mass spectrometric and quantum mecanical studies of charged particles in vapours over rubidium...

7/31/2019 Mass Spectrometric and Quantum Mecanical Studies of Charged Particles in Vapours Over Rubidium Iodide

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KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGYINSTITUT DES SCIENCES ET TECHNOLOGIE KIGALIAvenue de larme, B.P. 3900 Kigali, Rwanda Website: www.kist.ac.rwFax: +250 571925/571924 Tel: +250 576996/574698

TILE PAGE

FINAL YEAR PROJECT REPORT

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(((BBBSSSPPPHHHYYYSSS)))

PROJECT NO: PHY/01/09

NAMES: NKURUNZIZA EMMANUEL GS 20060793

AND

NDAYAMBAJE JACKSON GS 20060423

NAME OF SUPERVISOR: Prof. Alexander Pogrebnoi

and

Dr. Tatiana Pogrebnaya

ACADEMIC YEAR: 2009

T O W A R D S A B R I G H T E R F U T U R E

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http://www.kist.ac.rw/http://www.kist.ac.rw/http://www.kist.ac.rw/


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KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY

INSTITUT DE SCIENCE ET TECHNOLOGY DE KIGALI

FINAL YEAR PROJECT REPORT

FACULTY OF APPLIED SCIENCE

DEPARTMENT OF APPLIED PHYSICS

PROGRAM OF BACHELORS DEEGRE

CERTIFICATE

This is to certify that the project work entitled MASS SPECTROMETRIC AND

QUANTUM MECHANICAL STUDIES OF CHARGED PARTICLES IN VAPOURS

OVER RUBIDIUM IODIDE is a record of original work done by: NKURUNZIZA

Emmanuel(GS 20060793) together with NDAYAMBAJE Jackson (GS 20060423), underour supervision and guidance for partial fulfillment of the requirement for award of the

degree of Bachelors of Science in APPLIED MATHEMATICS of KIGAL INSTITUTE OF

SCIENCE AND TECHNOLOGY during the academic year of2009.

Signature:..

SUPERVISOR

Prof. Alexander Pogrebnoi

Signature:..

CO- SUPERVISOR

Dr.Tatiana Pogrebnaya

Date:../../2009 Date:../../2009

Submitted for the project examination at KIST on:../../.


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DECLARATION

We, NKURUNZIZA Emmanuel and NDAYAMBAJE Jackson, hereby declare that the

work entitled MASS SPECTROMETRIC AND QUANTUM MECHANICAL

STUDIES OF CHARGED PARTICLES IN VAPOURS OVER RUBIDIUM IODIDE

is our own work (our own contribution). The same work has never been submitted or

presented in any other University or Institute for academic purposes.

Signature:.. Signature.

NKURUNZIZA Emmanuel SUPERVISOR Prof Alexander Pogrebnoi

Date:../../2009 Date:../../2009

Signature:.. Signature.

NDAYAMBAJE Jackson CO-SUPERVISOR DrTatiana Pogrebnaya

Date:../../2009 Date:../../2009

Physics Department

Kigali Institute of Science and Technology (KIST)

P.O.Box 3900, Kigali


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DEDICATION

We dedicate our project to all people who provided any kind of contribution in order to give

-our spiritual life

-our physical life

-our family life

-our social life

-our academic life

-our romantic life

the meaning it has today.


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LIST OF CONTENTS

TILE PAGE ............................................................................................................................................. i

CERTIFICATE ....................................................................................................................................... ii

DECLARATION ................................................................................................................................... iii

DEDICATION ....................................................................................................................................... iv

LIST OF CONTENTS ............................................................................................................................ v

ABSRACT ............................................................................................................................................ vii

LIST OF TABLES, DIAGRAMS AND ILLUSTRATIONS .............................................................. viii

NOMACLATURE LIST ....................................................................................................................... ix

ACKNOWLEDGEMENTS .................................................................................................................... x

CHAPTER 1:GENERAL INTRODUCTION ........................................................................................ 1

1.1 Background ................................................................................................................................... 1

1.2 Mass spectrometers ....................................................................................................................... 2

1.2.1 Generalities on mass spectrometers .......................................................................................2

1.3. Quantum mechanics and statistical thermodynamics .................................................................. 5

CHAPTER 2: LITERATURE REVIEW ................................................................................................ 7

2.1 Advantages of ions existing in vapours over RbI ......................................................................... 7

2.2 Procedure on high temperature mass spectrometer ....................................................................... 8

2.2.1 Method of the electron impact (EI)........................................................................................9

(a) Determination of the molecular composition of the vapor ............................................................ 9

2.2.2 Method of the thermal ionization (TI) .................................................................................10

(a) The partial pressures of the ions .................................................................................................. 10

2.2.3 Calculation of enthalpies of the reactions ............................................................................11

2.3 Procedures quantum mechanical calculations ............................................................................. 11

2.3.1 The Hartree-Fock (HF) Self-consistent Field Method .........................................................12

2.3.2 Mller-Plesset (MPn) computation method .........................................................................16

2.3.3. Density Function Theory (DFT) computation method .......................................................19

CHAPTER 3: METHODS AND RESULTS ........................................................................................ 22


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3.1. Background ................................................................................................................................ 22

3.2. Experimental methods and results ............................................................................................. 22

3.2.1. Equipment ...........................................................................................................................22

3.2.2. Procedure of the studies ......................................................................................................25

3.2.3. Experimental results............................................................................................................25

3.3. Theoretical methods and results ................................................................................................. 27

3.3.1. Computational procedures ..................................................................................................27

CHAPTER 4: CALCULATION OF THERMODYNAMIC PROPERTIES ....................................... 33

4.1. Introduction ................................................................................................................................ 33

4 .2 .Calculation of thermodynamic functions of the ions ................................................................ 33

4.3. Calculation of dissociation enthalpies of the ions ...................................................................... 36

4.3. Results and discussion ............................................................................................................... 39

4.4 Conclusion and recommendation ................................................................................................ 40

REFERENCES ..................................................................................................................................... 41


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LIST OF TABLES, DIAGRAMS AND ILLUSTRATIONS

Page

Figure1.1: Main steps of measuring with a mass spectrometers ............................................................. 2

Figure 2.1: Flowchart of the HF SCF procedure................................................................................... 16

Figure 3.1: Schematic diagram of mass spectrometer ........................................................................ 23

Figure 3.2: Diagram of the EI/TI ion source ......................................................................................... 24

Table 3.1: Relative intensities of ion currents (T= 780 K) ................................................................... 26

Table 3.2: Relative intensities of ion currents (I) in the mass-spectrum of thermal emission forRbAg4I5 (copper cell, T= 790 K) ........................................................................................ 27

Table 3.3: Characteristic of Rb2I+ and RbI2

- ions and the energies of dissociations ............................. 29

Figure 3.3: Schematic arrangement of Rb2I+, Rb3I2

+, Rb3I2+, RbI2

- and Rb2I3- ions ............................. 29

Table 3.4: Characteristics of the Rb2I3- and Rb3I2

+ ions ........................................................................ 31

Table 3.5: Characteristics of the Rb2I3- pyramidal structure and Rb3I2

+ pyramidal structure ions........ 32

Table 4.1: Input file of calculation of thermodynamic properties of Rb3I2+ ......................................... 34

Figure 4.1: Flow-chart of calculating thermodynamic properties of ions ............................................. 35

Table 4.2: Thermodynamics functions for Rb2I+ linear at experimental temperature ......................... 35

Table 4.3: Thermodynamic functions for RbI2- linear at experimental temperature ............................. 36

Table 4.4: Thermodynamic functions for Rb3I2+ linear at experimental temperature ........................... 36

Table 4.5: Thermodynamic functions for of Rb2I3- linear at experimental temperature ..................... 36

Table 4.6: Enthalpy of dissociation for Rb2I+ ....................................................................................... 38

Table 4.7: Enthalpy of dissociation for RbI2- ........................................................................................ 38

Table 4.8: Enthalpy of dissociation for Rb3I2+ ...................................................................................... 39

Table 4.9: Enthalpy of dissociation for Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

- in gaseous phase .................... 39


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NOMACLATURE LIST

GAMESS: General atomic molecular electronic for structure systems

APCI: Atmospheric Pressure Chemical IonizationCI: Chemical Ionization

EI: Electron Impact

ESI: Electrospray Ionization

FAB: Fast Atom Bombardment

FD/FI: Field Desorption / Field Ionization

MALDI: Matrix Assisted Laser Desorption Ionization

TSP: Thermospray IonizationICP: Inductively coupled plasma

GD: Glow discharge

DIOS: Desorption ionization on silicon

DART: Direct Analysis in Real Time

SIMS: Secondary ion mass spectrometry

TIMS: Thermal ionization mass spectrometer

IAI: Ion Attachment IonizationTOF: Time of flight

MS: Mass spectrometer

HTMS: High temperature mass spectrometer

TI: Thermal ionization

EI: Electron impact

HFSCF: Hartree-Fock self consistent method

MO: Molecular orbit

MP: Mller-Plesset

DFT: Density functional method

SEM: Secondary electron multiplier

MIS: Magnetic ionization sensor

au : Atomique unit

: Angstrom

D: Debye


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ACKNOWLEDGEMENTS

This work, as exciting as it can appear, is never a result of our efforts only. The realization of

this report is a product of efforts from many people to whom we express our deepacknowledgements.

We particularly thank very much Prof. Alexander Pogrebnoi and Dr. Tatiana Pogrebnaya,educators in Kigali Institute of Science and Technology in the Department of AppliedPhysics who, despite his many tasks and responsibilities, they proposed this topic and kindlyaccepted to supervise this work. Their advice and their way of making themselves availablewitnessed their interest in the achievement of this work. We express our deepacknowledgements and gratitude to them and to their efforts.

Our deep acknowledgements are also addressed to all the Lectures of the Faculty of AppliedSciences, but particularly to those of the Department of Applied Physics who fullycontributed to our University trainings, among others:

Dr. OTIENO ONYANGO Frederick, Prof. Alexander Pogrebnoi and Dr. Tatiana

Pogrebnaya, Dr. KASHINJE Stanslas, Dr. DIRK Witthaut and Dr. KLAIN Jens formKaiserslautern (GERMANY), Dr. SAFARI Bonfils, Dr. NDUWAYO Lonard, Mr.MUSHINZIMANA Xavier, Mr. RAVI Kumhar, Mr. HABYARIMANA Fabien andothers.

Dr. KARANGWA Desire, Prof. SINHA Amrissathu, Dr. BASANZE Lonard. fromDepartment of Applied Mathematicsand others. We cannot forget Dr. Eng. ZIMULINDAFrancois Head of Department of Electronics & Electrical Engineering at KIST whosupported us. We thank you very much for your numerous encouragements. This is the resultof your conjugate efforts.

To all dear friends, dear BUTUNGE Pascal uncle ofNDAYAMBAJE Jackson, dearNIYIGENA Francis brother ofNKURUNZIZA Emmanuel and student in the departmentof Applied Mathematics in KIST, our classmates, for joy and endurance we shared during ourstudies. Our acknowledgements are also addressed to you.

Finally, to you all along this work, who went on helping me in different ways, I address toyou my sincere acknowledgements.

NKURUNZIZA Emmanuel and NDAYAMBAJE Jackson


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CHAPTER 1

GENERAL INTRODUCTION

1.1BackgroundIn the very recent past it have been proved that associating the experimental and theoretical

computation is the better way of calculating the thermodynamic properties of charged

particles. This is true because this pair of approaches is complementally between them. By

the time when experimental result doesnt inform about the isomeric structure of the ions, the

theoretical approaches consider that and on the other hand we know the accuracy of the

theoretical approaches by comparing its results and experimental. In this work, this pair of

approaches constructs a solid edifice in calculation of thermodynamic properties of selected

ions which are Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

-.

It has to be emphasized that ion in vapor over RbI are selected elevated temperature, and they

generate propellant force in ions thrust where the thrust force increase with the mass of the

ion and other plasma physics applications: this construct the 1 st motivation of our research

project.

A second motivating force is that tools which help in calculating thermodynamic properties

of ions are available and can be exploited in our project work. These tools of calculation can

be classified as:

Calculation of the thermodynamic properties of the ions

The treatment of the experimental data obtained using high temperature mass

spectrometric technique to find the energies of dissociation of the ion clusters.

In this chapter we are going to discuss some details about mass spectrometers with an

emphasis on how their mass spectrum may be used in calculating the thermodynamic

functions. Where the details in the methods used for ions will be discussed in chapter 2 and 3.

In this chapter we are going to explain why the quantum mechanical which will be used in

our project are numerical and iterative methods where some details will be part of chapter 2.

A part from surveying what other have done which are relative to our work in chapter 2 andthe details of our methods in chapter 3, the chapter 4 will display result of the calculations of


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some thermodynamic characteristic the Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

- ions and make

discussion, conclusions and recommendations for further work about these results. To start

let us try to understand the mass spectrometers because they are our main tool for

experimental data we used for the further treatment.

1.2Mass spectrometersIn our work, the mass spectrometer plays a very important role, because it is the instrument

that will be use to find experimental data which will help us to calculate thermodynamic

characteristics of the Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

- . Here the mass spectrometer will provide

the relative intensives of ions, results which are purely experimental (the details of how this is

done will be found in the next chapter).To start this procedure of calculation, we have to

understand first the generalities of mass spectrometers.

1.2.1 Generalities on mass spectrometers

Early spectrometry devices that measured the mass-to-charge ratio of ions were called mass

spectrographs which consisted of

instruments that recorded a

spectrum of mass values on a

photographic plate. A mass

spectroscope is similar to a mass

spectrograph except that the beam

of ions is directed onto a

phosphorous screen. The term mass

spectroscope was replaced by mass

spectrometer when the directillumination of a phosphor screen

was replaced by indirect measurements with an oscilloscope. Mass spectrometers are divided

into three fundamental parts, namely the ionization source, the analyzer, and the detectoras

shown on the figure 1.1

Briefly, the sample has to be introduced into the ionization source of the instrument. Once

inside the ionization source, the sample molecules are ionized, because ions are easier to

manipulate than neutral molecules. These ions are extracted into the analyzer region of the

Figure1. 1:Main steps of measuring with a mass

spectrometers


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mass spectrometer where they are separated according to their mass (m) -to-charge (z) ratios

(m/z). The separated ions are detected and this signal sent to a data system where the m/z

ratios are stored together with their relative abundance for presentation in the format of a m/z

spectrum.

(a) Sample introduction

The method of sample introduction to the ionization source often depends on the ionization

method being used, as well as the type and complexity of the sample.

(b) Methods of sample ionization

Many ionization methods are available and each has its own advantages and disadvantages.The ionization method to be used should depend on the type of sample under investigation

and the mass spectrometer available.

Nowadays we count more than 20 ionization methods where some of them are: Atmospheric

Pressure Chemical Ionization (APCI); Chemical Ionization (CI); Electron Impact (EI);

Electro spray Ionization (ESI)& Nan spray ionization; Fast Atom Bombardment (FAB);

Field Desorption / Field Ionization (FD/FI); Matrix Assisted Laser Desorption Ionization

(MALDI); Thermospray Ionization (TSP); Inductively coupled plasma (ICP); Glow

discharge(GD); desorption ionization on silicon (DIOS); Direct Analysis in Real Time

(DART); secondary ion mass spectrometry (SIMS); spark ionization; thermal ionization

(TIMS); Ion Attachment Ionization (IAI)etc. A complete list of currently used methods of

ionization is too long to be listed here and it is important to inform if it is either positive or

negative ionization which is concerned. This project limit itself in one method ofthermal

ionization and some details of this method will be discussed in the following chapter.

(c) Methods of mass analysis and separation of sample ions

The main function of the mass analyzeris to separate, orto resolve, the ions formed in the

ionization source of the mass spectrometer according to their mass-to-charge (m/z)ratios.

There are many types of mass analyzers, using either static or dynamic fields, and magnetic

or electric fields, and they have different features, including the m/z range that can be

covered, the mass accuracy, and the achievable resolution, but all operate according to the

differential equations describing the classical motion of charged particles in the


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electromagnetic fields. Each analyzer type has its strengths and weaknesses and there is a

compatibility of different analyzers with different ionization methods.

The more common mass analyzers are: sector field mass analyzer; time-of-flight analyzers

(TOF); Quadrupole mass analyzers subdivided in quadrupole ion trap or cylindrical ion trapmass spectrometer, and inlinear quadrupole ion trap; Fourier transform mass analyzer; Ion

cyclotron resonanceetc. But it should be informed that there are instruments that have more

than one analyzer and so can be used for structural and sequencing studies. Two, three and

four analyzers can be all been incorporated into commercially available instruments, and the

analyzers do not necessarily have to be of the same type, in this case the instrument is a

hybrid one and it is called tandem (MS-MS) mass spectrometer. More popular tandem mass

spectrometers include those of the quadrupole, magnetic sector-quadrupole, and morerecently, the quadrupole-time-of-flightgeometries. The reader is informed that all of this

above cannot be discussed in this project but the following chapter will put a stress on the

magnetic sector mass analyzerwhich was used for the investigation of the charged particles

in vapours over RbI.

(d)Detection and recording of ions

The final element of the mass spectrometer is the detector. The detector records either thecharge induced or the current produced when an ion passes by or hits a surface. A lot of

detectors including electron multiplier, Faraday cups, ion-to-photon detectors, Microchannel

Plate Detectors and others are also used because the number of ions leaving the mass

analyzer at a particular instant is typically quite small, considerable amplification is often

necessary to get a signal.

In a scanning instrument, the mass spectrum signal produced: the m/zvalues of the ions are

plotted against their intensitiesto show the number of componentsin the sample, the

molecular massof each component, and the relative abundanceof the various components in

the sample.


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(e) Data analysis

Mass spectrometry data analysis is a complicated subject matter that is very specific to the

type of experiment producing the data.Many mass spectrometers work in either negative ion

mode orpositive ion mode. It is very important to know whether the observed ions are

negatively or positively charged. This is often important in determining the nature of neutral

particles the ions are formed from.

Different types of ion sources result in different arrays of fragments produced from the

original molecules. An electron ionization source produces many fragments and mostly odd

electron species with one charge; whereas an electrospray source usually produces

quasimolecular even electron species that may be multiply charged. Tandem massspectrometry purposely produces fragment ions post-source and can drastically change the

sort of data achieved by an experiment. By understanding the origin of a sample, certain

expectations can be assumed as to the component molecules of the sample and their

fragmentations. A sample from a synthesis/manufacturing process will likely contain

impurities chemically related to the target component.

The greatest source of trouble when non-mass spectrometrists try to conduct mass

spectrometry on their own or collaborate with a mass spectrometrist is inadequate definition

of the research goal of the experiment. Adequate definition of the experimental goal is a

prerequisite for collecting the proper data and successfully interpreting it. Among the

determinations that can be achieved with mass spectrometry are molecular mass, molecular

structure, and sample purity. Each of these questions requires a different experimental

procedure. Simply asking for a "mass spec" will most likely not answer the real question at

hand. Advices for using accordingly the mass spectrometers are provided in references [1-9].

1.3. Quantum mechanics and statistical thermodynamics

In our project mass spectrometric results as will be shown in chapter three are not enough in

calculation of thermodynamic properties of the Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

-. In fact these

experimental results will have to be combined with thermodynamic properties of ions which

are calculated on the theoretical basis using quantum mechanics and statistical

thermodynamics.


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Quantum mechanics, as developed in chapter 2 will use numerical iterative methods of

calculation in order to compute the optimized internuclear distances and optimized vibration

frequencies of ions as indicated in chapter 3. Those values will be used as in put data in

statistics-thermodynamics calculations of thermochemical of ions as indicated in chapter 4.

In chapter 4, experimental values and theoretical values will be combined together in order to

calculate the enthalpy of dissociation of Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

- ions according to

0 00 0( ) (0)

(0) [( ) ln ]r pG T H

H T R KT

(3rd law of thermodynamics)

Where 0pK is the equilibrium constant of the reaction at standard pressure and at temperature

T found experimentally from mass spectrometric constant as indicated in chapter 2.

R is gas constant.

And( ) (0)G T H

T

is the reduced Gibbs energy. A value which is obtained by

succession of quantum and statistical thermodynamic calculations.

To start this procedure of calculation, let us review how mass spectrometric constant are usedto obtain 0ln pK and how quantum mechanics is developed to initiate statistical

thermodynamic calculations.


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CHAPTER 2

LITERATURE REVIEW

2.1 Advantages of ions existing in vapours over RbI

This was emphasized that this project investigate thermodynamic characteristic of ions in hot

vapors over RbI are chosen because of their property of being heavy ions and in a addition to

that they can resist to chemical and physical condition of propellant materials because they

are ionically bonded are not naturally radioactive. This is an important property in generating

propellant force using ionic thrusts where the thrust force increases with the mass of ion.

Wikipedia (the free internet encyclopedia) says:Ion thrusters utilize beams of ions

(electrically charged atoms or molecules) to create thrust in accordance with Newtons third

law. The method of accelerating the ions varies, but all designs take advantage of the

charge/mass ratio of the ions. This ratio means relatively small potential differences can

create very high exhaust velocities.

Normally ions of xenon, argon, bismuth, hydrogen, caesium and lithium are used ingenerating thrust; but comparing with Rb3I2

+; the heaviest among the four selected ions with

other ions which are normally used; Rb3I2+ is much heavier than usually used to creating

propelling force. According to Wikipedia [11], early often used hydrogen and Lithium in

generating thrust force but recent post technology have switched to Bismuth because of its

additional advantage of being relatively heavy.

Our research project is motivated by the fact that if ions over RbI are suitable for ions thrust

and other plasma physics applications, they could triple the thrusting force generated by

Bismuth ions which are currently in use. In this simple project we will not reach the level

where we can conclude if Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

+ ions are good propellants or not but

we will use the tools that we are going to discuss in the following part of this chapter in order

to calculate the enthalpies for reactions of the ions. Doing so it will be the very first initial

step which material scientists use to check if a given material is suitable for various

thermodynamic applications where propelling Aerospace is among them.


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2.2 Procedure on high temperature mass spectrometer

As it was said in chapter 1, that the second motivating force in our project work is that we

dispose tools of calculations of thermodynamic characteristics; the first tool is the mass

spectrometer which is the only experimental tool for such calculation. One of the most

universal and informative experimental methods of investigating the thermodynamic

characteristic is the method of High-Temperature Mass Spectrometry (HTMS).

This method HTMS was intensively developed and at present because the universally

recognized method, which occupies the leading position among other physical and chemistry

methods of investigating the high-temperature systems. In the following, we are going to seehow this method of HTMS is applied to a sector instrument and in the chapter 3 we will apply

the theory developed here in order to generalize relative intensities of ions over RbI which

are the only purely experimental data which are useful in our computation.

Heating of inorganic materials is accompanied, as is known, by two processes. The first of

them is sublimation, the passage of molecules during the gas phase. This can be achieved

both in the form the simple molecules and atoms and the form of association of molecules. At

temperatures sufficient for the ionization on the heated surface, the atoms and molecules can

also pass during the gas phase in the form of ions. Thus, the second possible process is the

process of the thermal emission of ions. The processes indicated can be depicted in the form

the diagrams

[ ] ,t

AB AB (2.1)

[ ] ,t

AB A B (2.2)

[ ] ( ) ,t

nn AB AB (2.3)

[ ] ,t

AB A B (2.4)

Simple ions in turn can interact with the molecules of the condensed phase, forming the ionic

associates

[ ] ( ) ,t

nB n AB B AB (2.5)


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[ ] ( ) .t

nA n AB A AB (2.6)

mass-spectrometric method makes it possible to study both the neutral and ionic components

of vapor. The neutral components of vapor is investigated, as a rule, by the method of

electron impact (EI), and ionic components with the method of thermal ionization (TI), which

is intensively developed in recent years. The survey of the experiments by methods (HTMS)

is brought by the authors of the work [10] and we are going briefly to review these methods.

2.2.1 Method of theelectron impact (EI)

The collisions between the charged particles and the molecules of gas occur with the passage

of the charged particles through the volume of a gas (mixture of both the neutral and ionic

components). If energy of the charged particles is small, then collisions bear elastic nature, i.e.

changes in the internal energy of the interacting particles does not occur. But if kinetic energy

of the charged particle is sufficiently great, then its part can pass into the internal energy of

the molecules of gas (mixture). Such collisions are inelastic. In HTMS with a study of the

neutral components of vapor is most frequently used the electron collision, i.e. the electrons

come out as the particles, which ionize the molecules (or atoms) of gas. The processes of

ionization can be produced according to the following basic channels:

a) Direct ionization with the formation of atomic ion

A + e A+(*) + 2e , (2.7)

b) Direct ionization with the formation of the molecular ions

AB +e (AB*) + e AB+(*) + 2e , (2.8)c) The dissociative ionization

AB + e (AB*) + e A+(*) + B(*) + 2 e , (2.9)

d) The formation of the ion pair

AB + e (AB*) + e A+ + B+ e , (2.10)

(a) Determination of the molecular composition of the vapor

The mass-spectrum obtained in the experiment reflects the implicitly qualitative and

quantitative composition of vapor. The determination of the mass number of ion and the

comparison of the probability of forming the isotopic varieties with the distribution of the


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intensities of lines in the mass-spectrum measured experimentally makes it possible to

unambiguously determine the chemical formula of ion. After the establishment of the

formulae of the ions it is necessary to determine molecules, from which these ions were

formed to conduct the interpretation of mass-spectrum, i.e., to quantitatively divide the

measured intensities of ion currents into the components, obliged by their origin to different

molecules. For the interpretation of mass-spectrum for example, a reduction in the energy of

the ionizing electrons, may be used. After the interpretation of mass-spectrum the calculation

of the partial pressures of the components becomes possible.

2.2.2 Method of the thermal ionization (TI)

The method of thermal ionization (TI) is at present intensively used during the study of the

ionic components in high-temperature vapor. Their basic difference consists in the fact that in

the method TI there is no need for the initiated (impact) ionization. Ions in this case are

formed as a result of the thermal ionization of the material inside the effusion cell and it is

necessary only to extract ions out of the cell by weak electric field. Thus, the essence of TI

method consists in the measurement of the relative intensities of ions, which exist in the

effusion cell under the thermodynamic equilibrium conditions.

(a) The partial pressures of the ionsBetween the ion current Ii and partial pressure (pi) there is a connection

pi = ionIi T1/2

M1/2, (2.11)

where ion is the coefficient, which characterizes the sensitivity of the instrument in the

regime of the thermal emission.

As an example for the reaction

Rb2I+ = Rb+ + [RbI] (2.12)

the equilibrium constant may be found through the ratio of the ion current intensities

0

2 2

(Rb ) (Rb )

(Rb I ) (Rb I )p

I MK

I M

(2.13)

Here 0pK is the dimensionless equilibrium constant of the reaction (2.12).


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2.2.3 Calculation of enthalpies of the reactions

The values of the pressures of the neutral components of vapour, can be used for the

calculations of the equilibrium constants of different chemical reactions, which include the

neutral components of vapor. The equilibrium constants of reactions, obtained at differenttemperatures, make it possible to treat experimental data according to the equation of the

isobar of the chemical reaction

2

00 )(ln

RT

TH

dT

Kd r . (2.14)

When the thermodynamic functions for all reactants are available, the use of the following

relationship is possible

]ln))0()(

[()0(0

000

KRT

HTGTHr

, (2.15)

where )()0()( 0

00

TT

HTG

the reduced energy of Gibbs. The method of calculation

on the basis of the relationship (2.15) was called in the literature calculation according to the

III law of thermodynamics.

The calculations of enthalpies of the reactions can also be done using the II law of

thermodynamics as follows:

ln

ln

ln

r p

p r r

r rp

G H T S

G RT K

RT K H T S

H SK

RT R

(2.16)

WhereG is Gibbs function

2.3 Procedures quantum mechanical calculations

First of all, we have to understand the reason why quantum mechanics intervene in our

calculation of thermodynamic characteristics of the Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

+ ions. The

reason is that we need to find the Gibbs free energy (using 3rd law) and the entropy (using 2nd

law), are values which cannot be found experimentally. To calculate these values quantummechanical calculations and statistical thermodynamic calculations are conducted in help


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12

with two computer programs: the first one is the GAMESS program which deals with the

calculations of the optimized geometric parameters and optimized frequencies of the ions. In

quantum mechanical calculations we will find that geometrical parameters and frequencies

are structural dependent according to linear, cyclic, or pyramidal structures.

Above all these geometric parameters and frequencies, which are output of GAMESS

program, are input of the second computer program aimed to compute thermodynamic

properties of ions; so far it is logic to say that thermodynamic properties are also structural

dependent. Up to this level, the reader should ask several equations about these computer

programs; especially on GAMESS program: what are the inputs of GAMESS program and

where are they coming from? What is the functioning method of this program and why are

they called quantum mechanical software? The following theory is trying to develop answerson these questions.

2.3.1 The Hartree-Fock (HF) Self-consistent Field Method

Hartree proposed in 1928 an iterative method called self consistent field (SCF) method. The

first step of the SCF method, one guesses the wave function for all the occupied Molecular

Orbitals (MOs). But in this Hartrees case it is the Atomic orbitals (AOs), since he was

working exclusively with atoms and uses these to construct the necessary one-electron

operators hi. hiis the one-electron Hamiltonian

2

1

1

2

Mk

i iikk

Zh

r

(2.17)

In fact this one-electron Hamiltonian was taken from the general Hamiltonian for molecules

which includes all involved interactions in the system of several electrons and nuclei which is

(2.18)

But this equation (2.18) has been reduced to (2.17) by applying two molecular orbital

approximations: theBornOppenheimer approximation and the Linear Combination of

Atomic Orbitals (LCAO) approach. Then the solution of one-electron Schrdinger equation

must satisfy the equation

22 2 22 2

2 2

k k li k

e k ik ij kli k i k i j k l

e Z eZ Z eH

m m r r r


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13

i i i ih (2.19)

in an atom, with its spherical symmetry, this is relatively straightforward, and Hartree was

helped by his retired father who enjoyed the mathematical challenge afforded by such

calculations provides a new set of, presumably different from the initial guess. So, the one-

electron Hamiltonians are formed a new using these presumably more accurate to

determine each necessary charge probability density associated to electrons, and the process

is repeated to obtain a still better set of.

Few years later, Fockproposed the extension of Hartrees SCF procedure to Slater

determinantal wave functions (see chapter 4, reference book [16]). Just as with Hartree

product orbitals, the HF MOs can be individually determined as eigenfunctions of a set of

one-electron operators, but now the interaction of each electron with the static field of all of

the other electrons (this being the basis of the SCF approximation) includes exchange effects

on the Coulomb repulsion. Some years later, in a paper that was critical to the further

development of practical computation, Roothaan described matrix algebraic equations that

permitted HF calculations to be carried out using a basis set representation for the MOs. We

will forego a formal derivation of all aspects of the HF equations, and simply present them in

their typical form for closed-shell systems (i.e., all electrons spin-paired, two per occupied

orbital) with wave functions represented as a single Slater determinant. More about Slater

determinant see chapter 4, reference [16]. This formalism is called restricted Hartree-Fock

method. The one-electron Fock operator is defined for each electron i as

1- - { }

2

2 zf v HF j

r

nucleik

i iiik

k

(2.20)

where the final term, the HF potential, is 2JiKi, and theJiand Kioperators are defined so as

to compute theJijand Kijintegrals (the procedure of calculatingJijand Kijintegrals is shownin chapter 4 [16]). To determine the MOs using the Roothaan approach, we solve the secular

equation

11 11 11 12 1 1

21 21 22 22 2 2

1 1 2 2

0

N N

N N

N N N N NN NN

F ES F ES F ES

F ES F ES F ES

F ES F ES F ES

,

(2.21)


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14

where we have to find its various rootsEj. In this case, the values for the matrix elements F

and Sare computed explicitly. Matrix elements Sare the overlap matrix elements. For a

general matrix element F(we here adopt a convention that basis functions are indexed by

lower case Greek letters, while MOs are indexed by lower-case Roman letters) we compute

21 1

2

1( | ) ( | )

2

nuclei

v kkk

F v Z vr

P v v

(2.22)

The notation g v where g is some operator which takes basis function as its argument,

implies a so-called one-electron integral of the form

( ) .vg v g dr (2.23)Thus, for the first term in Eq. (2.42) g involves the Laplacian operator and for the second

term g is the distance operator to a particular nucleus. The notation (|) also implies a

specific integration, in this case

12

1( | ) (1) (1) (2) (1) (2)vv dr dr

r (2.24)

where and represent the probability density of one electron and and the other.The exchange integrals (|) are preceded by a factor of 1/2 because they are limited to

electrons of the same spin while Coulomb interactions are present for any combination of

spins. The final sum in Eq. (2.22) weights the various so-called four-index integrals by

elements of the density matrix P. This matrix in some sense describes the degree to which

individual basis functions contribute to the many-electron wave function, and thus how

energetically important the Coulomb and exchange integrals should be (i.e., if a basis

function fails to contribute in a significant way to any occupied MO, clearly integralsinvolving that basis function should be of no energetic importance). The elements ofP are

computed as

2

occupied

i ii

P a a (2.25)

where the coefficients aispecify the (normalized) contribution of basis function to MO i

and the factor of two appears because with RHF theory we are considering only singlet wave

functions in which all orbitals are doubly occupied. While the process of solving the HF


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15

secular determinant to find orbital energies and coefficients are quite analogous to that of

effective Hamiltonian methods, it is characterized by the same paradox present in the

Hartree formalism. That is, we need to know the orbital coefficients to form the density

matrix that is used in the Fock matrix elements, but the purpose of solving the secular

equation is to determine those orbital coefficients. So, just as in the Hartree method, the HF

method follows a SCF procedure, where first we guess the orbital coefficients (e.g., from an

effective Hamiltonian method. For details on how this effective Hamiltonian method is used

in our discussion see chapter 4, reference [16]); and then we iterate to convergence. The full

process is described schematically by the flow chart in Figure 2.1

The energy of the HF wavefunction can be computed in a fashion analogous to the

following equation

2 2| | | |1

2

i ji i j

iji i j

E dr drr

(2.26)

Where i andj run over all the electrons, iis the energy of MO i from the solution of the one-

electron Schrdinger equation using the one-electron Hamiltonian defined by equation:

2

1

1{ }2

M

ki i iikk

zh v jr (2.27)

where the final term represents an interaction potential with all of the other electrons

occupying orbitals {j } and may be computed as

{ }j

iijj i

V j dr r

(2.28)

wherej is the charge (probability) density associated with electronj . The repulsive third

term on the r.h.s. of Eq. (2.27) is thus exactly analogous to the attractive second term, exceptthat nuclei are treated as point charges, while electrons, being treated as wave functions, have

their charge spread out, so an integration over all space is necessary.

In Eq. (2.26) we have replaced with the square of the wave function to emphasize how it is

determined (again, the double integration over all space derives from the wave function

character of the electronthe double integral appearing on the r.h.s. of Eq. (2.26) is called a

Coulomb integral and is often abbreviated asJij).


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16

Figure 2.1: Flowchart of the HF SCF procedure

2.3.2 Mller-Plesset (MPn) computation method

Even though the above theory of HF SCF method was proved to be a good approximation it

had to be corrected by introducing the theory of perturbation. See chapter 7, reference [8].

In fact RayleighSchrdinger perturbation theory provides a prescription for accomplishing

this. In the general case, we have some operator A that we can write as

(0)A VA (2.29)

where A(0)is an operator for which we can find eigenfunctions, V is a perturbing operator,

and is a dimensionless parameter that, as it varies from 0 to 1, maps A(0)into A. If we

expand our ground-state eigenfunctions and eigenvalues as Taylor series in, we have


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17

(0) (0) 02 3(0 ) 0 0 02 3

0 0 00 0 2 3

1 1

2! 3!

(2.30)

And

0 2 0 3 0

0 2 30 0 0

0 0 00 0 2 3

1 1

2! 3!

a a aa a

(2.31)

where (0)0a is the eigenvalue for(0)0 , which is the appropriate normalized ground-state

eigenfunction for A(0). For ease of notation, Eqs. (2.30) and (2.31) are usually written as

0 1 2 32 30 0 0 0 0

(2.32)

And

0 1 2 32 30 0 0 0 0a a a a a (2.33)

where the terms having superscripts (n) are referred to as nth-order corrections to the zeroth

order term and are defined by comparison to Eqs. (2.30) and (2.31). Thus, we may write

0

0 0V aA (2.34)As

0 1 2 32 300 0 0 0

0 1 2 0 1 2 32 3 3 2 300 0 0 0 0 0 0

VA

a a a a

(2.35)

Since Eq. (2.35) is valid for any choice of between 0 and 1, we can expand the left and

right sides and consider only equalities involving like powers of. Powers 0 through 3

require

00 0000 0aA

(2.36)

0 11 0 1 000 00 0 0 0

V a aA (2.37)

0 1 22 1 2 1 000 0 00 0 0 0 0

V a a aA (2.38)

0 1 2 33 2 3 2 1 000 0 0 00 0 0 0 0 0

V a a a aA (2.39)


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18

where further generalization should be obvious. Our goal, of course, is to determine the

various nth-order corrections. Equation (2.36) is the zeroth-order solution from which we

are hoping to build, while Eq. (2.37) involves the two unknown first-order corrections to the

wave function and eigenvalue.

To proceed, we first impose intermediate normalization of ; that is

00 0

1 (2.40)

By use of Eq. (2.32) and normalization of (0)0 , it must then be true that

000 0

|n

n (2.41)

Now, we multiply on the left by (0)0 and integrate to solve Eqs. (2.37)(2.39). In the caseof

Eq. (2.37), we have

0 10 0 0 0 0 1 0 000 00 0 0 0 0 0 0 0

V a aA

(2.42)

Using

0 0 1 00 00 0 0 0A A (2.43)

and Eqs. (2.36), (2.40), and (2.41), we can simplify Eq. (2.42) to

10 000 0

| |V a (2.44)

which is the well-known result that the first-order correction to the eigenvalue is the

expectation value of the perturbation operator over the unperturbed wave function.

As for (1)0 like any function of the electronic coordinates, it can be expressed as a linear

combination of the complete set of eigenfunctions ofA(0)

, i.e., 1 00

0i i

i

c

(2.45)

we can carry out analogous operations to determine the second-order corrections, then the

third-order, etc. The algebra is tedious, and we simply note the results for the eigenvalue

corrections, namely

20 0

020 0 00

0

| |j

jj

V

aa a

(2.46)


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And

0 00 0 0 0 0 0

0 0 0 03

0 0 0 000, 0

00

[ ]

)(

jkj j k k

j kj k

V V V V

aa a a a

(2.47)

We now consider the use of perturbation theory for the case where the complete operator

A is the Hamiltonian, H. Mller and Plesset (1934) proposed choices for A(0)and V with

this goal in mind, and the application of their prescription is now typically referred to by

the acronym MPn where n is the order at which the perturbation theory is truncated, e.g.,

MP2, MP3, etc. Note that in our project we use the perturbation theory truncated at order 2:

MP2.This MP2 method is not discussed in its details as HF method because it has a similar

functioning as HF with a specialty of inhering the perturbation theory for molecules in itscalculations; but MP2 computation results will be quite useful in our project especially in

chapter 3

2.3.3. Density Function Theory (DFT) computation method

The wave function is complicated. This function, depending on one spin and three spatial

coordinates for every electron (assuming fixed nuclear positions), is not, in and of itself,

particularly intuitive for systems of more than one electron. Indeed, one might approach theHF approximation as not so much a mathematical tool but more a philosophical one. We may

take advantage of our knowledge of quantum mechanics in asking about what particular

physical observable might be useful. What then is needed? The Hamiltonian depends only on

the positions and atomic numbers of the nuclei and the total number of electrons. The

dependence on total number of electrons immediately suggests that a useful physical

observable would be the electron density, since, integrated over all space, it gives the total

number of electronsN, i.e.,

( )N r dr (2.48)

Energy being separable into kinetic and potential components; If one decides a priori to try to

evaluate the molecular energy using only the electron density as a variable, the simplest

approach is to consider the system to be classical, in which case the potential energy

components are straightforwardly determined. The attraction between the density and the

nuclei is


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20

[ ( )] ( )| |

nucleik

nekk

ZV r r dr

r r

(2.49)

and the self-repulsion of a classical charge distribution is

1 21 2

1 2

( ) ( )1[ ( )]

2 | |ee

r rV r dr dr

r r

(2.50)

where r1 and r2 are integration variables running over all space.

The kinetic energy of a continuous charge distribution is less obvious. To proceed, we first

introduce the fictitious substance jellium. Jellium is a system composed of an infinite

number of electrons moving in an infinite volume of a space that is characterized by a

uniformly distributed positive charge (i.e., the positive charge is not particulate in nature, as it

is when represented by nuclei). This electronic distribution, also called the uniform electron

gas, has a constant non-zero density. Thomas and Fermi, in 1927, used fermions statistical

mechanics to derive the kinetic energy for this system as (Thomas 1927; Fermi 1927)

2 5

2 3 33

[ ( )] 3 ( )10

uegT r r dr (2.51)

Note that the various Tand Vterms defined in 3 above equations (2.49-2.51)are functions ofthe density, while the density itself is a function of three-dimensional spatial coordinates. A

function whose argument is also a function is called a functional, and thus the Tand Vterms

are density functionals. The ThomasFermi equations, together with an assumed variational

principle, represented the first effort to define a density functional theory (DFT); the energy

is computed with no reference to a wave function. However, while these equations are of

significant historical interest, the underlying assumptions are sufficiently inaccurate that they

find no use in modern chemistry (in ThomasFermi DFT, all molecules are unstable relative

to dissociation into their constituent atoms. . .)

By construction, HF theory avoids any self-interaction error and exactly evaluates the

exchange energy (it is only the correlation energy that it approximates); however, it is time-

consuming to evaluate the four-index integrals from which these various energies are

calculated. While Slater (1951) was examining how to speed up HF calculations he was

aware that one consequence of the Pauli principle is that the Fermi exchange hole is larger

than the correlation hole, i.e., exchange corrections to the classical inter-electronic repulsion

are significantly larger than correlation corrections (typically between one and two orders of


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21

magnitude). So, Slater proposed to ignore the latter, and adopted a simple approximation for

the former. In particular, he suggested that the exchange hole about any position could be

approximated as a sphere of constant potential with a radius depending on the magnitude of

the density at that position. Within this approximation, the exchange energy Ex is determined

as

143

39 3

[ ( )] ( )8

xE r r dr

(2.52)

Within Slaters derivation, the value for the constant is 1, and Eq. (2.52) defines so-called

Slater exchange. Given the differing values of in Eq. (2.52) as a function of different

derivations, many early workers saw fit to treat it as an empirical value, and computations

employing Eq. (2.52) along these lines are termed X calculations (or sometimes Hartree

FockSlater calculations in the older literature). Empirical analysis in a variety of different

systems suggests that = 34 provides more accurate results than either = 1 or = 23 . This

particular DFT methodology has largely fallen out of favor in the face of more modern

functionals, but still sees occasional use, particularly within the inorganic community.

This above theoretical description of DFT method of computation shows that there is a close

relation shows that there is a close relationship between this method and that of HF discussedin section 2.3.1 of this work. For more details on DFT see chap 8, reference book number

[16]. The above 3 methods of calculation; HF, MP2 and DFT are the 3 methods that will be

used in our GAMESS program for computation of optimized geometrical parameters of the

ions. More information about the description of GAMESS program is found in reference [14].

So far we have made a important step in briefly reviewing our second motivating force which

motivate our project which is tools of calculations. In the following we are going to describe

the experimental methods and survey the result obtained and we will implement quantum

mechanical calculation and display our results for Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

- ions.


32/51


33/51

23

Detector/display

SEM Cathode

Movable shutter

Effusion cell

Ion-optic

system

Electromagnet

Ionization

chamber

Collector of

electrons

Figure3. 1: Schematic diagram of mass spectrometer

b) Ionization processes

In the EC regime the neutral particles, effusing from by the of Knudsen's cell, heated by

resistance furnace, directed into the ionization chamber and are ionized due to the impact

with electrons of the controlled energy, which the cathode emits. Ions are drawn out by the

potential, applied to the extracting electrode (collimator), they are focused and they travelinto the system of the accelerating and deflecting lenses. The ion beam is separated according

to the mass-to-charge ratio in the magnetic field of the electromagnet; it passes through the

slit of the ion collector and then strikes the dynode surface of secondary electron multiplier

(SEM). The value of the amplified signal is measured by millimeter and is recorded with the

chart recorder. The development of mass-spectrum is ensured by a change of the magnetic

field intensity, which is achieved by a change of the current through the coils of the

electromagnet. For separation of useful signal from the background, whose presence iscaused by the ionization of residual gases, a movable shutter is provided.

To study the charged species existing in vapour over solids, the action of the instrument is

similar with the only difference that ions are formed as a result of thermal ionization inside

the effusion cell and are drawn out from it by the small potential, applied to the collimator.


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24

A vacuum system of mass spectrometer

The necessary vacuum is achieved by the magnetic-discharge pumps. This method of

evacuation makes it possible to reach the operating pressure (105106 Pa) without the

application of the freezing out traps. Fore-vacuum is created by rotary oil pump. The

measurement of pressure in the range between 103106 Pa is accomplished by two

magnetic- ionizing sensors (MIS) in

the region of the vaporizer and in the

region of the ion collector.

A combined EI/TI source possible to

work both in the regime of electron

impact, and in the regime of thethermal emission was used. The

diagram of the source of ions is given

in the figure Fig. 3.2. In the TI regime

the electron gun, ionization chamber

and shutter are under the potential of

the collimator. The system of

electrostatic lenses serves for thepreliminary acceleration of ions and

focusing of ion beam in the electric

field between the ionization chamber.

Shutter makes it possible to

completely overlap the molecular

beam of the molecules, which left

directly the effusion opening, and toseparate useful signal from the

background.

c) Measuring system and stabilization of temperature.

Heating effusion cell is produced by the resistance furnace, prepared from the molybdenum

wire with an overall section of 0.8-1 mm2

, reinforced by alundum. Radiation shield from thetantalum foil with a thickness of 0.1 mm is put on to the external surface of furnace. Smooth

Figure3. 2: Diagram of the EI/TI ion source

To the mass analyzer

Deflecting Plates

System of ionicoptic lenses

Electron gun

Ionization chamber

Collector

Extracting electrode

Cooling

Thermocouple

The cell

Movable shutter

Resistance oven

jacket

(collimator)


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25

adjustment and stabilization of the temperature of the cell was achieved by the use of a high-

precision temperature regulator. The temperature of the cell was measured by the tungsten-

rhenium thermocouple of that prepared from the wire with the diameter of 0.1-0.2 mm and

effusion cells welded-on to the housing with spot welding. The registration of thermal emf is

accomplished by a digital millivoltmeter. The calibration of thermocouples was executed on

the melting points of pure substances (Al, CsCl, CsBr, CsI). Further correction of the

indications of thermocouple is conducted on the assumption that correction it is equal to zero

at room temperature and linearly it grows with an increase in the temperature. Measuring

system and registration of ion currents: In the series mass spectrometer MI 1201 for

measuring the ion currents, the electrometric amplifier is used. The modified system includes

the secondary-electron multiplier and two electrometric amplifiers and makes it possible to

work in two regimes: (1) the electric current of collector is strengthened by electrometer and

dc amplifier, in this regime it is possible to measure the electrical signals from 10-7 to 10-14 A;

(2) electrical signal is initially strengthened by the secondary electron multiplier; the current

pulses, which enter from the output of that secondary electron multiplier, are amplified then

by electrometer in the regime of the measurement of average current. This regime makes it

possible to measure the currents up to 2 10-18 A. The record of mass-spectra was achieved

with the aid of the two-coordinate recording instrument.

3.2.2. Procedure of the studies

Effusion cell: In the experiments are used molybdenum cells with the diameter of the effusion

orifice of 0.6 mm and ratio of the evaporation area to the area of the orifice ~400. The cell

before the load of preparation was preliminarily cleaned by the mechanical removal of the

remainders of substance from the previous experiment with the subsequent boiling in the

distilled water. The drawing out potential difference between the cell and the collimator

varied in the range from 40 to 200 v. The identification of ions was achieved according to the

comparison of the distribution of the intensities of ions in the groups of the adjacent lines of

mass-spectrum with the distribution of isotopic varieties, calculated for the assumed

molecular formula of the ion.

3.2.3. Experimental results

The analysis of the thermal emission of ions from RbI, AgI and RbAg 4I5 of a study of

thermionic emission was investigated in the following temperature ranges: 492-878 K


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26

(RbAg4I5), 814-1009 K (RbI), 800-922 K (AgI). Let us note that for the RbAg 4I5 solid

electrolyte studies were carried out in the widest temperature interval (about 400 K). The

relative intensities of the ion currents of the ions emitted from the cell are given in table 3.1.

For all objects the most intensive ions are the atomic ions of rubidium of Rb+. Moreover it

turned out that the intensities of the ion currents of Rb+ above the system of RbAg4I5 3-5

orders are higher than above rubidium iodide (at the same temperatures). In the case of of

silver iodide the ion Rb+, being contaminant, has considerably higher intensity of ion current

(at 800 K, ~5 orders) in comparison with its own ion of Ag +. In its turn,I(Rb+) from AgI

exceeded the I(Rb+) from RbI in pure 2-3 orders. Thus, it is possible to conclude that in the

presence of iodide silvers are created favorable conditions for the thermoemission of the ions

of Rb+, even when rubidium it is contained in the form of an admixture.

The mass-spectrum of ions in the regime TI, obtained with the use of a copper effusion cell is

given in table 3.2. Just as in the case with the molybdenum cell (table. 3.1) together with the

simple ions are registered the ionic associates, which can be considered as the reaction

products of the atomic ions of Rb+ with the molecules of RbI and AgI. Negative ions in the

case with the copper cell were not discovered. The intensities of the ion currents of negative

ions were rather small (6- 8 orders less in comparison with Rb+). The most important special

feature of RbAg4I5 is the exceptionally low temperature, at which the emission of ions wasobserved: the threshold temperature of the appearance of ions was about 220 oC.

Table 3.1: Relative intensities of ion currents (T= 780 K)

Ion

RbAg4I5

550 K 630 K 844 K 878 K )

Rb+ 650 4.98 105 1.53 107 1.96 107

Rb2I+ 3 1.9 104 1.26 106 1.30 106

Rb+ 2RbI 1.5 9.00 102

I 40 100

RbI2 18.4


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27

Ion

AgI RbI

800 K 837 K 814 K 1009 K

Rb+ 3.04 106 2.30 107 1.45 104 6.44 107

Rb2I+ 2.84 103 2.98 105 1.44 104 1.55 108

Rb+ 2RbI 117 2.37 106

I 0.5 7 104 420

RbI2 1 104 144

Table 3. 2: Relative intensities of ion currents (I) in the mass-spectrum of thermal emission for RbAg4I5

(copper cell, T= 790 K)

Ion I Ion I

Rb+ 3.59106 Rb+2RbI 1.41102

Rb2I+ 1.30105

3.3. Theoretical methods and results

3.3.1. Computational procedures

The geometric parameters, normal vibration frequencies, of the ions present in vapor over

rubidium iodide, Rb2I+, Rb3I2

+, RbI2, Rb2I3

, have been calculated ab initio by the Hartree

Fock method and taking into account electron correlation (see section 2.3.). The main

equilibrium configuration of all ions was found to be the linear configuration ofDh

symmetry. Pentaatomic ions could also exist as two isomers, planar cyclic ofC2v symmetry

and bipyramidal ofD3h symmetry. Their energies were higher than that of theDh isomers,

and their contents in vapor were negligibly low. The energies and enthalpies of dissociation

of the ions with the elimination of the RbI molecule is calculated.


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28

The Rb2I+, Rb3I2

+ and RbI2 ionic associates were experimentally observed in saturated

vapour over rubidium iodide by high-temperature mass spectrometry (see section 3.2.3). The

determination of the thermochemical characteristics of ions from the mass spectrometry data

requires information about their structure and vibration frequencies.

With the goals of:

(1) to determine the most stable equilibrium geometric configuration, geometric parameters,

and normal vibration frequencies of the ions;

(2) to reveal the possible isomers among the alternative structures of the Rb3I2+ and Rb2I3

pentaatomic ions;

(3) then determination of the thermodynamic properties of the Rb2I+, Rb3I2+ and RbI2- ionic

associates will be calculated later in the next chapter.

(a)Methods of calculation

The calculations were performed using the HartreeFockRoothaan approximation, Mller

Plesset second-order (MP2) perturbation theory and the density functional theory method

DFT (as discussed in section 2.3.).

GAMESS is a powerful tool of calculation and optimization of structures of molecules and

ions.

(b)Results

The Rb2I+andRbI2

ions.

The calculated geometrical characteristics of the triatomic isoelectronic Rb2I+

and RbI2

ionsobtained in the HF, MP2, DFT approximations are listed in Table 3.3. According to these

results, the equilibrium configuration of both ions is a linear configuration ofDh symmetry

(see figures 3.1). The theoreticalRe(RbI)internuclear distances, normal vibration

frequenciesi, and intensities of IR spectral bandsAi obtained in different approximations

satisfactorily agree with each other.


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Table 3.3: Characteristic of Rb2I+

and RbI2-ions and the energies of dissociations

Figure3. 3: Schematic arrangement of Rb2I+, Rb3I2

+, Rb3I2

+, RbI2

-andRb2I3

-ions

Rb2I+ triatomic linear RbI2

- triatomic linear

Rb3I2+ pentaatomic linear

Rb2I3- pentaatomic linear

Property Rb2I+ RbI2

-

HF MP2 DFT HF MP2 DFT

Re 3.473 3.386 3.405 3.405 3.523 3.467

-E 58.801483 59.074256 59.529459 46.410327 46.783476 47.113892

1 16 18 14 22 22 22

2 16 18 14 22 22 22

3 78 82 80 62 67 64

4 113 120. 115 115 124 118

A1 0.413 0.405 0.365 0.352 0.336 0.317

A2 0.413 0.405 0.3865 0.352 0.336 0.317

Note:Re () is the equilibrium internuclear distance;E(au) is the total energy; i (cm1) is thenormal vibration frequency;Ai [D

2/(amu 2)] is the IR band intensity


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Rb3I2+ cyclic pentaatomic Rb2I3- cyclic pentaatomic

Rb3I2+ pyramidal pentaatomic Rb2I3

- pyramidal pentaatomic

The Rb3I2+

andRb2I3-ions.

Several geometric configurations were considered for the pentaatomic ions including linear

ofDhsymmetry, planar cyclic ofC2v symmetry, and bipyramidal ofD3hsymmetry (see

figure 3.3). For each configuration, geometric parameters were optimized and normal

vibration frequencies, IR spectrum band intensities, and energy stability were calculated. Themost stable structure had a linear configuration ofDhsymmetry.

The calculated characteristics of pentaatomic ions ofDhsymmetry obtained in the HF, MP2,

and DFT approximations are listed in Table 3.4. We see that the equilibrium internuclear

distances, vibrational frequencies, and IR spectrum band intensities obtained.


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Table3. 4: Characteristics of the Rb2I3-and Rb3I2

+ions

Property Rb3I2+ Rb2I3

-


Re1

3.431 3.350 3.431 3.467 3.388 3.409

Re2 7.000 6.826 7.005 7.063 6.885 6.940

-E 93.926006 94.416580 95.132497 81.535205 82.126126 82.717262

(1,2) 5 4 2 3 2 2

(3,4) 11 11 9 17 16 16

(5,6) 19 19 18 22 21 22

7 40 44 43 36 40 38

8 890 97 92 73 79 76

9 111 121 114 114 123 116

10 116 127 121 115 127 121

A(1,2) 0.111 0.081 0.151 0.031 0.031 0.04

A(5,6) 1 1 0.486 1 1 0.522

A8 0.011 0.021 0.029 0 0 0.011

A10 2 2 2 2 2 2

Note:Re (), is the equilibrium internuclear distance;E(au), is the total energy; i,(cm

1

), isthe normal vibration frequency;Ai [D

2/(amu 2)],and is the IR band intensity

In all these computer results as shown in tables 3.3 - 3.6 , the useful optimized parameters

for the ions which enables us to go forward in our calculations are optimized inter nuclear

distances Re and optimized vibration frequencies i Rb2I+, Rb3I2

+, RbI2 and Rb2I3

ions as

it is shown in the following chapter.


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Property Rb3I2+ Rb2I3


Re1 3.661 3.544 3.590 3.676 3.550 3.608

E 93.929638 94.426869 95.137779 81.536779 81.534425 82.719642

(1,2) 38 38 40 31 30 30

3 51 53 49 52 52 56

4 58 67 61 57 67 60

5 58 67 62 57 68 60

6 78 85 83 72 80 74

7 78 88 87 80 90 80

8 78 88 90 80 91 87

9 89 95 94 90 101 97

A1 0.150 0.147 0.173 0.069 30 0.030

A2 0.148 0.45 0.149 0.068 30 0.048

A3 0 0 0 0 52 0.002

A4 0 0 0.033 0 67 0.007

A5 0 0 0.010 0 68 0.010

A6 0.742 0.807 0.800 0.716 80 0.619

A7 0.743 0.709 0.659 0.800 90 0.734

A8 0.891 0.714 0.659 0.800 91 0.665

A9 0 0.012 0.003 0.004 101 0.028

Note:Re (), is the equilibrium internuclear distance;E(au), is the total energy; i,(cm1), is the

normal vibration frequency;Ai [D2/(amu 2)] and is the IR band intensity

Table3. 5: Characteristics of the Rb2I3-pyramidal structure and Rb3I2

+pyramidal structure ions


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CHAPTER 4

CALCULATION OF THERMODYNAMIC

PROPERTIES

4.1. Introduction

The geometric parameters and the normal vibration frequencies of the ions present in vapor

over rubidium iodide Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

-, were calculated using ab initio by the

HartreeFock method and taking into account electron correlation. The main equilibrium

configuration of all ions was found to be the linear configuration ofDh symmetry.

Pentaatomic ions could also exist as two isomers, planar cyclic ofC2v symmetry and

bipyramidal ofD3h symmetry. Their energies were higher than that of theDh isomers, and

their contents in vapor were negligibly low. In this Chapter we are going to use these

obtained results in the computer program which calculate the thermodynamic properties of

the ions and use them dynamic properties to calculate the enthalpies of dissociation of ions

over RbI in vapour at elevated temperature.

4 .2 .Calculation of thermodynamic functions of the ions

Thermodynamic properties of the ions which are: specific heat capacity (Cp), Gibbs energy

(G), Entropy (S), reduced Gibbs energy () and the function of enthalpyH(T) -H(0) are

calculated using statistical thermodynamics.Because of time constraints our project makes

further calculation for only the enthalpy of reaction by using the formula

r (0) [ ( ) ln ( )]r pH T T R K T

(3rd law of thermodynamics) (4.1)

Which shows that to find the enthalpy of dissociation for a given reaction only functions

are needed .

The function is calculated from statistical thermodynamics where its statistical sum or

partition function is


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1320 2

3 01

1111232 222 3 6

31 2

2 2 21

(2 )

8

1

88

n

kT

i

hvn k

kTk

mkT V Q

h

kTkT kT

ie

II Ieh h h

(4.2)

and =RlnQ (4.3)

The available computer program which was given to us by our supervisor and uses the

statistical thermodynamics together with the rigid rotator harmonic oscillator approximation

in order to compute the required thermodynamic properties of Rb2I+, RbI2

-, Rb3I2+ and Rb2I3

-

ions.

The input file of that program is as follows:

Table 4.1: Input file of calculation of thermodynamic properties of Rb3I2+

1

Rb3I2+ Dh for mp2

5

85.46725 126.9045 85.46725 126.9045 85.46725

-6.825 0.0 0.0 -3.475 0.0 0.0 0.0 0.0

0.0 3.475 0.0 0.0 6.825 0.0 0.0

2.0 1.0 400.0 1500.0 50.0

4.0 4.0 11.0 11.0 19.0 19.0 44.0 97.0

121.0 127.0

In the above input file (Table 4.1) the first row indicates that we deal with only one molecule,

the second row is the name of the file, the third row indicates that the molecule contains five

atoms, the 4th row indicates atomic masses of involved atoms, the 5th and 6th rows are

optimized internuclear distances according to MP2, the 7th row indicates the structure of the

molecule and the temperature range and the two last rows indicate vibration frequencies of

the atoms according to MP2. Here we restrict ourselves to linear structures because their


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results will be combined with experimental results for further treatment in order to get

enthalpies of dissociation of the ions.

This program which is named calculation of the thermodynamic functions of the ions or

molecules through molecular parameters computes thermodynamic properties as function oftemperature according to this simple flow-chat

Figure 4.1: Flow-chart of calculating thermodynamic properties of ions

The output of the above calculation involving the linear structure of Rb2I+, Rb3I2

+, RbI2- and

Rb2I3- according to given chemical reactions at experimental temperature are as indicated in

the tables 4.2-4.5. The units of Cp, , S, G are J/molK, the unit of H(T)-H(0) is J/mol.

Table 4.2:Thermodynamics functions for Rb2I+

linear at experimental temperature

T, K Cp S G H(T)-H(0)

600 62.272 335.950 396.022 364.738 36043.290

700 62.295 345.235 405.623 369.911 42271.686

800 62.309 353.315 413.943 374.906 48501.932

900 62.320 421.282 360.467 379.659 54733.412

Input:Optimizedmolecularparameters

Processing

:Statisticalthermodynamiccomputations

Output:Thermodynamicfunction of the ions:

, S, Cp, G


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4.3. Calculation of dissociation enthalpies of the ions

The results provided by the computer program which calculate thermodynamics properties of

the ions as shown in table 4.2 have to be used together with available experimental data

(these experimental results have been provided from a mass spectrometric experiment as it

was detailed in chapter 3 tables 3.13.2.) according to the equation (4.1) which is one of the

mathematical form of the 3rd law of thermodynamics.

Table 4.3: Thermodynamic functions for RbI2-linear at experimental temperature


600 112.148 554.970 664.052 607.681 65449.102

700 112.173 571.820 681.342 617.001 76665.242

800 112.190 586.467 696.322 626.001 87883.457

900 112.201 599.422 709.536 634.563 99103.057

1000 112. 611.035 721.358 642.661 110323.627

1100 112.216 621.558 732.053 650.310 121544.903

Table 4. 4:Thermodynamic functions for Rb3I2+

linear at experimental temperature


900 62.321 363.688 424.521 382.895 54749.859

1000 62.328 370.105 431.088 387.392 60982.353

1100 62.333 375.924 437.029 391.638 67215.450


700 112.166 562.877 672.321 607.991 76610.771

800 112.184 577.515 687.300 616.989 87828.319

900 112.197 590.461 700.514 625.550 99047.400

Table 4. 5: thermodynamic functions for of Rb2I3-linear at experimental temperature


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In equation 4.1 it is obvious that the calculation of enthalpy of reaction rHrequires to know

the values of functions for all species which appear in the concerning reaction.

For example:

The values of enthalpy of Rb2I+ = Rb+ + [RbI] reaction according to equation (4.1).

2( ) [ (Rb ) (RbI )] (Rb I )r cT

where cRbI is the reduced Gibbs energy of RbI in condensed phase, ln ( )pK T is an

experimental value of the equilibrium constant.

These calculations of the values of the rHwere computed using Microsoft Excell.

Through these tables, the reader may ask him/herself where the values of (Rb ) , [RbI] ,

(I ) are coming from?

The answer is that these values functions havebeen surveyed from the thermodynamic

data base [15] which is a sophisticated library for chemical-physicists because it provides the

thermodynamic properties of the species and the substance mentioned above. The disposed

thermodynamic data base none as explained in detail in [15] is on original-currentthermodynamic data base, Russian version which have been provided to use by the

supervisor of this research project. Note: In this world wide recognizable thermodynamic

data base which is currently in use, the values of enthalpy of reaction which are objects under

study in this project are not available.

The value for the heterophase reactions were recalculated for the gaseous reactions by adding

the enthalpy of sublimation sH(0 K) = 198 kJ/mol for RbI .The later also have been taken

from the data base [15].

The ion Rb2I3- whose values of the current intensity which would help in calculation of

ln ( )pK T are not registered experimentally; thus we adopted the values obtained

theoretically.

For each ion, note that uncertainties which appear in the table of results table 4.6 are taken

from experimental uncertainties in equilibrium constants.


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For Rb2I+ we have: Rb2I

+ = Rb+ + [RbI]

Table4. 6: Enthalpy of dissociation o for Rb2I+

N T, K lnKp (Rb+) ([RbI]) (Rb2I

+) r(T) rH(0)

gaseous

1 780 0.338 163.425 120.377 351.699 -67.897 142.8

2 781 0.539 163.452 120.444 351.779 -67.883 141.5

3 696 -0.037 161.057 114.524 344.863 -69.282 150.0

4 814 0.843 164.312 122.600 354.316 -67.404 137.4

5 760 0.546 162.885 119.032 350.083 -68.166 142.7

6 729 0.258 162.02 116.889 347.578 -68.669 146.4

7 761 0.569 162.913 119.100 350.163 -68.150 142.5

8 805 0.534 164.081 122.019 353.672 -67.572 140.0

9 814 0.615 164.312 122.600 354.316 -67.404 139.0

average 142.5

For RbI2- we have:

RbI2- = I- + RbI

mass spectrometric and quantum mecanical studies of charged particles in vapours over rubidium...

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