Atomic Physics Atomic Physics

Prof. C.A. KIWANGA (in memoriam)

African Virtual university Université Virtuelle Africaine Universidade Virtual Africana African Virtual University

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This document is published under the conditions of the Creative Commons http://en.wikipedia.org/wiki/Creative_Commons Attribution http://creativecommons.org/licenses/by/2.5/ License (abbreviated “cc-by”), Version 2.5. African Virtual University

Table of Contents

I. Atomic Physics______5

II. Prerequisite Course or Knowledge______5

III. Time______5

IV. Materials______5

V. Module Rationale______5

VI. Content______6 6.1 Overview ______6 6.2 Outline______7 6.3 Graphic Organizer______8

VII. General Objective(s)______8

VIII. Specific Learning Objectives______9

IX. Pre-assessment______10

X. Teaching and Learning Activities______15

XI. Glossary of Key Concepts______107

XII. List of Compulsory Readings______109

XIII. Compiled List of (Optional) Multimedia Resources______113

XIV. Compiled List of Useful Links______115

XV. Synthesis of the Module______119

XVI. Summative Evaluation______120

XVII. References______124

XVIII. Main Author of the Module ______125

XIX. File Structure ______125

African Virtual University

Notice

Foreword

This module has four major sections. The first one is theINTRODUCTORY section that consists of five parts vis: TITLE:- The title of the module is clearly described PRE-REQUISITE KNOWLEDGE: In this section you are provided with information regarding the specific pre-requisite knowledge and skills you require to start the module. Carefully look into the requirements as this will help you to decide whether you require some revision work or not. TIME REQUIRED: It gives you the total time (in hours) you require to complete the module. All self tests, activities and evaluations are to be finished in this specified time. MATERIALS REQUIRED: Here you will find the list of materials you require to complete the module. Some of the materials are parts of the course package you will receive in a CD-Rom or access through the internet. Materials recommended to conduct some experiments may be obtained from your host institution (Partner institution of the AVU) or you may acquire or borrow by some other means. MODULE RATIONALE: In this section you will get the answer to questions like “Why should I study this module as pre-service teacher trainee? What is its relevance to my career?”

The second is the CONTENT section that consists of three parts

OVERVIEW: The content of the module is briefly presented. In this section you will fined a video file (QuickTime, movie) where the author of this module is interviewed about this module. The paragraph overview of the module is followed by an outline of the content including the approximate time required to complete each section. A graphic organization of the whole content is presented next to the outline. All these three will assist you to picture how content is organized in the module. GENERAL OBJECTIVE(S): Clear informative, concise and understandable objectives are provided to give you what knowledge skills and attitudes you are expected to attain after studying the module. SPECIFIC LEARNING OBJECTIVES (INSTRUCTIONAL OBJECTIVES): Each of the specific objectives, stated in this section, are at the heart of a teaching learning activity. Units, elements and themes of the module are meant to achieve the specific objectives and any kind of assessment is based on the objectives intended to African Virtual University

be achieved. You are urged to pay maximum attention to the specific objectives as they are vital to organize your effort in the study of the module. The third section is the bulk of the module. It is the section where you will spend more time and is referred to as the TEACHING LEARNING ACTIVITIES. The gist of the nine components is listed below: PRE-ASSESSMENT: A set of questions, that will quantitatively evaluate your level of preparedness to the specific objectives of this module, are presented in this section. The pre-assessment questions help you to identify what you know and what you need to know, so that your level of concern will be raised and you can judge your level of mastery. Answer key is provided for the set of questions and some pedagogical comments are provided at the end. TEACHING AND LEARNING ACTIVITIES: This is the heart of the module. You need to follow the learning guidance in this section. Various types of activities are provided. Go through each activity. At times you my not necessarily follow the order in which the activities are presented. It is very important to note:

• formative and summative evaluations are carried out thoroughly • all compulsory readings and resources are done • as many as possible useful links are visited • feedback is given to the author and communication is done COMPILED LIST OF ALL KEY CONCEPTS (GLOSSARY): This section contains short, concise definitions of terms used in the module. It helps you with terms which you might not be familiar with in the module. COMPILED LIST OF COMPULSORY READINGS: A minimum of three compulsory reading materials are provided. It is mandatory to read the documents. COMPILED LIST OF (OPTIONAL) MULTIMEDIA RESOURCES: Total list of copyright free multimedia resources referenced in, and required for completion of, the learning activities is presented. COMPILED LIST OF USEFUL LINKS: a list of at least 10 relevant web sites that help you understand the topics covered in the module are presented. For each link, complete reference (Title of the site, URL),a screen capture of each link as well as a 50 word description are provided. SYNTHESIS OF THE MODULE: Summary of the module is presented. SUMMATIVE EVALUATION: Enjoy your work on this module. African Virtual University

I. Atomic Physics

By Prof.C.A.Kiwanga, The Open University of Tanzania

II. Prerequisite Course Or Knowledge

Before you start this Module, you are expected to be familiar with pre-university calculus, geometry and also to have done Physics modules Mechanics 1 & 2, Waves and Optics, Thermal Physics, Electricity 1 & 2 and Quantum Mechanics.

III. Time

You are expected to spend 120 hours of self study on this module. You should share the time allocation such that Learning Activities 1 and 3 take more time than Learning Activities 2 and 4. This works out to be 40 hours for Atomic Models, 20 hours for Electrical Discharges, 40 hours for Atomic Spectra and 20 hours for X-Rays.

IV. Material

The following list identifies and describes the equipment necessary for all of the activities in this module. The quantities listed are required for each group. 1. Computer (With Internet Access): - A personal computer with word pro- cessing and spreadsheet software 2. Periodic Table of Elements: - 3. Metre Stick: -

V. Module Rationale

Atomic physics may loosely be defined as the scientific study of the structure of the atom, its energy states, and its interactions with other particles and fields. Learning Atomic Physics is important not only for understanding the physics of the atom but also the technological applications thereof. For example, the fact that each element has its own characteristic “fingerprint” spectrum has contributed significantly to advances in material science and also in cosmology. African Virtual University

VI. Content

6.1 Overview

In this module you will learn about an important topic in physics, namely Atomic Physics. The subject matter of the module is a principal component of the so called Modern Physics, a scientific discipline that came into being in the late 19th century and early 20th century. You will be guided through the historical development of atomic theories, through the work of Dalton, Thompson, Rutherford and Bohr. These four scientists have a very special place in the development of Atomic Physics. The work by Dalton and Thompson laid the ground on which Rutherford and Bohr built upon to the extent that the models developed by the latter two scientists are usable to some extent today. Hence you will be required to solve problems relating to Rutherford’s and Bohr’s models of the atom. In Learning Activity 2 of this module you will be guided through the gas discharge phenomenon and the onset of cathode rays. This phenomenon was a puzzle to the scientists of the day but led to an important discovery of the electron, the first subatomic particle to be discovered. Towards the end of the Learning Activity you will be guided through Millikan’s oil drop experiment that led to the discovery that electric charge is particulate or quantized. In Learning Activity 3, you will be guided through the evolution of atomic spectra and learn about the uniqueness of an atomic spectrum for every element. The uniqueness of atomic spectra has scientific and technological implications. In Learning Activity 4, you will be guided through the origin of x-rays, the development of x-ray spectra and the uniqueness of x-ray spectrum for every element. Towards the end of the unit we discuss and solve problems using Moseley’s law and finally you will learn about the use of x-rays as an analytical tool. African Virtual University

6.2 Outline

Atomic Models (40 hours)

• Dalton’s and Thomson’s Models, • Rutherford’s alpha Scattering Experiment. • Rutherford’s Planetary Model of an Atom, • Bohr’s Model of an Atom ; • Bohr’s Postulates Electrical Discharges (20 hours)

• Discovery of Cathode Rays. • CRT “glow” Variation with Pressure, • Properties of Cathode Rays. Atomic Spectra (40 hours)

• Quantum Numbers • Angular Momenturm Coupling Schemes, • Vector Model of an Atom, • Zeeman Effect. • Fine Structure of Hydrogen Spectrum • Emission and Absorption Spectra. • Pauli Exclusion Principle. X-Rays (20 hours)

• Production Properties and Characteristic X-Ray Spectra, • X-Ray Diffraction, • Bragg Equation and Crystal Spectrometer • Atomic X-ray Spectra of Elements • Moseley’s Law. African Virtual University

6.3 Graphic Organizer

VII. General Objective(s)

The aim of this module is to guide the learner through a chronological development of Atomic Physics. The learner begins by studying the development of atomic models from Dalton, Thompson, Rutherford and finally Bohr. After atomic models the learner is guided through a phenomenon that led to the discovery of the electron and its negative charge. Gas discharge experiments also laid the ground as to how atoms could be excited. After completing this module you will be able to • Understand the development of atomic theories, • Solve problems related to emission and absorption spectra of atoms and • Describe production of x-rays and their interaction with matter

8 African Virtual University

VIII. Specific Learning Objectives (Instructional Objectives)

Content Learning objectives After Completing this section you would be able to:

Atomic Models (40 hours) • Describe the characteristics of Dalton and Thomson atomic mo- • Dalton’s and Thomson’s Models, dels • Rutherford’s Alpha Scattering Expe- • Solve problems related to the riment. alpha-scattering experiment • Rutherford’s Planetary Model of an • Solve problems using Bohr’s pos- Atom, tulates • Bohr’s Model of an Atom ; • Bohr’s Postulates Electrical Discharges (20 hours) • Explain the discharge phenomena • Discovery of Cathode Rays. under different pressures • CRT “glow” Variation with Pressure, • Put forward evidence that cathode • Properties of Cathode Rays. rays are electrons • Describe the setting and purpose of Millikan’s oil drop experiment Atomic Spectra (40 hours) • Quantum Numbers • Use the vector model of atom to • Angular Momenturm solve problems and explain proper- Coupling Schemes, ties • Vector Model of an Atom, • Explain the fine structure of spectra • Zeeman Effect. • Fine Structure of Hydrogen Spectrum • Emission and Absorption Spectra. • Pauli Exclusion Principle. X-Rays (20 hours) • Production Properties and Characte- • Explain the atomic origin of X-rays ristic X-Ray Spectra, • Distinguish characteristic X-Rays • X-Ray Diffraction, from Bremstrahlung radiation • Bragg Equation and Crystal Spectrometer • Use Bragg’s rule to solve problems • Atomic X-ray Spectra of Elements • Solve problems using Moseley’s Law • Moseley’s Law. African Virtual University 10

IX. Pre-assessment

Are you ready for Atomic Physics Module?

Dear Learner In this section, you will find self-evaluation questions that will help you test your preparedness to complete this module. You should judge yourself sincerely and do the recommended action after completion of the self-test. We encourage you to take time and answer the questions. Dear Instructor The Pre-assessment questions placed here guide learners to decide whether they are prepared to take the content presented in this module. It is strongly suggested to abide by the recommendations made on the basis of the mark obtained by the learner. As their instructor you should encourage learners to evaluate themselves by answering all the questions provided below. Education research shows that this will help learners be more prepared and help them articulate previous knowledge.

9.1 Self Evaluation Associated With Atomic Physics

Evaluate your preparedness to take the module on atomic physics. If you score greater than or equal to 60 out of 75, you are ready to use this module. If you score something between 40 and 60 you may need to revise your school physics on topics of mechanics, electromagnetism and modern physics. A score less than 40 out of 75 indicates you need to physics. All questions are in multiple choice format. The learner should choose the most appropriate alternative and award oneself 5 marks for each correct choice. 1. Before 1945, the atom was defined to be the smallest a) electrically charged particle b) divisible particle c) indistinguishable particle d) indivisible particle. 2. The colours of the rainbow are such that a) only the primary colours are present b) black and white colours are also present c) violet and red are at either edge of the spectrum d) none of the above is correct. African Virtual University 11

3. One essential apparatus in an experiment on the dispersion of white light is a) a double convex lens b) a rectangular glass block c) a curved mirror d) a triangular glass prism 4. X-rays are a) subatomic particles travelling at relativistic velocities b) produced when a metallic solid is heated to temperatures close to the respective melting point. c) at the short wavelength side of the electromagnetic spectrum. d) at the low frequency side of the electromagnetic spectrum 5. In classical physics a) an electron travels with an associated deBroglie wavelength b) a particle is associated with any wave phenomenon c) the Pauli exclusion principle applies d) none of the above is correct. 6. Phenomenological derivation of the Schrodinger equation was inspired by two equtions in classical physics a) wave equation and Newton’s second law of motion b) wave equation and Newton’s first law of motion c) Ampere-Maxwell equation and the wave equation d) none of the above equations. 7. One key result from quantum mechanics is a) the distinction of matter and wave phenomena b) the ultra violet catastrophe c) the non distinction of wave phenomena and moving subatomic particles d) the discovery of negative charge in cathode rays. 8. The partial differential equation for the hydrogen atom is best solved using a) cartesian coordinates b) cylindrical coordinates c) spherical polar coordinates d) none of the above coordinate systems. African Virtual University 12

9. A particle is performing circular motion with a tangential velocity v. If r is the radius of the circle, then the particle acceleration is given by a) v/r b) v2/r c) v/r2 d) v2/r2. 10. If the particle in Q.9 has mass m, the angular momentum L of the paricle is given by a) mv/r b) mv2/r c) mvr d) mv/r2 . 11. The angular momentum vector of the particle in Q.9 and Q.10, is given by r r r r r L r p , where p is the linear momentum. The z-component of is given L = r × p L by

a) L = xp − yp z y x b) L = yp − zp z z y c) L = xp − yp z x y d) L = zp − yp . z z y 12. A spherical positive charge Q has radius R. The magnitude of the electric field at a point a distance r < R from the centre is given by

r 2 a) E = Q 4πε 0 1 b) E = Q 4πε r 0 r c) E = Q 4πε 0 1 Q d) E = . 4πε r 2 0 African Virtual University 13

13. The quantisation of electromagnetic energy is summarized by the equation a) E = mc2 b) E = hω c) E = hν d) E = hc 14. An excited atom is one whose energy state is a) higher than that of the ground state b) lower than that of the ground state c) the same as that of the ground state d) such that none of the above is correct. 15. In terms of energy, violet light a) is more energetic than red light b) is less energetic than red light c) has the same energy as that of red light d) is such that none of the above is correct. 16. In terms of wavelength, the wavelenght of violet light a) is longer than that of red light b) is shorter than that of red light c) is equal to that of red light d) is such that none of the above is correct. 17. A particle of mass m carrying positive charge Q is left to drop between charged parallel plates. If the the electric field between the plates is E V/m acting upwards and the medium between plates is viscous causing a drag force bv, the relation between forces at balance is given by r r r a) mg = qE + bv r r r b) mg = qE - bv r r r c) mg = bqE ∧ v r r d) mg = bqE .v . 18) The condition for diffraction of light is that the wavelength a) is of the same order as that of the slit width b) is greater than that of the slit width c) is very much smaller than that of the slit width d) can assume any value relative the slit width. African Virtual University 14

19. Ionization energy is defined as being theenergy required to (a) remove an inner shell electron from a gaseous atom. (b) remove the outermost electron from a gaseous atom. (c) raise an electron from K-shell to M-shell in a gaseous atom. (d) neither of the above definitions.

20. The Binding energy of an atom is defined as being the energy required to (a) excite an inner shell electron. (b) completely remove an outer shell electron. (c) completely remove an inner shell electron. (d) implant an electron into an inner shell.

9.2 Answer Key:

1. d 11. a 2. c 12. c 3. d 13. c 4. c 14. a 5. d 15. a 6. a 16. b 7. c 17. b 8. c 18. a 9. b 19. b 10. c 20. c

9.3 Pedagogical Comment For The Learner:

The questions you have just done are meant to test your preparedness to take on this module. The module builds on the knowledge you already know and extends from there. Hence the percentage score is indicative of the level of preparedness of the learner. Any score of less than 50% implies a lot of catching up to be done before commencement of the module.. African Virtual University 15

X. Teaching And Learning Activities

Activity 1: Atomic Models

You will require 40 hours to complete this activity. In this activity you are guided with a series of readings, Multimedia clips, worked examples and self assessment questions and problems. You are strongly advised to go through the activities and consult all the compulsory materials and as many as possible among useful links and references.

Specific Teaching and Learning Objectives

• Describe the characteristics of Dalton and Thompson atomic models • Solve problems related to the alpha-scattering experiment • Solve problems using Bohr’s postulates

Summary of the Learning Activity

Learning Activity 1 lays the foundation for the whole module. The Activity begins by looking at the subject matter from an historical perspective. Atomic models by the founders of atomic physics, namely Dalton, Thompson, Rutherford and Bohr are presented. Lastly we introduce the concept of quantum numbers and also discuss Pauli’s Exclusion principle.

List of Required Readings

Reading 1: Atomic Models

Complete reference : From: wikipedia URL : http://en.wikipedia.org/wiki/Atomic_physics Accessed on the 20th April 2007 Abstract : This reading is compiled from wikipedia page indicated above and the links available in the page. Titles on Dalton’s model of the atom, Thompson’s plum pudding model, Rutherford’s alpha scattering experiment that led to the planetary model of an atom and quantum mechanics are discussed. Rationale : The material in this compilation is essential to the first activity of this module. African Virtual University 16

Reading 2: Bohr Model of Hydrogen Atom

Complete reference: http://musr.physics.ubc.ca/~jess/hr/skept/QM1D/node2.html Date Consulted: June 2007 Abstract : In three webpages the Bohr model of the hydrogen atom is presented concisely. You are advised to begin with the page referenced here and then use the next link to go to the derivation of the Bohr radius and click next again for calculation of energy levels. Rationale: The material is presented in a manner that it is easy to follow.

Reading 3: Theory of Rutherford Scattering

Complete reference: http://hyperphysics.phy-astr.gsu.edu/hbase/rutcon.html#c1 Date consulted: April 2007 Abstract: The physics of scattering as it relates to the Rutherford Model of the atom is beautifully presented. You will have to follow the outline as presented in this page and click on each link as presented in the outline. Rationale: The material presented in this link is essential and relevant to this course.

List of Relevant MM Resources

Reference: http://www.colorado.edu/physics/2000/index.pl Date consulted: December 2006 Description: A beautiful applet whereby you create your own atom. Upon entering the Physics 2000 Home page, click on Table of contents and then go to Science Trek and click on Electric Force. Place your cursor about 5 cm away from the proton. Click and drag the created electron at say 45° or greater towards the nucleus and let go. Then watch the electron make an elliptical orbit around the proton. You will be surprised at the number of non colliding “orbital electrons” you can create around the nucleus. Reference: http://www.weaowen.screaming.net/revision/nuclear/rsanim.htm Date consulted: April 2007 Description: A simulation of the Rutherford alpha particle scattering experiment against a gold target. In this simulation the nucleus is represented by a yellow dot and the alpha particle by a red dot which is smaller than the yellow dot. A scattering event is realized by the learner following the instructions regarding choice of the energy of the alpha particle, dragging the red dot and clicking the ‘fire’ bar. You must clear tracks and hits before the next scattering event. Should you get no response when click fire, try again. Implementation of one set of the instructions constitutes one African Virtual University 17

experiment. The next experiment starts by clicking the “next” bar to rest the position of the alpha particle. After several scattering events you need to clear tracks. The alpha particle energy is restricted between 8 and 25 MeV. Reference: http://www.physics.brown.edu/physics/demopages/Demo/modern/demo/ 7d5010.htm Date consulted: April 2007 Abstract: An animation of the experimental set up of Rutherford alpha scattering is shown. 400 alpha particles are fired at a thin gold foil. Reference: http://webphysics.davidson.edu/Applets/pqp_preview/contents/pqp_er- rata/cd_errata_fixes/section4_7.html Date Consulted: June 2007 Abstract: An animation on the Rutherford Scaterring in which you set your own values for number of alpha particles, kinetic energy, target nuclear charge and impact parameter. Référence : http://www.control.co.kr/java1/masong/absorb.html Date consulted: April 2007 Description: A Java applet for an absorption spectrum of a Bohr atom

List of Relevant Useful Links

Resource #1

Title: From Bohr’s Atom to Electron Waves URL: http://galileo.phys.virginia.edu/classes/252/Bohr_to_Waves/Bohr_to_Wa- ves.html Screen Capture:

Reactions to Bohr’s Model Bohr’s interpretation of the Balmer formula in terms of quantized angular momentum was certainly impressive, but his atomic model didn’t make much mechanical sense, as he himself conceded……

Description: A chronological account of the work by Niels Bohr that culminated in the quantisation of angular momentum. Rationale: The article is a lecture among several lectures in Modern Physics given by Prof Michael Fowler. You should link to the Physics 252 Home page and read as much as you can Lectures on Atoms, Particles and Waves. Date Consulted: April 2006 African Virtual University 18

Resource #2

Title: Chapter 27: Early Quantum Theory and Models of the atom http://www.google.com/search?q=cache:p4PiiJqdDkwJ:cherenkov.physics.iastate. edu/~mkpohl/teach/112/ch27.pdf+MODELS+OF+THE+ATOM&hl=en&ct=clnk &cd=79 Screen Capture:

Description: The article is Power Point presentation of Early Quantum Theory and Early models of the atmom:Thompson, Rutherford and Bohr. Rationale: The material is brief and sharp. You should read it. To access it first click on the url address given and the click on this link: http://cherenkov.physics.iastate. edu/~mkpohl/teach/112/ch27.pdf. Date Consulted: April 2006

Resource #3

Title: Atomic Physics URL: http://theory.uwinnipeg.ca/physics/bohr/node1.html Screen Capture: African Virtual University 19

Description: On this site you will find various links that will help you explore the Bohr model of the hydrogen atom and its extensions. This model was one of the greatest successes of early quantum theory, and spurred many further investigations which continue to this day. Rationale: The material in this resource is relevant to this module. . Date Consulted: April 2007

Resource #4

Title: Atomic Models and Spectra URL:http://online.cctt.org/physicslab/content/Phy1/lessonnotes/atomic/atomicmo- delsandspectra.asp Screen Capture:

Description: A chronological account of the work of Rutherford on alpha particle scattering and the emergence of the nucleus. Rationale: The material is good for you. . Date Consulted: April 2006 African Virtual University 20

Resource #5

Title: Rutherford Scattering URL: http://www.ux1.eiu.edu/~cfadd/1160/Ch29Atm/Ruthrfd.html Screen Capture:

Rutherford’s atomic scattering experiments Description: Concise notes on Rutherford Scattering Rationale: This article is part of a series of lecture notes in Atomic Physics. Follow the Links to get more material. . Date Consulted: April 2006 African Virtual University 21

Resource #6

Title: Atomic Structure Concepts URL:-http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/atomstructcon.html#c1 Screen Capture:

Description: This is very useful and almost a complete resource on the Physics of the Hydrogen atom. You follow the boxes sequentially, starting with the box Hydrogen energy levels in which you will be linked to the Bohr model etc. Rationale:This article provides the links to practically all the concepts relevant to this module. Date Consulted:- April 2006 African Virtual University 22

Detailed Description of the Activity (Main Theoretical Elements)

Introduction

In ancient Greece, there were two schools of thought regarding the structure of matter, namely Atomic theory conveying a particulate nature of matter and the continuous theory of matter proposed by Aristotle. The continuous theory of matter having been proposed by such a prominent person at the time overshadowed the atomic theory of matter for sometime.

Dalton’s Model of the Atom

John Dalton, at the beginning of the 19th century, proposed an atomic model that allowed limited quantitative study of the atom. Dalton’s model was that the atoms were tiny, indivisible, indestructible particles, like billiard balls, and that each one had a certain mass, size, and chemical beha- vour that was determined by what kind of element it was. Dalton’s model is silent about the composition and internal structure of the atom.

Thompson’s Model of the Atom

Towards the end of the 19th century much spectroscopic data had been gathered taking advantage of the development of photographic film, the gas discharge tube, and of the diffraction gratings. The characterisitic atomic spectrum for every element had been established. However a theoretical basis to explain the observation was lacking. J.J.Thompson having established that cathode rays were negatively charged, later given the name electrons, went on to assume that the electron is a part of the atom and proposed a model for the atom as a sphere full of an electrically positive subs- tance mixed with negative electrons “like the raisins in a cake”. Thompson’s model is frequently referred to as the “plumb pudding” model. In an African sense, one could picture a Thompson atom much like a spherically symmetric guava fruit. Thompson explained emission lines by suggesting that electrons radiated as they oscillated within the ‘positive pudding’. However, this could not explain the precise wavelength patterns emitted by different elements. African Virtual University 23

Rutherford’s Model of the Atom

Sir Ernest Rutherford proposed a model of the atom based on the results of alpha particle scattering that the atom consisted mainly of empty space with a tiny, positively charged nucleus, containing most of the mass of the atom, surrounded by negative electrons in orbit around the nucleus like planets orbiting the Sun. According to Maxwell’s electromagnetic theory, a charged particle in circular motion radiates energy and so an electron in a Rutherford’s atom should continuously lose energy as it moves in a planetary orbit and eventually should spiral down to the nucleus at the centre of the atom, which does not happen. Rutherford’s model though a much improved picture of the atom, but could not explain stability of the atom. Furthermore, according to classical physics, the energy emitted by an electron as it spirals down to the nucleus should have all frequencies, in other words the emitted spectrum should be continuous which is not the case. The emitted spectrum consist of lines in a dark background. Thus, Rutherford’s model could not explain the observed line spectra of elements. Bohr’s Model of the Atom Niels Bohr proposed an atomic model that would explain the discrepancies between the observed line spectra emitted by elements and the spectra predicted by the Ruther- ford’s atomic model.

Bohr proposed the following postulates

1. An electron in an atom moves in a circular orbit about the nucleus under the influence of the Coulomb force between the electron and the nucleus. r 2. An electron moves in an orbit for which its orbital angular momentum L is an integral multiple of . 3. An electron moving in an allowed orbit does not radiate electromagnetic energy. Thus, its total energy E remains constant. 4. Electromagnetic radiation is emitted if an electron, initially moving in an orbit of total energy E , discontinuously changes its motion so that it moves in an i orbit of total energy E . The frequency of the emitted radiation is equal to f the quantity E − E / h . ( i f ) African Virtual University 24

Electron Cloud Model of the Atom

The cloud model represents a sort of history of where the electron has probably been and where it is likely to be going. You can visualize a dot in the middle of a largely empty sphere to represent the nucleus while smaller dots around the nucleus to represent instances of the electron having been there. The collection of traces quickly begins to resemble a cloud.

Rutherford Scattering

Adapted from Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Ruther- ford_scattering Rutherford scattering is a phenomenon that was explained by Ernest Rutherford in 1911, and led to the development of the orbital theory of the atom. It is now exploited by the materials analytical technique Rutherford backscattering. Rutherford scattering is also sometimes referred to as Coulomb scattering because it relies on static electric (Coulomb) forces. A similar process probed the insides of nuclei in the 1960s, called deep inelastic scattering. Highlights of Rutherford’s Experiment • A beam of α particles were aimed at a thin gold foil. • Most of the particles passed through without deflection. • Others were deflected by various angles • Some were backscattered .

Sir Ernest Rutherford From these results Rutherford concluded that the majority of the mass was concentra- ted in a minute, positively charged region (the nucleus) surrounded by electrons. When a (positive) alpha particle approached sufficiently close to the nucleus, it was repelled strongly enough to rebound at high angles. The small size of the nucleus explained the small number of alpha particles that were repelled in this way. Rutherford showed, using the method below, that the size of the nucleus was less than about 10–14 m . African Virtual University 25

Scattering Theory

Main assumptions • Collision between a point charge but heavy nucleus with charge Q=Ze and a light projectile with charge q = ze is considered to be elastic, • Momentum and energy are conserved, • The particles interact by the Coulomb force; • The vertical distance the projectile is from the centre of the target, the impact parameter b, determines the scattering angle θ.

Fig. 1.1 Rutherford Scattering Geometry

The relationship between the scattering angle θ, the initial kinetic energy

1 K = mv2 and the impact parameter b is given by 2 0

zZ e2 b = cot θ / 2 1.1 2K 4πε ( ) 0 where z =2 for α -particle and Z = 79 for gold.

A Cursory Derivation of the Differential Cross section In Fig. 1.2 or 1.3, a particle that hits the ring between b and b + db is scattered into the solid angle dΩ between θ and θ + dθ. African Virtual University 26

By definition, the cross section is the proportionality constant

2πbdb = −σ θ 2π sinθdθ ( ) Hence,

⎛ dσ ⎞ dσ = 2πb db = ⎜ ⎟ dΩ 1.2 ⎝ dΩ ⎠

where dΩ = 2π sinθdθ The Differential Cross Section then becomes

dσ 2πb db = 1.3 dΩ 2π sinθdθ

From Eqns.1.1 and 1.3 we have

2 2 dσ ⎛ 1 ⎞ ⎛ qQ ⎞ 1 = ⎜ ⎟ ⎜ ⎟ 1.4 dΩ ⎝ 4πε ⎠ ⎝ 4K ⎠ sin4 θ / 2 0 α ( ) Eq.1.4, is called the Differential Cross section for Rutherford Scattering.

Fig.1.2 Schematic Geometry for Calculation of Scattering Cross Section African Virtual University 27

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/rutcon.html#c1 Fig.1.3 Detailed Geometrical Arrangements for Calculation of Scattering Cross Section In the above calculations, only a single α - particle is considered. In a scattering experiment, one must consider multiple scattering events and one measures the fraction of particles scattered through a given angle. For a detector at a specific angle with respect to the incident beam, the number of particles per unit area striking the detector is given by the Rutherford formula:

N nL Z 2k 2e4 N (θ ) = i 1.5 4r 2KE 2 sin2 θ / 2 ( )

Where Ni = number of incident α - particles, n = atoms per unit volume in target L = thickness of the target Z = atomic number of target e = electronic charge k = Coulomb’s constant r = target to detector distance, KE = kinetic energy of α - particles and θ = scattering angle. The predicted variation of detected alphas with angle is followed closely by the Geiger-Marsden data, shown in Fig. 1.4 below. African Virtual University 28

Fig.1.4 Verification of Rutherford’s Formula

Calculation of Maximal Nuclear Size

For head on collisions between alpha particles and the nucleus, all the kinetic energy

1 mv2 of the alpha particle is turned into potential energy and the particle is at rest. 2 The distance from the centre of the alpha particle to the centre of the nucleus (b) at this point is a maximum value for the radius, if it is evident from the experiment that the particles have not hit the nucleus.

Fig.1.5 Scattering with Different Impact Parameters African Virtual University 29

Applying the Coulomb potential energy between the charges on the electron and

1 1 q q nucleus, one can write: mv2 = 1 2 2 4πε b Rearranging: 0

1 2q q b = 1 2 1.6 4πε mv2 For an alpha particle: 0 • m (mass) = 6.7×10−27 kg −19 • q1 = 2×(1.6×10 ) C −19 • q2 (for gold) = 79×(1.6×10 ) C • v (initial velocity) = 2×107 m/s

Substituting these into Eqn.1.6, gives the value of the impact parameter of about 2.7×10−14m. The true radius is about 7.3×10−15 m.

The Bohr Model

From Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Bohr_model

Fig.1. 6 A Bohr Picture of the Hydrogen Atom

The Bohr model of the hydrogen atom, Fig.1.6, where a negatively charged electron confined toatomic shells encircle a small positively charged atomic nucleus, and that an electron jump between orbits must be accompanied by an emitted or absorbed amount of electromagnetic energy hν. The orbits that the electron travel in are shown African Virtual University 30

as grey circles; their radius increases as n2, where n is the principal quantum number. The 3→2 transition depicted here produces the first line of theBalmer series, and for hydrogen (Z = 1) results in a photon of wavelength 656 nm (red).

Expression for the Bohr Radius Consider the case of an ion with charge of the nucleus being Ze and an electron moving with constant speed v along a circle of radius r with the centre at the nucleus. The Coulomb force on the electron is

Ze2 F = 4πε r 2 The Coulomb0 force is balanced by the centripetal force, so that we have

Ze2 mv2 = 4 r 2 r πε0 h Using Bohr’s angular momentum quantization rule L = mrv = = h 2π

n2h2 th ε0 We have the n Bohr radius rn = 1.7 π mZe2 And the velocity of the electron in the nth orbit

Ze2 v = 1.8 n 2ε hn 0

The Classical Planetary Model

We compute the energy of the hydrogen atom and the frequency of the orbital motion of a Bohr atom. Energy

Total mechanical energy E = Ek + Ep ( kinetic + potential)

2 1 2 ⎛ −ke ⎞ E = mv + ⎜ ⎟ 1.9 2 r ⎝ ⎠ African Virtual University 31

1 where k = . 4πε 0 The orbital motion is maintained by the Coulomb force

ke2 mv2 ke2 = → mv2 = 1.10 2 r r r

We see from Eqns 1.9 and 1.10 that when an orbit is circular the kinetic energy is half the magnitude of the potential energy. Giving

1 ke2 ke2 E = − 2 r r

1 ke2 E = − 1.11 2 r

This equation shows that the total energy of the system is negative. As the orbital radius of the electron r increases, the energy E decreases approaching zero. Since the energy E is negative, the electron and proton form a bound system. For hydrogen E = -13.6 eV and r = 0.53 Å.

Frequency

ω v The orbital frequency f = = 1.12 2π 2π r where ω is the orbital angular speed of the electron. From Eqn. 1.10 we have

v ke2 = 3 r mr

Substituting this in Eqn(4) we have African Virtual University 32

1 ke2 f = 1.13 3 2π mr

For the H atom f = 7 x 1015 Hz, which is in the ultra violet region of the electromagnetic spectrum. If the electron radiates, the energy E will decrease becoming even more negative and from Eqn(3) the orbital radius r also decreases. The decrease in r in Eqn.1.13, gives rise to increase in the frequency f. So that we have a runaway effect that when energy is radiated E decreases, the orbital radius r decreases, which in turn gives rise to the orbital frequency f increasing and the radiated frequency continuously increasing. This planetary model predicts the electron to spiral inward toward the nucleus emit- ting a continuous spectrum. This process is calcuted to last not more than 1 x 10-8 s, a very short time indeed.

Task 1.1 Estimates using Thomson and Rutherford Models

Use the Thompson and the Rurtherford models of the atom to estimate the electric field on the surface of a gold atom (Thompson Model) and on the surface of the nucleus (Rutherford model), assume the atomic diameter to be 1×10-10 m and the nuclear diameter to be 1×10-15 m and also neglect the influence of the electrons.

Task 1.2 Derivation of Rutherford Scattering formula

Use the link given below to derive the Rutherford scattering formula, highlight the physical principles that are involved. http://hyperphysics.phy-astr.gsu.edu/hbase/rutcon.html#c1

Task 1.3 Niels Bohr’s Postulates

All four Niels Bohr’s postulate are said to have been ad hoc, inconsistent with exis- ting theory at time. Discuss. Formative Evaluation 1 1. Write an essay on the development of the model of the atom from Dalton to Bohr. 2. The proponents of Atomic theory of matter are gender imbalanced. Discuss. 3. How was the plum pudding model disapproved? 4. In the web-based literature given to you, there seems to have been a disagreement between Niels Bohr and Sir Ernest Rutherford. What was the disagreement about and how was it resolved? Is there any lifelong lesson to be learnt in this case? African Virtual University 33

5. In the figure shown below, what is the radius of the hydrogen atom Bohr orbit?

A Standing Wave Pattern Traced out by an Electron in an Orbit

6. (a)If the nuclear radius were 10 cm, what would be the diameter of the atom? (b) Repeat the calculation with the hypothetical nucleus assuming the radius of the earth, r = 6.4 x 106 m and compare the size of the hypothetical nucleus with the distance from earth to the moon 3.8 x 108 m. Ans: (a) 100,000 x 0.20 m= 24 km. (b) 6.4 x 1011 m 7. From the Bohr model, would you expect the energy of the electron to increase or decrease for larger orbits? Ans: To raise the electron further away from the nucleus requires more energy. Hence higher orbits have more energy. 8. Did Rutherford’s model explain (a) the stability of atoms? (b) why atoms emit discrete wavelengths? Elaborate your responses.

Assignment 1

1. List three assumptions used in the derivation of the Rutherford scattering differential cross section. 2. A 6.0 MeV α- particle is scattered at 40° by a gold nucleus. a. What is the corresponding impact parameter? b. If the gold foil is 3.0x10-7 m thick, what is the fraction of the 6.0 MeV beam of α - particles expected to be scattered by more than 45°? 3. Calculate the Bohr radius of a hydrogen atom in its ground state. Consult a physics book for required constants. 4. Calculate the ground state energy of Hydrogen as modelled by Niels Bohr. Electrons have negative energy. 5. Why is an orbit of radius 1 mm unlikely to be occupied by an electron in the Bohr model of the Hydrogen atom? Find the quantum number that characte- rises such an orbit. African Virtual University 34

6. Show on an energy level diagram for hydrogen, the quantum number corresponding to a transition in which the wavelength of the emitted light is 121.6 nm.

Teaching the Content in Secondary School 1

Depending on national physics curriculum, the basic knowledge on atomic models learnt in this Activity can be taught to High school students. African Virtual University 35

Activity 2: Electrical Discharges

You will require 20 hours to complete this activity. In this activity you are guided with a series of readings, Multimedia clips, worked examples and self assessment questions and problems. You are strongly advised to go through the activities and consult all the compulsory materials and use as many as possible useful links and references.

Specific Teaching and Learning Objectives

• Explain the discharge phenomena under different pressures • Put forward evidence that cathode rays are electrons • Describe the setting and purpose of Millikan’s oil drop experiment

Summary of the Learning Activity

In this learning activity you will learn about a phenomenon that baffled scientists in the 19th century. So called mysterious rays are observed when a large direct current voltage is applied across an evacuated glass tube that is equipped with at least two electrodes, a cathode or negative electrode and an anode or positive electrode in a configuration known as a diode. We shall also learn about an ingenious experiment that demonstrated the particulate nature of electric charge. African Virtual University 36

List of Required Readings

Reading 1: A Look Inside the Atom

Complete Reference: http://www.aip.org/history/electron/jjhome.htm Date Consulted: June 2007 Abstract: This is an account of the work by J.J.Thomson on Cathode rays that culminated in the discovery of the electron as a fundamental part of atom. Follow the links by clicking next. Rationale: The article is qualitative but very informative and relevant to this course.

Reading 2: Nobel Prize Lecture on Cathode Rays

Complete Reference: http://nobelprize.org/nobel_prizes/physics/laureates/1905/lenard-lecture.html Date Consulted: June 2007 Abstract: In the context of what you already know now, this is a light reading but informative article on cathode rays and misconceptions at the time. Rationale: The presentation is by a Physics Nobel Prize winner, Philipp Lenard, 1905. This is good motivational material for you.

Reading 3: The Millikan Oil Drop Experiment

Complete reference: http://hep.wisc.edu/~prepost/407/millikan/millikan.pdf Date Consulted: June 2007 Abstract: This is a good quantitative article on the practical aspects of the Millikan Oil Drop Experiment. Rationale: The material is good and relevant to the course.

African Virtual University 37

List of Relevant MM Resources

Reference: http://micro.magnet.fsu.edu/electromag/java/crookestube/ Date consulted: April 2007 Description: This applet enables you see how the tube glows with increased voltage. The applet can be operated by adjusting the Voltage slider bar to vary the electrical current within the tube. As the current level is increased, the electrons begin to ionize gases trapped within the tube causing them to begin glowing with a fluorescent blue color. As the ionizing electrons pass over the cross, a shadow appears on the one end of the vacuum tube. Reference : http://www.physchem.co.za/Static%20Electricity/Millikan.htm Date consulted: April 2007 Description: Condensed theory of Millikan’s Oil Drop Experiment and a virtual experiment is provided Reference: http://www68.pair.com/willisb/millikan/experiment.html Date Consulted: April 2007 Description: Millikan Oil Drop Experiment Applet Read the text in this link and then click “here” to watch a beautiful simulation of Millikan’s experiment. Drag the electric field bar to change the eletric field between plates and note the fectef on the oil drops. As the electric field increases more and more drops are attracted upwards to the positively charged plate. Reference: http://physics.nad.ru/Physics/English/top_ref.htm#mill Date consulted: April 2007 Description: This file contains animations of Science’s 10 Most Beautiful Experi- ments, Millikan’s experiment is number 3. Also Click on the video. African Virtual University 38

List of Relevant Useful Links

Resource #1

Title: Investigating Cathode Rays URL: http://schools.cbe.ab.ca/b858/dept/sci/teacher/zubot/Phys30notes/investnurays/investnurays.htm Screen Capture:

INVESTIGATING NEW RAYS

• Dalton, in 1808 proposed that matter is made of atoms. All substances were either made of single atoms or combinations of atoms (molecules).

• He thought that atoms were indivisible.

• In the 20th century, experiments showed that atoms were divisible. As a result, new particles and forces were found.

Description: Properties of cathode rays are investigated and illustrated. Rationale: This is a good article on properties of Cathode rays. You should find it highly informative. Date consulted: April 2007

Resource #2

Title: Cathode Rays URL: http://en.wikipedia.org/wiki/Cathode_ray Screen Capture:

A schematic diagram of a Crookes tube apparatus. A is a low voltage power supply to heat cathode C (a «cold cathode» was used by Crookes). B is a high voltage power supply to energize the phosphor-coated anode P. Shadow mask M is connected to the cathode potential and its image is seen on the phosphor as a non- glowing area.

Source: http://en.wikipedia.org/Image:Crookes Tube.svg. African Virtual University 39

Description: An encyclopediac presentation of Cathode rays covering definition, properties, history and applications. Rationale: This is a good article with a number of links containing materials relevant to the Learning activity. Date consulted: April 2007

Resource #3

Title:-The Cathode Ray Tube URL:- : http://www.physics.brown.edu/physics/demopages/Demo/modern/demo/ 7b3510.htm Screen Capture:

An older version of the Cathode Ray Tube

Description: A Cathode ray tube is described. Rationale: This article is part of a series of summaries of concepts in Atomic Phy- sics. Use the links to navigate to other relevant topics. Date consulted: April 2007 African Virtual University 40

Resource #4 Title:- The Oil Drop Experiment URL:- http://en.wikipedia.org/wiki/Oil-drop_experiment Screen Capture:

A Simplified scheme of Millikan’s oil-drop experiment.

Description: The Millikan Oil Drop Experiment is described including background, experimental procedure, theory and Feynman’s commentary on Millikan handling of data. Rationale: This is an encyclopediac presentation on The Millikan Oil Drop Expe- riment. You should find the links included in the article to be useful and complimen- tary. Date Consulted: April 2007 African Virtual University 41

Detailed Description of the Activity (Main Theoretical Elements)

Cathode Rays

Cathode rays are streams of electrons observed in vacuum tubes, i.e.evacuated glass tubes that are equipped with at least two electrodes, a cathode (negative electrode) and an anode (positive electrode) in a configuration known as a diode.

Properties of Cathode Rays

At atmospheric pressure, a spark does not extend much from the source, the cathode. However, under partial vacuum conditions sparking takes a longer distance.

Violet streamers at pressure p = 2.7 kPa

When air is pumped out of the tube, the electrodes, anode and cathode, are connected by one or more violet streamers, as illustrated in the figure above. At lower pressures, a pink glow fills the entire tube.

Continued pumping out, cause the pink glow to concentrate around the anode and a blue glow to concentrate around the cathode, as sketched in the figure below. The space between the glows is dark, called Faraday’s dark space. African Virtual University 42

Continued reduction in tube pressure, causes the dark space to expand and the colour at the electrodes to fade until the tube is dark, except for a faint glow around the anode, as sketched in the figure below. The dark region is called Crooke’s dark space.

Tube pressure p = 1.3 Pa or less

The glow in the tube is partly due to light emitted by gas atoms when electrons within them de-excite; it is also due to recombination of electrons and positive ions that occurs during collisisons of the particles. Striations are caused by alternate ionizations and recombinations in the tube. The dark bands, Faraday and Crooke’s dark spaces, are positions where ionizations are occurring mainly due to collisions between ions and neutral atoms. The gas atoms absorb energy which results in the excitation of electrons within them and also ionization of the atoms ; hence, there is no light emitted. The bright bands are places where light is being emitted either by de-excitation of electrons during recombination with positive ions or by the de-excitation of electrons within excietd atoms. Investigations on cathode rays revealed the following properties: 1. Cathode rays travel in straight lines and cast shadows sharp. 2. A paddle wheel placed in the path of the cathode rays turns, indicating that they are particles, travelling in the direction from the cathode to anode and have energy and momentum. 3. Cathode rays can be deflected by a magnetic field and also by an electric field, indicating that they are charged particles, carrying a negative electric charge. African Virtual University 43

4. Through measurements of the charge to mass ratio, reveals the identity of the particles regardless of the cathode material and the gas in the tube. 5. Thompson called the cathode ray particle, the Electron.

Millikan’s Oil-Drop Experiment

Adapted From Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Oil- drop_experiment

Robert A. Millikan in 1891

Experimental procedure

Simplified scheme of Millikan’s oil-drop experiment.

The diagram shows a simplified version of Millikan’s set up. A uniform electric field is provided by a pair of horizontal, parallel plates with a high potential difference between them. Drops of oil are allowed to drift between them. By varying the voltage, the drops can be made to rise or fall. A chosen drop is allowed to fall with the electric field turned off. The drag force acting on the drop is given by Stokes’ law: African Virtual University 44

F = 6πaηv where v is the terminal velocity (i.e. velocity in the absence of an electric field) of the falling drop, η is the viscosity of the air, and a is the radius of the drop. The weight of the drop

4 W = πa3ρg 3

The drop is in air, it experiences an upthrust

4 W = πa3dg up 3

The resultant downward force:

4 3 Wres = πa g(ρ − d) 3 where ρ and d are density of oil and air respectively. Now at terminal velocity, the resultant downward force is the drag force

4 πa3g(ρ − d) = 6πηav 2.1 3

1 ⎛ 9ηv ⎞ 2 => a = ⎜ ⎟ 2.2 2 ρ − d g ⎝ ( ) ⎠ African Virtual University 45

Source: http://www.phys.ufl.edu/~hill/teaching/2005/2061/ links/Millikan.pdf Fig. 2.1 Schematic diagram of the Millikan oil-drop apparatus.

Fig. 2.2 An oil droplet in the cloud carrying an ion of charge e falling at terminal speed, i.e. mg = bv.

If q is the charge on the drop and E is the electric field applied between the plates so that the drop begins to move upwards with a uniform velocity v1, then

4 The resultant upward force =E q − πa3 (ρ − d) g 3 4 3 = Therefore, E q − πa (ρ − d) g 6πηav1 3 From Eq.2.1 we have

E q = 6πηa v + v 2.3 ( 1 ) From Eqn. 2.1 and 2.2, Eqn. 2.3 becomes

1 6πη ⎡ 9vη ⎤ 2 q = ⎢ ⎥ v + v 2.4 E 2 ρ − d g ( 1 ) ⎣⎢ ( ) ⎦⎥ African Virtual University 46

Formative Evaluation 2

1. Explain how a lightning stroke is formed. 2. Using the magnetic field only, how does one know that cathode rays have negative charge? 3. An electron enters a mgnetic field of flux density B = 1 T with a velocity of 1x106 m/s at an angle 45° to the field. Determine the magnitude and direction of the force acting on the electron in the field. 4. How did Thompson determine that the cathode rays were the same regardless of the cathode materials and the gas in the tube? 5. What did Robert Milikan discover in his famous experiment?

Task 2.1 Group Discussion

Consult the link given below and discuss the matter raised in the article. http://www1. umn.edu/ships/ethics/millikan.htm. Are there any lifelong lessons to be learnt.

Task 2.2 The Thompson e/m Experimental Set Up

Source: http://schools.cbe.ab.ca/b858/dept/sci/teacher/zubot/Phys30notes/investnurays/investnurays.htm

A sketch of the Thompson apparatus used to determine the charge to mass ratio of an electron is shown above. (a) Describe how the path of the cathode rays is affected by (i) an electric field between deflecting coils directed in the negatve z-direction, (ii) a magnetic field between the magnetic coils directed in the y-direction.(b) Explain the physical principles applicable in a(i) and a(ii).(c) Identify two useful devices that were derived from the Thomson apparatus. African Virtual University 47

Assignment 2.1

1. The charge on one electron is about 1.6x10-19 C. Assuming an electric field of 3x104 Vm-1, estimate the radius of an oil drop for which its weight could be balanced by the electric force on the electron. 2. In the Thompson charge to mass ratio experiment, it is arranged such that the electron passes through a region in which the electric and magnetic fields are

e v perpendicular to each other. (a) Show that = , where v is the speed of m rB the electron, r is the radius of the circular path and B is the magnetic field. (b) Taking into consideration that in order for the electron to move in a circular rather than a helical path, the electric and magnetic forces must be equal, show

e E that = , where E is the electric field. 2 m rB

Teaching the Content in Secondary School 2

The material learnt in this activity can be taught in a High School with a minimal modification. African Virtual University 48

Activity 3: Atomic Spectra

You will require 40 hours to complete this activity. In this activity you are guided with a series of readings, Multimedia clips, worked examples and self assessment questions and problems. You are strongly advised to go through the activities and consult all the compulsory materials and use as many as possible useful links and references.

Specific Teaching and Learning Objectives

• Solve problems using Mosley’s Law • Use the vector model of atom to solve problems and explain properties • Explain the fine structure of spectra

Summary of the Learning Activity

In Learning activity 3 you will learn the uniqueness of emissions from different elements. Every element has its own characteristic “fingerprint” spectrum. This feature has a lot of scientific and technological implications.

List of Required Readings

Reading 1

Complete reference : URL: http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html Date consulted: June 2007 Abstract : Highly illustrated physics of the hydrogen atom, energy levels, electron transitions, fine and hyperfine structures all are very well discussed. Rationale: This article covers topics in line with this Learning Activity.

Reading 2: Emission Spectrum of Hydrogen

Complete reference : URL : http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/bohr.html Date Consulted: June 2007 Abstract: This article discusses the Emission Hydrogen Spectrum and includes solved practice problems. Rationale: This article covers topics in line with this module and the practice problems makes this reading very important. African Virtual University 49

Reading 3: Hydrogen Atom

Complete reference : An Introduction to the Electronic Structure of Atoms and Molecules URL: http://www.chemistry.mcmaster.ca/esam/Chapter_3/intro.html Date Consulted: June 2007 Abstract : This is section three of an article by Prof. Richard F.W. Bader Pro- fessor of Chemistry / McMaster University / Hamilton, Ontario. It discusses the hydrogen atom, the evolution of probability densities and hence orbitals and finaly the vector model of the hydrogen atom. Rationale: The material covered in this article is good and relevant to this Lear- ning Activity.

Reading 4: Mathematical Solution of the Hydrogen Atom

Complete reference : URL: http://www.mark-fox.staff.shef.ac.uk./PHY332/ato- mic_physics2.pdf Date Consulted: June 2007 Abstract : This article provides the methodology of solving the Hydrogen atom problem as a quantum mechanical problem. Rationale: The article is very relevant to this course as you will see how the three quantum numbers n, l, and m come out naturally.

Reading 5: Fine Structure of Hydrogen Atom

Complete Reference: http://farside.ph.utexas.edu/teaching/qmech/lectures/ node107.html Abstract: This article is part of a series of lecture notes in non relativistic quantum mechanics. Rationale: The material is good but requires a strong link with knowledge in quantum mechanics. African Virtual University 50

List of Relevant MM Resources

Reference: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BohrModel/ Flash/BohrModel.html Date consulted : April 2007 Abstract: Photon excitation of the Hydrogen atom is simulated . The excited electron returns to ground state accompanied by photon emission. The energy of the projectile photon ranges between 10.2 and 13.2 eV, just short of the ionization energy of 13.6 eV. The colour of the emitted line depends on the excitation energy; for example an excitation energy of 10.2 eV excites the electron from n = 1 to n=2, the de-excitation is accompanied by a red emitted line whereas excitation energy 13.2 eV excites the electron from n = 1 to n = 6 which gives rises rise to a series of lines, a violet line for a de-excitation from n = 6 to n = 1, a blue line for a de-excitation from n = 6 to n = 3 and a green line for a de-excitation from n =3 to n = 1.

List of Relevant Useful Links

Resource #1

Title: Modifications of the Bohr model URL:- http://theory.uwinnipeg.ca/physics/bohr/node5.html#SECTION002840000000000 000000 Abstract:- Despite the success of the Bohr model, there were some serious short- comings in the model. For example, on the experimental side detailed analysis of the emission spectra for hydrogen found a single emission line was actually at times composed of two or more closely spaced lines, a feature not present in the Bohr model. Hence a better theoretical base of the hydrogen atom was sought. Rationale: This article is part of a series of lecture notes in atomic physics. The material is relevant to this module. Date consulted: April 2007. African Virtual University 51

Resource #2

Title:- Bohr’s model of the Hydrogen Atom URL: http://www.ux1.eiu.edu/~cfadd/1160/Ch29Atm/Bohr.html Screen Capture:

Abstract: Having become convinced of the general validity of Rutherford’s nuclear model of the atm, Niels Bohr proposed the planetary model of atom which was able to explain to some extent the observed atomic spectrum of hydrogen. Rationale: This is part of a series of lecture notes on atomic physics. Follow the links to get more material. Date consulted: April 2007.

Resource #3

Title: Emission Line Spectrum, Absorption Line Spectrum and a Continous Spectrum URL:- http://www.physics.brown.edu/physics/demopages/Demo/modern/demo/7b1010.htm Screen Capture: African Virtual University 52

Abstract:- Emission line spectra from various gas spectrum tubes, Absorption linespectrum from a low pressure sodium gas and Continuous spectrum from a white light source are shown. Rationale:This article is part of a series of lecture notes in atomic physics. Follow the links for more material. Date consulted: April 2007.

Resource #4

Title: Spectra of Gas Discharges URL: http://laserstars.org/data/elements/index.html Screen Capture:

Hydrogen

Helium

Abstract The article shows Spectra of elements undergoing electrical discharge. Thirty six atomic spectra are shown in full colour. You will definetely enjoy wat- ching these spectra. Rationale: The material is most relevant to the Learning Activity. Date consulted: April 2007. African Virtual University 53

Detailed Description of the Activity (Main Theoretical Elements)

A Summarized Solution of the Schödinger Equation of The Hydrogen Atom

Reduction of a Two-Body Problem to a One-Body Problem

The Hydrogen atom is a two-body system interacting through Coulomb’s law. It can be reduced to a one - body system of reduced mass:

M m μ = p e M + m p e where Mp is the mass of the proton and me is the mass of the electron. The Schrödinger equation for the H-atom is therefore

2 ⎡−h 2 ⎤ ⎢ ∇ + V(r)⎥ψ (r ) = Eψ (r ) 2μ ⎣3.1 ⎦

We generalize the problem to include the case of a hydrogen like atom which consists of one electron moving around a nucleus of charge +Ze, so that the potential becomes

−Ze2 V (r ) = 4π ε r o The Laplacian Operator in Spherical Coordinates

Because of spherical symmetry of the potential function, Eq.3.1 is best handled in spherical co-ordinates r, θ and φ. The spherical co-ordinates are defined by the transformations given by Eq.3.2 Spherical co-ordinates are defined by the transformations:

African Virtual University 54

x = rsinθ cosφ ...... (3.2.1) y = rsinθ sinφ...... (3.2.2) z = rcos θ...... (3.2.3)

2 2 2 (3.2.4) r = x + y + z

⎛ z⎞ θ = cos−1 (3.2.5) ⎜ r ⎟ ⎝ ⎠ ⎛ y⎞ φ = tan−1 (3.2.6) ⎜ x ⎟ ⎝ ⎠

(r,θ,φ ) z

θ r y Y x φ

Fig.3.1 Position of a Particle in Two Coordinate Systems and the transformation from cartesian coordinates to spherical coordinates is facili- tated by the chain rule: African Virtual University 55

∂ ∂ r ∂ ∂θ ∂ ∂ φ ∂ = + + ∂ x ∂ x ∂ r ∂ x ∂θ ∂ x ∂ φ i i i i where xi represents x, y, or z. So that finally the Laplacian operator in spherical polar co-ordinates can be shown to be

⎛ ∂ 2 2 ∂ ⎞ 1 ⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂ 2 ⎤ ∇2 = + + sinθ + 3.3 ⎜ 2 ⎟ 2 ⎢ 2 2 ⎥ ⎝ ∂r r ∂r ⎠ r sinθ ∂θ ⎝⎜ ∂θ ⎠⎟ sin θ ∂φ ⎣ ⎦

Denoting the angular part of the Laplacian operator by L2 and the radial part by ℜ the Laplacian operator becomes:

1 ∇2 = ℜ + L2 2 and so the Schrodinger equation for the hydrogen atom then becomes:

2 ⎡ h ⎛ 1 2 ⎞ ⎤ ⎢− ⎜ ℜ + L ⎟ + V ( r )⎥ψ ( r ,θ ,φ ) = E ψ ( r ,θ ,φ ) 3.4 ⎢ 2μ ⎝ r 2 ⎠ ⎥ ⎣ ⎦ The method of separation of variables,

ψ ( r ,θ ,φ ) = R ( r )Y (θ ,φ ) leads to a radial differential equation and an angular dependent differential equation:

2μ Λ ℜ R ( r ) + ⎡ E − V (r )⎤ R ( r ) = R ( r ) 3.5 2 ⎣ ⎦ 2 h r 2 and L Y (θ ,φ ) = −ΛY (θ ,φ ) 3.6

The method of separation of variables can be repeated to the angular dependent differential equation by application of the product solution African Virtual University 56

Y θ ,φ = P θ Φ φ ( ) ( ) ( ) Which leads to two additional differential equations in θ and φ. The θ-dependent differential equation is given by

1 d ⎛ dP ⎞ m2P sinθ + Λ P − = 0 3.7 ⎜ ⎟ 2 sinθ dθ ⎝ dθ ⎠ sin θ where Λ = l l + 1 . ( ) And the φ - dependent differential equation is given by

d2Φ + m2Φ = 0 3.8 2 dφ Eq.3.8 can be solved readily to give

Φ φ = Aei m φ 3.9 ( )

The product solution Y θ ,φ = P θ Φ φ takes the functional form ( ) ( ) ( )

m l − m ⎛ 1 ⎞ 2 ⎛ d ⎞ l Y m η,φ = C m ei m φ η 2 − 1 3.10 l ( ) l ⎜ 1 2 ⎟ ⎜ dη⎟ ( ) ⎝ − η ⎠ ⎝ ⎠ which can be written in terms of Associated Legendre Functions defined as follows:

m/ 2 l − m m ⎛ 1 ⎞ ⎛ d ⎞ l P m η = −1 η2 − 1 3.11 l ( ) ( ) ⎜ 1 2 ⎟ ⎜ dη⎟ ( ) ⎝ − η ⎠ ⎝ ⎠ So that Eq.3.10 becomes African Virtual University 57

m Y m η,φ = −1 C m P m η eimφ 3.12 l ( ) ( ) l l ( )

The normalized spherical harmonic functions take the form

m l + m ! m (−1) 2l + 1 ( ) m i mφ Y η,φ = . P η e 3.13 l ( ) l l ( ) 2 l! 4π l − m ! ( ) And in full it becomes

l + m ! m/ 2 l − m m 1 2l + 1 ( ) imφ ⎛ 1 ⎞ ⎛ d ⎞ 2 l Y η,φ = e η − 1 l ( ) l 2 2 l! 4π l − m ! ⎝⎜ 1− η ⎠⎟ ⎝⎜ dη⎠⎟ ( ) ( )

Quantum Numbers

Two quantum numbers come out of the angular dependent differential equation,namely the orbital quantum number l and the magnetic quantum number m . The magnetic quantum number specifies the orientation of the angular momentum vector about the chosen axis of rotation and the orbital angular momentum quantum number specifies the shape of the probability density or orbital. The solution of the radial differential equation leads to a normalized radial solution

3 / 2 l ⎛ Z ⎞ ( n − l − 1)! ⎛ 2Zr ⎞ − Zr / na 2l +1 ⎛ 2Zr ⎞ R ( r ) = −2 e o L nl ⎜ na ⎟ 3 ⎜ na ⎟ n+l ⎜ na ⎟ ⎝ o ⎠ n ⎡( n + l )! ⎤ ⎝ o ⎠ ⎝ o ⎠ ⎣ ⎦

where n is the Principal Quantum number, a0 is the Bohr radius, Z is atomic number. The radial solutions are the Energy Eigenfunctions of the hydrogen atom. The energy eigenvalues are readily obtained from the definition of the principal quantum number n. African Virtual University 58

μ2k 2Z 2e4 n2 = h2 2μ E

kZ 2e2 h2 ∴ E n = − where a0 = . 2a n2 μke2 o

⎛ 2Zr ⎞ L2l +1 is the Associated Laguere Polynomial defined by n+l ⎜ na ⎟ ⎝ 0 ⎠

2l +1 n+l ⎛ d ⎞ ⎡ ⎛ d ⎞ ⎤ 2Zr L2l +1( ρ ) = ⎢eρ ρn+l e−ρ ⎥ where ρ = . n+l ⎜ ⎟ ⎜ ⎟ ( ) ⎝ dρ ⎠ ⎢ ⎝ dρ ⎠ ⎥ na0 ⎣ ⎦

Examples of Normalized Radial Solutions

1. For n = 1, l = 0:

3/ 2 ⎛ Z ⎞ − Zr / na ⎛ 2Zr ⎞ R 2 1.1.e o L1 10 = − ⎜ ⎟ 1 ⎜ ⎟ ⎝ na ⎠ ⎝ na ⎠ o o ⎛ 2Zr ⎞ d ⎡ d ⎤ Now L1 eρ e− ρ = - 1 1 ⎜ ⎟ = ⎢ ρ ⎥ ⎝ na ⎠ dρ dρ ( ) o ⎣ ⎦

3/ 2 ⎛ Z ⎞ − Zr / na => R 2 e o x( 1) 10 = − ⎜ ⎟ − ⎝ na ⎠ o

3/ 2 ⎛ Z ⎞ − Zr / a R 2 e o ∴ 10 = ⎜ ⎟ ⎝ a ⎠ o African Virtual University 59

2. For n = 2, l = 0:

3/ 2 0 ⎛ Z ⎞ 1 ⎛ 2Zr ⎞ − Zr / 2 a ⎛ 2Zr ⎞ R (r ) 2 e o L1 20 = − ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ⎝ 2a ⎠ 2.8 ⎝ 2a ⎠ ⎝ 2a ⎠ o o o ⎛ 2Zr ⎞ d ⎡ d2 ⎤ Now L1 eρ 2 e− ρ 2 ⎜ ⎟ = ⎢ 2 ρ ⎥ ⎝ 2a ⎠ dρ dρ o ⎣ ⎦ ⎛ 2Zr ⎞ 2Zr 2Zr ==> L1 2. 4 4 2 ⎜ ⎟ = − = − ⎝ 2a ⎠ 2a a o o o

3/ 2 ⎛ Z ⎞ 1 − Zr / 2 a ⎛ 2Zr ⎞ R (r ) 2 .1.e o 4 20 = − ⎜ ⎟ ⎜ − ⎟ ⎝ 2a ⎠ 4 ⎝ a ⎠ o o

3/ 2 ⎛ Z ⎞ ⎛ Zr ⎞ − Zr / 2 a R (r ) 2 e o ∴ 20 = ⎜ ⎟ ⎜ − ⎟ ⎝ 2a ⎠ ⎝ a ⎠ o o

3. For n = 2, l =1:

3/ 2 ⎛ Z ⎞ 1 ⎛ 2Zr ⎞ − Zr / 2 a ⎛ 2Zr ⎞ R (r ) 2 e o L3 21 = − ⎜ ⎟ 3 ⎜ ⎟ 3 ⎜ ⎟ ⎝ 2a ⎠ 2(3!) ⎝ 2a ⎠ ⎝ 2a ⎠ o o o

3/ 2 ⎛ Z ⎞ 1 1 ⎛ 2Zr ⎞ − Zr / 2 a ⎛ 2Zr ⎞ 2 e o L3 = − ⎜ ⎟ ⎜ ⎟ 3 ⎜ ⎟ ⎝ 2a ⎠ 6 2.6 ⎝ 2a ⎠ ⎝ 2a ⎠ o o o

3 3 ⎛ d ⎞ ⎡ ⎛ d ⎞ ⎤ Now L3 ⎢eρ 3e− ρ ⎥ = - 6 3 (ρ) = ρ ⎝⎜ dρ⎠⎟ ⎢ ⎝⎜ dρ⎠⎟ ( )⎥ ⎣ ⎦ 3/ 2 ⎛ Z ⎞ 1 1 ⎛ 2Zr ⎞ − Zr / 2 a ==> R r 2 e o 6 21 ( ) = − ⎜ ⎟ ⎜ ⎟ (− ) ⎝ 2a ⎠ 6 12 ⎝ 2a ⎠ o o

3/ 2 ⎛ Z ⎞ 1 ⎛ Zr ⎞ − Zr / 2 a R (r ) e o ∴ 21 = ⎜ ⎟ ⎜ ⎟ ⎝ 2a ⎠ 3 ⎝ a ⎠ o o African Virtual University 60

Degeneracy of Hydrogenic Energy Levels

Eigenfunctions belonging to the same eigenvalue are said to be degenerate. The energy En is only dependent on the principal quantum number n. But for each value of n, there are n values of l, l = 0, 1, ...... , n - 1. And for each value of l, there are (2l + 1) values of m. So that the total degeneracy of each energy level is the sum n−1 ∑ (2l + 1) = n2 l =0

The Total Hydrogenic Wavefunction

The total hydrogen wavefunction, except for time, is the product function:

ψ ( r ,θ ,φ ) = R ( r )Y m(θ ,φ ) nlm nl l

It will be noted that whereas the form of the eigenfunctions depends on the values of all three quantum numbers n, l, m, the energy eigenvalues depend only on the principal quantum number n.

Examples of Normalized Spatial Hydrogenic Wavefunctions:

1 1 0 −r / ao ψ 100 = R10( r )Y0 (θ ,φ ) = e π a 3 / 2 o 1 1 ⎛ r ⎞ 0 −r / 2ao ψ 200 = R 20Y0 = ⎜ 2 − ⎟ e 4 2π a 3 / 2 ⎝ a ⎠ o o 1 1 r 0 −r / 2ao ψ 210 = R 21Y1 = e cosθ 4 2π a 3 / 2 a o o 1 1 r 1 −r / 2ao iφ ψ 211 = R 21Y1 = e sinθ e , etc. 8 π a 3 / 2 a o o African Virtual University 61

The Radial Probability Density By definition the probability density of an electron in a hydrogenic eigenstate is given by the product

ψ * ψ = R * P * Φ* R P Φ nlm nlm nl lm m nl lm m

Thus, in its raw form, the probability density is a function of three variables which is rather difficulty to plot directly. Hence, the normal practice is to discuss the de- pendence of the probability density on each variable separately.

The radial probability density is defined by

π 2π P ( r )dr * r 2 sin drd d nl = ∫ ∫ ψ nlmψ nlm θ θ φ 0 0

π 2π r 2R * ( r )R ( r )dr P * P * sin d d = nl nl ∫ ∫ lm lmΦmΦm θ θ φ 0 0 The integrals over θ and φ are equal to unity because each of the functions P and Φ ( as well as R ) are separately normalized. Thus, the radial probability density is given by

P (r )dr = r 2 R * (r )R (r )dr 3.14 nl nl nl Whereas ψ * ψ r 2 sinθdrdθdφ gives the probability of finding an electron in nlm nlm 2 the volume element dτ = r sinθdrdθdφ , Eq.3.14 gives the probability of finding the electron anywhere with a radial coordinate between r and r + dr. Visualizing the hydrogen electron orbitals African Virtual University 62

Fig.3.2 Electron Probability Densities at Different Quantum Numbers Source: http://en.wikipedia.org/wiki/Image:HAtomOrbitals.png

In Fig.3.2, the image to the right shows the first few hydrogen atom orbitals (energy eigenfunctions). These are cross-sections of the probability density that are color- coded (black=zero density, white=highest density). The angular momentum quantum number l is denoted in each column, using the usual spectroscopic letter code («s» means l = 0; «p»: l = 1; «d»: l = 2). The main quantum number n (= 1, 2, 3, ...) is marked to the right of each row. For all pictures the magnetic quantum number m has been set to 0, and the cross-sectional plane is the xz-plane (z is the vertical axis). The probability density in three-dimensional space is obtained by rotating the one shown here around the z-axis. The «ground state», i.e. the state of lowest energy, in which the electron is usually found, is the first one, the «1s» state (n = 1, l = 0). An image with more orbitals is also available (up to higher numbers n and l). Note the number of black lines that occur in each but the first orbital. These are «nodal lines» (which are actually nodal surfaces in three dimensions). Their total number is always equal to n − 1, which is the sum of the number of radial nodes (equal to n - l - 1) and the number of angular nodes (equal to l). African Virtual University 63

Example 3.1 Show that the wavefunctions describing a 1s electron and a 2s electron are ortho- gonal. Solution: 1s electron n = 1, l = 0:

3/ 2 ⎛ Z ⎞ 1 Zr / a ψ = e− 0 1s ⎜ a ⎟ ⎝ 0 ⎠ π 2s electron n = 2, l = 0:

3/ 2 ⎛ Z ⎞ 1 ⎛ Zr ⎞ Zr / 2 a 2 e− 0 ψ 2 s = ⎜ ⎟ ⎜ − ⎟ a 4 a ⎝ 0 ⎠ π ⎝ 0 ⎠ The scalar product becomes

* 3/ 2 ∞ π 2π ⎡ ⎤ ⎛ Z ⎞ 1 Zr / a ψ * ψ dτ = ⎢ e− 0 ⎥ ∫ 1s 2 s ∫ ∫ ∫ ⎜ ⎟ all space 0 0 0 ⎢⎝ a0 ⎠ π ⎥ ⎣ ⎦

3/ 2 ⎡⎛ ⎞ 1 ⎛ ⎞ ⎤ Z Zr − Zr / 2 a0 2 x ⎢⎜ ⎟ ⎜ 2 − ⎟ e ⎥r sinθdφdθdr ⎢⎝ a0 ⎠ 4 2π ⎝ a0 ⎠ ⎥ ⎣ ⎦

2 ( Where the spherical element of volume dτ = r sinθdθdφ )

=> ψ * ψ dτ ∫ 1s 2 s all space

3 ⎛ ⎞ 1 ∞ π 2π ⎛ ⎞ Z Zr −3Zr / 2 a0 2 = ⎜ ⎟ ⎜ 2 − ⎟ e r sinθdφdθdr a 4 2 ∫ ∫ ∫ a ⎝ 0 ⎠ π 0 0 0 ⎝ 0 ⎠

3 ⎛ ⎞ 1 ∞ π ⎛ ⎞ 2π Z Zr 2 −3Zr / 2 a0 = ⎜ ⎟ ⎜ 2 − ⎟ r e sinθdθdr dφ a 4 2 ∫ ∫ a ∫ ⎝ 0 ⎠ π 0 0 ⎝ 0 ⎠ 0 African Virtual University 64

3 ⎛ ⎞ 1 ∞ ⎛ ⎞ π Z Zr 2 −3Zr / 2 a0 = ⎜ ⎟ ⎜ 2 − ⎟ r e dr sinθdθ a 2 2 ∫ a ∫ ⎝ 0 ⎠ 0 ⎝ 0 ⎠ 0

3 ∞ ⎛ ⎞ 1 ⎛ ⎞ π Z Zr 2 −3Zr / 2 a0 = ⎜ ⎟ ⎜ 2 − ⎟ r e dr (−cosθ) a 2 2 ∫ a 0 ⎝ 0 ⎠ 0 ⎝ 0 ⎠

3 ⎛ ⎞ 1 ⎡ ∞ ∞ ⎤ Z 2 −3Zr / 2 a0 Z 3 −3Zr / 2 a0 = ⎜ ⎟ ⎢2 r e dr − r e dr ⎥ a 2 ∫ a ∫ ⎝ 0 ⎠ ⎣⎢ 0 0 0 ⎦⎥

3 ⎡ ⎛ ⎞ ⎤ ⎛ Z ⎞ 1 2! Z 3! = ⎢2 − ⎜ ⎟ ⎥ ⎜ ⎟ ⎢ 3 4 ⎥ ⎝ a0 ⎠ 2 3 / 2 a0 ⎜ 3 / 2 ⎟ Z a0 ⎝ Z a0 ⎠ ⎣⎢ ( ) ( ) ⎦⎥ Hence

ψ * ψ dτ = 0 ∫ 1s 2 s all space

Example 3.2 An alternative form for determining the associated Laguerre polynomials is

2 n− l −1 i +1 ⎡ n + l !⎤ 2 l +1 ( ) i L (x) = −1 ⎣ ⎦ x n+ l ∑ ( ) i 0 n − l − 1− i ! 2l + 1+ i !i! = ( ) ( ) Use this relationship to find the Laguerre polynomial for n= 2 and l = 1 and check for consistency with a previous calculation for the same. African Virtual University 65

Solution: Substituting n = 2 and l = 1, we have

2 0 2 i +1 ⎡ 3 !⎤ 3 ( ) i (3!) L (x) 1 ⎣ ⎦ x ( 1) 6 3 = ∑(− ) = − = − i 0 0 − i ! 3+ i !i! 0!3!0! = ( ) ( ) Which agrees with a previous result for the same polynomial.

Example 3.3 The radial wave function describing an electron in a hydrogen like atom is given by

1/ 2 ⎛ 3 ⎞ 2Z / na n − l − 1 ! ( 0 ) ( ) − ρ / 2 l 2 l +1 R r = −⎜ ⎟ e ρ L ρ nl ( ) 3 n+ l ( ) ⎜ 2n⎡ n + l !⎤ ⎟ ⎝ ⎣( ) ⎦ ⎠ where ρ = 2Z / na r and a = 4πε h2 / µe2 ( 0 ) 0 0

What is the probability of finding a 1s electron at r > 0a ?

Solution: For a 1s electron n = 1, l = 0 and Z = 1, we have

na0 1 r = ρ = a0 ρ 2Z 2

3/ 2 ρ ⎛ 1 ⎞ − and the wavefunction R (r ) = 2 e 2 10 ⎜ a ⎟ ⎝ 0 ⎠

∞ * The probability will be given by P = ∫ R (r ) R (r )r 2 dr a 0 African Virtual University 66

* 3/ 2 3/ 2 2 ∞ ⎡ ⎛ 1 ⎞ ⎤ ⎡ ⎛ 1 ⎞ ⎤⎛ a ⎞ ⎛ a ⎞ ⎢ − ρ / 2 ⎥ ⎢ − ρ / 2 ⎥ 0 0 = ∫ 2⎜ ⎟ e 2⎜ ⎟ e ⎜ ρ⎟ d⎜ ρ⎟ 2 ⎢ ⎝ a0 ⎠ ⎥ ⎢ ⎝ a0 ⎠ ⎥⎝ 2 ⎠ ⎝ 2 ⎠ ⎣ ⎦ ⎣ ⎦

1 ∞ = ρ2 e− ρ dρ 2 ∫ 2

∞ 1 1 = ⎡e− ρ −ρ2 − 2ρ − 2 ⎤ = ⎡0 − e−2 −4 − 4 − 2 ⎤ = 5e−2 2 ⎣ ( )⎦ 2 ⎣ ( )⎦ 2 ➡ P = 0.6767.

Example 3.4 Determine < r > for a 2p wavefunction.

Solution

From ρ = 2Z / na r , for n = 2 and Z = 1, we have r = ρa . ( 0 ) 0 Hence,

∞ * r = R r rR r r 2 dr ∫ 21 ( ) 21 ( ) 0

3 3 ∞ ⎡ ρ ⎤ ⎡ ρ ⎤ 2 − 2 − 1 ⎛ 1 ⎞ 2 1 ⎛ 1 ⎞ 2 2 ⎢ e ⎥ a ⎢ e ⎥ a d a = ⎜ ⎟ ρ (ρ 0 ) ⎜ ⎟ ρ (ρ 0 ) (ρ 0 ) ∫ ⎢2 6 ⎝ a ⎠ ⎥ ⎢2 6 ⎝ a ⎠ ⎥ 0 ⎢ 0 ⎥ ⎢ 0 ⎥ ⎣ ⎦ ⎣ ⎦

a ∞ a = 0 ρ5e− ρ dρ = 0 5! = 5a . 24 ∫ 24 ( ) 0 0 African Virtual University 67

Vector Representation of Allowed Orbital Angular Momenta

The eigenvalue problem for the φ-dependent function is

ˆ LzΦΦΦ(()φ ) = mΦ(() φ ) where the quantum number m = 0, l-1, l-2,…(l-1),-l.

The eigenvalue problem for the θ ,φ dependent function is ( ) Lˆ2Y (θ ,φ) = Λh2Y (θ ,φ) where Λ = l (l + 1) , and the orbital quantum number l = 0, 1, 2, 3, …n -1 For a given value of l , the magnitude of orbital angular momentum r L = l( l + 1)h

The possible values of the component Lz can be represented schematically as the projections of a vector of length L = l(l + 1) on the z-axis. h We illustrate, in Fig.3.3, the allowed projections of orbital angular momentum for the cases of l = 1, 2, and 3.

Fig.3.3 Allowed Vector Projections for l = 1, 2 and 3 African Virtual University 68

The vector representing the total orbital angular momentum takes on any one of (2l+1) distinct orientations with respect to the chosen z-axis. It is allowed to have quantized components along the chosen axis. The property that there are a finite r number of distinct inclinations the vector L makes with any given axis, is sometimes called space quantization. The vector L should be thought of as covering a cone with the vector angle given by

m= L cosθ .

Spin

Definition of Spin All elementary particles, viz. protons, neutrons, electrons, etc. possess an Intrinsic Angular momentum called SPIN symbol S. There is no classical analogue that would permit a spin definition such as r r r S = r ∧ p s in a manner similar to the definition of the orbital angular momentum r r r L = r ∧ p .

1 The magnitude of S is h . Spin is an internal property of a particle, like mass or 2 charge. It constitutes an additional co-ordinate or degree of freedom in the quantum mechanical formulations. African Virtual University 69

Commutation Rules

These are exactly the same as those of orbital angular momentum,

⎡ ∧ ∧ ⎤ ∧ ie. ⎢S x ,S y ⎥ = ih S z , etc. ⎣ ⎦

∧ ∧ ⎡ 2 ⎤ ⎢S ,S z ⎥ = 0 , etc. ⎣ ⎦

⎡ ∧ ∧ ⎤ ∧ ⎢S z ,S + ⎥ = h S + , etc. ⎣ ⎦

Spin Wavefunctions or Spinors

These are denoted by sµ where s = 1/2 and µ = ±1/2 .

So that a spin up state will be denoted by

⎛1⎞ 1 1 χ = = up ⎜ 0⎟ 2 2 ⎝ ⎠ and a spin down state by

⎛ 0⎞ 1 1 χ = = ,− down ⎜1⎟ 2 2 ⎝ ⎠ ∧ ∧ 2 The spinors are simultaneous eigenfunctions of the spin operators S and S z :

1 1 1 ⎛ 1 ⎞ 1 1 3 1 1 ie. S 2 = + 1 h2 = h2 2 2 2 ⎝⎜ 2 ⎠⎟ 2 2 4 2 2

1 1 3 1 1 ➡ S 2 ,− = h2 ,− 2 2 4 2 2

African Virtual University 70

1 1 1 1 1 S = h z 2 2 2 2 2

1 1 1 1 1 and S ,− = − h ,− z 2 2 2 2 2

Thus the algebra of orbital angular momentum operators can be applied directly to that of the spin operators.

The Real Hydrogen Atom

In the discussion so far of the Hydrogen atom, a simplistic approach was adopted. Only the Coulomb interaction was included in the Hamiltonian. However, in a more realistic treatment, several corrections must be taken into account. These corrections include spin-orbit interaction, relativistic correction, and the nuclear hyperfine interaction. We now consider these effects in some detail.

Spin - Orbit Interaction

Angular Momenta and Magnetic Moments (Semi - Classical Picture) A current loop has associated with it a magnetic moment r r µ = IA r where I is the current and A is the vector area whose direction is perpendicular to the plane of the loop consistent with the right handed screw rule.

2 where A = π r And i = charge on electron × number of times per second electron passes a given point = ef where f is the frequency of rotation of the electron. Magnitude of the magnetic dipole moment

r µ = I A = (ef ) π r 2 ( ) African Virtual University 71

r Whose direction is opposite to the orbital angular momentum L because the electron has negative charge.

2m Now L = mvr = m(2π rf )r = 2mf π r 2 = µ e r e r Hence µ = − L . 3.15 2m

Since angular momentum is quantized we have

r ∧ l = mh l l In the first Bohr radius, m = 1and so Eq.3.15 becomes l

∧ r −eh l ∧ µl = = −µB l 3.16 2m

where µB is called the Bohr magneton and its value is given by

eh µB = 2m

It will be observed in Eq.3.16 that µl is directed antiparallel to the orbital angular momentum. The ratio of the magnetic moment to the orbital angular momentum is called the classical gyromagnetic ratio,

r µ e µ γ = rl = = B 3.17 l l 2m h

The spin angular momentum also has a magnetic moment associated with it. Its gyromagnetic ratio is approximately twice the classical value for orbital moments. ie.

r µs e γ = r = 3.18 s s m

African Virtual University 72

This means that spin is twice as effective as the orbital angular momentum in producing a magnetic moment. Eq.3.17 and 3.18 are often combined by writing

ge γ = 2m where the quantity g is called the spectroscopic splitting factor. For orbital angular momenta g = 1, for spin only g ≈ 2 (though experimentally g = 2.004). For states that are mixtures of orbital and spin angular momenta, g is non-integral.

1 Since s = h 2 the magnetic moment due to the spin of the electron is

r e h µs = γ s s = . = µB m 2

Thus, the smallest unit of magnetic moment for the electron is the Bohr magneton, whether one combines orbital or spin angular momentum. African Virtual University 73

The Larmor Frequency and The Normal Zeeman Effect

( Classical Treatment) We consider the effect of a weak magnetic field on an electron performing circular motion in a planar orbit. We assume the magnetic field is applied along the z axis and the angular momentum is oriented at an angle θ with respect to the z - axis, as shown in Fig.3.4 below.

Fig.3.4 The Precession of The Angular Momentum Vector in A Magnetic Field

r The torque on l is given by r r r τ = µ ∧ B 3.19 l l this is directed into the plane of the page, in the φ-direction. Now, the torque also equals the rate of change of the angular momentum, so we have r r dl r r r r τ l = = µl ∧ B = γ l l ∧ B 3.20 dt

But r dl = l sinθdφ

African Virtual University 74

so that the scalar form of Eq.3.20 becomes

dφ l sinθ . = γ lB sinθ 3.21 dt l

We define the precessional velocity by

dφ

ω L = dt

So that Eq.3.21 becomes

e ω = γ B = B 3.22 L l 2m

The angular velocity ωL is called the Larmor frequency. Thus, the angular momentum vector precesses about the z-axis at the Larmor frequency as a result of the torque produced by the action of a magnetic field on its associated magnetic moment. Using the Planck relation, the energy associated with the Larmor frequency is

ehB ∆E = ±ω L h = ± = ±µB B 3.23 2m where the signs refer to the sense of the rotation. It will be observed that this energy difference is the potential energy of a magnetic dipole whose moment is one Bohr magneton. Recall that the dipolar energy is given by r r ∆E = −µ.B In Eq.3.23, the positive sign corresponds to antiparallel alignment while the negative sign ( lower energy ) indicates parallel alignment. The overall effect of this energy associated with the Larmor frequency is that, if the energy of an electron having a moment µB is E0 in the absence of an applied field, then it can take on one of the energies

E ± µ B 0 B in a magnetic fieldB . African Virtual University 75

Thus, in a collection of identical atomic particles of the type discussed, a magnetic field produces a triplet of levels, called aLorentz triplet whose energies are E0, and E ± µ B . 0 B

This phenomenon is known as the Normal Zeeman effect. The Zeeman effect is in fact more complex than presented by the classical treatment. The electron spin is excluded in the classical model. Thus when a magnetic field is applied the spin and orbital angular moments will precess. The resulting energy level splittings cannot be explained classically and so require a quantum mechanical treatment. As a consequence of this inexplicable be- haviour, the more general Zeeman effect, including spin was historically misnamed as the anomalous Zeeman effect.

(a) Single Transition without an (b) Five transitions with an applied Magnetic Field applied external magnetic field

Fig.3.5 Transitions With and without a Magnetic Field African Virtual University 76

The Spin-Orbit Interaction - (Quantum Mechanical Treatment) In the introductory inclusion of spin in the Schrodinger wave function, it is assumed that the spin coordinates are independent of the coordinates of the configuration space. Thus, the total wavefunction is written as a product function

iE t / Ψ = ψ (r,θ,φ).χ(spin).e− n h total nlm

iE t / Ψ = R .e− n h l,m s,m 3.24 total nl l s

The assumption made above implies that there is no interaction between L and S,

⎡ ∧ ∧ ⎤ ie. ⎢ L , S ⎥ = 0 ⎣⎢ ⎦⎥ In this case, Ψ is an eigenfunction of both L and S and so m and m are good total z rz r l s quantum numbers; in other words, the projections of L and S are constants of the motion. r r But in reality there is an interaction by L and S called the Spin-Orbit interaction, r r expressed in terms of the quantity L . S . r r r r Since L . S does not commute with either L or S , Eq.3.24 is no longer correct and ml and ms cease to be good quantum numbers. We picture the spin-orbit interaction as the stationary spin magnetic moment interacting with the magnetic field produced by the orbiting nucleus. In the rest frame of the electron, there is an electric field r Ze ∧ (cgs) ε = 2 r r and a magnetic field

r ∧ r j ∧ r (cgs) H e = 2 r African Virtual University 77

∧ where r is directed from the nucleus toward the electron. r Assuming that v is the velocity of the electron in the rest frame of the nucleus, the current produced by the nuclear motion is r Ze r j = − v c in the rest frame of the electron.

Then

r ∧ r Ze v ∧ r 1 r r v H e = − 2 = − ∧ ε c r c

The spin moment of the electron precesses in this field at the Larmor frequency

r r e r r ω = γ H = − v ∧ ε 3.25 e e m c2 0 with the potential energy r v r r E = −µ .H = −ω .S 3.26 e s e e

Eqs.3.25 and 3.26 are valid in the rest frame of the electron.

Transformation to the rest frame of the nucleus introduces a factor of ½ - called the Thomas factor. [ This can be shown by calculating the time dilation between the two rest frames]. Hence, an observer in the rest frame of the nucleus would observe the electron to precess with an angular velocity of

r e r r ω = − v ∧ ε 3.27 L 2m c2 0 African Virtual University 78

and by an additional energy given by

1 r r 3.28 ∆E = − ω e .S 2

Eqs.3.27 and 3.28 can be put in a more general form by restricting V to be any central potential with spherical symmetry. So that

r ∧ ∂V r F = − r = −eε ∂r and so r r 1 ∂V r ∧ 1 1 ∂V r r 1 1 ∂V r v ∧ ε = v ∧ r = v ∧ r = − L e ∂r e r ∂r em r ∂ r 0 Eq. 3.27 then becomes

r 1 1 ∂V r ω = + L L 2m2 c2 r ∂ r 0 and the additional energy

1 1 ∂V r r ∆E = + L.S 3.29 2m2 c2 r ∂r 0 The scalar product r r L.S = mhs l For spin = ½,

r r 1 1 2 L.S = ml h. h = ml h 2 2 African Virtual University 79

The energy splitting then becomes

h2 m 1 ∂V ∆E = l 4m2 c2 r ∂r 0 For the Coulomb potential the energy splitting can be approximated by

D2 mZe2 c l 3.30 ∆E = 3 r where

h λ = is the Compton wavelength and c m c 0

h λ c D = or . c m c 2π 0 A useful result in computation is quoted without proof. The average value of 1/r3

1 Z 2 ie. = 3.31 r 3 a2 n2 l(l + 1/ 2)(l + 1) 0 for l ≠ 0.

So that the energy splitting becomes

D2m Z 3e ∆E = c l a2n2l( l + 1 / 2)( l + 1) 0 for l ≠ 0. African Virtual University 80

Angular Momentum Coupling Schemes

We have so far considered only the coupling of the spin and orbital momentum of a single electron by means of the spin-orbit interaction. We now consider the case of two electrons for which there are four constituent momenta.

The j - j Coupling Model

This model assumes that the spin-orbital interaction dominates the electrostatic interactions between the particles. Thus, we write for each particle r r r J = L + S 1 1 1

r r r and J = L + S 2 2 2 r r The total angular momentum is obtained by combining J and J : 1 2 r r r J = J + J 1 2 and

j = j + j , j + j − 1 ,...... , j − j 1 2 1 2 1 2

We illustrate j-j coupling by applying it to two inequivalent p-electrons. For each electron

1 3 j1 = j2 = or 2 2

Then the possible ways of combining these are shown in the Table 3.1. African Virtual University 81

Table 3.1 j-j Coupling of Two Inequivalent p - Electrons

j1 j2 j Spectral Terms Number of States in a Magnetic Field

3/2 3/2 3,2,1,0 ⎛ 3 3⎞ 16 , ⎝⎜ 2 2⎠⎟ 3,2 ,1,0

3/2 1/2 2,1 ⎛ 3 1⎞ 8 , ⎝⎜ 2 2⎠⎟ 2 ,1

1/2 1/2 1,0 ⎛ 1 1⎞ 4 , ⎝⎜ 2 2⎠⎟ 1,0

1/2 3/2 2,1 ⎛ 1 3⎞ 8 , ⎝⎜ 2 2⎠⎟ 2 ,1

36 states

In a weak magnetic field, each state of a given j will split into (2j+1) states corresponding to the allowed values of mj. Although the j-j coupling is used extensively for the description of the nuclear states observed in nuclear spectroscopy, it is not appropriate for many atomic systems because of the strong electrostatic and other interactions between the two electrons. The Russell-Saunders Coupling Scheme The Russell-Saunders model has been more successful in accounting for atomic spectra of all but the heavier atoms. The model assumes that, the electrostatic interaction, including exchange forces, between two electrons dominates the spin-orbit interaction. In this case, the orbital momenta and the spins of the two electrons couple separately to form r r r L = L + L 1 2 r r r and S = S + S 1 2 African Virtual University 82

The total angular momentum is given, as before, by r r r J = L + S For two inequivalent p-electrons we have: l = 2, 1, or 0 and s = 1 or 0. For each l and s, the j-values are

l + s , l + s − 1 ,...... , l − s

and for each j value there are (2j+1) values of mj. The combinations are given in the table below. It will be observed that, although the number of states is once again 36 in a weak magnetic field, their energies are not the same as those in the j-j coupling scheme. Table 3.2 : Russell-Saunders Coupling of Two Inequivalent p-Electrons Number of States in a l s j Spectral Terms Magnetic Field

2 1 3,2,1 3 D 15 1,2 ,3

1 D 2 0 2 2 5

1 1 2,1,0 3 P 9 0 ,1,2

1 P 1 0 1 1 3

3S 0 1 1 1 3

1 0 0 0 S0 1 36 states African Virtual University 83

The Lande g-Factor and The Zeeman Effect

The orbital and spin contributions to the magnetic moment are given by

r g e r ∧ µ = − l L = − g µ l l + 1 l l 2m l B ( ) 0

r g e r ∧ and µ = − s S = − g µ s s + 1 s s 2m s B ( ) 0 where gl = 1 and gs = 2.004 ≈ 2. r r Now, when L and S are coupled, we have r r r J = L + S r r r and µ = µ + µ l s

r µ r r µ = B (L + 2S ) 3.32 h r r It is evident from the expressions for J and µ that the total magnetic moment is not in general collinear with the total angular momentum, as illustrated in Fig.3.6.

Fig.3.6 The Total Magnetic Moment is not Collinear with the Total Angular Momentum African Virtual University 84

r r r r r Since L and S precess about J , it is apparent that µ also precesses about J . r r However, the effective magnetic moment, that is the component of µ along J , maintains the constant value,

r r r r r r r r r r r r µ.J µ L.J + 2S.J µ L.(L + S) + 2S.(L + S) µ = r = − B r = − B r j J h J h J

r r µ L2 + 2S 2 + 3L.S = − B r h J

2 2 3 2 2 2 µ L + 2S + (J − L − S ) = − B 2 r h J

r r 2 r r (where J 2 = L + S = L2 + 2L.S + S 2 ) ( )

µ (L2 + 4S 2 + 3J 2 − 3L2 − 3S 2 ➡ µ = − B r j h 2 J

µ r ⎛ 3J 2 + S 2 − L2 ⎞ µ = − B J j h ⎜ 2J 2 ⎟ ⎝ ⎠ µ r ⎛ J 2 + S 2 − L2 ⎞ µ = − B J 1+ j h ⎜ 2J 2 ⎟ ⎝ ⎠

µB ⎡ j( j + 1) + s(s + 1) − l(l + 1) ⎤ = − h j( j + 1).⎢1+ ⎥ h 2 j( j + 1) ⎣ ⎦ ⎡ j( j + 1) + s(s + 1) − l(l + 1) ⎤ j( j 1). 1 ∴ µ j = −µB + ⎢ + ⎥ 2 j( j + 1) ⎣ ⎦ African Virtual University 85

We define theLande g factor as

j( j + 1) + s(s + 1) − l(l + 1) g = 1+ 3.33 2 j( j + 1) and the effective magnetic moment becomes

µ = gµ j( j + 1) 3.34 j B

For zero spin, Eq.3.33 reduces to the classical case of g = 1 and for l = 0, g = 2. Now we are in a position to account for the so-called Anomalous Zeeman effect. r r In a weak magnetic field, the angular momentum J will precess about B such that the r projection of J along the field direction will be one of the allowed values of m h . j The corresponding magnetic moment along the field direction, taken to be the z- direction, will then be

µ = − gµ m z B j having a magnetic dipolar energy of

E = gm µ B 3.35 j B

In the classical case, g = 1, but in Eq.3.35, g depends upon the quantum numbers l, s, and j. We illustrate, in Table 3.3, by calculating the g-factor for an electron in a p-state and an s-state.

Table 3.3: g-Factor Calculations orbital state l j g p 1 3/2 4/3 p 1 1/2 2/3 s 0 1/2 2 African Virtual University 86

In a magnetic field B, such that µBB is less than the spin-orbit energy, j and mj are good quantum numbers and the energies of the states split as shown in the Table 3.4 and in Fig.3.7 below:

Fig.3.7 Zeeman Splitting for p and s States

Table 3.4: Calculations of Zeeman Splittings

Orbital state j mj g ∆E = gm j in units of µ B p 3/2 3/2 4/3 2 B p 3/2 1/2 4/3 2/3 p 3/2 -1/2 4/3 -2/3 p 3/2 -3/2 4/3 -2 p 1/2 1/2 2/3 1/3 p 1/2 -1/2 2/3 -1/3 s 1/2 1/2 2 1 s 1/2 -1/2 2 -1 African Virtual University 87

Thus, the so-called “anomalous” Zeeman effect is what would normally be expected for an electron having half-integral spin in a weak magnetic field. The “normal” or classical Zeeman effect cannot occur for a single electron in a weak magnetic field because of the spin term in Eq.3.33. However, in atoms in which the spins are paired so that the total spin is zero, the g-value for all spectroscopic states is the classical value and only three spectral lines are observed.

Atomic Spectra

When fine structure is ignored, it turns out that all wavelengths of atomic hydrogen are given by a single empirical formula, the Rydberg formula:

⎛ ⎞ 1 1 1 −3 = R ⎜ − ⎟ where R = 1.0967758 × 10 Å-1 λ ⎜ n2 n2 ⎟ ⎝ f i ⎠

Where nf = 1 and ni = 2,3,4.. gives the Lyman series (Ultra violet)

nf = 2 and ni = 3,4,5.. gives the Balmer series (Visible)

nf = 3 and ni = 4,5,6.. gives the Paschen series (infrared)

nf = 4 and ni = 5,6,7..gives the Brackett series(Far infrared) etc.

Fig. 3.8 Hydrogen Spectrum Source:http://www2.kutl.kyushu.ac.jp/seminar/MicroWorld1_E/Part4_E/P43_E/ Bohr_theory_E.htm African Virtual University 88

Continuous, emission, and absorption spectra

Fig.3.9 Continuous, Emission and Absorption Spectra Source: http://csep10.phys.utk.edu/astr162/lect/light/absorption.html

A continuous spectrum results when the gas pressure is high so that the gas emits light at all wavelengths. An absorption spectrum results when light passes through a cold and rarefied gas. An absorption spectrum is essentially a reversed emission spectrum of the same element that produced tha emission spectrum. The absorption and emission spectra of hydrogen are particularly useful is astrophysics because hydrogen is a adominant element in the universe.

The Pauli Exclusion Principle

To explain certain aspects of atomic spectra, Wolfgang Pauli determined that no 2 electrons can have all 4 quantum numbers alike. This is called the Pauli Exclusion Principle. The Pauli exclusion principle suggests that only two electrons with opposite spins can occupy an atomic orbital. Stated another way, no two electrons have the same 4 quantum numbers n, l, m, s. Pauli’s exclusion principle can be stated in some other ways, but the idea is that energy states have limited room to accommodate electrons. A state accepts two electrons of different spins. African Virtual University 89

In full orbitals (orbitals containing 2 electrons of opposite spin) one electron must be spin up, and the other spin down, and the electrons are said to be paired.

Electronic Configurations For Atoms With More Than One Electron

The Schroedinger wave equation was developed initially for hydrogen, an atom with only one electron. In such a case, all orbitals in each energy level have the same energy and are called Degenerate In atoms with more than one electron, the electrons repel each other, also the effective nuclear charge varies with the atomic number and the inner shell electrons screen the outer ones. As a result, the orbital energies are shifted somewhat as shown in the figure below. Variation of energy levels for atomic orbitals of some elements H Li Be B C N O F _2s_ _ _2p _ _ _ 2p _ 1s _ 2s _ _ _ 2p _ _ _ 2p _ 2s _ _ _ 2p _ 1s _ _ _ 2p _ 2s _ _ _ 2p _ _ 1s _ 2s _ _ _ 2s

_ 1s _ 2s

_ 1s

_ 1s Source: http://www.science.uwaterloo.ca/~cchieh/cact/c120/eleconfg.html

African Virtual University 90

The lower energy orbitals are filled before electrons are added to the next highest orbital.

Hund’s Rule

Hund’s rule suggests that electrons prefer parallel spins in separate orbitals of subshells. This rule guides us in assigning electrons to different states in each subshell of the atmic orbitals. In other words, electrons fill each and all orbitals in the subshell before they pair up with opposite spins. Pauli exclusion principle and Hund’s rule guide in figuring the electron configurations for all elements.

Task 3.1

1. Spin-orbit coupling splits all states into two except the s state. Why ? 2. Explain why the effective radius of helium atom is less than that of a hydrogen atom.

Formative Evaluation 3.1

1. Determine the shortest and longest wavelengths of Lyman series of hydrogen. 2. The study of atomic spectra was a kind of an industry towards the end of 19th century and at the beginning of the 20th century. Discuss 3. The longest wavelength in the Lyman series for hydrogen is 1215 Ǻ. Calculate the Rydberg constant. 4. Electrons of energy 12.2 eV are fired at hydrogen atoms in a gas discharge tube. Determine the wavelengths of the lines that can be emitted by the hydrogen. 5. Determine the magnetic moment of an electron moving in a circular orbit of radius r aout a proton. 6. Use the results from quantum mechanics to calculate the magnetic moments that are possible for n = 3. 7. Determine the normal Zeeman splitting of the cadmium red line of 6438 Ǻ when the atoms are placed in a magnetic field of 0.009 T. 8. Express L.S. in terms of J, L, and S. Given L = 1 and S = ½, calculate the possible values of L.S. 9. A beam of electrons enters a uniform magnetic field B = 1.2 T. Find the energy difference between electrons whose spin are parallel and anti parallel to the magnetic field. African Virtual University 91

Assignment 3.1

1. The normalized wavefunction for the ground state of a hydrogen-like atom with nuclear charge Ze has the form u(r ) = Aexp(−βr ) where A and β are constants and r the distance between the electron and the nucleus. Show the following:

β 2 Z h2 4πε (a) A2 = (b) β = where a = 0 π a 0 m e2 0 e

2 m ⎛ e2 ⎞ (b) the energy E Z 2 E , where E e = − 0 0 = 2 ⎜ ⎟ 2h ⎝ 4πε ⎠ (c) the expectation values of the potential and kinetic0 energies are 2E and -E respectively,

3a a (d) the expectation value r = 0 and (f) the most probable value of r is 0 2Z Z

2. For the state ψ210 of hydrogen atom, calculate the expectation values < r >, <1/ r> and and hence find the expectation values of the kinetic and potential energies. 3. Determine the shortest and the longest wavelengths of the Lyman series of the

hydrogen atom. ( Ans. λmax = 1215 Å and λmin = 912 Å) 4. Determine the second line of the Paschen series for hydrogen. (Ans. 12,820 Å) 5. Electrons of energy 12.2 ev are fired at hydrogen atoms in a gas discharge tube. Determine the walengths of the lines that can be emittted by the hydrogen. (Ans.6563 Å, 1215 Å 1026 Å )

r r 1 ⎡ ⎤ 2 6. Show that L .S = ⎣J (J + 1) − L ( L + 1) − S (S + 1)⎦ h . 2 r r 7. Calculate the possible values of L .S for L = 1 and S = ½. 8. Calculate the energy difference between the electrons that are parallel and antiparallel with a uniform magnetic field b = 0.8 Twhen a beam of electrons moves

eh perpendicular to the field. ( Hint ∆E = B ∆ms ) m African Virtual University 92

Activity 4: X-Rays

Specific Teaching and Learning Objectives

• Explain the atomic origin of X-rays • Distinguish characteristic X-Rays from Bremstrahlung radiation • Moseley’s relation and its use in solving problems • Use Bragg’s rule to solve problems

Summary of the Learning Activity

In Learning activity 4, you begin by reflecting on the origin of x-rays from an historical perspective. You learn further that each element has its own characteristic x-ray spectrum. This property is akin to a similar property you learnt in the previous unit and consequently has similar scientific and technological implications. African Virtual University 93

List of Required Readings

Reading 1: X-Ray Production

Complete reference : http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/ xtube.html Abstract :This article is part of a comprehensive series of articles on the physics of x-rays, covering all objectives of this Learning Activity. The opening article discusses x-ray production and the links discuss bremsstrahlung radiation, characteristic x-rays, Moseley law and X-ray diffraction. Rationale: The presentation by Hyperphysics is as always sharp and to the point. It is an essential reading. Date Consulted: June 2007

Reading 2: The Origin of Characteristic X-Rays

Complete reference : http://www4.nau.edu/microanalysis/Microprobe/Xray-Cha- racteristic.html Abstract : This article discusses characteristic x-ray production. The links to this page discuss continuum x-rays, electron shells, electron transitions , Moseley’s Law and other topics beyond the requirements of this course.. Rationale: This is good material and relevant to this course. Date consulted: June 2007

Reading 3:X-Ray Diffraction .

Complete reference : http://www.physics.upenn.edu/~heiney/talks/hires/whatis. html#SECTION00011000000000000000 Date Consulted: Junel 2007 Abstract : In this article, x-ray is concisely presented. Rationale: The article covers the contents of this activity

Reading 4: X-Ray Diffraction

Reference link: http://e-collection.ethbib.ethz.ch/ecol-pool/lehr/lehr_54_folie2.pdf Complete reference: http://www.google.com/search?q=cache:qLs7iI81agwJ:e-collection.ethbib.ethz.ch/ecol-pool/lehr/lehr_54_folie2.pdf+X-RAY+MOSLEY’S+LAW Abstract: This article contains Power Point Slides on practical aspects of X-Ray Diffraction, X-Ray Tube, X-Ray Spectrum and Mosley’s Law. To access the article start with the complete reference and then click on reference link. Rationale: The material is relavant to this activity. Please read it. African Virtual University 94

List of Relevant MM Resources

Reference: http://ie.lbl.gov/xray/mainpage.htm Date consulted: April 2007 Description: X-ray spectra of elements on the Periodic table. Spectra are drawn with a jave applet. Reference: http://www.eserc.stonybrook.edu/ProjectJava/Bragg/ Date consulted: April 2007 Description: A java applet of Bragg’s law and Diffraction. You should vary alter- nately the x-ray wavelength λ, the Bragg angle θ and the interplanner distance d and for each variation of a parameter study the effect thereof. Reference : http://www.mineralogie.uni-wuerzburg.de/crystal/teaching/iinter_ bragg.html Date consulted: April 2007 Description: An interactive tutorial Bragg Diffraction. Answer the questions for each interaction.

List of Relevant Useful Links

Resource #1

Title:-A Histrorical Overview of the Discovery of X-rays URL:- http://www.yale.edu/ynhti/curriculum/units/1983/7/83.07.01.x.html Screen Capture:

THE DISCOVERY OF X-RAYS In october of 1895, Wilhelm Conrad Ršntgen (1845-1923) who was professor of physics and the director of the Physical Institute of the University of Wurburg, became interested in the work of Hillorf, Crookes, Hertz, and Lenard. The previous June, he had obtained a Lenard tube from Muller and had already repeated some of the original experiments that Lenard had created. He had observed the effects Lenard had as he produced cathode rays in free air. He became so fascinated that he decided to forego his other studies and concentrate solely on the production of cathode rays.

Abstract: The article provides an historical presentation of the events that lead to the discovery of x-rays. It begins with the work by Dr.William Gilbert on magnetism in 1600 and culminates with the discovery of x-rays in 1895 by Roentgen. Rationale: The material is easy reading but relevant. It is good for you. Date consulted: April 2007 African Virtual University 95

Resource #2

Title:- Notes on the X-Ray Tube URL http://en.wikipedia.org/wiki/X-ray_tube Screen Capture:

Coolidge side-window tube (scheme) Abstract:- This is an encylopediac presentation of x-ray tubes and x-ray genera- tion. Rationale: The material including the links there in are most relevant to this Lear- ning Activity. Date consulted: April 2007 African Virtual University 96

Resource #3

Title: X-ray spectra of some elements on the Periodic Table URL:- http://ie.lbl.gov/xray/mainpage.htm Screen Capture: Germanium X-Ray Spectrum

Abstract: X-ray spectra of over 60 elements are interactively plotted. Click on the perioduc table any italicised element and then follow the instructions to ana- lyse the x-ray spectrum at hand. Rationale: The material is very good and relevant to this learning activity. Date consulted: April 2007 African Virtual University 97

Resource #4

Title:- Basic Diffraction Physics URL http://www-structmed.cimr.cam.ac.uk/Course/Basic_diffraction/Diffraction. html#diffraction Screen Capture:

Abstract: Basic x-ray diffraction physics is reviewed from a dififferent angle. Rationale: The material is good and relevant. Date consulted: April 2007

Resource #5

Title: URL: http://www.tulane.edu/~sanelson/eens211/x-ray.htm Screen Capture:

Abstract: A detailed account on X-Rays and x-ray production, continuous and characteristic x-ray spectra, x-ray diffraction and Bragg’s law. Rationale: This article is part of a lecture series on Earth & Environmental Sciences given by Prof. Stephen A. Nelson of University of Torornto, Canada. The material is good and relevant. Date consulted: April 2007 African Virtual University 98

Detailed Description of the Activity (Main Theoretical Elements)

Introduction

4.1: X-Ray tube

Adapted from Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/X-ray_ tube An X-ray tube is a vacuum tube designed to produce X-ray photons. The first X-ray tube was invented by Sir William Crookes. The Crookes tube is also called a discharge tube or cold cathode tube. A schematic x-ray tube is shown below.

Fig.4.1 A Schematic Diagram of an X-Ray Tube The glass tube is evacuated to a pressure of air, of about 100 pascals, recall that atmospheric pressure is 10 × 105 pascals. The anode is a thick metallic target, it is so made in order to quickly dissipate thermal energy that results from bombardment with the cathode rays.. A high voltage, between 30 to 150 kV, is applied between the electrodes; this induces an ionization of the residual air, and thus a beam of electrons from the cathode to the anode ensues. When these electrons hit the target, they are slowed down, producing the X-rays. The X-ray photon-generating effect is generally called the Bremsstrahlung effect, a contraction of the German “brems” for braking, and “strahlung” for radiation. The radiation energy from an xray tube consists of discrete energies constituting a line spectrum and a continuous spectrum providing the background to the line spectrum. African Virtual University 99

Fig.4.2 A More Detailed X-Ray Tube Head Two Types of X-Rays

The incident electrons can interact with the atoms of the target in a number of ways.

Continuous Spectrum

When the accelerated electrons (cathode rays) strike the metal target, they collide with electrons in the target. In such a colission part of the momentum of the incident electron is transferred to the atom of the target material, thereby loosing some of its kinetic energy, ∆K . This interaction gives rise to heating of the target. The projectile electron may avoid the orbital electrons of the target element but may come sufficiently close to the nucleus of the atom and come under its influence. The projectile electron we are tracking is now beyond the K-shell and is well within the influnce of the nucleus. The electron is now under the influence of two forces, namely the attractive Coulomb force and a much stronger nuclear force. The effect of both forces on the electron is to slow it down or decelerate it. The electron leaves the region of sphere of influence of the nucleus with a reduced kinetic energy and flies off in a different direction, because the vector velocity has changed. The loss in kinetic energy reappears as an x-ray photon, as illustrated in Fig. 4.3. During deceleration, the electron radiates an x-ray photon of energy hv = ∆K = K i − K f . The energy lost by incident electrons is not the same for all electrons and so the x-ray photons emitted are not of the same wavelength. This process of x-ray photon emission through deceleration is called Bremsstrahlung and the resulting spectrum is continuous but with a sharp cut-off wavelength. The minimum wavelength corresponds to an incident electron losing all of its energy in a single collision and radiating it away as a single photon. African Virtual University 100

hc If K is the kinetic energy of the incident electron, then K = hν = . The cut off λ wavelength depends solely on the accelerating voltage. min

hc hv = = eV where V is the accelerating voltage. max λ min

Fig.4.3 Deceleration of an Electron by a Positively Charged Nucleus

Characteristic X-Ray Spectrum

Because of the large accelerating voltage, the incident electrons can (i) excite electrons in the atoms of the target. (ii) eject tightly bound electrons from the cores of the atoms. Excitation of electrons will give rise to emission of photons in the optical region of the electromagnetic spectrum. However when core electrons are ejected, the subse- quent filling of vacant states gives rise to emitted radiation in the x-ray region of the electromagnetic spectrum. The core electrons could be from the K-, L- or M- shell. If K-shell (n=1) electrons are removed, electrons from higher energy states falling into the vacant K-shell states, produce a series of lines denoted as Kα , Kβ , ... as shown Fig.4.4. Transitions to the L shell result in the L series and those to the M shell give rise to the M series, and so on. Since orbital electrons have definite energy levels, the emitted x-ray photons also have well defined energies. The emission spectrum has sharp lines characteristic of the target element.

African Virtual University 101

Upon a close investigation of the x-ray lines L, M series and above shows that the lines are composed of a number of closely spaced lines as shown in Fig.4.5. split by the spin orbit interaction,.

Fig.4.4 X – Ray Transitions without Fine Structure

Fig.4.5 X – Ray Transitions with Fine Structure African Virtual University 102

Not all transitions are allowed. Only those transitions which fulfil the following selection rule are allowed: ∆l = ±1.

Fig.4.6 Characteristic X-Ray Emission Using a Molybdenum Target

The Moseley Relation

From experiment Mosley was able to show that the characteristic x-ray frequencies increase regularly with atomic number Z, satisfying the relation

ν1/ 2 = A Z − Z 4.1 ( 0 ) where Z is the atomic number of the target material and A and Z0 are constants that depend upon the particular transition being observed. The term (Z-Z0) is called the effective nuclear charge as seen by the electrons making transition to a given shell.

The frequency of the Kα line can be calculated approximately using Bohr atomic theory. The wavelength of lines emitted by hydrogenic atoms is given by the Rydberg formula.

⎛ ⎞ 1 2 1 = RZ ⎜ ⎟ 4.2 λ ⎜ n2 − n2 ⎟ ⎝ l u ⎠ where nu and nl are principal quantum numbers of the upper and lower states of the transition, Z is the atomic number of the one-electron atom. African Virtual University 103

For the Kα line the effective nuclear charge is (Z-1), nl = 1 and nu = 2, so that Eq.4.2 becomes

c 2 ⎛ 1 1 ⎞ ν = = cR Z − 1 − K α ( ) ⎜ 2 2 ⎟ λ ⎝ 1 2 ⎠ 3cR 2

ν K α = ( Z − 1) 4.3 4 A plot of ν1/ 2 versus Z yields a straight line. Eqn 4.3 is another way of expressing K α Moseleys relation.

X-Ray Diffraction

A plane of atoms in a crystal, also called a Bragg plane, reflects x-ray radiation in exactly the same manner that light is reflected from a plane mirror, as shown in Fig.4.7.

Fig.4.7 X-Ray Reflection From a Bragg Plane African Virtual University 104

Reflection from successive planes can interfere constructively if the path difference between two rays is equal to an integral number of wavelengths. This statement is called Bragg’s law.

Fig 4.8 Diffraction of X-Rays from Atomic Planes

From Fig. 4.8, AB = 2dsinθ so that by Bragg’s law, we have 2dsinθ = nλ 4. 4 where in practice, it is normal to assume first order diffraction so that n = 1. A given set of atomic planes gives rise a reflection at one angle, seen as a spot or a ring in a diffraction pattern also called a diffractogram. By varying the angle theta, the Bragg’s Law conditions are satisfied by different d- spacings in polycrystalline materials. Plotting the angular positions and intensities of the resultant diffracted peaks of radiation produces a pattern, which is characteristic of the sample. Where a mixture of different phases is present, the resultant diffractogram is formed by addition of the individual patterns. Based on the principle of X-ray diffraction, a wealth of structural, physical and chemical information about the material investigated can be obtained. A host of application techniques for various material classes is available, each revealing its own specific details of the sample studied. I am illustrating the X-Ray diffraction technique using a part of our own work on mineralogical studies of local minerals. I am presenting x-ray diffractograms of selec- ted iron sulphide samples from the Lake Victoria gold field, Tanzania. The technique used here is that of Powder Method whereby the sample is ground into powder and rotated in an x-ray beam. At any one orientation, only planes whose reflected x-rays interfere constructively will give rise to a signal in the detector. By rotating the sample in the x-ray beam a whole set of crystal planes will be brought into view. African Virtual University 105

The x-ray diffraction of the Nyamlilima sample, Fig.4.9, reveals that the sample comprises the mineral phases quartz, monoclinic pyrrhotite and pyrite whereas the sample from Mwamela, Nzega, Fig.4.10, consists of only hexagonal pyrrhotite and quartz mineral phases. Admittedly, the analysis of the Nyamlilima sample is incom- plete; there is an intense and yet unidentified reflection at θ 2 ≈ 30°.

The chemistry of the mineral phases mentioned above is as follows: quartz is SiO2, pyrite is FeS2 ( this mineral is notorious for deceiving inexperienced gold seekers and so it also bears the nickname of a poor man’s gold), pyrrhotite also known as magnetic pyrite, the chemical composition varies from FeS to Fe0.8S, where the monoclinic phase is the most ordered and the hexagonal phase the least ordered.

Fig.4.9 A Diffractogram of an Iron Sulphide Sample from Nyamlilima, Geita, Tanzania

Fig.4.10 A Diffractogram of an Iron Sulphide Sample from Mwamela, Nzega, Tanzania African Virtual University 106

Formative Evaluation 4

1. In the discovery of x–rays, it is said that “Roentgen replaced the screen with a photographic plate and employed his wife Bertha to place her hand on the photographic plate while he directed the rays at it for fifteen minutes. Frau Roentgen was taken back and somewhat frightened by the first x-ray plate of a human subject which enabled her to see her own skeleton”. Discuss. 2. A TV tube operates with a 24 kV accelerating voltage, what is the maximum

energy for x-rays from the TV set? Calculate the wavelength λmin for the continuous spectrum of x-rays emitted when 35 keV electrons fall on a molybdenum target. 3. Determine the wavelength of the K α line for molybdenum, Z = 42. 4. Determine the electronic configuration for an atom with Z = 20. 5. Obtain the ground state terms of He and Li. 6. In a cubic crystal, using x-rays of λ = 1.5 Å, a first order (100) planes reflection is observed at a glancing angle of 18°. What is the distance between the (100) planes.

Assignment 4

1. A TV tube operates with a 20 kV accelerating potential. What is the maximum energy of the x-rays produced? 2. The accelerating voltage of an x-ray tube is 60 kV. Calculate the minimum x-ray walength generated by tube. 3. Determine the wavelength of the Kα line for a molybdenum Z = 42 target. (Ans. λ = 0.721 Å) 4. An experiment measuring the Kα lines for iron and copper yields the following data:Fe : 1.94 Å and Cu : 1.54 Å. Calculate the atomic number of each of the elements. 5. In Figs.4.9 and 4.10, given that the x-ray wavelength λ = 1.54 Å and n = 1, calculate the d-values for the planes responsible for the reflection of the most intense ( 100%) lines of pyrrhotite and quartz. 6. A 0.083 eV neutron beam scatters from an unknown sample and a Bragg reflection peak is observed centred at 22°. Calculate the inter planar spacing.

Teaching the Content in Secondary School 4

The material in this Learning Activity can easily be adapted and taught to Secondary school students. African Virtual University 107

XI. Compiled List of all Key Concepts (Glossary)

Coulomb Scattering: A collision of two charged particles in which the Coulomb force is the dominant interaction. Source: http://www.answers.com/topic/coulomb-scattering Impact parameter : the shortest distance of a particle trajectory from the primary vertex in the transverse plane to the point where the particle decays. Source: http://hep.uchicago.edu/cdf/cdfglossary.html Scattering cross section - The area of a circle of radius b, the impact parameter. Dfferential Scattering Cross section: is defined by the probability to observe a scattered particle in a given quantum state per solid angle unit, such as within a given cone of observation. Source: http://en.wikipedia.org/wiki/Cross_section_(physics) Planetary orbit - the path that a planet makes around the sun under the influence of gravitational force. Source http://en.wikipedia.org/wiki/Orbit. Atomic shell – an arrangement of electrons in an atom, in compliance with appropriate physical laws. Bohr radius - The size of a ground state of hydrogen atom as calculated by Niels Bohr using a mix of classical physics and quantum mechanics. Source: http://education.jlab.org/glossary/bohrradius.html Rydberg constant - a constant that relates atomic spectra to the spectrum of hydrogen. Its value is 1.0977 × 107 per metre. Source: http://www.tiscali.co.uk/reference/encyclopaedia/hutchinson/m0025952.html Rydberg formula : is an impirical relation that gives all wavelengths of atomic Hydrogen. Quantum number- A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. Each quantum number specifies the value of a conserved quantity in the dynamics of the quantum system. Source: en.wikipedia.org/wiki/Quantum_number Quantisation of Angular Momentum- The Angular momentum quantum number can take only certain values in multiples of h . This phenomenon is also referred to as space quantisation African Virtual University 108

Angular Momentum Coupling:- The orbital and spin angular momentum of a particle can interact through spin-orbit interaction. The procedure of construc- ting eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. Source: http://en.wikipedia.org/wiki/Angular_momentum_coupling Stoke’s Law - an expression for the frictional force exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous r r r viscous fluid: F = −6π rηv where: F is the frictional force, r is the radius of the r particle, η is the fluid viscosity, and v is the particle’s velocity. Source: http://en.wikipedia.org/wiki/Stokes’_law Bremsstrahlung radiation – Radiation ( X-rays) produced by slowing down energetic electrons (or any charged particles) upon impact on a target (or an absor- ber). Source: http://www.ionactive.co.uk/glossary/Bremsstrahlung.html Bragg’s Law - the result of experiments into the diffraction of X-rays or neutrons off crystal surfaces at certain angles. Source: http://en.wikipedia.org/wiki/Bragg’s_law African Virtual University 109

XII. List of Compulsory Readings

Reading 1 Complete reference : Atomic Models From: wikipedia URL : http://en.wikipedia.org/wiki/Atomic_physics Accessed on the 20th April 2007

Abstract : This reading is compiled from wikipedia page indicated above and the links available in the page. Titles on Dalton’s model of the atom, Thompson’s plum pudding model, Rutherford’s alpha scattering experiment that led to the planetary model of an atom and quantum mechanics are discussed. Rationale: The material in this compilation is essential to the first activity of this module.

Reading 2 Complete reference: Bohr Model of Hydrogen Atom URL : http://musr.physics.ubc.ca/~jess/hr/skept/QM1D/node2.html DateConsulted: June 2007 Abstract : In three webpages the Bohr model of the hydrogen atom is presented concisely. You are advised to begin with the page referenced here and then use the next link to go to the derivation of the Bohr radius and click next again for calculation of energy levels. Rationale: The material is presented in a manner that it is easy to follow.

Reading 3 Complete reference: Theory of Rutherford Scattering URL : http://hyperphysics.phy-astr.gsu.edu/hbase/rutcon.html#c1 Date consulted: April 2007 Abstract: The physics of scattering as it relates to the Rutherford Model of the atom is beautifully presented. You will have to follow the outline as presented in this page and click on each link as presented in the outline. Rationale: The material presented in this link is essential and relevant to this course African Virtual University 110

Reading 4 A Look Inside the Atom Complete Reference: http://www.aip.org/history/electron/jjhome.htm Date Consulted: June 2007 Abstract: This is an account of the work by J.J.Thomson on Cathode rays that culminated in the discovery of the electron as a fundamental part of atom. Follow the links by clicking next. Rationale: The article is qualitative but very informative and relevant to this course.

Reading 5 Nobel Prize Lecture on Cathode Rays Complete Reference: http://nobelprize.org/nobel_prizes/physics/laureates/1905 lenard-lecture.html Date Consulted: June 2007 Abstract: In the context of what you already know now, this is a light reading but informative article on cathode rays and misconceptions at the time. Rationale: The presentation is by a Physics Nobel Prize winner, Philipp Lenard, 1905. This is good motivational material for you.

Reading 6 The Millikan Oil Drop Experiment Complete reference: http://hep.wisc.edu/~prepost/407/millikan/millikan.pdf Date Consulted: June 2007 Abstract: This is a good quantitative article on the practical aspects of the Millikan Oil Drop Experiment. Rationale: The material is good and relevant to the course.

Reading 7 Complete reference : URL: http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html Date consulted: June 2007 Abstract : Highly illustrated physics of the hydrogen atom, energy levels, electron transitions, fine and hyperfine structures all are very well discussed. Rationale: This article covers topics in line with this Learning Activity. African Virtual University 111

Reading 8 Complete reference : Emission Spectrum of Hydrogen URL : http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/bohr.html Date Consulted: June 2007 Abstract: This article discusses the Emission Hydrogen Spectrum and includes solved practice problems. Rationale: This article covers topics in line with this module and the practice problems makes this reading very important.

Reading 9 Hydrogen Atom Complete reference : An Introduction to the Electronic Structure of Atoms and Molecules URL: http://www.chemistry.mcmaster.ca/esam/Chapter_3/intro.html Date Consulted: June 2007 Abstract : This is section three of an article by Prof. Richard F.W. Bader Professor of Chemistry / McMaster University / Hamilton, Ontario. It discusses the hydrogen atom, the evolution of probability densities and hence orbitals and finaly the vector model of the hydrogen atom. Rationale: The material covered in this article is good and relevant to this Learning Activity.

Reading 10 Mathematical Solution of the Hydrogen Atom URL: http://www.mark-fox.staff.shef.ac.uk./PHY332/atomic_physics2.pdf Date Consulted: June 2007 Abstract : This article provides the methodology of solving the Hydrogen atom problem as a quantum mechanical problem. Rationale: The article is very relevant to this course as you will see how the three quantum numbers n, l, and m come out naturally.

Reading 11 Fine Structure of Hydrogen Atom Complete Reference: http://farside.ph.utexas.edu/teaching/qmech/lectures/node107.html Abstract: This article is part of a series of lecture notes in non relativistic quantum mechanics. Rationale: The material is good but requires a strong link with knowledge in quantum mechanics. African Virtual University 112

Reading 12 X-Ray Production Complete reference : http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xt be. html Date Consulted: June 2007 Abstract :This article is part of a comprehensive series of articles on the physics of x-rays, covering all objectives of this Learning Activity. The opening article discusses x-ray production and the links discuss bremsstrahlung radiation, characteristic x-rays, Moseley law and X-ray diffraction. Rationale: The presentation by Hyperphysics is as always sharp and to the point. It is an essential reading.

Reading 13 The Origin of Characteristic X-Rays URL : http://www4.nau.edu/microanalysis/Microprobe/Xray-Characteristic.html Date consulted: June 2007 Abstract : This article discusses characteristic x-ray production. The links to this page discuss continuum x-rays, electron shells, electron transitions , Moseley’s Law and other topics beyond the requirements of this course.. Rationale: This is good material and relevant to this course.

Reading 14 X-Ray Diffraction . URL : http://www.physics.upenn.edu/~heiney/talks/hires/whatis.html#SECTIO N00011000000000000000 Date Consulted: Junel 2007 Abstract : In this article, x-ray is concisely presented. Rationale: The article covers the contents of this activity

Reading 15 X-Ray Diffraction URL: http://e-collection.ethbib.ethz.ch/ecol-pool/lehr/lehr_54_folie2.pdf Complete reference: http://www.google.com/search?q=cache:qLs7iI81agwJ e-collection.ethbib.ethz.ch/ecol-pool/lehr/lehr_54_folie2.pdf+X RAY+MOSLEY’S+LAW Abstract: This article contains Power Point Slides on practical aspects of X-Ray Diffraction, X-Ray Tube, X-Ray Spectrum and Mosley’s Law. To access the article start with the complete reference and then click on reference link. Rationale: The material is relevant to this activity. Please read it. African Virtual University 113

XIII. Compiled List Of (Optional) Multimedia Resources

At least two, copyright free, relevant, resources other than a written text or a web site. These could be a video file, an audio file, a set of images, etc. For each -re source, Module Developers should provide the complete reference (APA style), as well as a 50 word abstract written in a way that motivates the learner to use the resources provided. The rationale for the resource provided should also be explained (maximum length : 50-75 words). An electronic version of each resource is required. Reference: http://www.colorado.edu/physics/2000/index.pl Date consulted: December 2006 Description: A beautiful applet whereby you create your own atom. Upon entering the Physics 2000 Home page, click on Table of contents and then go to Science Trek and click on Electric Force. Place your cursor about 5 cm away from the proton. Click and drag the created electron at say 45° or greater towards the nucleus and let go. Then watch the electron make an elliptical orbit around the proton. You will be surprised at the number of non colliding “orbital electrons” you can create around the nucleus. Reference: http://www.waowen.screaming.net/revision/nuclear/rsanim.htm Date consulted: April 2007 Description: A simulation of the Rutherford alpha particle scattering experiment against a gold target. In this simulation the nucleus is represented by a yellow dot and the alpha particle by a red dot which is smaller than the yellow dot. A scattering event is realized by the learner following the instructions regarding choice of the energy of the alpha particle, dragging the red dot and clicking the ‘fire’ bar. Implementation of one set of the instructions constitutes one experiment. The next experiment starts by clicking the “next” bar to rest the position of the alpha particle. After several scattering events you need to clear tracks. The alpha particle energy is restricted between 8 and 25 MeV. Reference:http://www.physics.brown.edu/physics/demopages/Demo/modern/demo/ 7d5010.htm Date consulted: April 2007 Abstract: An animation of the experimental set up of Rutherford alpha scattering is shown. African Virtual University 114

Reference: http://www.control.co.kr/java1/masong/absorb.html Date consulted: April 2007 Description: A Java applet for an absorption spectrum of a Bohr atom Reference: http://www.eserc.stonybrook.edu/ProjectJava/Bragg/index.html Date Consulted: April 2007 Description: The applet shows two rays incident on two atomic layers of a crystal, e.g., atoms, ions, and molecules, separated by the distance d. The layers look like rows because the layers are projected onto two dimensions and your view is parallel to the layers. The applet begins with the scattered rays in phase and interferring constructively. Bragg’s Law is satisfied and diffraction is occurring. The meter indicates how well the phases of the two rays match. The small light on the meter is green when Bragg’s equation is satisfied and red when it is not satisfied. The meter can be observed while the three variables in Bragg’s are changed by clicking on the scroll-bar arrows and by typing the values in the boxes. The d and q variables can be changed by dragging on the arrows provided on the crystal layers and scattered beam, respectively. African Virtual University 115

XIV. Compiled List of Useful Links

At least 10 relevant web sites. These useful links should help students understand the topics covered in the module. For each link, the complete reference (Title of the site, URL), as well as a 50 word description written in a way to motivate the learner to read the text should be provided. The rationale for the link provided should also be explained (maximum length : 50 words). A screen capture of each useful link is required.

Useful Link #1

Title: Atomic Models URL: http://mhsweb.ci.manchester.ct.us/Library/webquests/atomicmodels.htm Screen Capture:

Description: A well illustrated description of the atomic theories through time is given . Rationale: supplements the content in activity 1 Date Consulted: April 2007 African Virtual University 116

Useful Link #2

Title: Atomic Spectra of Hydrogen URL: http://physics.gmu.edu/~mary/Phys246/10Spectrophotometer.pdf Screen Capture:

Description: A good description of hydrogen spectrum is available at this link . Date Consulted: May 2007

Useful Link #3

Title: Hydrogen Atom URL: http://en.wikipedia.org/wiki/Hydrogen_atom Screen Capture:

Description: The physics of hydrogen atom is described in this article. Rationale: Hydrogen atom is a good starting point for the description of atomic spectra in general. Therefore it is essential to have fundamental grasp of the physics and there fore additional reading material of this kind is necessary. Date Consulted: May 2007 African Virtual University 117

Useful Link #4

Title: How light is made from the ordered motion of electrons in atoms and molecules URL: http://zebu.uoregon.edu/~soper/Light/atomspectra.html Screen Capture:

Description: How ordered motion of electrons gives rise to discreet energy levels and hence light is provided here. Rationale: Relevant to Activity three of the module. Date Consulted: 2007-05-28 African Virtual University 118

Useful Link #5

Title: NIST Atomic spectra Database URL: http://physics.nist.gov/PhysRefData/ASD/index.html Screen Capture:

Description: This database provides access and search capability for NIST critically evaluated data on atomic energy levels, wavelengths, and transition probabilities that are reasonably up-to-date. A table of ground levels and ionization energies for the neutral atoms is given. You can also find here links to related databases of NIST. Date Consulted: May 19, 2007 African Virtual University 119

XV. Synthesis of the Module

Atomic Physics

In this module you have learnt about an important topic in physics, namely Atomic Physics. The subject matter of the module is a principal component of the so called Modern Physics a scientific discipline that came into being in the late 19th century and early 20th century. You you have been guided through the historical development of atomic theories, through the work of Dalton, Thompson, Rutherford and Bohr. These four scientists have a very special place in the development of Atomic Physics. The work by Dalton and Thompson laid the ground on which Rutherford and Bohr built upon to the extent that the models developed by the latter two scientists are usable to some extent today. Hence you have acquired skills to solve problems relating to Rutherford’s and Bohr’s models of the atom. In Learning Activity 2 of this module you have been guided through the gas discharge phenomenon and the onset of cathode rays. This phenomenon was a puzzle to the scientists of the day but led to an important discovery of the electron, the first subatomic particle to be discovered. Towards the end of the Learning Activity you have been guided through Millikan’s oil drop experiment that led to the discovery that electric charge is particulate or quantised. In Learning Activity 3, you have been guided through the evolution of atomic spectra and learnt about the uniqueness of an atomic spectrum for every element. The uniqueness of atomic spectra has scientific and technological implications. In Learning Activity 4, you have been guided through the origin of x-rays, the development of x-ray spectra and the uniqueness of x-ray spectrum for every element. Towards the end of the unit we discussed and have solved problems using Mosely’s law. Finally you have learnt about the use of x-rays as an analytical tool. African Virtual University 120

XVI. Summative Evaluation

1. (a) Bohr’s atomic model is based on four postulates. State them and give their mathematical representation. (b) Derive an expression for the radius of the nth orbit of the electron in a hydrogenic atom of atomic number Z, where n denotes a principal quantum number.(c ) Calculate the radius of the Ground state orbit for hydrogen. 2. Describe how J.J.Thompson measured the charge to mass ratio of the electron

q E and derive the exprsession = where the symbols have the satandard 2 meaning. m B R 3. (a) Distinguish between orbital angular momentum and spin angular momentum. Hence define the total angular momentum of an electron in an atom. r (b) Consider the two ways in which and S vectors add to form the vector r r r J when l = 1 and s = ½ . If the angle between L and S is θ, show that

j( j + 1) − l( l + 1) − s( s + 1) cosθ = 2 l( l + 1)s( s + 1)

4. A hydrorgen atom state is known to have the quantum number l=3.(a) What are

the possible n, ml, ms quantum numbers? (b) What are the quantum numbers

n,l, ml, ms for the two electrons of the helium atom in its ground state? (c ) State Pauli exclusion. Use the principle to determine the quantum state of the outermost electron in the magnesium atom (Z = 12). 5. (a)Distinguish between excitation energy and ionization potential. Illustrate your answer by referring to the hydrogen atom. (b) Suppose an electron from an inner shell is completely removed from an atom. How does the required energy compare with the ionization potential of the atom? Explain. (c) A sodium ion is neutralized by capturing an electron of energy 1 eV. What is the wavelength of the emitted radiation if the ionization potential of sodium is 5.4 volts? 6. (a)In the investigation of structure of the atom, Rutherford performed one important experiment. Give a brief description of the experiment. What was the main conclusion from the experiment? (b) What is the closest distance of approach that a 5.3 MeV alpha particle can make with an initially stationary gold nucleus? African Virtual University 121

(ZAU = 79) 7. A beam of 100 keV electrons is incident on a Mo (Z=42) target. Binding energies of the core electrons for K and L shells in Mo are given in the table below:

Shell K LI LII LIII Orbital 1s 2s 2p 2p Binding Energy , keV 20.000 2.866 2.625 2.520

Calculate the wavelengths of the Kα X-rays emitted.

Answer Key

1. (a) Bohr postulates: Postulate 1: Coulomb force balanced by centripetal force,

4 meqe 1 postulate 2: L = n h ; postulate 3: E n = − ; 8h2 2 n2 ε0 postulate 4: ∆E = E − E . i f

2 ε0nh (b) Bohr radius rn = . (c) In the ground state n = 1, so that the Bohr m Ze2 π e radius is r1.

2. In the Thompson tube the electric force is balanced by the magnetic force. r 3. (a) Orbital angular momentum L is due to rotation of electron in its orbit. Spin r angular momentum S has no classical analogue. Total angular momentum is r r the vector sum of L and S . r r (b) Vector sum of L and S African Virtual University 122

Applying cosine rule to the triangle made by vectors J, L, and S we have

J 2 − L2 − S 2 cosθ = 2L S

But L = l( l + 1) ; S = s( s + 1) and J = j( j + 1) h h h

j( j + 1) − l( l + 1) − s( s + 1) Upon substitution we have cosθ = . 2 l( l + 1)s( s + 1)

4.(a) If l = 3, then ms = ±1/2 , ml = 0, ±1, ±2, ±3, n = 4.

(b) Quantum Electron 1 Electron 2 number n 1 1 l 0 0

ml 0 0

ms +1/2 -1/2

(c ) No two electrons can occupy an energy state definied by the same quantum numbers. African Virtual University 123

Outermost electrons

Quantum Electron 1 Electron 2 number n 3 3 l 0 0

ml 0 0

ms +1/2 -1/2

5. (a) Excitation energy is energy required to raise an atom from one lower energy state to a higher level, whereas Ionisation potential is energy required to completely remove an outermost electron from the atom. (b) Ionisation potential is less Energy required to remove an electron from an inner shell. (c ) Energy of Emitted radiation = (5.41 - 1) eV, hence λ = hc/E = 2.82 x10-7 m.

4πε T 6 (b) Using cot θ / 2 = 0 b where T = 5.3 MeV and Z = 79. ( ) 2 2Ze

3.795 => b = cot(θ / 2) this expression is adequate since θ is unknown. ε0 7. Transitions are subject to the selection rule ∆l = ±1.

o c hc hc 12.4 keV . λ = = = = A where E is the initial energy and E is ν hν E − E E − E i f i f i f the final energy.

Module Assessment

The sum of the Scores in the Tasks, Assignements in the four Learning Activities should consistitute 40% of the total score in the module and the Summattive Eva- luation should consistitute 60%. African Virtual University 124

XVII. References

Foot C.J.(2005) Atomic Physics, Oxford University Press, Chapters 1 and 2.. Willmont, J.C. (1975), Atomic Physics, Wiley. Beiser A., (2004) Applied Physics, 4th ed., Tata McGraw_Hill edition, New Delhi, India. Bernstein, J.Fishbane, P.M. and Gasiorowicz, Modern Physics, Prentice Hall. Anderson, E.E. 1971, Modern Physics and Quantum Mechanics, W.B.Saunders Co. Philadelphia. Cohen-Tannoudji,C., Diu, B., Laloe, F. 1977, Quantum Mechanics, John Wiley and Sons Inc., Paris. Gasiorowicz, S. 1974, Quantum Physics, John Wiley and Sons Inc., New York: Liboff, R.L. 1980, Introductory Quantum Mechanics, Addison-Wesley Publishing Co. Inc., New York. Landau, L.D. and Lifshiftz, E.M., 1958, Quantum Mechanics Non-Relativistic- Theory, Addison-Wesley Publishing Co. Inc., London. Merzbacher,E., 1961, Quantum Mechanics, John Wiley and Sons Inc., New York Rae, A.I.M., 1986, Quantum Mechanics, Adam Hilger/English Language Book Society, Bristol. African Virtual University 125

XVIII. Main Author of the Module

About the author of this module

Christopher Amelye KIWANGA Associte Professor of Physics The Open Universty of Tanzania P.O.Box 23409 DAR ES SALAAM TANZANIA., E-mail: [email protected] , [email protected] . Breif Biography: I am a Graduate from Lancaster University, UK where I gained the Ph.D and M.Sc in Physics while the B.Sc I obtained from the University Dar es Salaam, Tanzania. For my Ph.D and M.Sc I worked in Physical Electronics having written a thesis on Field Electron Emission on Surfaces Coated with Selenium and a dissertation on Chromium Diffusion in Gallium Arsenide respectively. Upon return to Tanzania, I worked on Applications of γ-Radiation to the analysis of Sulphides from the Lake Victoria Gold Field. I have taught at the University of Dar es Salaam for 29 years and at the Open Uni- versity of Tanzania for six years todate. You are always welcome to communicate with the author regarding any question, opinion, suggestions, etc in respect of this module.

XIX. File Structure

Name of the module (WORD) file : Atomic Physics.doc

Name of all other files (WORD, PDF, PPT, etc.) for the module. Compulsory readings Atomic Physics.pdf

Abstract : The seven compulsory readings proposed for this module are compiled in one pdf file.