my masters thesis

Modelling and Classification of the Hyperfine Structure of

Atoms and Ions

This thesis is submitted to the Department of Physics, University of Karachi

under the supervision of Dr. Zaheer-Uddin as requirement of the degree of

Masters of Science

By Shahbaz Nihal

Department of Physics

University of Karachi

Pakistan

In the name of ALLAH

Allah causeth the revolution of the day and the night. Lo! herein is

indeed a lesson for those who see.

SURAH

ALLAH the Most Beneficial and the most

Merciful


indeed a lesson for those who see.

SURAH 24, AYAT 44-45

the Most Beneficial and the most


CERTIFICATE

This is to certify that Mr. Shahbaz Nihal of class MSc. (Physics) has

completed his thesis of 6 credit hours entitle “Modelling and Classification of

the Hyperfine Structure of Atoms and Ions” in the year 2008.

Internal Examiner

Dr. Zaheer Uddin

Assistant Professor

Department of Physics

University of Karachi

Dated: _____________

Dedication

I would like to dedicate my thesis to my Mother. She is the one person who

has always been a source of inspiration for me. I have and always will look

up to her personality for motivation and courage.

ACKNOWLEDGMENTS

First of all I would like to thank the All Mighty Allah for giving the wisdom, the opportunity and

strength to initiate and successfully complete this endeavour.

After the thanking almighty I would like to:

• Thank Dr. Zaheer Uddin for his sincere and knowledgeable guidance not only on work

related issues but also on personal matters.

• Thank my family for their constant support and encouragement during my work.

• Acknowledge the help and guidance of Mirza Ghazanfar Ahmed on technical issues in

C#. Particularly in graphics related work which was an important part of this work.

• Appreciate the irreplaceable support of my colleagues specially my Manager Ms. Saima

Haque (M. Phil Anthropolgy, University of Cambridge) for her support throughout my

Masters program.

And lastly but definitely not the least I would like to thank my friends Rida Usman, Saira Bibi,

Umair Javed Khalid and Jaleel Shah for their support and encouragement.

Table of Contents

Abstract 1

Chapter 1: Introduction

1.1 Spectroscopy 2

1.2 History of Spectroscopy 2

1.3 Computer Based Spectroscopy 3

1.4 Importance of Spectroscopy 3

Chapter 2: Introduction to Atomic Structure

2.1 Classical and Semi-Classical Models 5

2.1.1 The Rutherford’s Atomic Model 5

2.1.2 Bohr’s Atomic Mode 6

2.2 Quantum Mechanical Model 7

2.2.1 The Wave Mechanics (Schrodinger’s Scheme) 7

2.2.2 The Matrix Mechanics (Heisenberg’s Matrix Mechanics) 8

Chapter 3: Quantum Mechanical Description of Atom

3.1 One Electron System 9

3.2 The Spin of Electron 13

3.3 The Fine Structure of the Spectrum 14

3.4 N Electron System 15

3.4.1 The Central Field Approximation 15

3.4.2 Spin-Orbit Interaction 16

3.4.2.1 L-S Coupling 16

3.4.2.2 J-J Coupling 18

3.2.3 Pauli’s Exclusion Principle 20

3.2.4 Electronic Configuration in an N Electron System 21

Chapter 4: Line Broadening Effects and Line Profiles

4.1 Natural Broadening 22

4.2 Doppler Broadening 24

4.3 Pressure Broadening 26

Chapter 5: The Hyperfine Interaction and Hyperfine Structure 5.1 Hyperfine Interaction 28

5.1.1 Magnetic Dipole Hyperfine Interaction 29

5.1.2 Electric Quadruple Hyperfine Structure 31

5.1.3 Intensity of Components in Hyperfine Structure 32

5.1.4 Important Points About the Hyperfine Structure 33

5.2 Line Broadening Effects on Hyperfine Structure: Combined Line Profile of

Hyperfine Structure

33

5.3 Experimental Techniques in Hyperfine Studies 37

5.3.1 Doppler Free Spectroscopy 37

5.3.1.1 Collimated Atomic Beam Spectroscopy 37

5.3.1.2 Saturation Spectroscopy 38

5.3.2 Doppler Limited Spectroscopy 39

5.3.2.1 Two Photon Spectroscopy 39

5.3.2.2 Optogalvanic Spectroscopy

40

Chapter 6: Programming Platform

6.1 Software Framework 42

6.2 Microsoft .Net Framework 42

6.2.1 Base Class Library (BCL) 42

6.2.2 Common Language Runtime (CLR) 44

6.3 C# (C-sharp) 44

6.4 Working in C# Environment 45

6.4.1 MSIL (Microsoft Intermediate Language) 46

6.4.2 JIT Compilers 46

6.5 Visual Studio 46

6.5.1 Visual Studio Code Editor and InteliSense 48

6.5.2 Designer 48

6.5.3 Other Accessibility Tools 49

Chapter 7: Modelling the Hyperfine Structure

7.1 Overview of the Simulation 50

7.2 The Simulation 50

7.2.1 The Main Window-The Central Command 51

7.3 The Simulate Window 57

7.4 Drawing The Hyperfine Structure 59

Chapter 8: Analysing the Hyperfine Structure

8.1 Classification of The Hyperfine Structure 61

8.1.1 Finding a New Level by Combination of Wavenumbers 61

8.1.2 Determination of New Level by Fluorescence Lines 62

8.1.3 Determination of New Level by Analysis of the Hyperfine Structure 63

8.2 Energy Corrections in Hyperfine Levels 64

8.2.1 The Fitter Program 64

8.2.2 Theoretical Background of the Fitter 65

8.2.3 Energy Value Corrections in the Fitter 65

References 67

1 Modelling and Classification of the Hyperfine Structure of Atoms and Ions

Abstract

Computer simulations have become an important part of today’s spectroscopic studies. Not only

do they help experimenters analyse results, they are also important for information and data

management. The use of computer simulations becomes especially important in the study of

hyperfine interactions within atoms and ions where there are many levels to be managed.

Furthermore formula and equations used for the determination of hyperfine structure are

extremely lengthy and hence very time consuming.

The aim of this work is to develop a computer simulation through which one can model as well

as manage and classify the hyperfine structure of atom or ion of any element.


Chapter 1: Introduction

1.1 Spectroscopy

Spectroscopy is the study of properties of matter by examining its interaction with

electromagnetic fields. Spectroscopy is a very broad field of Experimental Physics. Originally

the concept of spectroscopy was limited to the use of visible electromagnetic radiation. However

during the late 20th

century the concept of spectroscopy extended greatly to comprise any

measurement as a function of either wavelength or frequency (or energy).

1.2 History of Spectroscopy

The study of spectroscopy officially started with the study of white light by Sir Issac Newton

using a glass prism. Nearly 100 years after Newton’s work, Fraunhauffer discovered and

measured the absorption lines in Sun’s spectrum. In 1860 Bunsen and Kirchhoff confirmed the

connection between wavelength of light emitted and the nature of the element or compound used

for source. He showed that each element and compound emits a unique spectrum of energy, thus

pioneering the field of Spectroscopic Analysis. This led to the discovery of many new elements

(like, Rubidium, Cesium, Thallium and Indium). The presence of Helium on Sun by Lockyer in

1868 was confirmed on the basis of this proposal.

Realising that spectroscopy could be used to probe the structure of atoms and molecules, Balmer,

Rydberg and others group spectral lines in some reasonable fashion. Simple equations were

given to generate the wavenumber of a spectral line in the spectrum of an element. In 1896

Zeeman observed the spitting of spectral lines in magnetic field which were interpreted by

Lorentz in terms of oscillating charged particles. During the same period a major advancement

was made in this field by Neil Bohr. Using a semi-classical version of Rutherford’s atomic

model, Bohr proposed a theory of atomic structure establishing the link between the spectra of an

element and its atomic structure. Bohr’s theory gave good results for one electron atoms (e.g. H,

Li+2

etc.) however extension to more complicated atoms and molecules gave incorrect results.

The short comings of the semi-classical model of atomic structure were eradicated with the

establishment of Quantum Mechanics.

Quantum Mechanics opened many fields of research within Spectroscopy through which

spectroscopic studies of many complex atoms, ions and molecules became possible. In addition,

the determination of transition probabilities, study of hyperfine structure, phenomena of isotope

shift and many more became possible. It also opened gates to the development of microscopic

and radio frequency spectroscopy. An important milestone in spectroscopic studies was the


invention of LASER. The ability of laser to produce highly coherent and monochromatic light

made possible the study of individual transition in the spectrum of an element.

1.3 Computer Based Spectroscopy

The use of computers in Theoretical and Applied Physics is becoming increasingly important.

Advanced control engineering has made it possible for experimenters to collect accurate data

even in extreme physical conditions. While hardware support allows experimenters to gather

massive amounts of data, software programs help them analyse it. Computer simulations help

experimenters to extrapolate the outcome of an experiment beyond the capabilities of apparatus

used to collect the data [1].

Computer simulations also play an important role in today’s spectroscopic studies. Computer

based data recording and software techniques provide means of accurate data processing and

collection. The use of computer based computational method becomes especially important in

the study of hyperfine structure of elements, where calculations are lengthy and hence time

consuming.

1.4 Importance of Spectroscopy

Determination of Atomic Structure

Neil Bohr made the first attempt to explain the spectrum of elements in terms of the structure of

its atoms. He used Planck’s hypothesis of quantized energy to explain the emission of radiation

in atoms. Although Bohr’s theory could not be extended to atoms having more than one electron,

his idea was, nevertheless, crucial for future studies in this field. By studying the light emitted

(or absorbed) by atoms during de-excitation (or excitation) one can extract information about the

energy levels involved in the transition thus providing direct information about the structure of

the atom.

Invention of laser enabled scientist to study individual excitation within atoms. As advanced

experimental techniques became available, various effects within the atoms, like fine and

hyperfine splitting of spectral lines became visible providing us a chance to probe further into the

atom.

Study of the Hyperfine Structure

Hyperfine interaction, which is the subject of our work, occurs as a result of the weak interaction

between the electron and the nucleus of an atom. The interaction results in the splitting of fine


structure of an atom. The study of these structures finds application in many areas of Applied

Science like Astrophysics, Nuclear Technology, standards for time and length etc.

Study of Molecular Structure

Molecular Spectroscopy mainly consists of determination of the vibration and rotation modes of

the atoms within the molecule. One aspect of Molecular spectroscopy which is particularly

useful in the field of Chemistry is the Nuclear Magnetic Resonance (or NMR). The NMR

technique is extremely useful in the study of the structure of the molecules and determination of

relative positioning of atoms within a molecule. Electron Spin Resonance (or ESR) is another

method for the determination of molecular structure. It is quite similar in process to NMR

techniques except for the fact that in ESR, the use of the spin of the outer electron of an atom is

made rather than the spin of the nucleus.

Sample Analysis

Ever since the relation between the emitted radiation and the atomic structure was established,

spectroscopy has become an indispensable tool for studying the composition of sample.

Spectroscopic method of compound analysis is better than the conventional methods of chemical

analysis, it’s a lot more accurate and less amount of material is required for the process.

The same process can also be used to test the purity of a sample. The spectrum of the given

sample is compared with the spectrum of the pure compound. The impurity present in the sample

also contributes to its overall spectrum. By comparison of the two spectrums (the pure and the

sample’s) and the intensity of the lines one can estimate the nature as well as the amount of

impurity present.

X-Ray Spectroscopy

X-ray spectroscopy involves the use of high energy X-rays to study the inner structure and the

electronic configuration of a crystal. The diffraction patterns of the X-rays that are diffracted

from the target crystal are studied to determine useful information about the material.

Modelling and Classification of


During the 19th

century various attempts were made to account for the spectroscopic data

obtained on the basis of the internal

this purpose which, on a broad scope

• Classical and Semi-classical Models

• The Quantum Mechanical Model

2.1 Classical and Semi-Classical Models

2.1.1 The Rutherford’s Model

Rutherford proposed that the electrons in an atom are distributed outside the nucleus and are

continuously in motion around it. Scattering of alpha particle from thin gold foil (Rutherford’s

scattering) was in favour of this model. However Rutherford’s model suffers from the following

two deficiencies:

• It provided no means of preventing an electron from

continuously emitting energy.

• Thus making it theoretically unstable

Figure 2.1: Rutherford’s atomic model of a lithium atom

and Classification of the Hyperfine Structure of Atoms and Ions


arious attempts were made to account for the spectroscopic data

internal structure of the sample. Various models were presented for

on a broad scope, can be divided into the following categories:

classical Models

The Quantum Mechanical Model

Classical Models

The Rutherford’s Model



s in favour of this model. However Rutherford’s model suffers from the following

no means of preventing an electron from falling into the nucleus while

emitting energy.

theoretically unstable

Figure 2.1: Rutherford’s atomic model of a lithium atom. Image courtesy Wikipedia

5 of Atoms and Ions

arious attempts were made to account for the spectroscopic data

Various models were presented for

can be divided into the following categories:



s in favour of this model. However Rutherford’s model suffers from the following

into the nucleus while

Image courtesy Wikipedia


2.1.2 Bohr’s Atomic Model

In order to resolve the issues raised on Rutherford’s atomic model, Bohr proposed a semi-

classical model by combining Rutherford’s idea and the concept of photons as discreet energy

packets. Bohr put forth the following points:

• Electrons are not distributed around the nucleus at arbitrary distances. Rather they reside

in orbits of fixed energy (i.e. stationary states).

• Only those orbits were possible for electrons around the nucleus for which the angular

momentum of the electron is integral multiple of ħ, i.e. � � � ħ • Electrons can move between these “prescribed” stationary states either through

absorption or emission of photons (energy).

The conclusions Bohr derived from these assumptions are summarized in table 1. Various

spectral series (Lyman, Balmer, Paschen Brackett, Pfund etc.) of the hydrogen spectrum were

studied through Bohr’s atomic model and were well accounted for.

While Bohr’s semi-classical model successfully explained the spectrum of the one electron

systems (e.g. hydrogen), the theory could not be extended to more complicated atoms or simple

diatomic molecules. This was primarily due to lack of explaination for interaction between more

than one electron in an orbit or motion under the influence of two or more nuclei.

• The radius of the nth

orbit of an atom with atomic number Z is given as:

� � ��4��

Where,

�� 4��

• The speed of the orbiting electron in the nth orbit: �� 2��

• The Bohr energy of the electron is given by:

�� 2�� • Frequency of the emitted or absorbed photons during transition: �� 1�� 1��

*In the above equations: K is the Coulomb’s constant ( � ��), Z is the atomic

number of the atom and h is the Planck’s constant.

Table 1: Results of Bohr’s semi-classical atomic model


A number of corrections were proposed in Bohr’s model among which the most notable are the

Somerfield’s model (also called the Bohr-Sommerfeld), the reduced mass and the relativistic

Bohr model. Although these corrections refined the results of the theory, it was still unable to

explain the energy spectrum of atoms with more than one electron or of simple diatomic

molecules.

2.2 Quantum Mechanical Model

Quantum Mechanics was postulated with a completely original approach rather than putting

assumptions on to the Classical/Newtonian Physics. The entire structure of Quantum Mechanics

is based on certain postulates. These postulates have slightly evolved over time but their

fundamental idea remains the same:

• The state of any system (for example a particle or an atom in a particular environment) is

specified through its wave function. The wave function itself depends on space and time.

In QM, the wave function is the only thing that one can know about the system. The

wave function of a system, therefore, contains everything that one could ever want to

know about it.

• In order to derive information about the system (technically speaking about the

observable of the system) out of a wave function, it is operated upon by an appropriate

function. Operation of a quantum mechanical operator on a system’s wave function

yields a value (called its Eigenvalue).

• The significance of eigenvalues of the operator is that these are the only values of

observable in which we can find the system when the observable of the system is

experimentally evaluated.

The current version of Quantum Mechanics emerged from two independent approaches.

However it should be mentioned that the above stated postulates are applicable on both:

• The Wave Mechanics (Schrödinger’s Scheme)

• The Matrix Mechanics (Heisenberg’s Matrix Mechanics)

2.2.1 The Wave Mechanics (Schrödinger’s Scheme)

In the Wave Mechanics formulation, the dynamics of a particle are defined by a wave equation

called the Schrödinger’s equation. Schrödinger’s equation is a second order differential equation.

In the wave mechanics formulation, the operator is an algebraic function (differential, additive,

multiplicative etc) which operates on a wave function. The value(s) generated by the application


of the operator (the Eigenvalue) is then the possible values that can be observed for the operator.

However there are certain restrictions on the possible wave functions for a system:

The function must be:

• Single-valued

• Continuous

• Finite everywhere

• Should vanish appropriately at infinity.

The application of Schrödinger’s equation on physical situation generally yields a spectrum of

states (Eigenvectors or Eigenfunctions) and eigenvalues (this basic feature of Quantum

Mechanics introduces the concept of no-prior-knowledge in Quantum Mechanics which has been

a burning question between physicist and philosophers). As we will see in the following sections

that QM can account for many interactions which Classical Mechanics cannot account for.

2.2.2 The Matrix Mechanics (Heisenberg’s Matrix Mechanics)

Matrix Mechanics was the first complete description of Quantum Mechanics. In matrix

mechanics, the operator of an observable is an infinite square matrix (A vector is represented as a

matrix with only a single row). When a matrix operates on a vector, in general a new vector is

produced, in a way similar to an operator in Schrödinger’s scheme.

In 1927 Schrödinger and Eckard proved that the Heisenberg and Schrödinger scheme are

mathematically equivalent. Throughout this publication we will employ the Schrodinger’s wave

equation for analysis.


Chapter 3: Quantum Mechanical Description of Atoms

3.1 One Electron System

For wave function , the Schrodinger’s time independent equation is given as:

!� ħ�2� "� # $%&'( � � 3.1

Where � ħ)

�*"� # $%&' is called the quantum mechanical Hamiltonian operator. The nucleus of

the atom is considered a point of charge Ze, where Z is the atomic number of the atom. The

motion of the electron around the nucleus is primarily governed by the electrostatic potential

around the nucleus. Therefore,

$%&' � � ��4+��

Furthermore, in order to account for the motion of the nucleus, we use the reduced mass which is

given by,

� � �,-�, # -

Where M is the mass of the nucleus and �,is the mass of the electron. Since the potential

function depends only on the scalar distance r, therefore it is reasonable to work in the spherical

polar co-ordinates. After substituting the Laplacian operator ("�) of equation 3.1, in spherical

polar coordinates, the Schrodinger equation can be written as,

� ħ�2� . 1�� //� �� / /�� # 1�� sin 3 //3 �sin 3 / /3� # 1��45��3 !/� /6�(7 # $%&' � � 3.2

We assume that the solution of the above equation, the wave function of the electron, can be

separated into radial and angular components that is:

%&,9,:' � �%&';%9,:' Furthermore,

;%9,:' � Θ%θ'Φ%<'


After replacing the solution the wave function in equation 3.2 and doing mathematical

manipulation, we get:

=�Φ=>� � ��?�6 3.3

� 1sin 3 ==3 �sin 3 =Θ=3� # �?� 145��3 � @%@ # 1'3 3.4

1�� == �� =�=� � @%@ # 1'�� 2�ħ� $%&'� # 2��

ħ� � � 0 3.5

After solving equation 3.3 we get,

Θ%9' � �B*CD

Where the range of �? becomes,

�? � 0,E1,E2,E3,…

The equation involving θ becomes,

Θ%9' � HI?*C%JKL9' Where, I?*%JKL9' is the associated Legendre function defined as,

I?*C%M' � %1 � N�'|*C|� � ==N�|*C| I?%M'

And I?%M' is called the Legendre Polynomial, given by the Rodrigues formula,

I?%M' � 12?@! � ==N�? %N� � 1'?

One thing is immediately clear from the solution of Θ%9': @ must be a non-negative integer

otherwise Rodrigues formula (I?%M'', which involves the differential operator applied “@” times,

does not make sense at all. Moreover, if |�?| R @, then the associated Legendre function (I?*C%M') is equal to zero and hence, ultimately, %&,9,:' is zero. Therefore no solution exists for |�?| R @. The possible values of �? then are restricted to the following range,

�? � �@,�@ # 1,�@ # 2… , @ � 2, @ � 1, @


In order to find a solution to the radial function �%&'we notice that the probability of finding the

electron should be zero at infinity, i.e. �%&' S 0 as � S ∞, this fact is used to calculate the energy

of the electron in the nth

orbit from the nucleus. The energy comes out to be,

� � � ��2%4T�'�ħ��

Where n is an integer satisfying the relation � U @ # 1. For positive energy there is no such

restriction because for positive energies the electrons will no longer be bound to the atom’s

potential. Picture 3.1 shows plot of first few radial functions of the hydrogen atom.

Therefore, to summarize, electron in an atom is characterized by three quantum numbers. Each

value of � (called the principle quantum number) contains � values for @ (azimuthally quantum

number) from 0,1,2,… %� � 1' and for each value of @ there are 2@ # 1 possible values available

to �? (called the magnetic quantum number). As will be discussed later, the value of the

magnetic quantum number defines the orientation of the electron with respect to an external

magnetic field, which is usually taken as the z axis. States with different values of ml are

degenerate i.e. they have the same energy. The degeneracy in the values of �? is a consequence

of the fact that in the absence of field, there is no preferred direction in an atom. The @ degeneracy is a direct result of the coulomb potential and is peculiar to hydrogen atom. When the

spin-orbit interaction and the relativistic effects are taken into account, this degeneracy is

partially removed.


Figure 7.6: Some radial function of hydrogen atom


3.2 The Spin of Electron

1922, an experiment was performed by Otto Stern and Walther Gerlach aimed at measuring the

spatial separation, if any, between two parts of a silver atom beam passed through

inhomogeneous magnetic field. The result of the experiment could only be explained if it is

postulated that electrons had an intrinsic spinning motion in addition to the orbital motion around

the nucleus. At the time of its proposal no physical meaning could be given to the spin of the

electron in terms of it actually spinning around its own axis (for reasons either violating the

Special Theory of Relativity or against well established facts). A full theoretical description of

the spin was given by Paul Dirac using the relativistic version of Schrodinger’s equation

proposed by Oskar Klein and Walter Gordon called the Klein-Gordon equation.

Paul Dirac proposed that spin is an intrinsic property of every fundamental (and hence

composite) particle. The spin angular momentum of a particle is given as,

V � W4%4 # 1'ħ Where, s is called the spin quantum number. Its value is intrinsic to a particle (equally

fundamental as mass) and is equal to ½ for electrons, protons and neutrons. For a given particle

of spin 4, there are 24 # 1 possible orientations in an external magnetic field, given as:

�X � �4,�4 # 1,�4 # 2,… , 4 � 2, 4 � 1, 4

For electron (spin ½) there are only two possible alignments with external magnetic field:

�X � 1 2⁄ ,� 1 2⁄

Therefore �X becomes the fourth quantum number required to completely and correctly identify

an electron in atoms. Due to the spin of the electron, it possesses a magnetic dipole moment

which is given as:

Z? � �[?\]@ Where, \] is called Bohr’s magneton which is defined as,

\] � �ħ2�

@ is the generailsed angular momentum (either orbital or spin) and [? is the generalized constant

relating the magnetic moment to the angular momentum.

Due to its magnetic moment, an electron will interact with any external magnetic field. The

presence of an external magnetic field creates an energy separation between the two spin

orientations of the electron hence lifts the degeneracy in �X.


3.3 The Fine Structure of the Spectrum

The fine structure of the spectrum of atoms is the splitting of the spectral lines into fine lines

with very small separation. They result from relativistic effects, the interaction between the spin

and orbital motion of the electron etc. The complete Hamiltonian for one electron systems, with

the above mentioned effects considered, is given then:

12�� ^_ # �̀ ab� # �ħ2�` c. e f a � g�8�i`� � �ħ�8��`� e. e6 � �ħ4��`� c. e6 f _

From the above equation we see that in addition to the non-relativistic Hamiltonian � ��*) ^_ #,j ak� # ,ħ�*j c. e f ab, we get three additional terms appearing as perturbations to the non-

relativistic equation.

The Relativistic term

g�8�i`�

The relativistic term introduces relativistic corrections in the energy.

The Spin-Orbit Term

�ħ4��`� c. e6 f _

The spin-orbit interaction term arise due to the interaction between the spin and the orbital

motion of the electron. The interaction is basically between the magnetic field, created by the

orbital motion, and the spin of the electron. This results in the above mentioned term being added

to the non-relativistic Hamiltonian.

The Darwin Term

�ħ�8��`� e. e6

The Darwin term appears for the s-orbital due to a phenomenon in Dirac theory called

zitterbewegung, whereby the electron does not move smoothly but instead undergoes extremely

rapid small-scale fluctuations, causing the electron to see a smeared-out Coulomb potential of the

nucleus.[2]


3.4 N Electron System

Schrodinger’s equation (both relativistic and non relativistic) are single particle equations i.e.

they can only model the dynamics of a single particle. In a one electron system, one only needs

to take into account the interaction between the electron and the nucleus. However, in a multi-

electron system the interaction with other electrons within the system must also be addressed.

Due to this many new effects occur in N electron systems which were either negligible or absent

in one electron systems.

Unfortunately, almost none of the multi-electron systems are exactly solvable. Therefore one

must resort to different approximations in order to roughly sketch the properties of multi-electron

systems.

3.4.1 The Central Field Approximation

In moving from one electron systems to N electron systems, the following interaction must be

taken into account:

• The kinetic energy of the electron and their potential energy in the electrostatic field of

the nucleus

• The electrostatic force of repulsion between electrons within the atom

• The interaction between the spin and orbital motion of the electron

• Small effects such as spin-spin interaction between the electrons, the relativistic effects,

radiative corrections and nuclear corrections.

In Central Field approximation the small effects mentioned above are neglected due to their

small contribution in the actual solution, they are treated as perturbation. However, the relative

importance of various effects within the nucleus varies a great deal from element to element and

unfortunately there is no satisfactory solution. Nevertheless, the Central Field approximation

serves as the first hand tool for analyzing multi-electron systems.

In Central Field Approximation, the nucleus is considered point-like and infinitely massive and

hence at rest with respect to the moving electrons. This is not a unreasonable approximation as

the nucleus is very much heavier than the electrons therefore is essentially at rest. Due to the

repulsive force between the electrons within the atom, a repulsive term occurs in the

Hamiltonian. The Hamiltonian of an N electron system is thus given as:


l �m!� ħ�

2�"B� � ��4T��B(nBo� # m ��4T��Bp

nBqpo� 3.6

The electron therefore moves in an effective potential of the central nucleus and other r � 1

electrons. It is further assumed that the potential of the inter-electron repulsion is considered

spherically symmetric. This makes the radial and angular variables separable as in case of one

electron system. The angular part then has the same solution as the hydrogen but the radial part

no longer has the simple coulomb form.

For the solution of the Hamiltonian, 3.6, Hartree model of self-consistent fields is used in which

we make initial guess for the spherical asymmetric potential and follow an iterative process until

the final result converges to a self-consistent values of the potential and wavefunction .

3.4.2 Spin-Orbit Interaction

Spin-orbit interaction is the interaction between the spin and the orbital motion of the electron.

This effect becomes particularly important when there are more than one electron outside the

central electron core (consisting of completely filled orbitals). Due to this reason there are large

numbers of energy levels giving rise to complex energy spectrum.

One can include the spin-orbit interaction of the electron by adding the Hamiltonian of spin-orbit

interaction in equation 3.6. The total Hamiltonian then becomes

l �m!� ħ�2� "B� � ��4T��B # s%&t'uB. vB(nBo� # m ��4T��Bp

nBqpo� 3.7

Where,

s%&t' � 12\�`� !1�B /w%&t'/�B (

From equation 3.7 one can divide the spin-orbit interaction into two types depending on the

atomic number of the atom in question

3.4.2.1 L-S Coupling

For atoms with small atomic number (Z < 30) the electron-electron repulsion becomes stronger

than the LS coupling, therefore the LS coupling is considered perturbation to the Hamiltonian

independent of LS coupling. This is because of the fact that the sum of s%&t'uB. vB term in

equation 3.7 vanishes for completely filled electronic orbits (∑ �?t � 0'B leaving only few


electrons in the valence orbit capable of producing spin-orbit interaction. This is called the

Russell-Saunders coupling [3]

In LS coupling the spin of individual electron couples together to form total spin angular

momentum S (∑ VBB � V), similarly total orbital angular momentum sum up to form the total

orbital angular momentum ((∑ �BB � �). Both the total spin and angular momentum combine to

form the total orbital angular momentum vector J (J = L + S). Both L and S precess about the

total angular momentum J and the total angular momentum precesses around the external

magnetic field (usually taken to be the Z axis, figure 3.2). Furthermore, the Hamiltonian with

spin-orbit coupling does not commute with individual spin and orbital angular momentum

(�B and VB). However, it does commute with total spin (S) and orbital angular momentum (L),

hence the total angular momentum (J).

Figure 3.1 Orbital Angular Momentum, L, and Spin Angular Momentum, S, couple to form

Total Angular Momentum J. Both L and S couple precess about the J.


Figure 3.2: Precession of the Total Angular momentum in the presence of an external magnetic

field. IT customarily taken to be along the Z axis.

3.4.2.2 J-J Coupling:

On the other hand in heavy atoms, due to the involvement of many electron (hence large number

of orbits), the L.S term dominates due to larger number of electrons outside the filled core.

Therefore, the electron-electron coulomb repulsion term (∑ ,)��&tynBqpo� ' is treated as

perturbation to the Hamiltonian including the spin-orbit interaction.

In J-J coupling, each orbital angular momentum Li tends to combine with each individual spin

angular momentum Si, originating individual total angular momentum zB. These then add up to

form the total angular momentum J (Figure 3.3).

z � mzBB �m%�B # VB'B

The sum of the total angular momentum precesses about the external magnetic field or the Z axis

(figure 3.4).


Figure 3.3: Showing the coupling between individual spin (Si) and orbital angular momentum

(Li). The figure on the right shows coupled angular momentum precessing about the total angular

momentum.

In the case of J-J coupling, the total Hamiltonian does not commute with the individual spin and

orbital angular momentum (Li and Si) but not with the total (L and S).


Figure 3.4: The sum of the individual angular momentum J precesing about the Z axis which is

usually taken as the direction of external magnetic field.

3.2.3 Pauli’s Exclusion Principle

Pauli’s exclusion principle states that no two fermions can occupy the same state in a system

simultaneously. Originally the principle was proposed for electrons in atoms but then it was later

extended to all particle having half-integral spins. For a N electron system, Pauli’s exclusion

principle states that, no two electrons in a atom can have same set of all four quantum numbers

i.e. �, @,�? , �X. In addition, the wave function of electron in atoms is also labelled by their

parity. Parity is defined as the inversion of coordinates of space i.e.

I !N{|( � !�N�{�|( 3.8

P is the parity operator. A wave function is said to have even parity if it does not change its sign

under the transformation 3.8 and odd if it does. Since electrons are indistinguishable particles

therefore the Hamiltonian of an electron in atom should have either an even or odd parity. This

also serves as “good” quantum number i.e. it can be used to label states within an atom.


3.2.4 Electronic Configuration in an N-electron System

For any given value of n (principle quantum number) there are n possible value of the @ (the

azimuthal quantum number), for each value of @ there are 2@ # 1 values of �? and for each of

these there are two possible values of �X. The total number of electrons in state � is thus 2��.

Different states within an atom are denoted by the s, p, d, f,…, after } the sequence become

alphabetical. The first four alphabets do not have any theoretical interpretation; they are

historically named afer the first series identified in the spectra of the alkali metals: sharp,

principal, diffuse and fundamental.

These alphabets correspond to 0, 1, 2, 3… values of the azimuthal quantum number respectively.

The principle quantum number is written as the co-efficient of the alphabetical symbol while the

number of electrons in each is written as its power.

As one move from lower level states to higher lying levels (in terms of energy), the orbital start

overlapping each other; a situation called configuration mixing or configuration interaction.

However, since the states are also labelled by their parity status, only states of same parity are

mixed.


Chapter 4: Line Profiles and Broadening Processes

When studying the energy spectrum of elements, it is worth noticing that the absorption and

emission of energy from atoms does not occur at a fixed frequency (or wavelength). Instead, a

spread in the absorbed or emitted frequency occurs about the centre frequency (the frequency

which the atom would emit or absorb if not perturbed by any broadening phenomena). Accurate

measurement of intensity and wavelengths require knowledge of the profiles of the spectral lines.

These profiles are of interest in themselves as they can give important information about the

condition of the source and interaction between the atoms and the collisions process. Line profile

is defined as the distribution of intensity about the centre frequency.

Line broadening mechanism, on a broad level, can be divided into homogenous and

inhomogeneous broadening. Homogenous broadening occurs when the probability of absorption

or emission of radiations is same for all atoms in a sample this broadens the line profile in the

same way, giving rise to symmetric broadening about the line centre. In-homogenous broadening

produces asymmetric line profiles about the centre frequency. Three main processes contribute to

the spread in the frequency emission: Natural broadening, Doppler broadening, and Pressure

broadening (also called Collisional broadening).

4.1 Natural Broadening

We know that energy is absorbed or emitted by an atom in moving between energy level, say Ea

and Eb. The frequency emitted or absorbed is given by Bohr’s hypothesis ~�� %�� ' ħ⁄ .

However, the frequency as seen by the detecting instrument is not exactly as given by the

formula. This is because an electron in any energy state, except for its ground state, decays with

a finite lifetime �. By the uncertainty principle, the energy of such a level cannot be precisely

determined, but must be uncertain by an amount of order ħ �⁄ which is called the natural width

of the level.

�B � m 1HBpp

Where, HBp is called the radiative transition rate or radiative transition probability. By uncertainty

principle, the energy width of a level ∆� has a minimum uncertainty in its life time ∆� is given

as: ∆�∆� � ħ Taking ∆� � �B, the energy width can be given as:

∆�B � ħ�B � ħmHBpp


If an electron is moving from an energy level b to a, the effective width of the transition is the

sum of the width of both the upper and lower levels. ∆�� ∆�� # ∆��

∆�� ħ�mH�BB #mH�pp � � �∆��

The Line Profile of natural broadening can then be given as:

�%�' � �� 14�%� � ��'� # ^ 14�b�

Where, �� the resonant frequency where the intensity is maximum i.e. the centre frequency. A

plot of the frequency and the corresponding intensity about the centre frequency ��is shown in

figure 4.1. This form of the line profile is characteristic of Lorentzian, or Cauchy-Lorentzian,

distribution ^� � �%��')��)�b. It is obvious from the graph that natural line profile is

homogenously broadened about the centre frequency. An important factor in line profiles is the

width of the frequency interval at the half the maximum intensity.

Figure 4.1: Graph of intensity �%�' versus frequency showing a Lorentzian distribution of

intensity about the centre frequency

The phenomenon of natural line broadening has its roots in the theoretical foundations of

Quantum Mechanics therefore it cannot be eliminated from the system. The life time of excited


states is of the order 10−6

to 10−9

have a line width of 0.1 to 100 MHz. A similar situation exists in absorption; the absorption line

is broadened due to finite life t

extent of the line broadening effect is very small as compared to other line broadening effects

particularly the Doppler broadening. Therefore for our work we choose to consider only the

effects of Doppler broadening on spectral lines.

4.2 Doppler Broadening

Doppler Effect is the change in the wavelength of the emitted light when the source of light is

moving with respect to the detector

the effect is called red shift while in case of negative shift the effect is called

Doppler broadening of spectral lines depends upon of the motion of the emitting atom, it is most

prominent when the sample is in gaseous form where thermal motion the atoms exhibit a full

range of velocities given by Maxwell’s velocity distribution

Figure 4.2 Distribution of particle speed for oxygen particles at

Celsius (273K, 293K and 873K respectively). Speed distribution can be derived from

Maxwell-Boltzmann distribution.

The shifted frequency due to the


9 sec. Hence a transition from such state to the ground state will


is broadened due to finite life time of the excited state. However, in most practical cases the



s of Doppler broadening on spectral lines.


moving with respect to the detector (or vice versa). If the shift in the wavelength is positive, then

while in case of negative shift the effect is called


ent when the sample is in gaseous form where thermal motion the atoms exhibit a full

range of velocities given by Maxwell’s velocity distribution (figure 4.2).

Distribution of particle speed for oxygen particles at -100, 20 and 600 degrees


Boltzmann distribution. Image courtesy Wikipedia

The shifted frequency due to the motion of the atom towards the detector is given as:

24 of Atoms and Ions

sec. Hence a transition from such state to the ground state will


ime of the excited state. However, in most practical cases the




. If the shift in the wavelength is positive, then

while in case of negative shift the effect is called blue shift. Since


ent when the sample is in gaseous form where thermal motion the atoms exhibit a full

100, 20 and 600 degrees


motion of the atom towards the detector is given as:


� � �� 1 E �̀ # ^�̀b� E�� For non-relativistic speed, which is a fairly good approximation in most cases, the above formula

reduces to: � � �� ^1 E �̀b 4.1

For atom in thermal equilibrium, by Maxwell velocity distribution, the number of atoms having

speeds between � and � # =� is giving as:

I%�'=� � � �2�� Ng � !��2��(=� 4.2

Putting 4.1 in 4.2,

I%�'=� � � �`�2�� Ng � !�`�%� � ��'�2�� (=�

After including the contribution from all velocity components of all atoms one reaches the

following equation for intensity profile:

�%�' � ��Ng � .`%� � ��'�� 7� 4.3

�� * and is called most probable speed at temperature T. An important parameter in the

line profile is the Full Width at Half Maximum (FWHM). It is the width of the profile at half the

maximum intensity and for Doppler profiles it is called the Doppler width, give as:

�~� � ^��̀b�8��@�2�

Equation 4.3 then becomes:

�%�' � ��Ng � �%� � ��'0.6�~� �� 4.4

Equation 4.4 is a characteristic equation of Gaussian distribution function.


Figure 4.3: Combined Doppler profiles from atoms travelling in different directions. Image

courtesy of Oxford University Press

Table 4.1 shows a comparison between the Natural and Doppler broadening for some laser

transitions. It can be seen that Doppler broadening dominates the natural broadening by a

significant amount, therefore one can neglect the natural line broadening in comparison with

Doppler broadening.

Table 4.1: Natural Broadening VS Doppler Broadening

Laser Wavelength

(nm)

Natural Broadening

(MHz)

Doppler Broadening

(MHz)

Neon (He-Ne) 632.8 14 1500

Argon ion 488.0 450 2700

Cadmium (He-Cd) 441.6 45 1100

Copper 510.5 0.36 2300

4.3 Pressure Broadening

An atom in an excited energy state can also move to lower energy state via non-radiative energy

transfer. The principal mechanism of such transfer is the collision of the excited atoms with other


atoms. Because of which this type of broadening mechanism is also called collisional or pressure

broadening. The collisional broadening is actually the result of inelastic collisions between

atoms. It is very prominent in plasmas and gas discharges because of the long range Coulomb

interaction between charged particles.

The extent of the collisional broadening, as compared to other broadening mechanisms, primarily

depends upon the physical conditions of the sample. Due to this reason studying pressure

broadening is an important part of stellar spectroscopy where studying the stellar atmosphere

provides details of its structure. The extent of the pressure broadening also depends upon the

experimental setup and the technique used.

The collisional broadening produces a Lorentzian profile, the same as that of natural broadening.

In order to avoid pressure broadening, the sample should be kept at low temperature. By

changing the pressure and observing the corresponding change in the width the information

about the collisions occurring in the gas can be obtained.


Chapter 5: The Hyperfine Interaction and Hyperfine

Structure

In an atom the most important interaction between the nucleus and the electrons is the coulomb

interaction � ,)& . All other interactions with the nucleus are categorised under hyperfine

interactions. The hyperfine interaction in an atom can be divided into two categories on the basis

of their effect on the energy levels in an atom.

• Hyperfine interaction causing shift in the energy: In this type of hyperfine interaction

the energy of the orbits is simple shifted by certain amount. This type of interaction can

be further divided into two categories:

o Mass Effect: The shift in energy that occurs within different isotopes of the same

element.

o Volume Effect: The change in energy because of the distribution of charges

inside the nucleus belonging to the same element.

• Hyperfine interaction causing splitting of energy levels: In this type of hyperfine

interaction, the energy level of the atoms split into further very fine energy levels. The

number of these levels depends upon the total quantum number of the orbit energy level

and the spin quantum number of the nucleus. This type of hyperfine interaction occurs

because of:

o Magnetic Dipole Interaction

o Electric Quadruple Interaction

In practise the former type is called the isotope shift while the later is termed as hyperfine

interaction.

5.1 Hyperfine Interaction

The hyperfine interaction occurs due to the interaction of the electromagnetic multi-pole moment

of the nucleus with the surrounding electrons. General symmetry arguments of parity and time

reversal invariance imply that the number of possible multi-pole nuclear moments is 2k. The only

non-vanishing nuclear multi-pole moments are the magnetic moments of odd k and the electric

moments of even k, namely the magnetic dipole (k=1), electric quadrupole (k=2), magnetic

octupole (k=3) and so on. The most important among these are the magnetic dipole moments

(caused by the spin of the nucleus) and electric quadruple moment (caused by the departure from

spherical distribution of charges in the nucleus).


5.1.1 Magnetic dipole Hyperfine Interaction

Protons and neutrons are made up of particles called quarks. These quarks posses ½ spin, similar

to an electron, as a result of which protons and neutrons possess an inherent spin of ½. The

intrinsic spins of each proton and neutron in the nucleus add up to form the total spin of the

nucleus (represented by the symbol I). The eigenvalues of ¡� are given as �%� # 1'ħ, where I is

called the nuclear spin quantum number. The projection of this nuclear on any axis is given by �¢ � ��, �� # 1, �� # 2,… , � � 2, � � 1, �. If the spin of the nucleons add up to an integer then

the nucleus is called a boson (and obeys Bose-Einstein statistics). On the other hand if the spins

add up to a half integer then the nucleus is called a fermion (and obeys Fermi-Dirac statistics).

The total Hamiltonian of magnetic hyperfine interaction can be written as,

l � l� # l£�

Where, l� is the unperturbed, zero-order Hamiltonian including the Coulomb interaction and the

relativistic correction discussed in the previous sections and l£� is the Hamiltonian of hyperfine

interaction. Because the magnetic moment of the nucleus is much smaller than that of electron,

the Hamiltonian l£� is treated as perturbation in l�. The magnetic dipole moment of the

nucleus is given as:

Zn � [¢\n� Here, [¢ is a dimensionless factor called nuclear g factor or nuclear Lande factor, µ¤ is called

nuclear magneton, it analogous to Bohr’s magneton and is related to it as:

\n � \]1836 � 5.05082 f 10��¦ z �§

The magnetic field due to the dipole moment of the nucleus interacts with the orbital (L) as well

as spin (S) angular momentum of the electron. It is therefore convenient to split l£� into

Hamiltonians of the orbital and spin interactions:

l£� � l� # l� 5.1

l�, the Hamiltonian of interaction with orbital motion, can be given as:

l� � � 5ħ�� a. e


Where a%&' � � ¨�� ©n f e^�&b� is the magnetic potential of a point dipole located at the

origin. This equation leads to:

l� � \�4 2ħ� [¢\]\n 1�i u. ª 5.2

Next we find the contribution of the interaction of the nuclear magnetic dipole moment with the

spin motion of the electron. The magnetic field of the vector potential is given as:

« � e f a � � \¬4 �©n" �1�� e%©n . e' 1�� The energy of the spin interaction with magnetic field is given as:

l� � �©X. « � 2\]v.«ħ

l� � � \�4 2ħ� [¢\]\n �v. ¡"� �1�� %v. e'%¡. e' 1�� 5.3

Putting 5.2, 5.3 back in 5.1 we get,

l£� � \�4 2ħ� [¢\]\n � 1�i u. ª � v. ¡" �1®� # %v. e'%¡. e' 1�� It is convenient to introduce a quantity called the total angular momentum of the atom i.e.

electron + nucleus. It is denoted by F and defined as,

¯ � ¡ # ° The quantization of the total angular momentum follows the same scheme as that of orbital and

spin angular momentum i.e.

¯� � ±%± # 1'ħ and -´ � �±,… , ± The possible values of the quantum number F is given by,

± � |� � z|, |� � z| # 1,… , %� # z' � 1, %� # z' The shift in the energy due to first order perturbation is then given as,

∆� � H2 µ±%± # 1' � �%� # 1' � z%z # 1'¶ � H�2


Where A is given as:

H � \�4 4[¢\]\n 1z%z # 1'%2� # 1' �\i%��'i ��= � � ±%± # 1' � �%� # 1' � z%z # 1'

5.1.2 Electric Quadrupole Hyperfine Structure

So far we have been assuming that the distribution of charges inside the nucleus is spherically

symmetric. However, this is not always the case. The Electric Quadrupole moment is defined as

the deviation of charge distribution in the nucleus from spherical distribution. The electric

quadruple moment is a symmetric, second-order tensor. If ®· is the coordinate of a proton with

respect to the centre of mass of the nucleus, if ¸·� � ¸�, ¸·� � ;· , ¸·i � �· are the coordinate

of the Pth

proton, then:

¹Bp �mº3¸·B¸·p � �Bp�·�k· 5, » � 1,2,3

The electric quadrupole moment is customarily defined as the average of the component ¹ � ¹ii of the state ¼|�, -¢ � �½: ¹ � ¾�,-¢ � �¿m%3�·� � �·�'· ¿�, -¢ � �À

For spherical distribution of charge it is clear that �·� � 3�·� hence Q is equal to zero i.e. there is

no electric quadrupole moment for spherical charge distributions. If the nuclear charge

distribution is elongated along the direction of I, a configuration called prolate, then Q > 0. On

the other hand Q < 0 for flattened nuclei a configuration called oblate. The Hamiltonian of the

electric dipole interaction is given as:

lÁÂ � Ã 32 ¡. °%2¡. ° # 1 � ¡�°'2�%2� � 1'%2z � 1'

Where the quadrupole coupling constant B is given by,

Ã � ¹ Ä/�$�/|� ½ Here,

Ä/�$�/|� ½ � ¾», �p � »Å /�$�/|� Å»,�p � »À


Is the average gradient of the electrin field produced by the electron at the nucleus.

The shift in the energy produced due to first order perturbation is given as:

∆� � Æ»�±-´ÇlÁÂÇ»�±-´È � Ã432�%� # 1' � 2�%� # 1'z%z # 1'�%2� � 1'%2z � 1'

The total shift in energy due to the magnetic dipole moment and the electric quadrupole moment

can be given as:

∆� � H�2 # Ã432�%� # 1' � 2�%� # 1'z%z # 1'�%2� � 1'%2z � 1'

From which we get:

� � �j # ÉÊHÊ # ËÊÃÊ � É?H? # Ë?Ã? 5.4

Where,

É � 12 µ±%± # 1' � �%� # 1' � z%z # 1'¶ � �2

Ë � 34�%� # 1' � 2�%� # 1'z%z # 1'2�%2� � 1'%2z � 1'

The constants A and B are called the hyper fine constants.

It is worth noting that the shift in energy is independent of -´ and therefore is (2F+1) fold

degenerate. This degeneracy is removed by the application of external magnetic field. This is

called the Zeeman Effect in Hyperfine Structure.

5.1.3 Intensity of Components in Hyperfine Structure

The relative intensities of the individual hyperfine components are given by[6]:

¡%¯ÌS¯Í' � %2±Ê # 1'%2±? # 1'2� # 1 ÎzÊ ±Ê �±? z? 1Ï� 5.5

The term ÎzÊ ±Ê �±? z? 1Ï is the 6J symbol.


5.1.4 Important Points about the Hyperfine Structure

• The term hyperfine structure refers to the spectral lines due to transitions between the

entire arrays of the levels that result from the splitting of the upper and/or lower levels

involved in the transitions.

• The number of hyperfine components corresponding to a fine structure energy level is the

smaller of the (2J+1) and (2I+1). These components are said to form a hyperfine

multiplet. The case where the number of hyperfine component is (2I+1) is particularly

important because then the spin of the nucleus of the atom can be determined directly by

counting the number of the hyperfine structure.

• The selection rule for transitions in F is 0,E1 with transition from 0 to 0 is being

forbidden.

• Transition between same values of J but different values of F is permissible but they are

very weak, in the microwave region therefore they are best observed by using stimulated

emission techniques.

• The energy (hence frequency) separation between various hyperfine transitions is very

small.

• Within an hyperfine multiplet the ratio of the sums of the intensities of all transitions

from two states with quantum numbers F and F’ are in the ratio of their statistical weight

i.e. %2± # 1' %2±′ # 1'⁄

5.2 Line Broadening Effects on Hyperfine Structure: Combined Line

Profile of Hyperfine Structure

In the study of hyperfine structure of elements the effects of Doppler Broadening usually

dominates other broadening effects discussed in the previous chapter. Furthermore Doppler

broadening affects every single transition between the energy levels involved in the structure.

The energy separation between various hyperfine transitions is extremely small. This causes the

Doppler profile of transition to overlap onto each other. For example, observe the individual

Doppler profiles of the hyperfine transitions between Ju=Jl=6.5 for different Doppler widths in

figure 5.1. The vertical lines are the Line Centres: It is the theoretical (ideal) frequency emitted

by the transition, also called the central frequency. The curve is the Doppler profile of the line.

Observe that the Doppler profiles of the lines overlap each other, particularly along the right

hand side of the figure where the line centres are very close to each other. Also the extent of the

overlap increases with the Doppler width due to the increase in the width of the emission. Due to

overlapping of the Doppler profiles, the intensity of the emission at a particular frequency

becomes the sum of the intensities due to all the line centres at that frequency. The combine

Doppler profile of the same hyperfine structure is plotted in Figure 5.2. The equation of intensity

at any frequency � along the axis then becomes:


�� m ��?? ?B�, j,�Ð&,X� m��Ng � �%� � ��'0.6�~� �

� 5.6

Figure 5.1: Individual Doppler profiles of hyperfine transitions at 500 MHz (upper) and 1000

MHz (lower) Doppler widths.


Notice that at 500 MHz Doppler width, the actual hyperfine structure of the transitions is

partially blocked due to the broadening of the spectral lines. This is most prominent for the lines

at the right hand side of the graph where they are closely packed. Now if we increase the

Doppler width from 500 MHz to 1000 MHz (also in figure 5.2), you can see that the combined

Doppler profile has engulfed nearly the entire hyperfine structure and very little can be said

about the individual spectral lines. At even higher Doppler widths, the envelop of the combined

Doppler profile increases even further and the structural information of the hyperfine structure

becomes even more vague.


Figure 5.2: The combined Doppler profile of the transition in figure 5.1

From these plots the importance of the Doppler width in the study of hyperfine structures

becomes clear: Lower the Doppler width, better will be the resolution of the spectrum. As a final

evidence consider the devastating case of 2000 MHz Doppler width: All the information of the

structure is lost (figure 5.2 a).


Figure 5.2 a: Doppler width of 2000 MHz

Over the time various techniques have been developed to reduce the impact of Doppler

broadening on the hyperfine structure. These techniques on a broad scope can be divided into

Doppler free and Doppler limited spectroscopy. We will explain a few of these techniques in

what follows.

5.3 Experimental Techniques in Hyperfine Study

5.3.1 Doppler Free Spectroscopy

5.3.1.1 Collimated Atomic Beam Spectroscopy In Collimated Atomic Beam spectroscopy a beam of the sample under study is converted into a

highly collimated beam. The collimated beam of atom is crossed by a narrow band, single mode

laser. Since the effect of Doppler width on the incident light is seen only along the direction of

travel of the atom, it is reduced by the collimation ratio of the atomic beam (collimation ratio is

defined as the ratio between the length of travel and the change in diameter of the beam). A

schematic experimental setup is shown in figure 5.3. Main characteristics of the Collimated

Beam Spectroscopy are:

• Atomic beam can be created out of almost any sample therefore this technique can be

applied to large number of samples.


• The Doppler width is proportional to the collimation ratio for which one can easily reach

values above 100. With collimation ratio above 100 the Doppler profile is is about 5MHz.

• In collimated beams, the number of collision between the atoms is significantly reduced

due to small spread in the velocities thus reducing the extent of pressure broadening.

Figure 5.3: A schematic experimental setup of Collimated Atomic Beam spectroscopy.

5.3.1.2 Saturation Spectroscopy Saturation spectroscopy is a very clever technique for obtaining high resolution, Doppler free

spectroscopy. The basis of the saturation spectroscopy, like collimated beam, spectroscopy, is the

fact that the Doppler broadening is observed by atoms moving along the direction of the incident

laser beam. Figure 5.4 shows experimental setup for Saturation Spectroscopy.

The saturation spectroscopy technique involves the use of two laser: one laser, called the bleach

or saturation laser, is a highly intense laser while the second laser, called the probe laser, is much

less in intensity (about 10 times than the saturation laser). In gases atoms exhibit an entire range

of velocities given by Maxwell’s velocity distribution. The basic idea of saturation spectroscopy

is to excite a particular velocity group in the Maxwell Distribution by the saturation laser. The

Saturation laser is intense enough to severely deplete the ground state of the atoms of the

selected velocity group. When the Probe laser is made to pass through the sample it detects the

hole burned in the velocity distribution and registers an increased intensity. As the frequency is

scanned by the laser, the probe beam shows an increased intensity, notifying that a resonance

frequency and a transition.


Figure 5.4: Experimental setup for Saturation Spectroscopy

5.3.2 Doppler Limited Spectroscopy

5.3.2.1 Two Photon Absorption Spectroscopy In two photon spectroscopy an atom is made to absorb a second photon while in excitation

because of the first photon. The first photon takes the atom from the ground state (energy ��) to

intermediate level (energy ��) while the second photon takes it from the intermediate level to the

final state (energy ��). If the frequency of the two photons is �� and �� then the total energy

difference can be given as:

∆� � %�� ' # %�� ' � �� # ��

If the sources of both photons are carefully aligned then the Doppler broadening from one

photon cancels the broadening of the second photon (see figure 5.5). Thus the line broadening is

theoretically reduced to the natural broadening of the transition. In addition to provide method

for studying Doppler free hyperfine interaction, two photon spectroscopy also provides the

following two benefits:

• The resultant transition by two photon absorption can lie in the far ultraviolet region

while the excitation source can be in the ultraviolet region.

• The transitions due to two photon absorption are not parity forbidden. This is because the

individual transitions are parity forbidden (even to odd or odd to even) making the entire

transition permissible to similar parity transitions (i.e. even to even or odd to odd).

Two Photon Spectroscopy seems to be the most ideal method for studying the Doppler free

hyperfine structure. However, the low absorption coefficient for simultaneous absorption of two

photons renders this method not of general use.


Figure 5.5: Schematic diagram of two photon spectroscopy. Image courtesy Department of

Chemistry, University of Kobe

5.3.2.2 Optogalvanic Spectroscopy

The Optogalvanic Spectroscopy technique is based on the change in the current produced in a

discharge tube when laser tuned to resonant frequency is passed through it. In Optogalvanic

Spectroscopy a high intensity laser is incident on a discharge of gas. If the laser is tuned to one of

the resonant frequencies it excites the atoms to higher energy levels. During the life time of the

excitation of the atoms the chances that the atom will be ionized increases. The incident laser is

scanned through the frequency range. Every time the laser is tuned to one of the resonant

frequencies an increase in the current is recorded. This marks the point where a transition has

occurred. Figure 5.6 is a standard set-up for studying the optogalvanic spectroscopy[4][5].


Figure 5.6: Schematic diagram of experimental setup of Optogalvanic spectroscopy.


Chapter 6: Programming Platform

6.1 Software Framework

A software framework is an abstraction which provides general codes written to be useable by a

wide variety of programs. These codes can also be over-ridden selectively for a particular use in

a program. A software framework in many respects is similar to a software library from which

various predefined functions can called, used and manipulated for different purposes.

6.2 Microsoft .Net Framework

Microsoft .Net framework is a powerful programming platform developed by Microsoft

Corporation. It is a software framework that provides the programmer with a large variety of pre-

defined controls. The .Net Framework is provided free of cost by Microsoft and can be easily

installed on Windows operating system. It includes libraries for both console based applications

as well as Windows based applications. The .Net framework provides tools in a number of areas,

including user interface, data access, database connectivity, cryptography, web application

development, numeric algorithms, and network communications etc.

There are two main features of the .Net Framework: The Base Class Library (BCL) and the

Common Language Runtime (CLR).

6.2.1 Base Class Library (BCL)

The real beauty of the .Net Framework lies in its Base Class Library structure. The BCL provides

a systematic and elegant way of calling pre-made library functions and managing custom made

functions. A BCL supports the following Hierarchy:

• Namespace

o Sub-namespace

� Class

• Sub-Class

o Function

This hierarchy cannot be broken i.e. every function must belong to a sub-class, every sub-class

must be under a class and every class must be associated to a namespace. A Function may


directly be associated with a class. Some commonly used BCL contained in the .Net Framework

2.0 package are:

Namespace Name Description

System

This namespace includes the core needs for

programming. It includes base types like

String, DateTime, Boolean, and so forth,

support for environments such as the console,

math functions, and base classes for

attributes, exceptions, and arrays.

System.Collections

Defines many common containers or

collections used in programming, such as

lists, queues, stacks, hashtables, and

dictionaries. It includes support for generics.

System.ComponentModel

Provides the ability to implement the run-time

and design-time behavior of components and

controls. It contains the infrastructure "for

implementing attributes and type converters,

binding to data sources, and licensing

components"

System.Data

This namespace represents the ADO.NET

architecture, which is a set of computer

software components that can be used by

programmers to access data and data services.

System.Drawing

Provides access to GDI+ graphics

functionality, including support for 2D and

vector graphics, imaging, printing, and text

services.

System.IO

Allows you to read from and write to

different streams, such as files or other data

streams. Also provides a connection to the

file system.


System.Text

Supports various encodings, regular

expressions, and a more efficient mechanism

for manipulating strings (StringBuilder).

System.Threading

Helps facilitate multithreaded programming.

It allows the synchronizing of "thread

activities and access to data" and provides "a

pool of system-supplied threads".

System.Windows.Forms Provides tools for creating windows forms

All of the above mentioned namespace have numerous classes and sub-classes which can be used

for various purposes. So for example if I want to calculate the result of a number raised to certain

power in .Net I would give the following command:

Systems.Math.Pow(x,y)

This tells the language reader (CLR explained later) to access the System namespace, and in that

the Math class and within the Math class, call the function Pow and give it the two arguments

(values passed on to the function) x and y. This has the same meaning as: xy.

6.2.2 Common Language Runtime (CLR)

The most important concept of the .Net Framework is the existence and functionality of the .Net

Common Language Runtime (CLR), and also called .net Runtime for short. It is a framework

layer that resides above the OS and handles the execution of the all the .Net applications. The

CLR provides the environment of a virtual machine to programs based on the .Net Framework.

This means that a .Net based application need not to consider the actual capabilities of the CPU

that will execute the program. The CLR also provides other important services such as security,

memory management, and exception handling which are essential for making a stable and user

friendly application. Figure 6.1 shows the relationship between a .Net application, CLR and

Windows operating system.[7]

6.3 C # (C-Sharp)

C# is designed to be a platform-independent. It's syntax is similar to C and C++ syntax, and C#

is designed to be an object-oriented language. There are, for the most part, minor variations in

syntax between C++ and C#. C# is one of the many languages that are compatible with .Net


Framework and can create application that can run in the CLR environment. C# is a much

simpler language than its predecessor C and C++ languages, yet it is an extremely powerful and

flexible language. C# has the advantage of being the only language designed from the ground up

for the .NET Framework and may be the principal language used in versions of .NET that are

ported to other operating systems. To keep languages such as the .NET version of Visual Basic

as similar as possible to their predecessors yet compliant with the CLR, certain features of the

.NET code library are not fully supported. By contrast, C# is able to make use of every feature

that the .NET Framework code library has to offer.

Figure 6.1: Program compilation and execution flow chart for .Net applications

6.4 Working in C# Environment

C#, as part of the .NET framework, is compiled to Microsoft Intermediate Language (MSIL).

MSIL allows C# to be platform independent and runs using just in time compiling. Therefore

programs running under .NET gain speed with repeated use. Furthermore, because the other

languages that make up the .NET platform compile to MSIL, it is possible for classes to be

inherited across languages. The MSIL is what allows C# to be platform independent.


6.4.1 MSIL (Microsoft Intermediate language)

When we compile our .Net program using any .Net compliant language (such as C#) our source

code does not get converted into the executable binary code (the language of the machine) but to

an intermediate language called Microsoft Intermediate Language (MSIL) which is interpreted

by CLR. MSIL is operating system and hardware independent code. Upon program execution

this MSIL is converted into binary executable.

6.4.2 JIT Compilers (JITs)

When our intermediate language (IL) compile code needs to executed, the CLR invokes the JIT

compiler, which compile the IL code to native executable code that is designed for the specific

machine and OS (operating system). JIT compilers in many ways are different from traditional

compilers as they compile the intermediate language to native code only when desired; e.g. when

a function a function is called, the IL of the function’s body is converted to native code just in

time (hence the name of the compiler). So the part of code that is not used by that particular run

is never converted to native code. If some IL code is converted to native code, then the next time

its needed, the CLR ruses the same (already compile) copy without re-coupling. So if a program

runs for some time (assuming that all or most of the functions get called), then it won’t have any

just-in-time performance penalty.

As JIT compilers are aware of the of the specific processor and operating system at runtime, they

can optimize the code extremely efficiently resulting in very robust application. Also, since JIT

compiler knows the exact current state of executable code, they can also optimize the code by in-

lining small functions calls (like replacing body of small function when it’s called in a loop,

saving the function call time).

6.5 Visual Studio

Visual Studio basically is a language editor developed by Microsoft Corporation to cater to a

vast variety of programming languages. It is an Integrated Development Environment (IDE)

which, apart from C#, is capable of managing C++, J#, ASP.NET, Java etc. It can be used to

develop console and Graphical user interface applications along with Windows Forms

applications, web sites, web applications, and web services in both native code together with

managed code for all platforms supported by Microsoft Windows, Windows Mobile, Windows

CE, .NET Framework, .NET Compact Framework and Microsoft Silverlight.

Visual Studio includes a code editor supporting IntelliSense as well as code refactoring. The

integrated debugger works both as a source-level debugger and a machine-level debugger. Other


built-in tools include a forms designer for building GUI applications, web designer, class

designer. Visual Studio supports languages by means of language services, which allow any

programming language to be supported (to varying degrees) by the code editor and debugger,

provided a language-specific service has been authored. Built-in languages include C/C++ (via

Visual C++), VB.NET (via Visual Basic .NET), and C# (via Visual C#). Support for other

languages such as Chrome, F#, Python, and Ruby among others has been made available via

language services which are to be installed separately. A screen shot of Visual Studio is given in

figure 6.2.

Figure 6.2: A screen shot of Microsoft Visual Studio Express

Some of the features of Visual Studio are given below:


6.5.1 Visual Studio Code Editor and InteliSense:

Visual Studio, like any other IDE, includes a code editor that supports syntax highlighting and

code completion using IntelliSense for not only variables, functions and methods but also

language constructs like loops and queries. IntelliSense is supported for the included languages,

as well as for XML and for Cascading Style Sheets and JavaScript when developing web sites

and web applications. Autocomplete suggestions are popped up in a modeless list box, overlaid

on top of the code editor. The code editor is used for all supported languages. The Visual Studio

code editor also supports setting bookmarks in code for quick navigation. Other navigational aids

include collapsing code blocks and incremental search, in addition to normal text search and

regex search.

Visual Studio features background compilation (also called incremental compilation). As code is

being written, Visual Studio compiles it in the background with in order to provide feedback

about syntax and compilation errors, which are flagged with a red wavy underline. Warnings are

marked with a green underline. Background compilation does not generate executable code,

since it requires a different compiler than the one used to generate executable code

6.5.2 Designer

Visual Studio includes a host of visual designers to aid in the development of applications. These

tools include:

WinForms Designer

The WinForms designer is used to build GUI applications using WinForms. It includes a palette

of UI widgets and controls (including buttons, progress bars, labels, layout containers and other

controls) that can be dragged and dropped on a form surface. Layout can be controlled by

housing the controls inside other containers or locking them to the side of the form. Controls that

display data (like textbox, list box, grid view, etc.) can be data bound to data sources like

databases or queries. The UI is linked with code using an event-driven programming model.

Class designer

The Class Designer is used to author and edit the classes (including its members and their access)

using UML modelling. The Class Designer can generate C# and VB.NET code outlines for the

classes and methods.

Data designer

The data designer can be used to graphically edit database schemas, including typed tables,

primary and foreign keys and constraints. It can also be used to design queries from the graphical

view.


6.5.3 Other Accessibility Tools

Open Tabs Browser The open tabs browser is used to list all open tabs and switch between them

Properties Editor

The Properties Editor tool is used to edit properties in a GUI pane inside Visual Studio. It lists all

available properties (both read-only and those which can be set) for all objects including classes,

forms, web pages and other items.

Object Browser

The Object Browser is a namespace and class library browser for Microsoft .NET. It can be

used to browse the namespaces (which are arranged hierarchically) in managed assemblies.

Solution Explorer

In Visual Studio parlance, a solution is a set of code files and other resources that are used to

build an application. The files in a solution are arranged hierarchically, which might or might not

reflect the organization in the file system. The Solution Explorer is used to manage and browse

the files in a solution.

While the full version of Visual Studio has many features that make programming in C# very

simple, its smaller version, called Visual Studio Express, is limited in its abilities. Nevertheless it

is extremely powerful and proved to be more than enough for our work.


Chapter 7: Modelling the Hyperfine Structure

7.1 Overview of the Simulation

In the study of hyperfine structures, the number of spectral lines involved in the excitation of

atoms and ions becomes very huge therefore it becomes very difficult to manage them manually.

Furthermore, the calculation involving the prediction of wavenumber and intensity of the

hyperfine components are extremely length and complicated. Doing these calculations manually

is a very tedious task and can include a large human error factor. The basic purpose of this

simulation is to help the experimenter manage these levels with least difficulty

We have designed this simulation on very generic assumptions so that it can be applied to any

atom or ion. All the data that is used in this simulation, for any element, comes from the

following three files:

• A DAT file that contains information of the atomic levels of the element.

• A DAT file that contains information of the first ionic levels of the element.

• A wavelength file (DAT file) which contains all known lines whether classified or

unclassified. The wavelength table has more than 12000 entries.

All the files above mentioned are in CSV (Comma Separated Variable) format i.e. fields within

the files are separated by a comma (,). The usefulness of this simulation in the study of hyperfine

structures becomes obvious when one considers that the wavelength file contains over 17000

lines. Although currently we only work with Praseodymium and Tantalum, we hope to

collaborate with other research groups in gathering data for other elements and include it in our

work.

7.2 The Simulation

When the program is executed it asks the user to select the element on which he/she wishes work

Figure 7.1.


Figure 7.1: Selecting element for analysis

7.2.1 The Main Window-The Central Command

The drop down menu lists all the elements available for analysis i.e. elements whose wavelength

and levels files are available. After selecting the element the main window of the simulation is

loaded (figure 7.2). This window is the central command of the simulation to which all the other

forms and functions are linked.

Figure 7.2: The main windows form of the simulation


The main window is fundamentally divided into two sections. The right hand side section shows

the details of the suggested levels while the left hand side shows the selected classification and

the relevant details. The description of the various components of the components (numbered

above) is as follows:

1. Menu Bar: The menu bar consists of a number of menus that provides various data

analysis tools useful for hyperfine study.

a. Program Menu: The program menu allows you to save the changes that you

have made to the wavelength file. It also provides you the option of exiting the

simulation.

b. Mark Line Menu: The Mark Line menu provides two options:

i. Mark Line: The Mark Line option allows you the bookmark a specific

classification in the wavelength list.

ii. Goto Marked Line: Goto Marked Line takes you to the bookmarked line

from any classification.

c. The Scaling Menu: The scale menu lets you set the scale of the graphic panels14

and 13.

d. The Settings Menu: Allows you to set various parameters for your analysis. The

parameters include: the Doppler width, the nuclear magnetic moment, difference

in the wavenumber that can be tolerated for transition etc.

e. The Search Menu: The menu allows you to set whether you want to work with

ionic and/or atomic levels. If you choose atomic then only the atomic levels will

be searched and same for ionic. You can also select if the transition

principle ∆z � 0,E1 should be followed while searching for suggested

transitions.

f. The Transition Menu: It allows you to see the transition list for the selected

classification.

g. The Classifications Menu: Allows you to select whether the classification

routine should be followed or not.

h. The Show Levels Menu: Shows a list of atomic or ionic levels for the selected

element. From the show level window you can filter the levels according to your

convenience and can also copy or save for other purposes (figure 7.3).

i. The Seek New Levels Menu: Provides various tools for analysing new levels.

j. Reload Levels: Reloads the levels is used for reloading the atomic and ionic

levels.

k. The Delete Menu: Can either delete a particular wavelength classification or a

particular level from the list.

l. FT Spectrum: Displays the Fourier Transform spectrum for the selected

wavelength classification.


2. Wn-Wl Converter: This is a real-time calculator that can convert wavenumber to

wavelength both vacuum and air. It can also calculate the difference of two wavenumber

and wavelengths. Figure 7.4 shows a screen shot of the Wn-Wl Converter.

3. Wavelength Classification Section: This section displays the details of the selected

classification from the classification file including

a. Information of the wavenumber, angular quantum number (J), the parity,

hyperfine constants (A and B) of both upper and lower levels.

b. The wavenumber value of the given wavelength (converted by the Peck-Reed

formula) and the difference in the wavenumber of the levels and also the

difference between the two.

It also shows the wavelength of the previous and the next classification in the wavelength

file (figure 7.5).

Figure 7.3: Screen shot of the show level window showing the atomic levels of

Praseodymium (Pr).


Figure 7.4: Screenshot of the Wn-Wl Converter showing the converted value and the

difference.

Figure 7.5: Close-up of component 3 of the Main form showing the details of the

selected wavelength classification as well as next and previous classification. Also

showing the Previous, Next and Goto Lamda buttons


4. These three buttons labeled Next, Previous and Goto Lambda allow you to move

through various wavelengths in the wavelength files.

a. Next: Moves to the next wavelength in the classification table and updates the

suggestions (component 8) accordingly.

b. Previous: Moves to the previous wavelength in the classification table and

updates the suggestions (component 8) accordingly.

c. Goto Lamda: Allows you to jump directly to a particular wavelength in the

classification file and updates the suggestions (component 8) accordingly. If the

given value of wavelength is not available in the wavelength file, it will jump to

the nearest classification available. However component 8 is strictly updated

according to the provided value of wavelength.

5. Details: Shows the details of the hyperfine structure of the selected classification.

Basically it is much more detailed version of component 14. If all the hyperfine constants

of the transition is not known then a message box will appear stating so (figure 7.6).

6. Change Entry: Allows the user to change the details of the currently selected

classification in the wavelength file.

7. Insert Line: Allows the user to insert a new line into the classification file at any later

time.

8. Transition Suggestion: It is a list box which shows the results of the level search in the

atomic and ionic levels. These are the transitions suggested for a given classification

(figure 7.7). Each item shows summarised information of the suggested transition. This

list box is updated every time the user moves between the classifications. It is also

updated when the wavenumber difference value (in the Settings menu) and the selection

in the Search menu is changed. The list box allows you to move through various

suggestions by either clicking on one of the suggested transitions or by clicking the

Previous (<<) and the Next (>>) button located below it (component 10). Moving

through the suggestion updates the component 15 below it and draws the combined

Doppler profile of the selected suggestion.


Figure 7.6: Screen shot of the Details windows

9. Shows the number of suggestions in the Transition Suggestion box, both for atomic and

ionic levels.

10. Allows the user to move between the suggested transitions. User can also move between

these transitions by clicking on the specific suggestion.

11. Simulate: Allows you to study the suggested transition in detail (explained in detail

later). Only one instance of the simulate window can be opened at a time.

12. Detail: Shows the details of the hyperfine structure of the suggested transition that is

selected. Basically it is much more detailed version of component 15. Its appearance

similar to the figure 7.6.

13. Select for Entry: Allows you to insert the selected suggestion in the wavelength list.


Figure 7.7: Suggestion list box showing the suggested transitions between the levels

in the atomic and ionic level files. It also shows summarised information of the upper

and lower levels.

14. It is a preview of the hyperfine structure of the selected classification. If the values of the

selected classification are not known then this area will notify it.

15. Displays a preview of the hyperfine structure of the selected suggestion in the transition

suggestion list box. This graph is updated when the selection in the Transition Suggestion

list box is changed, when the nuclear spin quantum number is changed or when the

Doppler width is changed. It also shows the parameters of the transition along the bottom

of the area (figure 7.7)

7.3 The Simulate Window

Simulate is the most important feature of the simulation. It allows you to analyse the hyperfine

structure of the selected transition in the Transition Suggestion list box. It opens a new window

(figure 7.8) where the snap shot of the selected transition (component 15) is drawn on a large

scale with better resolution. In this window that individual components of the hyperfine structure

are also clearly visible. It also provides sliders and text boxes to change the parameters of the

transition (nuclear spin, Doppler width, angular momentum and the hyperfine constants) and

observe the changes in the hyperfine structure accordingly. Details of the various tools in the

Simulate window are described below.


Figure 7.7: Preview of the suggested transition between J 3.5 and 4.5 (as shown at the

bottom left side of the preview).

.

igure 7.8: Screenshot of the Simulate window.


1. This is the graph area where the hyperfine structure of the selected transition is plotted.

The graph is updated in real time i.e. the hyperfine structure in the graph area will

respond instantaneously to changes in the values of hyperfine constant on the right hand

side.

2. Value of the hyperfine constant A of the upper transition level. The value in this field can

be entered directly. User can also use the slider below it change the value in steps.

Clicking on the slider arrows at the sides change the value by 5 units.

3. Value of the hyperfine constant B of the upper transition level. The value in this field can



4. Value of the hyperfine constant A of the lower transition level. The value in this field can



5. Value of the hyperfine constant B of the lower transition level. The value in this field can



6. The angular momentum quantum number of the upper transition level. This value is to be

entered directly.

7. The angular momentum quantum number of the lower transition level. This value is to be

entered directly.

8. Value of the nuclear quantum number.

9. Enter the value of Doppler width of the hyperfine structure.

10. Clicking on this button updates the graph area, component 1, after changing the transition

parameters.

11. Displays a list of all possible transitions of the hyperfine fine structure in a new form.

12. Values of the Total Angular Momentum for the upper and lower levels of transition.

7.4 Drawing The Hyperfine Structure

Drawing the hyperfine structure of a transition is a two stage process.

• In this first step the details of the transition (the upper and lower angular momentum

quantum number, nuclear spin quantum number and the hyperfine constants A and B)

are used to calculate the frequency and the intensity of the spectral lines. For this we use

equation number 5.5 and 5.4. If the graph field is a part of the Simulate or the Details

window then spectral lines of the hyperfine structure are drawn on the graph field

otherwise for preview these lines are not drawn on the field. For convenience of the user

the value of the frequency is also written on the top of the respective spectral line.


• In the second step, the program scans across the entire frequency axis (the X-axis) from

least value to the maximum value in steps of 10 units. At each interval, value of intensity

is calculated by evaluating the sum of the intensity at that point due to all line centres

using equation 5.6.


Chapter 8: Analyzing the Hyperfine Structures

8.1 Classification of Hyperfine Structure

8.1.1 Finding a New Level by Combination of Wavenumbers

If a particular line in the wavelength file is unclassified then the classification program doesn’t show any

suggestion on selecting that wavelength in the wavelength file. This shows that there is an unknown level

is involved in the transition. This unknown level can be found by using the wave number of the

unclassified line. The basic principle is the same as Ritz combination principle[8].

• We first assume certain parity for the new and unknown level either odd or even.

• Then the list of level with opposite parity to the assumed parity is analysed because transition are

forbidden to levels with opposite parities. If the unknown level is assumed to be odd (or even),

then in the list of even (or odd), the level with the lowest energy is taken and the wave number of

the unclassified line is added. This gives the energy of the hypothetical new level in wavenumber.

• Then the wavenumber of all other unclassified lines from the line is added to other even level

energies. If the energy obtained in this way coincides with the energy of the hypothetical level

within a given uncertainty, then the even levels with wavelength are displayed.

• This procedure is then repeated for the next higher even level. In this way a number of

hypothetical levels are calculated. The J-value of these hypothetical levels can be easily found by

the selection rules.

• The next step in the analysis is to simulate the investigated unclassified line fixing the hyperfine

constants of the levels involved (or assumed to be involved) of the lower even levels.

• If a simulation is impossible then the hypothetical level is rejected.

• If the simulation is possible, then the hyperfine constants of the hypothetical levels are

eliminated.

• To make sure that the hypothetical level really does exists, other unclassified lines which appear

as decay from the hypothetical level, are taken and their simulated hyperfine structures are

compared with the corresponding hyperfine structures in FT spectra (figure 8.1).

• If one or more hyperfine structures are identical then the level existence is confirmed. If no odd

levels fulfil this condition this means that our initial assumption of the parity is flawed. Now we

must search by assuming that the parity is even. The procedure for finding new level with even

parity is same.


8.1.2 Determination of New Level by Fluorescence Lines

Figure 8.1:


8.1.2 Determination of New Level by Fluorescence Lines

Figure 8.1: Screenshot of a FT Spectrum

62 of Atoms and Ions


New energy level can also be found using information obtained from laser spectroscopy, if the

unclassified line is excited by laser light. The fluorescence lines are fed in the search procedure through

the window which accepts the fluorescence lines observed during the experiment. The search procedure is

again based on Ritz combination principle. It is a bit different from above method in a sense that it only

uses the current line and lines from the data based in a certain wavelength range around the observed

fluorescence lines[8].

8.1.3 Determination of New Level by Analysis of the Hyperfine Structure

Sometimes it is possible to have as idea about the transition by looking at the hyperfine structure of an

unclassified line. The hyperfine structure of the unknown upper and known lower level is then found by

simulation. Then a lower level is searched in the level list with the hyperfine constants close to the pair

obtained by simulation. If the simulation is perfect the lower level can be obtained from the list. The

addition of centre of gravity wavenumber of the line and wavenumber of the lower level gives the energy

of the upper unknown level. After introducing this level in the level list, the transition list from this level

generated by the classification program is investigated to see if some other unclassified lines are also

decay channels from this level. If this occurs, its hyperfine structure verifies whether this unclassified line

is a real decay from this level. If this is so this confirms the new level[8].

Unclassified Line leading to discovery of new level

8.2 Energy Corrections in Hyperfine Levels


8.2.1 The Fitter Program

For accurate evaluation of the hyperfine constants and centre of gravity wavenumber of the line, the

hyperfine structures are recorded as intensity distribution function. To evaluate the hyperfine constants

and centre of gravity of a laser spectroscopic recorded line or of a line extracted from FT spectra a

program is specially developed designed, it’s called The Fitter.

Fitter was designed at Hamburg, Germany. In order to extract the information in the hyperfine structure, a

mathematical model of intensity distribution is computed which best fits the intensity distribution as a

function of time (with equidistant time intervals). Although the time interval is constant, due to scanning

behaviour of the laser (broadening effects) the spectrum is not equidistant in frequency. By a

linearization process the Fitter converts the measured intensity measured as a function of time (�%Ð') into

function of frequency (�%�') in which frequency is equally spaced.

Figure 8.2: A screen shot of the main window of the Fitter program

8.2.2 Theoretical Background of the Fitter


The intensity of the hyperfine structure is introduced theoretically by the intensity formula given in the

article in chapter 5. The physical conditions are introduced as boundary conditions. The model is based on

the variation of these parameters generally according to the method of least square methods.

�VV � mµ��%�' � ��%��, �Ñ'¶�

The function ESS is a squared error sum, ��%�' is the measured intensity at a given frequency point k and ��%��, �Ñ' is the calculated intensity at the corresponding position and the vector a is a set of parameters.

The above equation can be expanded using Taylor series to obtain a number of non-linear and

inhomogeneous equations are obtained. The number of such equations is exactly equal to the number of

parameters in vector a. The solution of these equations is obtained in such a way that the deviation from

measured intensity becomes least and the solution also gives a new set of parameters of vector a. This

new vector is used as starting value for next iteration and this procedure is repeated until the abort criteria

are reached.[8]

8.2.3 Energy Values Correction in Fitter

Finding the corrected value of energy of the hyperfine structures, our simulation is used in conjunction

with the Fitter Program. The procedure for it is as follows[9]:

• From the Simulation program we first look for the transition from zero level, by selecting the

option ‘select lines with energy levels’ located in the Select menu. Then by using next or previous

button one can move to the lines which involve the level of interest.

• The FT spectrum and classification program must match. If the hyperfine structure has good

signal-to-noise ratio then one can use it for fitting.

• The hyperfine structure of interest is extracted from the FT spectrum by clicking on the rage to be

selected.

• This information will be saved with the name of the line.

• Parameterdatie program linearizes the .DAT file of the line. It makes two files with the extension

.lin and .par. The message is displayed in the window.

• Once the file is satisfied it is opened in the fitter program and it is saved with the same name but

the extension .ein in the EIN folder.

• Using the value of the transition parameters z�, zÊ, H�, HÊ, Ã�, ÃÊ and the graph is fitted over the

theoretically obtained graphs.

The spectrum is linearized because the marker etalon notes the intensity as a function of time, which is

equally space, and must be converted to equally spaced in frequency.


Figure 8.3: A screen shot of one of the fitter graph in the Fitter program

The difference of the two levels must be zero if that is so then the energy found out is correct. If otherwise

then it has two possibilities either the wavenumber is wrong or the energy of the upper and lower level is

incorrect. By using this program we have done the correction of these energy levels and found out the

accurate value. This program also converts the centre of gravity wavelength to centre of gravity

wavenumber.


References

Papers and Articles

[1] Department of Physics,City College,Kolkata, Curr. Sci., 2005, 89, 1978

[2] Randal C Telfer Johns, Hopkins University May 6 1996

[3] Russel-Saunders , Astrophysics J., 61, 38 (1925)

[4] Penning, F. M., Physica 8, 137, (1928)

[5] Kenty, C., Phys. Rev. 80, 95 (1950)

[6] Sobel’man, I. I., “An introduction to the theory of atomic spectra” Pergamon, Oxford (1972)

[7] Common Language Infrastructure (CLI) 4th

edition (June 2006)

[8] Master’s Thesis Classification of Physics Structure, Ms Rubecca Sikander, Department

of Physics, University of Karachi, 2007

[9] Project report of Energy Corrections to Hyperfine levels of Tantalum II, Department of

Physics, University of Karachi

Books

1 Laser Fundamentals, Second Edition

William T. Silfvast

2 Introduction to Quantum Mechanics, Second Edition

David J. Grifiths

3 Physics of Atoms and Molecules B.H. Bransden

C.J. Joachain

4 Modern Physics

Von Newman,

5 The Atomic Theory EU Condon

GH Shortely’

6 Atoms and Molecules

Mitchel Weissbluth

7 Laser Spectroscopy: Basic Concepts and

Instrumentation

W. Demtröder

my masters thesis

Documents