinal adiabatic approximation calculation.It should be stressed here that the calculations which were made for this thesis weredone in the adiabatic approximation, which means that we did the calculations only forthe lowest Landau level (n = 0).

3.2 Numerical techniques

3.2.1 Finite-element method and B-splines

The finite-element method is a numerical technique whose goal is to find approximatesolutions of partial differential equations (PDE) or integral equations. The solution ap-proach is based either on eliminating the differential equation completely, or renderingthe PDE into an approximating system of ordinary differential equations. Then thesedifferential equations are solved numerically using standard techniques, such as Runge-Kutta methods.The finite-element method is our tool for the calculation of the longitudinal wave func-tions gmν(z) of the N -electron atom in intense magnetic fields.As basis functions for the finite elements we make use of B-splines. A spline is a functionwhich consists of piecewise polynomials with certain connection constraints.To solve the Hartree-Fock equations for the longitudinal wave functions, we make use offinite-elements and B-spline interpolation.

19


Therefore we divide the z-axis into M finite-elements. We can choose between 3 differ-ent types of finite-element partitions: A quadratic widening, a cubic widening, and aquadratic-linear widening. Since the structure of the wave functions is richer near thenucleus, we make use of the quadratic widening.The longitudinal part of the wave functions can be expanded for each electron in termsof B-splines:

gi(z) =∑l

α(i)l Bl(z) (3.20)

The problem of finding solutions of the Hartree-Fock equations (3.42) is then reduced to

the determination of the expansion coefficients α(i)l for each electron and its longitudinal

wave function gi(z). Expressing the total energy as a real-valued function of all expan-sion coefficients and minimizing with respect to the expansion coefficients will lead to asystem of inhomogeneous linear equations (3.93) [6]

Aiαi = bi i = 1, ..., N . (3.21)

where the matrices are given by

(Ai)ij =M∑m=1

[∫Im

b′i(z)b′j(z) + c(z)bi(z)bj(z)dz

], (3.22)

and the vectors are given by

(bi)j =M∑m=1

[∫Im

hi(z)bj(z)dz

], (3.23)

whereas ci(z) and hi(z) are defined in the Hartree-Fock equations.Typically we use B-splines of the order 6, and 15-20 finite elements. The maximumintegration radius zmax is variable and chosen in such a way that all longitudinal wavefunctions have decayed exponentially.

3.3 Adiabatic approximation Hartree-Fock procedure

3.3.1 Hartree-Fock equations

The adiabatic approximation case corresponds to the vector of the Landau amplitudes:

tad := (1, 0, ..., 0) . (3.24)

20


This means that we restrict our considerations and calculations to the lowest Landaulevel (n = 0). Thus the wave function becomes a single product of a lowest Landau levelstate and the longitudinal part of the wave function:

ψmi,νi(z, ρ, φ) = gνi,mi(z) Φ0mi(ρ, φ) . (3.25)

The Hartree-Fock equations reduce to a system of 1-dimensional integro-differential equa-tions for the longitudinal wave functions [6], [7]:[− d2

dz21

+ V (0,0)mi

(z1)− εi+∑j=1j 6=i

∫ ∞−∞

gmjνj(z2) gmjνj(z2)U (00,00)mimj

(z1, z2) dz2

]gmiνi(z1)

=∑j=1j 6=i

gmjνj(z1)

∫ ∞−∞

gmjνj(z2) gmiνi(z2)A(00,00)mimj

(z1, z2) dz2 .

(3.26)

The potential functions V 0,0mi

(z), U(00,00)mimj (z1, z2), A

(00,00)mimj (z1, z2) include the transversal

Landau statesΦnm(r⊥) =< r|nm > (3.27)

for the non-adiabatic approximation case n = 0. The effective electron-nucleus andelectron-electron potentials are given by [6]:

V (0,0)mi

(z) :=< 0mi|−2Z

|r||0mi >

− 2Z

∫Φ∗0mi(r

⊥) Φ0mi(r⊥)

|r| dr⊥ , (3.28)

U (00,00)mi,mj

(z1, z2) :=(1)< 0mi|(2) < 0mj|2

|r1 − r2||0mi >

(1) |0mj >(2)

2

∫ ∫Φ∗0mi(r

⊥1 ) Φ∗0mj(r

⊥2 ) Φ0mi(r

⊥1 ) Φ0mj(r

⊥2 )

|r1 − r2|dr⊥1 dr

⊥2 , (3.29)

A(00,00)mi,mj

(z1, z2) :=(1)< 0mi|(2) < 0mj|2

|r1 − r2||0mi >

(2) |0mj >(1)

2

∫ ∫Φ∗0mi(r

⊥1 ) Φ∗0mj(r

⊥2 ) Φ0mi(r

⊥2 ) Φ0mj(r

⊥1 )

|r1 − r2|dr⊥1 dr

⊥2 . (3.30)

The magnetic field strength parameter β is contained in the effective potentials in termsof the scaling of the unit of length for the z-coordinates. The following scaling propertieshold [6]:

V (0,0)mi

(z) 7−→√β V (0,0)

mi(√β z) (3.31)

21


U (00,00)mimj

(z1, z2) 7−→√β U (00,00)

mimj(√β z1,

√β z2) (3.32)

A(mimj)(00,00)(z1, z2) 7−→

√β A(00,00)

mimj(√β z1,

√β z2) (3.33)

When the potentials are given for β = 1, the potentials for β 6= 1 are given by theright-hand side of the terms.A more compact description of the Hartree-Fock equations follows with the definition ofthe integral functions [12]:

Y (00,00)mimj

(z1) :=

∫ ∞−∞

gmjνj(z2) gmjνj(z2)U (00,00)mimj

(z1, z2) dz2 , (3.34)

X(00,00)mimj

(z1) :=

∫ ∞−∞


(z1, z2) dz2 , (3.35)

for the integrals of the direct electron-electron potential and the electron-electron ex-change potential:− d2

dz2+ V (0,0)

mi(z)− εi +

∑j=1j 6=i

Y (00,00)mimj

(z)

gmiνi(z) =N∑j=1j 6=i

gmjνj(z)X(00,00)mimj

(z) . (3.36)

These equations have to be solved self-consistently for every electron with the quantumnumbers (mi, νi). The summations over the integral functions represent the Coulombinteractions with the other electrons of the quantum numbers (mj, νj).

3.3.2 Parity attribute

The z-parity is a symmetry of the Hamiltonian and thus also a symmetry of the longi-tudinal wave functions:

gmiνi(−z) = (−1)νigmiνi(z) . (3.37)

Therefore the integrations can be reduced to integrals over the positive z-axis. Theintegral functions Y 00,00

mimj(z) are even and can be calculated over the positive z-axis:

Y (00,00)mimj

(z1) =

∫ ∞−∞

(gmjνj(z2))2 U (00,00)mimj

(z1, z2) dz2

=

∫ ∞0

(gmjνj(z2))2 U (00,00)mimj

(z1, z2) dz2

+

∫ ∞0

(gmjνj(−z2))2 U (00,00)mimj

(z1,−z2) dz2

=

∫ ∞0

(gmjνj(z2))2

[U (00,00)mimj

(z1, z2) + U (00,00)mimj

(z1,−z2)

]dz2

= Y (00,00)mjνj

(−z1) . (3.38)

22


The integrals functions X00,00mimj

(z) can also be calculated over the positive z-axis, foreither even or odd parity:

X(00,00)mimj

(z1) =

∫ ∞−∞


(z1, z2) dz2

=

∫ ∞0


(z1, z2) dz2

+

∫ ∞−∞

gmjνj(−z2) gmiνi(−z2)A(00,00)mimj

(z1,−z2) dz2

=

∫ ∞0

gmjνj(z2) gmiνi(z2)

[A(00,00)mimj

(z1, z2)

+ (−1)(νi+νj)A(00,00)mimj

(z1,−z2)

]dz2

= (−1)νi+νjX(00,00)mimj

(−z1) . (3.39)

It can easily be shown, that for integrals of even functions G(z) the following equationsare fullfilled:

G(z) = G(−z) →∫ ∞−∞

G(z)dz = 2

∫ ∞0

G(z)dz . (3.40)

This rule is applicable for the occurring even products of functions

G(z) =

gmiνi(z) gmiνi(z)

gmiνi(z) ddz2

gmiνi(z)

gmiνi(z)V(0,0)mi (z) gmiνi(z)

gmiνi(z)Y(00,00)mimj (z) gmiνi(z)

gmiνi(z)X(00,00)mimj (z) gmiνi(z) .

(3.41)

3.3.3 Hartree-Fock-Finite-Elements equation

The longitudinal Hartree-Fock equation is given by [6]:[− d2

dz2+ V (0,0)

mi(z)− εi +

∑j=1j 6=i

Y (00,00)mimj

(z)

]︸︷︷︸

ci(z)

gi(z) =∑j=1l 6=i

gj(z)X(00,00)mimj

(z)

︸︷︷︸hi(z)

. (3.42)

They are of the same form as the Helmholtz-equation, which is given by:

− g′′i (z) + ci(z) gi(z) = hi(z) . (3.43)

23


For every electron i, the inhomogeneity hi(z) and the absolute term ci(z) enter into thefinite-element equations:

hi(z) :=∑j=1j 6=i

gj(z)X(00,00)mimj

(z) , (3.44)

ci(z) := V (0,0)mi

(z)− εi +∑j=ij 6=i

Y (00,00)mimj

(z) . (3.45)

For the application of the finite-element method, the transition into a finite-dimensionalsubspace is necessary. For this purpose, we divide the z-axis into M finite elements anddefine it by the resulting (M + K − 1) B-splines. The longitudinal wave functions cannow be expanded in terms of B-splines:

gi(z) =M+K−1∑k=1

αik Bk(z) . (3.46)

For each electron i we obtain a system matrix Ai and a system vector bi:

ai(Bk, Bl) :=M∑m=1

[∫Im

B′k(z)B′l(z) + ci(z)Bk(z)Bl(z)dz

]= (Aadi )kl , (3.47)

li(bk) :=M∑m=1

[∫Im

hi(z)Bk(z)dz

]= (badi )k . (3.48)

The Hartree-Fock-Finite-Element equations are a system of N linear equations for the(M +K − 1) coefficients of the B-spline expansion [6]:

Aadi αadi = badi i = 1, ..., N . (3.49)

These equations are solved iteratively for each electron, starting with a wave functioncalculated using Runge-Kutta methods.

24

3.4 Hartree-Fock-Roothaan method

3.3.4 Energy equations

A multiplication followed by an integration for the Hartree-Fock equations leads to theenergy equations for each electron:

εi =

∫ ∞−∞

gmiνi(z)

[− d2

dz2+ V (0,0)

mi(z)

]gmiνi(z) dz

+∑j=1j 6=i

∫ ∞−∞

gmiνi(z)Y (00,00)mimj

(z) gmiνi(z) dz

+∑j=1j 6=i

∫ ∞−∞

gmjνj(z)X(00,00)mimj

(z) gmiνi(z) dz

= ε(KV )i + ε

(U)i + ε

(A)i . (3.50)

The single-particle energy εi includes the contributions of the kinetic energy and theeffective electron-nucleus potential ε

(KV )i , the direct electron-electron potential ε

(U)i , and

the electron-electron exchange potential ε(A)i .

The repulsive electron-electron interaction yields a positive contribution, whereas theexchange interaction yields a negative contribution:

εi < 0, ε(KV )i < 0, ε

(U)i > 0, ε

(A)i < 0 . (3.51)

The total energy of the system is given by:

E =N∑i=1

ε(KV )i +

1

2

(ε

(U)i + ε

(A)i

). (3.52)


3.4.1 Expansion into Landau states

The Landau amplitudes t are determined by fixing the longitudinal wave functions gmν .With the expansion in orthonormal Landau states

ψmiνi(ρ, ϕ, z) =

NL∑ni=0

tnimiνi Φnimi(ρ, φ) gmiνi(z)

∼= gmiνi(z)

NL∑ni=0

tnimiνi Φnimi(ρ, φ)

=:< r|miνi > , (3.53)

25


the premises for the Hartree-Fock-Roothaan method are fulfilled. For the single-electronstates

|miνi >=

NL∑ni=0

tnimiνi |nimiνi > (3.54)

follows because of the orthogonality of the Landau states with different Landau quantumnumbers ni and the normalization of the longitudinal wave functions gmiνi(z):

< nimiνi|n′imiνi >= δnin′i ,

NL∑ni=0

|tnimiνi |2 = 1 . (3.55)

3.4.2 Hartree-Fock-Roothaan equations

The Fock matrix for the ith electron with the quantum numbers (miνi) is obtained fromthe Fock operator F and the basis states |nimiνi > of the single Landau channels:

< nimiνi|F |n′imiνi > = Fnin′i

=(1)< nimiνi|h(1)|n′imiνi >(1)

+N∑j=1j 6=i

(1) < nimiνi|(2)| < mjνj|2

|r1 − r2||mjνj >

(2) |n′imiνi >(1)

−N∑j=1j 6=i

(1) < nimiνi|(2)| < mjνj|2

|r1 − r2||mjνj >

(1) |n′imiνi >(2)

(3.56)

Here, h(1) is the sum of the single-electron parts of the Hamiltonian H in equation (3.17).The Hartree-Fock summation of the two-particle terms ranges over all the other electronsof the configuration. In this way effective Coulomb potentials appear. Now we cannotrestrict ourselves to the adiabatic potentials with n = 0. We also need the non-adiabaticpotentials in the higher Landau channels (n > 0) [6]:

V(ni,n

′i)

mi (z) :=< nimi|−2Z

|r||n′imi >

− 2Z

∫Φ∗nimi(r

⊥) Φn′imi(r⊥)

|r|dr⊥ , (3.57)

U(ninj ,n

′in′j)

mimj (z1, z2) :=(1)< nimi|(2) < njmj|2

|r1 − r2

|n′imi >(1) |n′jmj >

(2)

= 2

∫ ∫Φ∗nimi(r

⊥1 ) Φ∗njmj(r

⊥2 ) Φn′imi

(r⊥1 ) Φn′imi(r⊥2 )

|r1 − r2|dr⊥1 dr

⊥2 , (3.58)

26


A(ninj ,n

′in′j)

mimj (z1, z2) :=(1)< nimi|(2) < njmj|2

|r1 − r2

|n′imi >(2) |n′jmj >

(1)

= 2

∫ ∫Φ∗nimi(r

⊥1 ) Φ∗njmj(r

⊥2 ) Φn′imi

(r⊥2 ) Φn′imi(r⊥1 )

|r1 − r2|dr⊥1 dr

⊥2 . (3.59)

The same scaling relations for the magnetic field strength parameter as in the adiabaticcase, hold:

V(ni,n

′i)

mi (z) 7−→√β V

(ni,n′i)

mi (√β z) , (3.60)

U(ninj ,n

′in′j)

mimj (z1, z2) 7−→√β U

(ninj ,n′in′j)

mimj (√β z1,

√β z2) , (3.61)

A(ninj ,n

′in′j)

mimj (z1, z2) 7−→√β A

(ninj ,n′in′j)

mimj (√β z1,

√β z2) . (3.62)

The matrices for the ith electron are a combination of the 5 parts of the Hartree-Fockequation. For the single matrix elements follows:

Fnin′i = F(TS)

nin′i+ F

(K)

nin′i+ F

(V )

nin′i+ F

(U)

nin′i+ F

(A)

nin′i, (3.63)

which corresponds, in matrix notation, to:

F = F TS + FK + F V + FU + FA . (3.64)

Here, the first 3 terms are the matrix elements of the single-electron Fock operator, givenby:

F(TS)

nin′i= 4βni · δnin′i , (3.65)

F(K)

nin′i= −

∫ ∞−∞

gmiνi(z)d2

dz2gmiνi(z) dz · δnin′i , (3.66)

F(V )

nin′i= −

∫ ∞−∞

(gmiνi(z))2 V(nin

′i)

mi (z) dz , (3.67)

F (TS) represents the combined transversal and spin energy, F (K) the longitudinal kineticenergy, and F (V ) the Coulomb energy in the field of the nucleus.The calculation of the two-particle Fock matrix elements is much more extensive, becausewe have to sum over the other electrons and their Landau channels, too.The direct Coulomb energy of the ith electron is given by the Fock matrix F (U):

F(U)

nin′i=∑j=1j 6=i

∑ni,nj=0

tnjmjνj tn′jmjνj(1) < nimiνi|(2) < njmjνj|

2

|r1 − r2||n′jmjνj >

(2) |n′imiνi >(1)

=∑j=1j 6=i

∑ni,nj=0

tnjmjνj tn′jmjνj

∫ ∞−∞

∫ ∞−∞

(gmiνi(z1))2 (gmjνj(z2))2 U(ninjn

′in′j)

mimj (z1, z2) dz1 dz2 .

(3.68)

27


The contribution of the exchange energy F (A) is given in an analogous way, by:

F(A)

nin′i= −

∑j=1j 6=i

∑ni,nj=0

tnjmjνj tn′jmjνj(1) < nimiνi|(2) < njmjνj|

2

|r1 − r2||n′jmjνj >

(1) |n′imiνi >(2)

= −∑j=1j 6=i

∑ni,nj=0

tnjmjνj tn′jmjνj

·∫ ∞−∞

∫ ∞−∞

gmiνi(z1) gmjνj(z2) gmjνj(z1) gmiνi(z2)A(ninjn

′in′j)

mimj (z1, z2) dz1 dz2 .

(3.69)

The z-parity properties can be used, as in the case of the adiabatic approximation, toreduce the integrations to the positive z-axis.An iteration of the transversal calculation for the ith electron consists of the two steps:Construction of the Fock matrix

Fi = F(TS)i + F

(K)i + F

(V )i + F

(U)i + F

(A)i , (3.70)

and the solution of the corresponding eigenvalue problem, i.e. the Hartree-Fock-Roothaanequations [6]:

Fiti = εiti i = 1, ..., N . (3.71)

The eigenvector to the lowest eigenvalue εi delivers the new Landau amplitude vectorti for the Fock matrix Fi of the following integration. In the first step, the integrationbegins with the adiabatic approximation amplitude vector ti = tad = (1, 0, ..., 0) for eachelectron.

3.4.3 Energy equations

The Hartree-Fock-Roothaan iterations are repeated until the total energy converges. Asin the longitudinal calculation, the integration of the Hartree-Fock equations yields thesingle-particle energies, which again yield the total energy. Since the integration leadsto the Fock matrices again, the energy equations are given by

E =∑i=1

∑ni,nj=0

tnimiνitn′imiνi

(F

(TS)

nin′i+ F

(K)

nin′i+ F

(V )

nin′i+ F

(U)

nin′i+ F

(A)

nin′i

)

=N∑i=1

ε(TS)i + ε

(K)i + ε

(V )i + ε

(U)i + ε

(A)i . (3.72)

In contrast to the adiabatic approximation longitudinal energy equation, the transversalenergy equation now includes the contribution from the higher Landau levels (n > 0).

28

3.5 The general Hartree-Fock-Finite-Elements method


3.5.1 Self-consistence of the general method

The Hartree-Fock method consists of two parts: A determination of the longitudinalwave functions in adiabatic approximation (adiabatic HFFE-method), and a second cal-culation, where the Landau amplitudes are calculated (HFR method), based on thedetermined longitudinal wave functions.The total method is doubly self consistent, since the longitudinal calculation again con-siders the Landau amplitudes of the transversal calculation (general HFFE method).

3.5.2 Variational calculation

The general equation of variation for the energy functional in the Hartree-Fock methodis:

δ

[< ψtot, H, ψtot > −

N∑i=1

εi < ψi, ψi >

]= 0 . (3.73)

The ansatz for the longitudinal wave functions requires definite Landau amplitudes,which means that they are not varied:

ψmiνi(z, ρ, ϕ) = gmiνi(z)

NL∑ni=0

tnimiνi Φnimi(ρ, ϕ) . (3.74)

With the Hamiltonian of an N -electron atom, given by equation (3.17)

H =N∑i=1

h(i)T + h

(i)S + h

(i)K + h

(i)V +

N∑i,j=1i<j

2

|ri − rj|(3.75)

the energy functional becomes:

< Ψ|H|Ψ > =N∑i=1

∫ψ∗miνi(r)

(h

(1)T + h

(1)S + h

(1)K + h

(1)V

)ψ∗miνi(r)dr

+1

2

∑i,j=1i 6=j

∫ ∫ψ∗miνi(r1)ψ∗mjνj(r2)

2

|r1 − r2|ψmiνi(r1)ψmjνj(r2) dr1 dr2

− 1

2

∑i,j=1i 6=j

∫ ∫ψ∗miνi(r1)ψ∗mjνj(r2)

2

|r1 − r2|ψmiνi(r2)ψmjνj(r1) dr1 dr2 .

(3.76)

29


Since the amplitudes are not varied, this equation can be simplified even before theexecution of the variation:

< ψmiνi(r), ψmiνi(r) > =< gmiνi(z), gmiνi(z) > ·NL∑

ni,n′i=0

tnimiνitn′imiνi < Φnimi(r⊥),Φn′imi

(r⊥) >︸︷︷︸= δnin′i

=< gmiνi(z), gmiνi(z) > ·NL∑ni=0

(tnimiνi)2

︸︷︷︸=1

=< gmiνi(z), gmiνi(z) > .(3.77)

In a similar way the functional of the transversal and the spin parts is proportional tothe longitudinal scalar product:

∫ψ∗miνi(r) h

(1)TSψmiνi(r) dr

=< gmiνi(z), gmiνi(z) > ·∫ ( NL∑

ni=0

tniΦ∗nimi

(r⊥)

)h

(1)TS

(NL∑ni=0

tn′imiνiΦ∗n′imi

(r⊥)

)︸︷︷︸=

Pn′i(4βn′i)tn′

imiνi

Φn′imi

(r⊥)

dr⊥

=< gmiνi(z), gmiνi(z) > ·NL∑ni,n′i

tnimiνitn′imiνi (4βn′i)

∫Φ∗nimi(r

⊥) Φn′imi(r⊥)dr⊥︸︷︷︸

= δnin′i

=< gmiνi(z), gmiνi(z) > · 4βNL∑ni=0

(tnimiνi)2ni︸︷︷︸

=:ε(TS)i

.

(3.78)

Here εTSi is the contribution of the transversal and spin part, to the single electron en-

ergy. In adiabatic approximation it follows: ε(TS)i → 0 .

In the functional of the longitudinal kinetic energy, the Landau amplitudes do not ap-

30


pear:

∫ψ∗miνi(r) h

(1)K ψmiνi(r) dr

= −∫ψ∗miνi(r)

∂2

∂z2ψmiνi(r) dr

= −∫gmiνi(z)

d2

dz2gmiνi(z) dz ·

NL∑ni,n′i=0

tnimiνi tn′imiνi

∫Φ∗nimi(r

⊥) Φn′imi(r⊥)dr⊥︸︷︷︸

δnin′i

= −∫gmiνi(z)

d2

dz2gmiνi(z) dz ·

NL∑ni=0

(tnimiνi)2

= −∫gmiνi(z)

d2

dz2gmiνi(z) dz

=< gmiνi(z),− d2

dz2gmiνi(z) > . (3.79)

The functional of the Coulomb energy in the field of the nucleus, requires a redefinitionof the corresponding effective potential

∫ψ∗miνi(r) h

(1)V ψmiνi(r) dr =

∫ψ∗miνi(r)

−2Z

|r|ψmiνi(r) dr

=

∫(gmiνi(Z))2

(NL∑ni=0

tnimiνi Φ∗nimi(r⊥)

)−2Z

|r|

NL∑n′i=0

tn′imiνi Φn′imi(r⊥)

dr

=

∫(gmiνi(z))2

NL∑ni,n′i=0

tnimiνi tn′imiνi V(ni,n

′i)

mi (z)

dz

=< gmiνi(z), Vmi(z), gmiνi(z) > .(3.80)

The effective potential Vmi(z) contains the contributions of the Landau channels, accord-ing to their amplitudes. In the adiabatic approximation, it is essential that Vmi(z) →V 0,0mi

(z).The calculation of the two-electron terms is carried out analogously.

31


The functional of the electron-electron interaction is given by:∫ ∫ψ∗miνi(r1)ψ∗mjνj(r2)

2


=

∫ ∫(gmiνi(z1))2 (gmjνj(z2))2 ∑

ni,n′i,nj ,n′j=0

tnimiνi tn′imiνi tnjmjνj tn′jmjνj U(ni,nj ,n

′i,n

′j)

mimj (z1, z2)

︸︷︷︸

=:Umimj (z1,z2)

dz1 dz2

=< gmiνi(z1) gmjνj(z2), Umimj(z1, z2) gmiνi(z1) gmjνj(z2) > , (3.81)

and the functional of the electron-electron exchange interaction by:∫ ∫ψ∗miνi(r1)ψ∗mjνj(r2)

2


=

∫ ∫gmiνi(z1) gmjνj(z2) gmjνj(z1) gmiνi(z2) ∑

ni,n′i,nj ,n′j=0

tnimiνi tn′imiνi tnjmjνj tn′jmjνj A(ni,nj ,n

′i,n

′j)

mimj (z1, z2)

︸︷︷︸

=:Amimj (z1,z2)

dz1dz2

=< gmiνi(z1) gmjνj(z2), Amimj(z1, z2) gmiνi(z2) gmjνj(z1) > . (3.82)

In both cases we obtain functionals by a redefinition of the effective potentials.The variation of the energy functional is thus:

δ < Ψ|H|Ψ > = δ

[N∑i=1

< gmiνi(z), (ε(TS)i − d2

dz2+ Vmi(z)) gmiνi(z) >

]

+1

2δ

∑i,j=1i 6=j

< gmiνi(z1) gmjνj(z1), Umimj(z1, z2) gmiνi(z1) gmjνj(z2) >

− 1

2δ

∑i,j=1i 6=j

< gmiνi(z1) gmjνj(z2), Amimj(z1, z2) gmiνi(z2) gmjνj(z1) >

.

(3.83)

This is equivalent to the terms in the adiabatic approximation calculation of KLEWS[7] and PROESCHEL [12]. The only difference is the extra transversal energy ε

(TS)i and

32


the redefinition of the effective potentials with respect to the Landau amplitudes. Thevariation leads to:

δ < Ψ|H|Ψ > =N∑i=1

< δgmiνi(z), (ε(TS)i − d2

dz2+ Vmi(z))gmiνi(z) >

+∑i,j=1i 6=j

< δgmiνi(z1) gmjνj(z2), Umimj(z1, z2) gmiνi(z1) gmjνj(z2) >

−∑i,j=1i6=j

< δgmiνi(z1) gmjνj(z2), Amimj(z1, z2) gmiνi(z2) gmjνj(z1) > . (3.84)

The factor 2 in the two-electron scalar products accounts for the fact that the summationtakes place over all orbitals, and that in the variation of the two-electron states eachorbital appears twice.For the variational equation, which takes into account the orthogonality constraints,follows after carrying out the scalar products:

0 =N∑i=1

[∫ ∞−∞

δgmiνi(z)

[ε

(TS)i − d2

dz2+ Vmi(z)− εi

]gmiνi(z) dz

+∑i,j=1i6=j

∫ ∞−∞

δgmiνi(z) gmiνi(z)Ymimj(z) dz

−∑i,j=1i6=j

∫ ∞−∞

δgmiνi(z) gmjνj(z)Xmimj(z) dz . (3.85)

By analogy with the adiabatic approximation case, we have introduced the integralfunctions Ymimj(z) and Xmimj(z).

3.5.3 Hartree-Fock-Finite-Elements equation and energy equation

The longitudinal Hartree-Fock equation (with fixed) Landau amplitudes, is given by[6]:− d2

dz2+ ε

(TS)i + Vmi(z)− εi +

∑j=1j 6=i

Ymimj(z)

gmiνi(z) =N∑j=1j 6=i

gmjνj(z)Xmimj(z) (3.86)

The functions include the fixed Landau amplitudes and converge, in the adiabatic case,to their corresponding counterparts.

33


In the energy equations, we have to consider the transversal energy ε(TS) of the higherLandau channels:

εi = ε(TS)i +

∫ ∞−∞

gmiνi(z)[− d2

dz2+ Vmi(z)

]gmiνi(z) dz

+∑i,j=1i 6=j

∫ ∞−∞

gmiνi(z)Ymimj(z) gmiνi(z) dz

−∑i,j=1i6=j

∫ ∞−∞

gmiνi(z)Xmimj(z) gmiνi(z) dz

= ε(TS)i + ε

(KV )i + ε

(U)i + ε

(A)i . (3.87)

The criteria for convergence of the general Hartree-Fock-Finite-Elements method refersto the total energy of the system:

E =N∑i=1

+ε(TS)i + ε

(KV )i +

1

2

(ε

(U)i + ε

(A)i .

)(3.88)

The general Hartree-Fock equation leads (as in the adiabatic approximation case) di-rectly to the Hartree-Fock-Finite-Elements equation. It is of the same form as the generalHelmholtz equation. After the redefinitions

hi(z) :=∑i,j=1i 6=j

gj(z) Xmimj(z) , (3.89)

ci(z) := ε(TS) + Vmi(z)− εi +∑i,j=1i 6=j

Ymimj(z) (3.90)

of the inhomogeneity and the constants in the Helmholtz equations, and the redefinitionof the bilinear forms and the linear form, we obtain a system matrix and a system vectorfor each electron in the B-spline basis:

(Ai)kl =M∑m=1

[∫Im

B′k(z)B′l(z) + ci(z)Bk(z)Bl(z)dz

](3.91)

(bi)k =M∑m=1

[∫Im

hi(z)Bk(z)dz

](3.92)

They take into account the influence of the higher Landau states. The general HartreeFock-Finite-Elements equation

Aiαi = bi i = 1, ..., N . (3.93)

reduces to (3.49), for each electron i in the adiabatic approximation case ti = tad =(1, 0, ..., 0).

34

4 Oscillator strengths

The analysis of atomic spectra requires not only the calculation of energy differencesbetween different states, but also the determination of the oscillator strengths for eachpossible transition. Once energies and wave functions of states have been obtained in theHartree-Fock calculations, we can determine the oscillator strengths of dipole transitionsbetween these states.


For the investigation, we regard two states of an atom, Ψ and Ψ′. Ψ represents the initialstate of the transition, and Ψ′ the final, excited state of the atom. As stated before, thestate Ψ is determined by the Slater determinant of the single-particle wave functions:

Ψ(r1, ..., rN) =1√N !

∣∣∣∣∣∣∣∣∣∣ψ1(r1) · · · ψ1(rN). .. .. .

ψN(r1) · · · ψN(rN)

∣∣∣∣∣∣∣∣∣∣. (4.1)

The single-particle wave functions are given by the ansatz:

ψi(z, ρ, ϕ) = gmiνi(z)

NL∑ni=0

tnimiνiΦnimi(ρ, ϕ) (4.2)

The state for each particle i = 1, ..., N is characterized by a set of quantum numbers(mi, νi). The second state, which is also part of the transition, is given by a Slaterdeterminant, too:

Ψ′(r1, ..., rN) =1√N !

∣∣∣∣∣∣∣∣∣∣ψ′1(r1) · · · ψ′1(rN). .. .. .

ψ′N(r1) · · · ψ′N(rN)

∣∣∣∣∣∣∣∣∣∣, (4.3)

35


where in this case the single-particle wave functions are given by the primed quantities:

ψ′i(z, ρ, ϕ) = g′m′iν′i(z)

NL∑n′i=0

t′n′im′iν′iΦ′n′im

′i(ρ, ϕ) . (4.4)

The dipole matrix element of a dipole transition between the two states is defined as:

p(q)Ψ′,Ψ =< Ψ′|

N∑i=1

r(q)i |Ψ > . (4.5)

Here, r(q)i represent the components of the position operator of the ith electron. The

three components correspond to the three types of a dipole transition, namely:

• Linear polarizationr0i = zi , (4.6)

• Right handed circular polarization

r+1i = − 1√

2(xi + iyi) , (4.7)

• Left handed circular polarization

r0i = +

1√2

(xi − iyi) . (4.8)

The modulus squared of the dipole matrix element p(q) is the so-called dipole strength:

d(q)Ψ′Ψ = |p(q)

Ψ′Ψ|2 . (4.9)

After the multiplication with the energy difference of the two states and the factor 2me~2 ,

the dipole strength leads to the dimensionless oscillator strength:

f(q)Ψ′Ψ =

2me

~2~ωΨ′Ψ d

(q)Ψ′Ψ =

2me

~2(EΨ′ − EΨ) d

(q)Ψ′Ψ . (4.10)

In atomic units, this equation reduces itself to:

f(q)Ψ′Ψ = (EΨ′ − EΨ) d

(q)Ψ′Ψ . (4.11)

We can reduce the many-particle matrix elements of the states Ψ,Ψ′ to the matrixelements of the single particle wave function ψi, ψ

′i.

< Ψ′|N∑j=1

r(q)j |Ψ >=< Ψ′|Ψ >

N∑i,j=1

< ψ′i|r(q)j |ψj > (< ψ′i|ψj >)−1 (4.12)

36

4.2 Matrix elements

Once the overlap matrix elements < ψ′i|ψj > and the single electron dipole matrix

elements < ψ′i|r(q)j |ψj > are known, we can calculate the oscillator strength with the

help of these equations.Transitions with q = 0, represent, classically speaking, dipole radiation parallel to thez-axis, whereas transitions with q = ±1 right-handed circularly respectively left-handedcircularly polarized radiation.

4.2 Matrix elements

As a difference to the inventions in Ruder et al. [14], the calculations are performedin usual atomic units, and not in Z-scaled atomic units. The overlap matrix elementsbecome, taking into account the orthonormality of the Landau states:

< ψ′i|ψj > =< g′m′iν′i |gmiνi > ·NL∑n′i=0

NL∑nj=0

t′n′im′iν′itnjmjνj < n′im′i|njmj >︸︷︷︸

δn′injδm′

imj

= δm′imj < g′m′iν′i |gmjνj > ·

[NL∑n=0

t′nm′iν′itnm′jν′j

]. (4.13)

For the adiabatic approximation case (NL = 0), with the Landau amplitude tad =(1, 0, ..., 0), the last term in brackets is equal 1.A similar calculation leads to the single-electron dipole matrix elements for the transi-tions q = 0:

< ψ′i|r(0)|ψj > =< g′m′iν′i |Z|gmjνj > ·NL∑n′i=0

NL∑nj=0

t′n′im′iν′itnjmjνj < n′im′i|njmj >︸︷︷︸

δn′injδm′

imj

= δm′imj < g′m′iν′i |z|gmjνj > ·

[NL∑n=0

t′nm′iν′itnm′jν′j

]. (4.14)

The spherical components of the dipole operator with circular polarization leads to achange in the Landau and the magnetic quantum number. From the description in termsof raising and lowering operators, for the Landau states, follows in atomic units,

r(+1)|nm >=−1√2β

[√n+ 1|(n+ 1)(m+ 1) > +

√n−m|n(m+ 1) >

], (4.15)

r(−1)|nm >=1√2β

[√n|(n− 1)(m− 1) > +

√n−m+ 1|n(m− 1) >

]. (4.16)

37


By inserting these relations into the definition of the single-electron dipole matrix ele-ments, one obtains for q = +1 the relation:

< ψ′i|r(+1)|ψj > =< g′m′iν′i |gmjνj > ·(−1)√

2β

NL∑n′i=0

NL∑nj=0

t′n′im′iν′itnjmjνj

·√nj + 1 < n′im

′i|(nj + 1)(mj + 1) > +

√nj −mj < n′im

′i|nj(mj + 1) >

= δm′i(mj+1) ·−1√2β

< g′m′iν′i |gmjνj >

·

[NL∑n=0

√n+ 1t′(n+1)m′iν

′itnm′jν′j +

NL∑n=0

√n−mjt

′nmiνi

tnmjνj

].

(4.17)

In the adiabatic approximation case (NL = 0) the last term in the brackets reduces itselfto√|mj|. An analogous calculation for the transitions with q = −1 leads to:

< ψ′i|r(−1)|ψj > =< g′m′iν′i |gmjνj > ·1√2β

NL∑n′i=0

NL∑nj=0

t′n′im′iν′itnjmjνj

· √nj < n′im′i|(nj − 1)(mj − 1) > +

√nj −mj + 1 < n′im

′i|nj(mj − 1) >

= δm′i(mj−1) ·1√2β

< g′m′iν′i |gmjνj >

·

[NL∑n=0

√nt′(n−1)m′iν

′itnm′jν′j +

NL∑n=0

√n−mj + 1t′nmiνitnmjνj

].

(4.18)

Here, in the adiabatic approximation case, the term in the brackets reduces itself to√|mj − 1| =

√|m′i|.

38

5 Algorithm and implementation

This chapter gives a brief overview about the program and the HFFER method itself.Starting with the discussion of the initial wave functions, I will introduce the paralleliza-tion and the program flow. We will finally see that with this method we can producedata of atomic systems in intense magnetic fields, in a fast and accurate way.

5.1 Initial wave functions

The HFFER method starts with a set of B-spline coefficients for the description of theadiabatic approximation initial wave functions. For the longitudinal wave function ofeach electron, a spline interpolation is carried out with B-splines of the order K = 6 onthe M finite elements taking into account the two derivatives at the boundaries z = 0and z = zmax.

However, finding appropriate initial wave functions is the most crucial step. The qual-ity of the initial wave functions decides whether the calculations will converge or not.Since the development of our program, different ansatzes were used, with varying success.

The first ansatz for an initial wave function was carried out by KLEWS [7]. A polynomialansatz was used

gi(z) =

(νi∑j=0

bijzj

)exp(ai|z|) . (5.1)

The problem with this ansatz is the free parameter ai. This parameter has to be variedfor different atoms and different magnetic field strengths to force convergence of thecalculations. This description is not very useful for the calculation of a great set ofatomic data for different atoms in a great range of magnetic field strengths. Especiallyfor the calculation of excited states, the parameter ai is not very helpful, since it doesnot change the form of the wave functions in a proper way, but only the strength of theexponential decay of the wave function.

A second ansatz was made by ENGEL [6] who used approximate wave functions of thehydrogen atom in intense magnetic fields, namely wave functions of a truncated Coulomb

39


potential. This ansatz was suggested by CANUTO & KELLY [3] for the first time.The wave functions can be expressed in analytical form and according to the level scheme,we have to distinguish between tightly bound states (m ≤ 0, ν = 0) and hydrogen-likestates (m ≤ 0, ν > 0).For tightly bound states, a Gaussian ansatz is used:

g(z) ∝ exp

[−λ2

22

a2l

]. (5.2)

Here, λ is a variational parameter which is determined by minimizing the expectationvalue of the correct Hamiltonian of the hydrogen atom. This leads to a transcendentalequation. which is solved numerically using the Brent method (cf. [2], [3], [6]).For hydrogen-like states, Whittaker functions are used:

g(z) ∝ Wα, 12(ξ)

[2z

αa0

+2d

αa0

](5.3)

These are the solutions of the one-dimensional Schrodinger equation in a truncatedCoulomb potential, which is in our case, generated by the effective potentials. The cut-off parameter d and the parameter α of the Whittaker functions can be determined bythe condition that the expectation values of the truncated Coulomb potential and theexact Coulomb potential coincide.

Wave functions of tightly bound states (i.e. the ground states) can be calculated in avery fast and accurate way. However, calculations for excited states (especially for anti-symmetric wave functions) and intense magnetic fields depend on the right estimation ofthe effective nuclear charge. Here, only a small range of effective nuclear charge valuesguarantee the convergence of the calculations. In most of the cases, the Whittaker func-tions are no longer appropriate initial choices, and the calculations therefore no longerconverge. This can be seen in Figure 5.1 and Figure 5.2.

Figure 5.1 shows calculations for Li+1. Here, we carry out calculations for doubly excitedstates, where both electrons perform a ∆ν = 1 transition, with the initial quantumnumbers <0,0|-1,0> and the final quantum numbers <0,1|-1,1>. The effective nuclearcharges for each electron are varied up to 3, since three protons are present in the nucleusof lithium. Calculations with convergence are marked with a cross. Free spots indicatenon convergent calculations.

Figure 5.2 illustrates a similar plot. This time, the calculations were made for Li+1, ata magnetic field strength of B = 1× 109 Tesla.

The ansatz that is used in this thesis generates initial wave functions with the helpof Runge-Kutta methods. Here, Runge-Kutta solves the Schrodinger equation for thehydrogen atom. Runge-Kutta is a well known iterative method in numerical analysisto solve ordinary differential equations. These wave functions distinguish themselves

40

5.1 Initial wave functions

0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5 3

Eff

ect

ive

nu

cle

ar

cha

rge

: 2

nd

ele

ctro

n

Effective nuclear charge: 1st electron

Helium like Lithium, at B = 10^8 Tesla

Figure 5.1: Convergence diagram for Li+1, at a magnetic field strength of B = 1 × 108

Tesla

0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5 3

Eff

ect

ive

nu

cle

ar

cha

rge

: 2

nd

ele

ctro

n

Effective nuclear charge: 1st electron

Helium like Lithium, at B = 10^9 Tesla

Figure 5.2: Convergence diagram for Li+1, at a magnetic field strength of B = 1 × 109

Tesla

41


through their high stability throughout a great range of magnetic field strengths and evenfor excited states. No free parameters have to be considered and even multiply excitedstates can be calculated in an accurate way, while convergence is always achieved

5.2 Program flow

The program flow can be divided into two main sequences: The HFFE- and the HFR-part. Those two parts consist of a cycle of iterations, until the convergence of the totalenergy. The program starts with the input of the free parameters. These consist ofthe quantum numbers (m, ν), the magnetic field strength B, the atom number Z, theelectron number n, the number of finite elements M , the maximum Landau numberNL and the integration range r. Then, the potentials and the adiabatic approximationwave functions are calculated. After a complete B-spline interpolation of the initial wavefunctions and the determination of the coefficients αstarti , the HFFE process begins.The adiabatic approximation wave functions coefficients tadi and αstarti enter the initialHFFE iteration, with tadi = (1, 0, ..., 0). In the HFFE stage, the Hartree-Fock equations

Figure 5.3: Flow chart of the algorithm (adopted from ENGEL [6]).

42

5.2 Program flow

then are solved to determine the B-spline expansion coefficient vectors αi of the orbitals.The converged values αconvi are handed over to the HFR stage, where the Hartree-Fock-Roothaan equations are solved to obtain new converged Landau amplitudes tconvi , whichin turn serve as ”frozen” input for the next HFFE stage of the iteration. Both the HFFEand the HFR stages are parallelized. The process stops as soon as the absolute energyvalues of two successive iteration steps differ less than the preadjusted convergence cri-teria.After the calculation of the states and their energies, the oscillator strengths of the cor-responding dipole transitions can be calculated.It should be mentioned again, that the calculations in this thesis were performed onlyin adiabatic approximation. This means that the calculations were made including onlythe lowest Landau state (NL = 0), which means that the Hartree-Fock-Roothaan stagecan be omitted. The reason for this purpose is to minimize the computing time, and theproduction of as many data as possible. The energy differences between the adiabaticapproximation and the non-adiabatic calculations range in the magnitude of a few per-cent for magnetic field strength parameters βZ = β

Z2 > 10.

43


44

6 Calculations and results

We will present the numerical results that have been made this far. The results coverthe calculation of energies and oscillator strengths for singly excited atoms in intensemagnetic fields. They cover the first 6 elements in every possible ionization state, for6 different magnetic field strengths. As it has been mentioned before, the calculationswere made under adiabatic circumstances, which means that the HFR part has beenomitted and only the lowest Landau level was considered. A more exact calculation canbe realized by taking the higher Landau levels into account. To economize the timeand the resources, we decided to make the calculations of adiabatic nature. Since thedifferences between the adiabatic and the non-adiabatic calculations lie within a fewpercent, the adiabatic calculations should satisfy our need to draw conclusions whetherthe observations that have been made can be interpreted as atomic transitions in theneutron star atmosphere.

6.1 Helium

6.1.1 He

We start with the results of our calculations for helium, because this allows us to comparewith the results of other groups, to test the accuracy of the calculations. To avoid pagesfull of tables, we directly plot our results. The energies and oscillator strengths for eachpossible transition of neutral helium are shown in the following figures and Grotriandiagrams.

The Grotrian diagram for relevant transitions from the ground state of helium, atB = 108 Tesla, to excited states are shown in Figures 6.1 and 6.2. In Figure 6.1 itis the outer electron (m = −1, ν = 0) which makes the transition, in Figure 6.2 it is theinner electron (m = 0, ν = 0). It can be seen that only (∆m = 0) transitions possesssizeable oscillator strengths, while (∆m = ±1) transitions are strongly suppressed. TheFigure also includes (∆m = −1) transitions from states where the electron initially oc-cupies an (e.g. thermally) excited tightly bound state. Again the oscillator strengths ofthese transitions are small (in the order of 10−2 to 10−3, at most).

45


m=0m=0m=0

m=−1m=−1m=−1

m=−2m=−2m=−2m=−3m=−3m=−3

m=−4m=−4m=−4ν =0ν = 0ν = 0

ν =1ν = 1ν = 1

ν =2ν = 2ν = 2

ν =3ν = 3ν = 3

147.21(0.233)

155.91(2.778· 10−2)

152.04(5.068· 10−5)

43.00(1.469· 10−2)

150.71(1.069· 10−4) 104.41

(0.234)

112.93(2.635· 10−2)

109.28(1.856· 10−5)

18.08(7.008· 10−3)

108.63(3.066· 10−5) 86.41

(0.245)

94.86(2.632· 10−2)

91.36(1.022· 10−5)

10.46(4.801· 10−3)

90.96(1.286· 10−5) 76.01

(0.254)

84.41(2.634· 10−2)

80.74(7.342· 10−6)

Figure 6.1: Grotrian diagram for relevant transitions of the outer electron in helium, atB = 108 Tesla. Calculated in adiabatic approximation, transition energies ineV (and the respective oscillator strengths) are shown next to the arrows.

46

6.1 Helium

m=0m=0m=0

m=−1m=−1m=−1

m=−2m=−2m=−2m=−3m=−3m=−3

m=−4m=−4m=−4ν =0ν = 0ν = 0

ν =1ν = 1ν = 1

ν =2ν = 2ν = 2

ν =3ν = 3ν = 3

271.50(0.123)

280.12(1.625· 10−2)

275.50(9.199· 10−6)

127.77(0.103)

136.28(1.221· 10−2)

132.58(1.836· 10−5)

31.58(6.232· 10−3)

131.64(3.776· 10−5) 96.51

(9.960· 10−2)

104.75(1.144· 10−2)

100.68(1.075· 10−5)

Figure 6.2: Grotrian diagram for relevant transitions of the inner electron in helium, atB = 108 Tesla. Calculated in adiabatic approximation, transition energies ineV (and the respective oscillator strengths) are shown next to the arrows.

Figure 6.3 shows the oscillator strength spectra of relevant transitions of He at 6 dif-ferent magnetic field strengths. We recognize numerous transitions with large oscillatorstrengths in the range of several eV. They result from transitions between hydrogen-likestates (ν > 0), where transition energies are small. Obviously their relative size de-creases, compared to other strong transitions, as the magnetic field is increased.

At higher energies, the oscillator strength spectrum of neutral He shows 5 relevant tran-sition lines, each with an immediate neighbor at significant lower oscillator strength.These 5 lines belong to transitions from ν = 0 to ν = 1 (i.e. ∆ν = 1 and ∆m = 0).Their immediate neighbors at lower oscillator strengths belong to the same initial statetransitions, but this time from ν = 0 to ν = 3 (i.e. ∆ν = 3 and ∆m = 0). The rela-tively high oscillator strengths for these transitions can be explained by the fact that theoverlap between the states with ν = 0 and ν = 1 is relatively high compared to other

47


configurations. We could add several other lines to the plots by considering higher mag-netic levels with m < −4, from where additional transitions with ∆ν = 1 and ∆ν = 3can take place. Assuming that their, e.g. thermal, occupation is small, we do not con-sider them. Also with increasing magnetic quantum number the corresponding ∆ν = 1and ∆ν = 3 transitions (with ∆m = 0) are shifted to lower energies.

Since the transition lines lie at small energies even a significant increase of the magneticfield strength does not put them in the range of 700 eV, where the first of the absorptionfeatures in 1E 1207.4-5209 is observable. The significant transition lines do not reachthe relevant energies until a magnetic field strength of about B = 5× 109 Tesla. Thesemagnetic field strengths do not match our estimations. Also relativistic effects shouldbe considered in this range of magnetic field strengths, which are not considered in thesecalculations.We can therefore rule out helium as a candidate for the absorption features, observed in1E 1207.4-5209.Table 6.1 lists the relevant transitions that can be seen in Figure 6.3 The first block liststransitions of the inner electron. The second block belongs to transitions of the outerelectron.

Number Initial State <mi, νi> Final State <mf , νf> ∆E f1 <0,0|1,0> <0,1|1,0> 271,50 0,1232 <0,0|1,0> <0,3|1,0> 280,12 0,0163 <-2,0|1,0> <-2,1|1,0> 127,77 0,1034 <-2,0|1,0> <-2,3|1,0> 136,28 0,0125 <-3,0|1,0> <-3,1|1,0> 96,51 0,0946 <-3,0|1,0> <-3,3|1,0> 104,75 0,014

7 <0,0|1,0> <0,0|1,1> 147,21 0,2338 <0,0|1,0> <0,0|1,3> 155,91 0,0289 <0,0|2,0> <0,0|2,1> 104,41 0,234

10 <0,0|2,0> <0,0|2,3> 112,93 0,02611 <0,0|3,0> <0,0|3,1> 86,41 0,24512 <0,0|3,0> <0,0|3,3> 94,86 0,02613 <0,0|4,0> <0,0|4,1> 76,01 0,25414 <0,0|4,0> <0,0|4,3> 84,41 0,026

Table 6.1: Relevant transitions and their oscillator strengths. He at a magnetic fieldstrength of 108 Tesla

48

6.1 Helium

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electrons: 2 protons: 2 B: 1.00E+07 Tesla thermal energy [eV]: 150

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Figure 6.3: Spectra of He for 6 different magnetic field strengths.49


6.1.2 He+1

In the following we take a look at the transition energies and oscillator strengths of singlyionized helium (Table 6.2). In the case of singly ionized helium we can see 4 character-istic transition lines, each of them belonging to the ∆ν = 1 transition with ∆m = 0.Since the binding energy for the single electron is larger than in neutral helium, the 4relevant transition lines are increased in energy in respect to neutral helium. These linesdo have the same neighbors as in the case of helium, which represent the same initialstate transitions with ∆ν = 3, instead of ∆ν = 1.Figure 6.5 shows oscillator strength spectra of He+1 for 6 different magnetic field strengths.As for neutral helium, also singly ionized helium can be ruled out as a possible candi-date for the observed spectra, since the transition energies do not reach the requiredmagnitude up to a magnetic field strength of about 109 Tesla.

Number Initial State <mi, νi> Final State <mf , νf> ∆E f1 <0,0> <0,1> 365,62 0,4122 <0,0> <0,3> 403,08 0,0393 <-1,0> <-1,1> 243,60 0,2324 <-1,0> <-1,3> 279,15 0,0205 <-2,0> <-2,1> 196,29 0,1866 <-2,0> <-2,3> 230,48 0,0157 <-3,0> <-3,1> 169,13 0,1648 <-3,0> <-3,3> 202,18 0,013

Table 6.2: Relevant transitions and their oscillator strengths. He+1 at a magnetic fieldstrength of 108 Tesla

The Grotrian diagram in Figure 6.4 shows the energies and oscillator strengths of rele-vant transitions for singly ionized helium. Since most of the transitions have negligibleoscillator strengths, only the ∆ν = 1 and ∆ν = 3 transitions are plotted. Thermallyexcited state were taken into account, but are irrelevant because of their decreasingtransition energies. Here, the single electron can also occupy the state <-1,0>, since itis not reserved by a second electron.

After a first analysis of helium, we can draw a careful conclusion: The ∆ν = 1 and∆ν = 3 transitions (with ∆m = 0) show the greatest oscillator strengths because of therelative large overlap between these states.Therefore we will restrict our further examinations to exactly these kind of transitionsand neglect other possible transitions, since their transition probabilities are negligible.

It should be mentioned here, that our results agree well with those of MEDIN & LAI[8] and RUDER et al. [14]. This demonstrates the efficiency and accuracy of our

50

6.2 Lithium

calculations, and gives confidence in the results of the further calculations that havebeen made in the course of this thesis.

m=0m=0m=0

m=−1m=−1m=−1

m=−2m=−2m=−2m=−3m=−3m=−3

m=−4m=−4m=−4ν =0ν = 0ν = 0

ν =1ν = 1ν = 1

ν =2ν = 2ν = 2

ν =3ν = 3ν = 3

365.62(0.412)

403.08(0.039)

243.60(0.232)

279.15(0.020)

196.29(0.186)

230.48(0.015)

169.13(0.164)

202.18(0.013)

Figure 6.4: Grotrian diagram for relevant transitions of the single electron in He+1, atB = 108 Tesla. Calculated in adiabatic approximation, transition energiesin eV (and the respective oscillator strengths) are shown next to the arrows.

6.2 Lithium

6.2.1 Li

As next possible candidate, we examine the spectra of lithium and its possible ionizationstates. Considering the results that have been found in the case of helium, we expecthigher transition energies for lithium and further atoms and ions with higher nuclear

51


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Figure 6.5: Spectra of He+1 for 6 different magnetic field strengths.52

6.2 Lithium

charge. These expectations are confirmed by the results that can be seen in Table 6.3and Figure 6.6. The energetically highest transition line belongs to the ∆ν = 3 transitionof the innermost electron. For B = 108 Tesla, this happens at an energy of about 418 eV.Next to this line we can see the corresponding ∆ν = 1 transition for the same electron,from the same initial state with the quantum numbers <0,0>. This is understandable,since the innermost electron is subjected to the greatest Coulomb forces of the nucleus.The second relevant line, at an energy of 224 eV, belongs to the ∆ν = 1 transition ofthe central electron with the initial quantum numbers <-1,0>. Next to it, we can seeagain the ∆ν = 3 transition, at 232 eV, with an oscillator strength smaller by about onemagnitude. These lines are followed by the transition of the outermost electron, with thequantum numbers <-2,0>, and several other lines from thermally excited states. Evenin the case of B = 109 Tesla, most of the transitions lie in the range of 200 - 300 eV.

This leads to the conclusion that also neutral lithium can be ruled out as a possiblecandidate for the line features, observed in 1E 1207.4-5209.

Number Initial State <mi, νi> Final State <mf , νf> ∆E f1 <0,0|-1,0|-2,0> <0,1|-1,0|-2,0> 417,99 0,0672 <0,0|-1,0|-2,0> <0,3|-1,0|-2,0> 426,00 0,009

3 <0,0|-1,0|-2,0> <0,0|-1,1|-2,0> 224,03 0,1434 <0,0|-1,0|-2,0> <0,0|-1,3|-2,0> 232,21 0,0195 <0,0|-3,0|-2,0> <0,0|-3,1|-2,0> 123,74 0,1296 <0,0|-3,0|-2,0> <0,0|-3,3|-2,0> 131,77 0,0167 <0,0|-4,0|-2,0> <0,0|-4,1|-2,0> 94,91 0,1258 <0,0|-4,0|-2,0> <0,0|-4,3|-2,0> 102,70 0,015

9 <0,0|-1,0|-2,0> <0,0|-1,0|-2,1> 144,46 0,21310 <0,0|-1,0|-2,0> <0,0|-1,0|-2,3> 152,59 0,02711 <0,0|-1,0|-3,0> <0,0|-1,0|-3,1> 104,22 0,21112 <0,0|-1,0|-3,0> <0,0|-1,0|-3,3> 112,15 0,02613 <0,0|-1,0|-4,0> <0,0|-1,0|-4,1> 86,33 0,22114 <0,0|-1,0|-4,0> <0,0|-1,0|-4,3> 94,21 0,02615 <0,0|-1,0|-5,0> <0,0|-1,0|-5,1> 75,97 0,23314 <0,0|-1,0|-5,0> <0,0|-1,0|-5,3> 83,82 0,026

Table 6.3: Relevant transitions and their oscillator strengths. Li at a magnetic fieldstrength of 108 Tesla

53


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Figure 6.6: Spectra of Li for 6 different magnetic field strengths.54

6.2 Lithium

6.2.2 Li+1

Singly ionized lithium produces similar spectra as neutral helium, since the remaining 2electrons occupy the same states and therefore are able to perform the same transitions.We take a look at the spectra in Figure 6.7, where our attention is mainly focusedon the spectrum with the magnetic field strength of 108 Tesla. The highest transitionline belongs again to the ∆ν = 3 transition of the inner electron, closely followed bythe corresponding ∆ν = 1 transition. Several other lines in the range of 150 - 350 eVbelong to other possible transitions of the inner and outer electron, by taking thermallyexcited states into account. As recently as a magnetic field strength of 5 × 108 Tesla,the energetically strongest lines reach the range of 800 - 900 eV, where though thecorresponding oscillator strengths decrease by almost 50 %.Despite the fact that the electrons in ionized lithium experience stronger Coulomb forces,the bigger part of the spectrum still rests in the range of 300 - 500 eV, at B = 5× 108

Tesla. Forcing the spectrum into the range of 700 eV leads to magnetic field strengthsof about 5× 109 Tesla. Apart from the fact that relativistic effects should be consideredat such high magnetic field strengths, the oscillator strengths are vanishingly small.

Therefore, singly ionized lithium is definitely not a proper candidate for the observedfeatures in 1E 1207.4-5209.

Number Initial State <mi, νi> Final State <mf , νf> ∆E f1 <0,0|-1,0> <0,1|-1,0> 515,71 0,3032 <0,0|-1,0> <0,3|-1,0> 551,55 0,0343 <-2,0|-1,0> <-2,1|-1,0> 245,65 0,0804 <-2,0|-1,0> <-2,3|-1,0> 278,61 0,0085 <-3,0|-1,0> <-3,1|-1,0> 193,80 0,0626 <-3,0|-1,0> <-3,3|-1,0> 225,12 0,006

7 <0,0|-1,0> <0,0|-1,1> 296,19 0,4388 <0,0|-1,0> <0,0|-1,3> 320,12 0,0449 <0,0|-2,0> <0,0|-2,1> 218,27 0,305

10 <0,0|-2,0> <0,0|-2,3> 250,71 0,02911 <0,0|-3,0> <0,0|-3,1> 181,88 0,26312 <0,0|-3,0> <0,0|-3,3> 213,41 0,02313 <0,0|-4,0> <0,0|-4,1> 159,71 0,24014 <0,0|-4,0> <0,0|-4,3> 190,51 0,020

Table 6.4: Relevant transitions and their oscillator strengths. Li+1 at a magnetic fieldstrength of 108 Tesla

55


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Figure 6.7: Spectra of Li+1 for 6 different magnetic field strengths.56

6.2 Lithium

6.2.3 Li+2

Doubly ionized lithium produces almost a perfect copy of the singly ionized heliumspectrum. The only difference is the shift of the transition lines to higher energies,which again is caused by the stronger Coulomb forces.Table 6.5 lists relevant transitions and their oscillator strengths, while Figure 6.8 showsthe spectra of doubly ionized lithium, for 6 different magnetic field strengths. The mostinteresting aspect in the spectra of doubly ionized Lithium is the fact that, in the case ofB = 1×108 Tesla, one of the strongest transition lines (at 603 eV) is almost in the rangeof the observable absorption feature. By increasing the magnetic field strength, we canshift this line further into the range of higher energies. As it can be seen in Figure 6.8,even at a magnetic field strength of B = 5×108 Tesla, this line shows still an interestingoscillator strength of about 0.3 - 0.35.So, this is the first line in the hitherto examined spectra which shows relevant energieswith non-negligible oscillator strengths. The rest of the spectrum still lies under thethreshold of 700 eV, even for B = 5 × 108 Tesla, while the oscillator strengths becomeinsignificant with increasing magnetic field strength. Therefore the residual part ofthe spectrum (apart from the ∆ν = 1 transition from the ground state) is of littleimportance, at least for the explanation of the absorption features in 1E 1207.4-5209.

After our first two examinations of helium and lithium, we draw another careful conclu-sion: Regarding the spectra of lithium and its several ionization states, we can see thatespecially the energetically strongest transition lines (which belong to the ∆ν = 1 and∆ν = 3 transitions of the innermost electron) possess significant oscillator strengths de-spite of increasing ionization. Not until very high magnetic field strengths up to 5× 109

Tesla, these oscillator strengths remain strong and hold considerable values.

Number Initial State <mi, νi> Final State <mf , νf> ∆E f1 <0,0> <0,1> 603,81 0,4912 <0,0> <0,3> 683,59 0,0453 <-1,0> <-1,1> 388,95 0,1404 <-1,0> <-1,3> 462,73 0,0115 <-2,0> <-2,1> 308,05 0,0876 <-2,0> <-2,3> 377,72 0,0077 <-3,0> <-3,1> 262,39 0,0668 <-3,0> <-3,3> 328,92 0,005

9 <0,0> <1,0> 221,78 0,038

Table 6.5: Relevant transitions and their oscillator strengths. Li+2 at a magnetic fieldstrength of 108 Tesla

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Figure 6.8: Spectra of Li+2 for 6 different magnetic field strengths.58

6.3 Beryllium

6.3 Beryllium

6.3.1 Be

The next candidate in the table of elements is beryllium. Consisting of 4 electrons, wesee instantly that the spectrum is richer than in the case of lithium, since 1 more electronis able to perform additional transitions. The spectrum itself is slightly shifted to higherenergies, in comparison to lithium. Here again, the strongest transition lines are buildupon the transitions of the innermost electron from the initial state <0,0|-1,0|-2,0|-3,0>to the final state <0,3|-1,0|-2,0|-3,0>, and from the initial state <0,0|-1,0|-2,0|-3,0> tothe final state <0,1|-1,0|-2,0|-3,0>, respectively. These transitions take place at energiesin the range of 600 eV (Table 6.6). The following transitions in energetic order arethe ones, where the second innermost, the third innermost and the outermost electronperform the same transitions with ∆ν = 1. Once again each of these lines has a neighborright next to him, which belongs to the transition from the same initial state, but thistime with ∆ν = 3. These ∆ν = 3 transitions are increased in energy but possess loweroscillator strengths, which are decreased by one order of magnitude.The bigger part of the spectrum lies in the range of 50 - 150 eV, at B = 1× 108 Tesla.They belong to transitions from thermally excited states, but are insignificant since theirenergies are far to low.

The relevant transitions, their energies, and the corresponding oscillator strengths arelisted in Table 6.6. Here, every block represents transitions of a certain electron, begin-ning with the innermost. The Table belongs to the spectrum of beryllium at a magneticfield strength of 1× 108 Tesla, which can be seen in Figure 6.9 .

An increase of the magnetic field strength shifts the energetically strongest transitionlines into the range of 1 keV. However, this shift causes an extreme decline of the corre-sponding oscillator strengths.The increase of the magnetic field strength has a marginal effect on the remaining tran-sitions. Even at a field strength of B = 1× 109 Tesla, most of the spectrum still rests inthe range of 200 - 600 eV.

Considering that the bigger part of the spectrum is energetically low and that the energet-ically high transitions possess insignificant oscillator strengths, this should be evidencethat none of the transition lines (generated by neutral beryllium) match the neutronstar absorption features. Hence, beryllium is no good candidate for the generation ofthe features in 1E 1207.4-5209.

We will further examine the spectra of ionized beryllium. Looking back at the behaviorof ionized lithium, we expect higher transition energies. Here, the oscillator strengths ofthe innermost electron should not decline by an increase of the magnetic field strength,as we have seen in the case of lithium.

59


Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0|-1,0|-2,0|-3,0> <0,1|-1,0|-2,0|-3,0> 591,50 0,0472 <0,0|-1,0|-2,0|-3,0> <0,3|-1,0|-2,0|-3,0> 599,26 0,0073 <-4,0|-1,0|-2,0|-3,0> <-4,1|-1,0|-2,0|-3,0> 129,82 0,0714 <-4,0|-1,0|-2,0|-3,0> <-4,3|-1,0|-2,0|-3,0> 137,48 0,009

5 <0,0|-1,0|-2,0|-3,0> <0,0|-1,1|-2,0|-3,0> 302,80 0,0976 <0,0|-1,0|-2,0|-3,0> <0,0|-1,3|-2,0|-3,0> 310,60 0,0147 <0,0|-5,0|-2,0|-3,0> <0,0|-5,1|-2,0|-3,0> 98,08 0,0688 <0,0|-5,0|-2,0|-3,0> <0,0|-5,3|-2,0|-3,0> 105,48 0,009

9 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,1|-3,0> 205,10 0,15210 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,3|-3,0> 212,98 0,02113 <0,0|-1,0|-4,0|-3,0> <0,0|-1,0|-4,1|-3,0> 121,14 0,13714 <0,0|-1,0|-4,0|-3,0> <0,0|-1,0|-4,3|-3,0> 128,81 0,01815 <0,0|-1,0|-5,0|-3,0> <0,0|-1,0|-5,1|-3,0> 93,78 0,13314 <0,0|-1,0|-5,0|-3,0> <0,0|-1,0|-5,3|-3,0> 101,23 0,017

15 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,0|-3,1> 141,58 0,20416 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,0|-3,3> 149,33 0,02717 <0,0|-1,0|-2,0|-4,0> <0,0|-1,0|-2,0|-4,1> 103,59 0,19818 <0,0|-1,0|-2,0|-4,0> <0,0|-1,0|-2,0|-4,3> 111,14 0,02619 <0,0|-1,0|-2,0|-5,0> <0,0|-1,0|-2,0|-5,1> 85,99 0,20920 <0,0|-1,0|-2,0|-5,0> <0,0|-1,0|-2,0|-5,3> 93,47 0,02621 <0,0|-1,0|-2,0|-6,0> <0,0|-1,0|-2,0|-6,1> 75,68 0,22022 <0,0|-1,0|-2,0|-6,0> <0,0|-1,0|-2,0|-6,3> 83,16 0,026

Table 6.6: Relevant transitions and their oscillator strengths. Be at a magnetic fieldstrength of 108 Tesla

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Figure 6.9: Spectra of Be for 6 different magnetic field strengths.61


6.3.2 Be+1

The spectra of singly ionized beryllium are illustrated in Figure 6.10. The relativehigh oscillator strengths are outstanding. In comparison to neutral beryllium, they arealmost doubled up in intensity. The common shift to higher energies is also apparent.Unfortunately only two transition lines range nearby the 700 eV threshold (at B = 1×108

Tesla). They belong to transitions of the innermost electron (Table 6.7).We can force the whole spectrum to higher energies, by increasing the magnetic fieldstrength. However, the corresponding oscillator strengths decline hereby.Another important aspect can be found in the same plot, at a magnetic field strength of1× 109 Tesla. Here, the two strongest transition lines appear around 1400 eV, followedby a pair of transitions around 700 eV. Once again, the two strongest lines belong totransitions of the innermost electron, while the lines around 700 eV belong to transitionsof the central electron.

The two lines at 0.7 and 1.4 keV coincide with the two absorption features in the spec-trum of 1E 1207.4-5209. However, the oscillator strengths are insignificant, especiallythose of the energetically stronger lines. Furthermore, a magnetic field in the range of1 × 109 Tesla is about one magnitude larger than current calculations predict, whichmakes singly ionized beryllium a questionable candidate for the generation of the ob-served neutron star spectrum.

Though the present calculations do not match the absorption features of 1E 1207.4-5209,we carefully approach spectra in the right order of magnitude by analyzing atoms andions with higher nuclear charge.

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Figure 6.10: Spectra of Be+1 for 6 different magnetic field strengths.63


Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0|-1,0|-2,0> <0,1|-1,0|-2,0> 682,43 0,2412 <0,0|-1,0|-2,0> <0,3|-1,0|-2,0> 717,62 0,0293 <-3,0|-1,0|-2,0> <-3,1|-1,0|-2,0> 237,26 0,0244 <-3,0|-1,0|-2,0> <-3,3|-1,0|-2,0> 268,26 0,002

5 <0,0|-1,0|-2,0> <0,0|-1,1|-2,0> 383,58 0,3716 <0,0|-1,0|-2,0> <0,0|-1,3|-2,0> 417,28 0,0417 <0,0|-3,0|-2,0> <0,0|-3,1|-2,0> 225,18 0,1728 <0,0|-3,0|-2,0> <0,0|-3,3|-2,0> 256,41 0,0179 <0,0|-4,0|-2,0> <0,0|-4,1|-2,0> 181,97 0,138

10 <0,0|-4,0|-2,0> <0,0|-4,3|-2,0> 211,74 0,013

11 <0,0|-1,0|-2,0> <0,0|-1,0|-2,1> 265,52 0,45012 <0,0|-1,0|-2,0> <0,0|-1,0|-2,3> 297,39 0,04713 <0,0|-1,0|-2,0> <0,0|-1,0|-3,1> 203,50 0,33514 <0,0|-1,0|-2,0> <0,0|-1,0|-3,3> 233,96 0,03315 <0,0|-1,0|-2,0> <0,0|-1,0|-4,1> 172,41 0,29616 <0,0|-1,0|-2,0> <0,0|-1,0|-4,3> 202,11 0,02817 <0,0|-1,0|-2,0> <0,0|-1,0|-5,1> 152,95 0,27518 <0,0|-1,0|-2,0> <0,0|-1,0|-5,3> 182,08 0,025

Table 6.7: Relevant transitions and their oscillator strengths. Be+1 at a magnetic fieldstrength of 108 Tesla

6.3.3 Be+2

The spectra of doubly ionized beryllium are more clearly arranged, since only 2 elec-trons are able to perform transitions. As expected, the spectra are shifted towards higherenergies, due to increased Coulomb forces. The trend is evident that multiply ionizedatoms possess higher and more robust oscillator strengths, as can be seen in Figure 6.11.Here, the oscillator strengths for the ∆ν = 1 transitions of both electrons range in thedimension of 0.4 - 0.6, at a magnetic field strength of 1×108 Tesla. These are the highestoscillator strengths that have been measured up to this point. The transition energiesand the corresponding oscillator strengths are listed in Table 6.8. Even an increase ofthe magnetic field strength does not have a serious impact on the oscillator strengths.Especially the line which belongs to the ∆ν = 1 transition of the outer electron (from theinitial state <0,0|-1,0> to the final state <0,0|-1,1>) possesses a significant transitionenergy, in the range of 800 - 900 eV, at a magnetic field strength of 5×108 Tesla (Figure6.11).

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Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0|-1,0> <0,1|-1,0> 769,58 0,4232 <0,0|-1,0> <0,3|-1,0> 849,03 0,0423 <-2,0|-1,0> <-2,1|-1,0> 360,89 0,0384 <-2,0|-1,0> <-2,3|-1,0> 429,88 0,003

5 <0,0|-1,0> <0,0|-1,1> 446,24 0,5586 <0,0|-1,0> <0,0|-1,3> 518,44 0,0507 <0,0|-2,0> <0,0|-2,1> 331,20 0,2848 <0,0|-2,0> <0,0|-2,3> 398,97 0,0239 <0,0|-3,0> <0,0|-3,1> 275,34 0,205

10 <0,0|-3,0> <0,0|-3,3> 340,20 0,01611 <0,0|-4,0> <0,0|-4,1> 240,78 0,16812 <0,0|-4,0> <0,0|-4,3> 303,33 0,012


In the same case, the strongest transition lines (which belong to transitions of the innerelectron) range around 1400 eV. Unfortunately, this is not the proper distance betweenboth lines, to explain both absorption features.Moreover, the transition lines are highly sensitive to a variation of the magnetic fieldstrength. Even a small change in the magnetic field strength leads to an increase of thetransition energies in the range of several hundred eV.We assume a varying magnetic field strength over the surface of the neutron star [14].In the case of doubly ionized beryllium, this would lead to broad absorption features,which are not present in the spectrum of 1E 1207.4-5209.

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6.3 Beryllium

6.3.4 Be+3

The spectra of multiply ionized beryllium with one remaining electron are very simple,and illustrated in Figure 6.12. Be+3 possesses only 1 dominant transition line, whichagain belongs to the ∆ν = 1 transition from the initial state <0,0>. This line possessesrelevant energies and oscillator strengths (Table 6.9).As already mentioned, we assume a varying magnetic field strength over the surface of theneutron star [14]. Like in the case of Be+2, the transitions in Be+3 are highly susceptibleto a change of the magnetic field strength. A variation of the magnetic field strengthin the order of one magnitude leads to an energy shift of about 1 keV (Figure 6.12).This would lead to broad absorption features, again. Therefore, this ionization state ofberyllium is not a proper choice for the explanation of the neutron star spectrum.

Summing up the results up to this point we might claim, with the utmost probability,that none of the modelings for the first 4 elements match the observable absorptionfeatures of 1E 1207.4-5209.

However, we can state that the assumptions that we have drawn before, have beenapproved:Oscillator strengths of multiply ionized atoms remain strong despite the variation of themagnetic field strength. However, we have to consider that a variation of the magneticfield strength leads to great energetic shifts, for highly ionized atoms.

Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0> <0,1> 851,28 0,5472 <0,0> <0,3> 985,65 0,0483 <-1,0> <-1,1> 534,40 0,0744 <-1,0> <-1,3> 655,92 0,0065 <-2,0> <-2,1> 417,85 0,0346 <-2,0> <-2,3> 530,98 0,002

9 <0,0> <-1,0> 331,76 0,056


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6.4 Boron

6.4 Boron

6.4.1 Boron

We can take a look at the spectra of Boron in Figure 6.14 and trace back the transitionslines with the aid of Table 6.10, which lists the transition energies and their oscillatorstrengths.

At a magnetic field strength of 1 × 108 Tesla, only two transition lines lie in the rangeof 700 - 800 eV. These again, belong to transitions of the innermost electron. However,these lines possess insignificant oscillator strengths, less than 0.05. Thus, they are un-likely responsable for the detectable absorption features of 1E 1207.4-5209.We can force more transitions into the relevant energy scale, by increasing the magneticfield strength. However, this does not prevent the rapidly decline of the correspondingoscillator strengths. Furthermore, even at magnetic field strengths of about 1×109 Tesla,most of the spectrum still ranges between 200 - 600 eV, and is therefore of no relevance.Without greater analysis, we can easily add neutral Boron to the list of irrelevant atoms.

Figure 6.13 is an exemplary plot of the wave functions for the electrons in Boron. Theplot shows Boron in a ground state (left) and in a singly excited state (right) of theinnermost electron, at a magnetic field strength of 1× 108 Tesla

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Figure 6.13: Wave functions of boron at a magnetic field strength of 1 x 108 Tesla. Left:The ground state. Right: A singly excited state of the innermost electron,with m = 0 and ν = 1

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Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0|-1,0|-2,0|-3,0|-4,0> <0,1|-1,0|-2,0|-3,0|-4,0> 789,64 0,0402 <0,0|-1,0|-2,0|-3,0|-4,0> <0,3|-1,0|-2,0|-3,0|-4,0> 797,45 0,006

3 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,1|-2,0|-3,0|-4,0> 391,12 0,0754 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,3|-2,0|-3,0|-4,0> 398,75 0,012

5 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,0|-2,1|-3,0|-4,0> 261,86 0,1156 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,0|-2,3|-3,0|-4,0> 269,51 0,0177 <0,0|-1,0|-5,0|-3,0|-4,0> <0,0|-1,0|-5,1|-3,0|-4,0> 126,70 0,0908 <0,0|-1,0|-5,0|-3,0|-4,0> <0,0|-1,0|-5,3|-3,0|-4,0> 134,12 0,0129 <0,0|-1,0|-6,0|-3,0|-4,0> <0,0|-1,0|-6,1|-3,0|-4,0> 96,65 0,085

10 <0,0|-1,0|-6,0|-3,0|-4,0> <0,0|-1,0|-6,3|-3,0|-4,0> 103,82 0,01411 <0,0|-1,0|-7,0|-3,0|-4,0> <0,0|-1,0|-7,1|-3,0|-4,0> 81,80 0,08812 <0,0|-1,0|-7,0|-3,0|-4,0> <0,0|-1,0|-7,3|-3,0|-4,0> 88,88 0,011

13 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,0|-2,0|-3,1|-4,0> 193,54 0,15914 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,0|-2,0|-3,3|-4,0> 201,21 0,02315 <0,0|-1,0|-2,0|-5,0|-4,0> <0,0|-1,0|-2,0|-5,1|-4,0> 118,91 0,14116 <0,0|-1,0|-2,0|-5,0|-4,0> <0,0|-1,0|-2,0|-5,3|-4,0> 126,31 0,01917 <0,0|-1,0|-2,0|-6,0|-4,0> <0,0|-1,0|-2,0|-6,1|-4,0> 92,69 0,13618 <0,0|-1,0|-2,0|-6,0|-4,0> <0,0|-1,0|-2,0|-6,3|-4,0> 99,87 0,01819 <0,0|-1,0|-2,0|-7,0|-4,0> <0,0|-1,0|-2,0|-7,1|-4,0> 79,52 0,14120 <0,0|-1,0|-2,0|-7,0|-4,0> <0,0|-1,0|-2,0|-7,3|-4,0> 86,63 0,018

21 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,0|-2,0|-3,0|-4,1> 138,77 0,19922 <0,0|-1,0|-2,0|-3,0|-4,0> <0,0|-1,0|-2,0|-3,0|-4,3> 146,26 0,02723 <0,0|-1,0|-2,0|-3,0|-6,0> <0,0|-1,0|-2,0|-3,0|-6,1> 85,44 0,20024 <0,0|-1,0|-2,0|-3,0|-6,0> <0,0|-1,0|-2,0|-3,0|-6,3> 92,65 0,02625 <0,0|-1,0|-2,0|-3,0|-7,0> <0,0|-1,0|-2,0|-3,0|-7,1> 75,23 0,21126 <0,0|-1,0|-2,0|-3,0|-7,0> <0,0|-1,0|-2,0|-3,0|-7,3> 82,42 0,026

Table 6.10: Relevant transitions and their oscillator strengths. B at a magnetic fieldstrength of 108 Tesla

70

6.4 Boron

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Figure 6.14: Spectra of B for 6 different magnetic field strengths.71


6.4.2 B+1

The spectra of singly ionized Boron (Figure 6.15), shows what we already expect. Hereagain, the transition lines reveal a shift to higher energies, due to stronger Coulombbinding. The oscillator strengths show an important increase in comparison to neutralBoron. In most of the cases, the oscillator strengths almost doubled up their intensities.At a magnetic field strength of B = 5 × 107 Tesla, two lines are located nearby 700eV. These two lines belong to transitions of the innermost electron, with ∆ν = 1 and∆ν = 3, respectively. Up to a magnetic field strength of B = 1×109 Tesla, these remainthe only transition lines in the significant energy range, between 0.7 - 1.4 keV. Thus, therest of the spectrum is nonrelevant.

At a magnetic field strength of B = 5×107 Tesla, the ∆ν = 1 transition of the innermostelectron possesses an oscillator strength of about 0.3. At a magnetic field strength ofB = 1 × 108 Tesla, this line shifts into the range of 900 eV (Table 6.11), holding anoscillator strength of 0.21. Considering the gravitational red shift (which is not takeninto account in the present calculations), this line could be of interest.However, an increase of the magnetic field strength up to B = 5× 108 Tesla moves thisline into the energy range of 1500 eV.If the assumption of a varying field strength is accurate, this line would be a questionablecandidate since it would lead to broad absorption features, too.

Thus, singly ionized Boron is not a plausible candidate for the generation of the ob-served neutron star spectrum.

Table 6.11 lists the relevant transitions for singly ionized Boron, at a magnetic fieldstrength of 1× 108 Tesla.

72

6.4 Boron

Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0|-1,0|-2,0|-3,0> <0,1|-1,0|-2,0|-3,0> 871,65 0,2142 <0,0|-1,0|-2,0|-3,0> <0,3|-1,0|-2,0|-3,0> 907,60 0,026

5 <0,0|-1,0|-2,0|-3,0> <0,0|-1,1|-2,0|-3,0> 469,71 0,3246 <0,0|-1,0|-2,0|-3,0> <0,0|-1,3|-2,0|-3,0> 503,29 0,0387 <0,0|-4,0|-2,0|-3,0> <0,0|-4,1|-2,0|-3,0> 221,36 0,0928 <0,0|-4,0|-2,0|-3,0> <0,0|-4,3|-2,0|-3,0> 251,38 0,0109 <0,0|-5,0|-2,0|-3,0> <0,0|-5,1|-2,0|-3,0> 178,42 0,073

10 <0,0|-5,0|-2,0|-3,0> <0,0|-5,3|-2,0|-3,0> 206,91 0,00711 <0,0|-6,0|-2,0|-3,0> <0,0|-6,1|-2,0|-3,0> 155,14 0,06612 <0,0|-6,0|-2,0|-3,0> <0,0|-6,3|-2,0|-3,0> 182,88 0,006

13 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,1|-3,0> 332,18 0,40614 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,3|-3,0> 364,49 0,04515 <0,0|-1,0|-4,0|-3,0> <0,0|-1,0|-4,1|-3,0> 211,75 0,22316 <0,0|-1,0|-4,0|-3,0> <0,0|-1,0|-4,3|-3,0> 241,63 0,02317 <0,0|-1,0|-5,0|-3,0> <0,0|-1,0|-5,1|-3,0> 173,50 0,18318 <0,0|-1,0|-5,0|-3,0> <0,0|-1,0|-5,3|-3,0> 202,02 0,01819 <0,0|-1,0|-6,0|-3,0> <0,0|-1,0|-6,1|-3,0> 152,28 0,16720 <0,0|-1,0|-6,0|-3,0> <0,0|-1,0|-6,3|-3,0> 180,11 0,01621 <0,0|-1,0|-7,0|-3,0> <0,0|-1,0|-7,1|-3,0> 138,04 0,15922 <0,0|-1,0|-7,0|-3,0> <0,0|-1,0|-7,3|-3,0> 165,39 0,015

23 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,0|-3,1> 246,16 0,46024 <0,0|-1,0|-2,0|-3,0> <0,0|-1,0|-2,0|-3,3> 276,63 0,05025 <0,0|-1,0|-2,0|-4,0> <0,0|-1,0|-2,0|-4,1> 192,83 0,35326 <0,0|-1,0|-2,0|-4,0> <0,0|-1,0|-2,0|-4,3> 221,90 0,03627 <0,0|-1,0|-2,0|-5,0> <0,0|-1,0|-2,0|-5,1> 165,07 0,31628 <0,0|-1,0|-2,0|-5,0> <0,0|-1,0|-2,0|-5,3> 175,29 0,02829 <0,0|-1,0|-2,0|-6,0> <0,0|-1,0|-2,0|-6,1> 147,42 0,29730 <0,0|-1,0|-2,0|-6,0> <0,0|-1,0|-2,0|-6,3> 175,24 0,02831 <0,0|-1,0|-2,0|-7,0> <0,0|-1,0|-2,0|-7,1> 134,85 0,28532 <0,0|-1,0|-2,0|-7,0> <0,0|-1,0|-2,0|-7,3> 162,32 0,026

Table 6.11: Relevant transitions and their oscillator strengths. B+1 at a magnetic fieldstrength of 108 Tesla

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Figure 6.15: Spectra of B+1 for 6 different magnetic field strengths.74

6.4 Boron

6.4.3 B+2

Multiply ionized atoms possess large and robust oscillator strengths. This trend isevident and can be further confirmed by the examination of doubly ionized Boron.Table 6.12 lists the transitions at a magnetic field strength of B = 1 × 108 Tesla. Thecorresponding plot is illustrated in Figure 6.16.

Our attention is instantly focused on two plots in Figure 6.16.At B = 5×107 Tesla, the energetically strongest transition lines are situated around 700eV. They belong to transitions of the innermost electron. Here, the ∆ν = 1 transitiondistinguishes itself through its high oscillator strength of about 0.5. Right next to thisline, we detect the corresponding ∆ν = 3 transition, but with an insignificant oscillatorstrength.At B = 5× 108 Tesla, we recognize two lines, also situated around 700 eV. These linesbelong to transitions of the outermost electron. Here again, the ∆ν = 1 transitionpossesses a remarkable oscillator strength of about 0.4, followed again by the negligible∆ν = 3 transition.Considering their energy values, at the respective magnetic field strengths, these linesare highly interesting.Unfortunately, the variation of the magnetic field strength leads to a great shift of thespectrum. With the help of the energetically strongest transition line, we can see that thechange of the magnetic field strength by one order of magnitude leads to an energy shiftof about 1 keV. As already mentioned, this would lead to broad absorption features.

Paying particular attention to this aspect, we assume that doubly ionized Boron isunlikely the origin of the absorption features of 1E 1207.4-5209.

75


Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0|-1,0|-2,0> <0,1|-1,0|-2,0> 948,49 0,3792 <0,0|-1,0|-2,0> <0,3|-1,0|-2,0> 1029,12 0,039

3 <0,0|-1,0|-2,0> <0,0|-1,1|-2,0> 539,11 0,5184 <0,0|-1,0|-2,0> <0,0|-1,3|-2,0> 612,56 0,0495 <0,0|-3,0|-2,0> <0,0|-3,1|-2,0> 321,43 0,1416 <0,0|-3,0|-2,0> <0,0|-3,3|-2,0> 386,72 0,0127 <0,0|-4,0|-2,0> <0,0|-4,1|-2,0> 264,27 0,0988 <0,0|-4,0|-2,0> <0,0|-4,3|-2,0> 325,93 0,0809 <0,0|-5,0|-2,0> <0,0|-5,1|-2,0> 231,65 0,081

10 <0,0|-5,0|-2,0> <0,0|-5,3|-2,0> 290,66 0,00611 <0,0|-6,0|-2,0> <0,0|-6,1|-2,0> 208,65 0,07112 <0,0|-6,0|-2,0> <0,0|-6,3|-2,0> 266,35 0,005

13 <0,0|-1,0|-2,0> <0,0|-1,0|-2,1> 382,17 0,59014 <0,0|-1,0|-2,0> <0,0|-1,0|-2,3> 449,81 0,05415 <0,0|-1,0|-3,0> <0,0|-1,0|-3,1> 298,59 0,35616 <0,0|-1,0|-3,0> <0,0|-1,0|-3,3> 362,31 0,03017 <0,0|-1,0|-4,0> <0,0|-1,0|-4,1> 254,26 0,27618 <0,0|-1,0|-4,0> <0,0|-1,0|-4,3> 315,55 0,02219 <0,0|-1,0|-5,0> <0,0|-1,0|-5,1> 225,65 0,23420 <0,0|-1,0|-5,0> <0,0|-1,0|-5,3> 285,07 0,01821 <0,0|-1,0|-6,0> <0,0|-1,0|-6,1> 205,09 0,20822 <0,0|-1,0|-6,0> <0,0|-1,0|-6,3> 262,94 0,01523 <0,0|-1,0|-7,0> <0,0|-1,0|-7,1> 189,29 0,190


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6.4.4 B+3

The spectra of triply ionized boron are very simple (Figure 6.17). Only 2 transitionlines have to be considered. These, as usual, belong to the ∆ν = 1 transitions for bothelectrons, with the initial quantum numbers <0,0|-1,0>. These 2 transition lines possesssignificant energies and oscillator strengths (6.13).At a magnetic field strength of B = 5 × 107 Tesla, the ∆ν = 1 transition of the innerelectron already lies within the range of 700 - 800 eV. The corresponding oscillatorstrength, above 0.5, is remarkable. The second relevant line, which belongs to the∆ν = 1 transition of the outer electron, passes the 700 eV threshold somewhere betweenB = 1×108 and B = 5×108 Tesla. At B = 5×108 Tesla, this line possesses an oscillatorstrength of about 0.45, though the energy already lies around 1200 eV.Both lines possess significant energies and oscillator strengths, in the range of B = 1×108

Tesla. Moreover, the oscillator strengths of triply ionized boron remain strong up toB = 5 × 108 Tesla. If this ionization state of boron is present in the neutron staratmosphere, it should be observable. However, we can detect the same sensitivity ofthe transition energies relating to the variation of the magnetic field strength. Thisagain should lead to broad absorption features, which are not present in spectrum of1E 1207.4-5209. Furthermore, none of the lines produced by our calculations for triplyionized boron are detectable in the observed neutron star spectrum of 1E 1207.4-5209.Thus, our calculations of atomic data and our modelings of neutron star atmospheres,make triply ionized boron a questionable element for the explanation of the absorptionfeatures in the spectrum of 1E 1207.4-5209.

Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0|-1,0> <0,1|-1,0> 1027,21 0,5022 <0,0|-1,0> <0,3|-1,0> 1163,49 0,047

3 <0,0|-1,0> <0,0|-1,1> 594,41 0,6344 <0,0|-1,0> <0,0|-1,3> 714,80 0,0535 <0,0|-3,0> <0,0|-3,1> 365,59 0,1436 <0,0|-3,0> <0,0|-3,3> 470,88 0,0107 <0,0|-4,0> <0,0|-4,1> 318,48 0,1058 <0,0|-4,0> <0,0|-4,3> 419,06 0,0079 <0,0|-5,0> <0,0|-5,1> 285,44 0,084

10 <0,0|-5,0> <0,0|-5,3> 382,15 0,005


78

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6.4.5 B+4

The spectra of Boron with one remaining electron can be seen in Figure 6.18. Thespectra are as simple as possible, since the single electron performs just a handful ofpossible transitions. We can observe a single significant transition line (see Table 6.14),which belongs to the ∆ν = 1 transition of the ground state. This line passes the 700eV threshold at a magnetic field strength of B = 5 × 107 Tesla. Though the oscillatorstrength remains intense for all magnetic field strengths, the energy varies in the rangeof 1 keV while increasing the magnetic field strength by one order of magnitude. Thissensitivity is detectable throughout highly ionized atoms, and remains a major problemin our calculations and modelings of neutron star atmospheres. Moreover, this ionizationstate of boron is not able to explain both absorption features, since only the ∆ν = 1transition possesses significant energies and oscillator strengths. Therefore, this ion isnot supposed to deliver an adequate explanation of the absorption features of 1E 1207.4-5209.

Number Initial State <mi,νi> Final State mf ,νf ∆E f1 <0,0> <0,1> 1103,13 0,5902 <0,0> <0,3> 1302,46 0,0503 <-1,0> <-1,1> 678,14 0,035

9 <0,0> <-1,0> 451,25 0,076


6.5 Carbon

We take a look at the rich spectra of carbon (Figure 6.19). Here, we can only detect twotransition lines that range over the threshold of 700 eV, at a magnetic field strength ofB = 1× 108 Tesla. The bigger part of the spectrum remains in the range 100 - 600 eV,even in the case of B = 1× 109 Tesla. Forcing the spectrum to higher energies leads toa decline in the corresponding oscillator strengths. Even the two relevant lines, whichbelong to the ∆ν = 1 and the ∆ν = 3 transition of the innermost electron, do not exceedan oscillator strength of 0.2.Most of the transitions in carbon possess negligible energies. Furthermore, lines thatpass the 700 eV threshold offer insignificant oscillator strengths (0.1, at most).Thus, a carbon atmosphere is unlikely the origin of the absorption features in the spec-trum of 1E 1207.4-5209.

80

6.5 Carbon

Number State1 <m,ν> State2 m,ν ∆E f1 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,1|-1,0|-2,0|-3,0|-4,0|-5,0> 1009,19 0,0432 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,3|-1,0|-2,0|-3,0|-4,0|-5,0> 1017,41 0,007

3 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,1|-2,0|-3,0|-4,0|-5,0> 490,24 0,0664 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,3|-2,0|-3,0|-4,0|-5,0> 497,90 0,0105 <0,0|-6,0|-2,0|-3,0|-4,0|-5,0> <0,0|-6,1|-2,0|-3,0|-4,0|-5,0> 130,69 0,0146 <0,0|-6,0|-2,0|-3,0|-4,0|-5,0> <0,0|-6,3|-2,0|-3,0|-4,0|-5,0> 137,91 0,003

7 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,1|-3,0|-4,0|-5,0> 322,76 0,0958 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,3|-3,0|-4,0|-5,0> 330,30 0,0159 <0,0|-1,0|-6,0|-3,0|-4,0|-5,0> <0,0|-1,0|-6,1|-3,0|-4,0|-5,0> 128,17 0,058

10 <0,0|-1,0|-6,0|-3,0|-4,0|-5,0> <0,0|-1,0|-6,3|-3,0|-4,0|-5,0> 135,41 0,00811 <0,0|-1,0|-7,0|-3,0|-4,0|-5,0> <0,0|-1,0|-7,1|-3,0|-4,0|-5,0> 97,50 0,05412 <0,0|-1,0|-7,0|-3,0|-4,0|-5,0> <0,0|-1,0|-7,3|-3,0|-4,0|-5,0> 104,47 0,00813 <0,0|-1,0|-8,0|-3,0|-4,0|-5,0> <0,0|-1,0|-8,1|-3,0|-4,0|-5,0> 82,16 0,05614 <0,0|-1,0|-8,0|-3,0|-4,0|-5,0> <0,0|-1,0|-8,3|-3,0|-4,0|-5,0> 89,03 0,008

15 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,0|-3,1|-4,0|-5,0> 239,12 0,12916 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,0|-3,3|-4,0|-5,0> 246,66 0,01917 <0,0|-1,0|-2,0|-6,0|-4,0|-5,0> <0,0|-1,0|-2,0|-6,1|-4,0|-5,0> 124,01 0,10218 <0,0|-1,0|-2,0|-6,0|-4,0|-5,0> <0,0|-1,0|-2,0|-6,3|-4,0|-5,0> 131,25 0,01419 <0,0|-1,0|-2,0|-7,0|-4,0|-5,0> <0,0|-1,0|-2,0|-7,1|-4,0|-5,0> 95,30 0,09520 <0,0|-1,0|-2,0|-7,0|-4,0|-5,0> <0,0|-1,0|-2,0|-7,3|-4,0|-5,0> 102,27 0,01321 <0,0|-1,0|-2,0|-8,0|-4,0|-5,0> <0,0|-1,0|-2,0|-8,1|-4,0|-5,0> 80,90 0,09822 <0,0|-1,0|-2,0|-8,0|-4,0|-5,0> <0,0|-1,0|-2,0|-8,3|-4,0|-5,0> 87,79 0,013

23 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,0|-3,0|-4,1|-5,0> 185,11 0,16524 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,0|-3,0|-4,3|-5,0> 192,62 0,02425 <0,0|-1,0|-2,0|-3,0|-6,0|-5,0> <0,0|-1,0|-2,0|-3,0|-6,1|-5,0> 116,81 0,14426 <0,0|-1,0|-2,0|-3,0|-6,0|-5,0> <0,0|-1,0|-2,0|-3,0|-6,3|-5,0> 124,01 0,02027 <0,0|-1,0|-2,0|-3,0|-7,0|-5,0> <0,0|-1,0|-2,0|-3,0|-7,1|-5,0> 91,56 0,13828 <0,0|-1,0|-2,0|-3,0|-7,0|-5,0> <0,0|-1,0|-2,0|-3,0|-7,3|-5,0> 98,54 0,01929 <0,0|-1,0|-2,0|-3,0|-8,0|-5,0> <0,0|-1,0|-2,0|-3,0|-8,1|-5,0> 78,71 0,14330 <0,0|-1,0|-2,0|-3,0|-8,0|-5,0> <0,0|-1,0|-2,0|-3,0|-8,3|-5,0> 85,61 0,019

31 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,0|-3,0|-4,0|-5,1> 136,03 0,19732 <0,0|-1,0|-2,0|-3,0|-4,0|-5,0> <0,0|-1,0|-2,0|-3,0|-4,0|-5,3> 143,33 0,02833 <0,0|-1,0|-2,0|-3,0|-4,0|-6,0> <0,0|-1,0|-2,0|-3,0|-4,0|-6,0> 101,58 0,18734 <0,0|-1,0|-2,0|-3,0|-4,0|-6,0> <0,0|-1,0|-2,0|-3,0|-4,0|-6,0> 108,64 0,02635 <0,0|-1,0|-2,0|-3,0|-4,0|-7,0> <0,0|-1,0|-2,0|-3,0|-4,0|-7,0> 84,72 0,14536 <0,0|-1,0|-2,0|-3,0|-4,0|-7,0> <0,0|-1,0|-2,0|-3,0|-4,0|-7,0> 91,71 0,02637 <0,0|-1,0|-2,0|-3,0|-4,0|-8,0> <0,0|-1,0|-2,0|-3,0|-4,0|-8,1> 74,64 0,14538 <0,0|-1,0|-2,0|-3,0|-4,0|-8,0> <0,0|-1,0|-2,0|-3,0|-4,0|-8,3> 81,60 0,026

Table 6.15: Relevant transitions and their oscillator strengths. C at a magnetic fieldstrength of 108 Tesla

81


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6.5 Carbon

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Figure 6.19: Spectra of C for 6 different magnetic field strengths.83


84

7 Conclusion and Outlook

7.1 Conclusion

In the course of this thesis, we have performed extensive calculations of atomic data formodelings of neutron star atmospheres. The careful analysis of the calculations leads usto following important conclusions:

• Our code represents a fast and accurate way for the routine production of energiesand oscillator strengths of atoms in intense magnetic fields.

• We can draw the conclusion that only ∆ν = 1 and ∆ν = 3 transitions, respectively,are relevant, since other possible transitions possess negligible oscillator strengths.Moreover, oscillator strengths of ∆ν = 3 transitions are smaller by one order ofmagnitude, compared to the ∆ν = 1 transitions.

• The trend is evident, that the ionization leads to large and robust oscillatorstrengths. However, at very high ionization the energies of the transitions reacthighly sensitive to the variation of the magnetic field strength across the neutronstar surface. As we have seen in the case of highly ionized boron, even a smallincrease of the magnetic field strength leads to a shift of the transition energies byone order of magnitude. Therefore, highly ionized atoms are no good candidatesto explain the features, observed in 1E 1207.4-5209.

7.2 Outlook

The first 5 elements, in every possible ionization state, and neutral carbon are ques-tionable candidates for the explanation of the features, observed in 1E 1207.4-5209.We therefore recommend that future investigations focus their attention on medium Z-atoms. Here, the binding energies will be increased.Looking for significant oscillator strengths, slightly ionized atoms should deliver inter-esting transition lines with non-negligible oscillator strengths, which remain unaffectedby the variation of the magnetic field strength across the neutron star surface.Also, as this thesis has demonstrated, essentially only ∆ν = 1 transitions need to be

85

7 Conclusion and Outlook

regarded.These facts confine the range of atoms and ions considerably.

It is clear that much additional work will be required before a complete understandingof this phenomenon occurs. However, we were able to produce relevant data and drawimportant conclusions. This study certainly will stimulate further investigations in thisfield.

86

Bibliography

[1] D. Braess. Finite Elemente. Springer Verlag, 2003.

[2] W. Brent. Algorithms for minimisation without derivatives. PhD thesis, PrenticeHall, New Jersey, 1972.

[3] V. Canuto and D. Kelly. Hydrogen atom in intense magnetic field. AstrophysicalSpace Science, 1972.

[4] V. Canuto and J. Ventura. Quantizing magnetic fields in astrophysics. Fundamentalsof Cosmic Physics, 1972.

[5] C. Cohen-Tannoudji. Quantum Mechanics I. Hermann, 1977.

[6] D. Engel. Hartree-Fock-Roothaan-Rechnungen fur Vielelektronen-Atome inNeutronenstern-Magnetfeldern. PhD thesis, Universitat Stuttgart, 2007.

[7] M. Klews. Diskretisierungsverfahren zur Untersuchung von Atomen in zeitabhangi-gen elektrischen Feldern und in extrem starken Magnetfeldern. PhD thesis, Univer-sitat Tubingen, 2003.

[8] Z. Medin and D. Lai. Density-functional-theory calculations of matter in strongmagnetic fields. Physical Review A, 2006.

[9] K. Mori and C. Hailey. Atomic calculation for the atmospheres of strongly magne-tized neutron stars. Astrophysical Journal, 2006.

[10] K. Mori and C. Hailey. Detailed atmosphere modelling for the neutron star 1e1207.4-5209. Astrophysical Journal, 2006.

[11] NASA. Chandra x-ray observatory, 2009.

[12] P. Proschel. Hartree-Fock-Rechnungen an Atomen in extrem starken magnetischenFeldern. PhD thesis, Universitat Erlangen-Nurnberg, 1982.

[13] C. Roothaan. New developments in molecular orbital theory. Rev. Mod. Physics,1951.

[14] H. Ruder, G. Wunner, H. Herold, and F. Geyer. Atoms in strong magnetic fields.Springer Verlag, 1994.

[15] L. Schiff and H. Snyder. Theory of the quadratic zeeman effect. Physical Review,1939.

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[16] L. L. und E.M. Lifschitz. Lehrbuch der Theoretischen Physik, Quantenmechanik.Akademie Verlag, 1988.

[17] Wikipedia. Landau-niveau — wikipedia, die freie enzyklopadie, 2009. [Online;Stand 20. Oktober 2009.

[18] Wikipedia. Pulsar — Wikipedia, The Free Encyclopedia, 2009. [Online; accessed20-October-2009].

88

Acknowledgment

Last but not least, I would like to thank following persons:

• My beloved family. For their their encouragement, their patience and their readi-ness to make sacrifices to my benefit

• My closest friends. For their generosity and their worries about me.

• Prof. Dr. Gunther Wunner. For his genial and kind support, his professional andhuman supervision, and the interesting master thesis.

• Dr. Jens Harting. For his assistance, his time and his patience.

• The whole 1. Institute of Theoretical Physics, at the University of Stuttgart.Without your help and the wonderful atmosphere, this work would not have comethis far.

89

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