Atomic Physics
Prepared by: Prof.C.A.Kiwanga, The Open University of Tanzania
NOTICE
2
TABLE OF CONTENTS
Topic
Page number
ACTIVITY 1: Atomic Models
14
1. Dalton’s and Thomson’s Models,
19
2. Rutherford’s Alpha Scattering Experiment.
19
3. Rutherford’s Planetary Model of an Atom
20
4.Bohr’s Model of an Atom
20
ACTIVITY 2: Electrical Discharges
1. Discovery of Cathode Rays.
2. CRT “glow” Variation with Pressure
3.
Properties of Cathode Rays
30
35
35
36
4.Millikan’s Oil Experiment
37
ACTIVITY 3: Atomic Spectra
1. Quantum Numbers
41
2. Angular Momenturm Coupling Schemes,
3. Vector Model of an Atom
4. Zeeman Effect.
5. Fine Structure of Hydrogen Spectrum
48
66
55
62
63
6. Emission and Absorption Spectra
7. Pauli Exclusion Principle.
72
ACTIVITY 4: X-Rays
78
1
Production Properties and Characteristic X-Ray Spectra,
2.
X-Ray Diffraction,
3.
Bragg Equation and Crystal Spectrometer
4.
Atomic X-ray Spectra of Elements
5.
Moseley’s Law.
75
83
87
88
81
87
3
FOREWORD
This module has four major sections.
The first one is the INTRODUCTORY section that consists of five parts vis:
6.
TITLE:- The title of the module is clearly described
7.
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.
8.
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.
9.
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.
10.
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:
11.
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.
12.
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.
13.
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 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.
4
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:
14.
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.
15.
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
16.
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.
17.
COMPILED LIST OF COMPULSORY READINGS: A minimum of three
compulsory reading materials are provided. It is mandatory to read the
documents.
18.
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.
19.
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.
20.
SYNTHESIS OF THE MODULE: Summary of the module is presented.
21.
SUMMATIVE EVALUATION:Enjoy your work on this module.
5
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
MATERIALS
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 processing
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.
VI
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.
6
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 sub- atomic
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.
6.1 OUTLINE
1.
Atomic Models
22.
23.
24.
25.
26.
b. Electrical Discharges
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
1.
(20 hours)
Discovery of Cathode Rays.
CRT “glow” Variation with Pressure,
Properties of Cathode Rays.
c.Atomic Spectra
Quantum Numbers
(40 hours)
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
37.
38.
39.
40.
41.
(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
(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.
7
6.2 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
VIII. Specific Learning Objectives (Instructional Objectives)
Content
Atomic Models (40 hours)
1.
Dalton’s and Thomson’s Models,
43. Rutherford’s Alpha Scattering Experiment.
44. Rutherford’s Planetary Model of an Atom,
45. Bohr’s Model of an Atom ;
46. Bohr’s Postulates
42.
2.
Electrical Discharges (20 hours)
Discovery of Cathode Rays.
48. CRT “glow” Variation with Pressure,
49. Properties of Cathode Rays.
47.
Atomic Spectra (40 hours)
50. Quantum Numbers
3.
Learning objectives
After Completing this section you
would be able to:








Describe the characteristics of Dalton and
Thomson atomic models
Solve problems related to the alphascattering experiment
Solve problems using Bohr’s postulates
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
Use the vector model of atom to solve
problems and explain properties
Explain the fine structure of spectra
Angular Momenturm Coupling Schemes,
52. Vector Model of an Atom,
53. Zeeman Effect.
54. Fine Structure of Hydrogen Spectrum
55. Emission and Absorption Spectra.
51.
56.
Pauli Exclusion Principle.
4.
X-Rays
(20 hours)
Production Properties and Characteristic XRay Spectra,
58. X-Ray Diffraction,
59. Bragg Equation and Crystal Spectrometer
60. Atomic X-ray Spectra of Elements
61. Moseley’s Law.


57.


Explain the atomic origin of X-rays
Distinguish characteristic X-Rays from
Bremstrahlung radiation
Use Bragg’s rule to solve problems
Solve problems using Moseley’s Law
9
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.
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
10
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.
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 L  r  p ,
where p is the linear momentum. The z-component of L is given by
a) Lz  xp y  ypx
b) Lz  ypz  zp y
c) Lz  xpx  yp y
d) Lz  zpz  yp 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
11
a) E 
r2
Q
4 0
1
Q
b) E 
4 0 r
r
Q
c) E 
4 0
1 Q
d) E 
.
4 0 r 2
13) The quantisation of electromagnetic energy is summarized by the equation
a) E = mc2
b) E = h
c) E = h
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
a) mg  qE  bv
b) mg  qE - bv
c) mg  bqE  v
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.
19. Ionization energy is defined as being the energy required to
12
(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
6. a
11. a
16. b
2. c
7. c
12. c
17. b
3. d
8. c
13. c
18. a
4. c
9. b
14. a
19. b
5. d
10. c
15. a
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..
13
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.
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.
14
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
1. 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.
2. 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
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.
3. 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.
4. Reference:
http://webphysics.davidson.edu/Applets/pqp_preview/contents/pqp_errata/cd_errata_fi
xes/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.
5. 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
15
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_Waves.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
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/~m
kpohl/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:
16
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/atomicmodelsandspectra.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.
.
17
Date Consulted:- April 2006
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
Resource #6
Title: Atomic Structure Concepts
URL:-http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/atomstructcon.html#c1
Screen Capture:
18
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
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 behavour 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 substance 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.
19
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 Rutherford’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.
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 Ei , discontinuously changes its motion so that it moves in an orbit of total
energy E f . The frequency of the emitted radiation is equal to the quantity
 Ei  E f  / h .
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.
20
RUTHERFORD SCATTERING
Adapted from Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Rutherford_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  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 concentrated 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 .
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 .
21
Fig. 1. 1 Rutherford Scattering Geometry
1
2
The relationship between the scattering angle , the initial kinetic energy K  mv02
and the impact parameter b is given by
b
zZ e 2
cot  / 2 
2 K 4 0
1.1
where z =2 for  -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 d between  and  + d.
By definition, the cross section is the proportionality constant
2 bdb     2 sin d
Hence,
 d
d  2 b db  
 d

 d

1.2
where d   2 sin d
22
The Differential Cross Section then becomes
2 b db
d

d  2 sin  d
1.3
From Eqns.1.1 and 1.3 we have
2
2
d  1   qQ 
1

 

4
d   4 0   4 K  sin  / 2 
1.4
Eq.1.4, is called the Differential Cross section for Rutherford Scattering.
Fig.1.2 Schematic Geometry for Calculation of Scattering Cross Section
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  - 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.
23
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   
N i nLZ 2 k 2e 4
4r 2 KE 2 sin2  / 2 
1.5
Where Ni = number of incident  - 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  - particles and
 = scattering angle.
The predicted variation of detected alphas with angle is followed closely by the Geiger-Marsden data,
shown in Fig. 1.4 below.
Fig.1.4: Verification of Rutherford’s Formula
Calculation of Maximal Nuclear Size
1
mv 2 of the
2
alpha particle is turned into potential energy and the particle is at rest. 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.
For head on collisions between alpha particles and the nucleus, all the kinetic energy
24
Fig.1.5 Scattering with Different Impact Parameters
Applying the Coulomb potential energy between the charges on the electron and nucleus, one can
1
1 q1q2
write:
mv 2 
2
4 0 b
Rearranging:
b
1
2q1q2
4 0 mv 2
1.6
For an alpha particle:




m (mass) = 6.7×10−27 kg
q1 = 2×(1.6×10−19) C
q2 (for gold) = 79×(1.6×10−19) 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.
25
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 to
atomic 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 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 the Balmer
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
F
Ze 2
4 0 r 2
The Coulomb force is balanced by the centripetal force, so that we have
mv 2
=
r
4 0 r 2
Ze 2
Using Bohr’s angular momentum quantization rule
th
We have the n
Bohr radius
rn 
 0 n 2 h2
 mZe 2
L  mrv 
h

2
1.7
26
And the velocity of the electron in the nth orbit
Ze 2
vn 
2 0 hn
1.8
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)
  ke 2 
1
2
E  mv  
 r 
2


1
where k 
.
4 0
1.9
The orbital motion is maintained by the Coulomb force
ke 2
r2

mv 2
r
 mv 2 
ke 2
r
1.10
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 ke 2 ke 2
E

2 r
r
E
1 ke 2
2 r
1.11
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
The orbital frequency
f 

v

2 2 r
1.12
27
where  is the orbital angular speed of the electron. From Eqn. 1.10 we have
v

r
ke 2
mr 3
Substituting this in Eqn(4) we have
f 
For the H atom
spectrum.
1
2
ke 2
mr 3
1.13
f = 7 x 1015 Hz, which is in the ultra violet region of the electromagnetic
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 emitting 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 existing 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?
28
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?
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.
2.
3.
4.
5.
6.
List three assumptions used in the derivation of the Rutherford scattering differential
cross section.
A 6.0 MeV - 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  particles expected to be scattered by more than 45?
Calculate the Bohr radius of a hydrogen atom in its ground state. Consult a physics
book for required constants.
Calculate the ground state energy of Hydrogen as modelled by Niels Bohr. Electrons
have negative energy.
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 characterises such an orbit.
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.
29
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.
30
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.
31
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/lenardlecture.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.
List of Relevant MM Resources
1. 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.
2. 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
3. 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
32
electric field bar to change the eletric field between plates and note the effect on
the oil drops. As the electric field increases more and more drops are attracted
upwards to the positively charged plate.
4. 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
Experiments, Millikan’s experiment is number 3. Also Click on the video.
List of Relevant Useful Links
Resource #1:
Title:- Investigating Cathode Rays
URL:http://schools.cbe.ab.ca/b858/dept/sci/teacher/zubot/Phys30notes/investnurays/in
vestnurays.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:
33
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 phosphorcoated anode P. Shadow mask M is
connected to the cathode potential and its
image is seen on the phosphor as a nonglowing area.
Source: http://en.wikipedia.org/Image:Crookes Tube.svg.
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.h
tm
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
Physics. Use the links to navigate to other relevant topics.
Date consulted: April 2007
34
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 decsribed 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
Experiment. You should find the links included in the article to be
useful and omplimentary.
Date Consulted: April 2007
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
35
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.
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
36
either by de-excitation of electrons during recombination with positive ions or by the deexcitation 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.
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.
37
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:
F  6 av
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   a3 g
3
The drop is in air, it experiences an upthrust
4
Wup   a 3dg
3
The resultant downward force:
4
Wres   a 3 g    d 
3
where  and d are density of oil and air respectively.
Now at terminal velocity, the resultant downward force is the drag force
4 3
 a g    d   6 av
3
2.1
1
=>

2
9 v
a  

 2   d  g 
2.2
Source:
http://www.phys.ufl.edu/~hill/teaching/2005/2061
/links/Millikan.pdf
Fig. 2.2 An oil droplet in the
Fig. 2.1 Schematic diagram of the Millikan oil- cloud carrying an ion of
charge e falling at terminal
drop apparatus.
speed, i.e.
mg = bv.
38
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 =Eq   a 3    d  g
3
4
Therefore, Eq   a 3    d  g = 6 av1
3
From Eq.2.1 we have
Eq  6a  v  v1 
2.3
From Eqn. 2.1 and 2.2, Eqn. 2.3 becomes
1
2
6 
9v
q

  v  v1 
E  2   d  g 
2.4
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 1x10 6 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
39
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.
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 perpendicular to
e
v
each other. (a) Show that
, where v is the speed of the electron, r is the radius

m rB
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
e
E
 2 , where E is the electric field.
magnetic forces must be equal, show that
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.
40
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.
41
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 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 4: Mathematical Solution of the Hydrogen Atom
Complete reference :
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 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.
List of Relevant MM Resources
1. 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.
42
List of Relevant Useful Links
Resource #1:
Title:- Modifications of the Bohr model
URL:- http://theory.uwinnipeg.ca/physics/bohr/node5.html#SECTION002840000000000000000
Screen Capture:
Abstract:- Despite the success of the Bohr model, there were some serious
shortcomings 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.
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
43
Screen Capture:
Abstract:- Emission line spectra from various gas spectrum tubes, Absorption line
spectrum 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 watching
these spectra.
44
Rationale: The material is most relevant to the Learning Activity.
Date consulted: April 2007.
Detailed Description of the Activity (Main Theoretical Elements)
A Summarized Solution of the Schödinger Equation of
Atom
The Hydrogen
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 p me
M p  me
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 2



V(r)

 (r )  E (r )
2



3.1
We generalize the problem to include the case of a hydrogenlike atom which consists of
one electron moving around a nucleus of charge +Ze, so that the potential becomes
V (r ) 
 Ze 2
4   r
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,  and .
The spherical co-ordinates are defined by the transformations given by Eq.3.2
Spherical co-ordinates are defined by the transformations:
45

(r,  )
x = rsin cos ...........(3.2.1)
y = rsin sin..............(3.2.2)
z = rcos ...................(3.2.3)
z

r
y
x
r  x2  y2  z 2
Y

X
z
r
 y
  tan 1  
x
  cos 1  
(3.2.4)
(3.2.5)
(3.2.6)
Fig.3.1 Position of a Particle in Two
Coordinate Systems
and the transformation from cartesian coordinates to spherical coordinates is facilitated by
the chain rule:
r 
 
 




 xi  xi  r  xi    xi  
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 
   2 
 sin 

2
2
r

r
sin




r
r

 sin   



2
Denoting the angular part of the Laplacian operator by L2 and the radial part by
the Laplacian operator becomes:
2   
3.3
,
1 2
L
2
and so the Schrodinger equation for the hydrogen atom then becomes:
 2

1

  2 L 2   V ( r ) ( r , , )  E ( r , , )


r

 2 

3.4
46
The method of separation of variables,
 ( r , , )  R( r )Y (  , )
leads to a radial differential equation and an angular dependent differential equation:
 R( r ) 
2
2
 E  V (r ) R( r ) 

r2
3.5
R( r )
and L2 Y (  , )    Y (  , )
3.6
The method of separation of variables can be repeated to the angular dependent differential
equation by application of the product solution
Y  ,   P    
Which leads to two additional differential equations in  and . The -dependent differential
equation is given by
1 d 
dP 
m2 P
sin



P

0
sin  d 
d 
sin2 
3.7
where   l  l  1 .
And the  - dependent differential equation is given by
d 2
d
2
 m 2  0
3.8
Eq.3.8 can be solved readily to give
    Ae i m 
3.9
The product solution Y  ,   P     takes the functional form
Yl  ,    C e
m
m
l
i m
m
2
 1   d 

 

2 
d

1




 
l m

2
 1
l
3.10
which can be written in terms of Associated Legendre Functions defined as follows:
 1 

Pl     1 
2 
1 
m
m
m/2
 d 


 d 
l m

2
 1
l
3.11
47
So that Eq.3.10 becomes
Ylm  ,     1 C lm Plm  e im
m
3.12
The normalized spherical harmonic functions take the form
(1) m
Yl  ,    l
2 l!
m
2l  1
.
4
l  m ! m
Pl  e i m
l  m !
3.13
And in full it becomes
1 2l  1
Yl  ,    l
2 l! 4
m
l  m ! im  1 
e
l  m !  1   2 
m/2
 d 


d



l m

2

1
l
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
 Z 
Rnl ( r )  2 

 na 
3/ 2
l
( n  l  1 )!  2 Zr   Zr / na 2 l 1  2 Zr 
Ln l 
 e

3 
n  ( n  l )!  na 
 na 
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.
 2 k 2 Z 2e 4
n  2
2 E
2
 En  
kZ 2e 2
2a n2
where
a0 
2
 ke 2
.
48
 2 Zr 
L2nll1 
 is the Associated Laguere Polynomial defined by
 na0 
L2nll1 ( 
 d 
)

 d 
2 l 1 
 d 
e 

  d  

n l

n l  
e

 
where  

2 Zr
.
na0
Examples of Normalized Radial Solutions:
1. For n = 1, l = 0:
 Z
R10  2
 na



3/ 2
 2Zr 

1.1.e  Zr / na L11 
 na 
 2Zr  d   d

 
Now L11 
e
e    = - 1


 na  d  d

=>
 Z 

R10  2
 na 
Z
R10 2 
 a 

3/ 2
e  Zr / na x (1)
3/ 2
e  Zr / a
2. For n = 2, l = 0:
 Z
R20 (r )  2
 2a 



3/ 2
0
1  2 Zr   Zr / 2 a 1  2 Zr 

 e

L2 
2.8  2a 
 2a  
Now
 2Zr  d   d 2 2   
 
L 
 e 
e
2

 2a  d  d
==>
 2Zr 
2Zr
2Zr
  2.
L12 
4 
4
2a
a
 2a  
1
2
 Z 

R20 (r )  2
 2a 
3/ 2
 2Zr

1
.1.e Zr / 2 a 
 4 
4
 a

49
 Z 

R20 (r )  
 2a 
3/ 2

Zr 
 2  e  Zr / 2 a
a 

3. For n = 2, l =1:
 Z
R21 (r )  2
 2a 



 Z
 2
 2a 
Now
 d 
L      
 d 
3
3
3
3/ 2



3/ 2
1  2Zr   Zr / 2 a 3  2Zr 


e
L3 
2(3!) 3  2a 
 2a o 
1 1  2Zr   Zr / 2 a 3  2Zr 

e

L3 
6 2.6  2a 
 2a  
   d 3 3 
e    e
  d 
 Z 

==> R21 r   2
 2a 
 Z 

R21 (r )  
2
a
 
3/ 2

3/ 2



=-6
1 1  2Zr  Zr / 2 a

e
 6
6 12  2a 
1  Zr   Zr / 2 a
 e
3  a 
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   n 2
l 0
The Total Hydrogenic Wavefunction
The total hydrogen wavefunction, except for time, is the product function:
 nlm ( r , , )  Rnl ( r )Ylm (  , )
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.
50
Examples of Normalized Spatial Hydrogenic Wavefunctions:
 100  R10 ( r )Y00 (  , ) 
1
 a
3/ 2
er / a

r   r / 2a
 2 e
a 
4 2 a

1
1 r  r / 2a
 R 21Y10 
e
cos 
3/ 2 a
4 2 a
1
 200  R20Y00 
 210
1
 211  R21Y11 
1
3/ 2
1
1
8  a
3/ 2
r  r / 2a
e
sin e i
a
, etc.
The Radial Probability Density
By definition the probability density of an electron in a hydrogenic eigenstate is given by the
product
* *
 *nlm nlm  R*nl Plm
 m Rnl Plm 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 dependence of the
probability density on each variable separately.
The radial probability density is defined by
 2
Pnl ( r )dr  
  nlm nlm r
*
2
sin drd d
0 0
 2

r 2 R*nl ( r
)Rnl ( r )dr 
 Plm Plm m m sin d d
*
*
0 0
The integrals over  and  are equal to unity because each of the functions P and 
( as well as R ) are separately normalized.
Thus, the radial probability density is given by
Pnl (r )dr  r 2 Rnl* (r ) Rnl (r )dr
3.14
51
Whereas
*
 nlm
 nlmr 2 sin drdd gives the probability of finding an electron in the volume
element d = r 2 sin drdd , 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
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).
52
Example 3.1:
Show that the wavefunctions describing a 1s electron and a 2s electron are orthogonal.
Solution:
1s electron n = 1, l = 0:
Z
 1s   
 a0 
3/ 2
1
e Zr / a0

2s electron n = 2, l = 0:
 2s
Z
  
 a0 
3/ 2
1 
Zr 
 2  e Zr / 2 a0
a0 
4  
The scalar product becomes
 Z 3 / 2
*
 1s 2 s d      

a
allspace
0 0 0
 0 
 Z 3 / 2
x  
 a0  4
  2

e Zr / a0 


*
1
1
2


Zr 
 2  e Zr / 2 a0  r 2 sin dddr
a0 


( Where the spherical element of volume d  r 2 sin dd )
=>

 2 s d
*
1s
all space
3
  2
Z

1
Zr 
 2  e 3 Zr / 2 a0 r 2 sin dddr
  



a0 
 a0  4 2 0 0 0 
3

2
Z

1
Zr  2 3 Zr / 2 a0


  
2

r
e
sin

d

dr
   a0 
0 d
 a0  4 2 0 0 
3

 Z  1 
Zr  2 3Zr / 2 a0


  
2

r
e
dr
  a0 
0 sin d
 a0  2 2 0 
3
 Z  1 
Zr 

 2  r 2 e 3 Zr / 2 a0 dr  cos  0
  

a0 
 a0  2 2 0 
3

 Z  1   2 3Zr / 2 a0
Z 3 3Zr / 2 a0 
  
2
r
e
dr

re
dr 
 
a0 0
 a0  2  0

53

Z 1 
2!
Z
3!

  
 
2
3
4 
 a0  2  3Z / 2a0  a0  3Z / 2a0  
3
Hence

 2 s d  0
*
1s
allspace
Example 3.2:
An alternative form for determining the associated Laguerre polynomials is
L2nll1 ( x) 
nl 1
i 1
  1
i 0
n  l !2
xi
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.
Solution:
Substituting n = 2 and l = 1, we have
0
L33 ( x)    1
i 0
i 1
3!2 xi  (1) (3!)2  6
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
 2Z / na0 3 n  l  1! 

Rnl r   
3





2
n
n

l
!


1/ 2
where
  2Z / na0 r and
e   / 2  l L2nll1   
a0  40  2 / e 2
What is the probability of finding a 1s electron at r > a0 ?
Solution:
For a 1s electron n = 1, l = 0 and Z = 1, we have
54
na0
1
  a0 
2Z
2
r
1
R10 (r )  2 
 a0 
and the wavefunction
3/ 2
e


2

P   Rr  Rr r 2 dr
*
The probability will be given by
a0
  1 3 / 2

  2  e   / 2 
a

2
  0

*
  1 3 / 2
 a  2  a 
2  e   / 2  0   d  0  
  a0 
 2   2 

1
 2 e   d
2 2

 
1
 e   2  2   2
2




2


1
0  e 2  4  4  2  5e 2
2
P = 0.6767.
Example 3.4:
Determine < r > for a 2p wavefunction.
Solution
From   2Z / na0 r , for n = 2 and Z = 1, we have r = a0 .
Hence,

r   R21 r  rR21 r r 2 dr
*
0

1  1

 
 2 6  a0
0


3
2

 e



2


 a  1  1
 0  2 6  a0


 


2
 e   a0 2 d  a0 



3
2

a
a
 0   5e   d  0 5!  5a0 .
24 0
24
Vector Representation of Allowed Orbital Angular Momenta
The eigenvalue problem for the -dependent function is
55
L̂z    m   where the quantum number m = 0, l-1, l-2,…(l-1),-l.
The eigenvalue problem for the  ,  dependent function is
L̂2Y  ,    2Y  ,  where   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
L  l( l  1 )
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.
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
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 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 cos
 cos 
m
.
l (l  1)
56
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
  
S  r  ps
in a manner similar to the definition of the orbital angular momentum
  
L  r  p.
1
 . Spin is an internal property of a particle, like mass or charge. It
2
constitutes an additional co-ordinate or degree of freedom in the quantum mechanical
formulations.
The magnitude of S is
Commutation Rules
These are exactly the same as those of orbital angular momentum,
ie.

  
S
x ,S y  i S z , etc.


 2  
 S ,S z  0 , etc.



  
S
z ,S    S  , etc.


Spin Wavefunctions or Spinors
These are denoted by
s
where s = 1/2 and  = 1/2 .
So that a spin up state will be denoted by
1 
11
 up    
0 2 2
and a spin down state by
57
 0
1
1
 down     ,
1  2 2


The spinors are simultaneous eigenfunctions of the spin operators S 2 and S z :
ie.

and
11
11  1 1
3
11
   1 2
 2
22
22  2 2
4
22
S2
S2
1 1
3
1 1
,
  2 ,
2 2
4
2 2
Sz
11
1 11
 
22
2 22
Sz
1 1
1 1 1
,
   ,
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


  IA

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.
where
A  r2
And i = charge on electron × number of times per second electron passes a given point = ef
58
where f is the frequency of rotation of the electron.
Magnitude of the magnetic dipole moment
 
  IA   ef   r 2
Whose direction is opposite to the orbital angular momentum L because the electron has
negative charge.
L  mvr  m  2 rf  r  2mf  r 2 
Now
Hence
2m

e
e
L.
2m

3.15
Since angular momentum is quantized we have


l  ml  l
In the first Bohr radius, ml  1 and so Eq.3.15 becomes


e l
l 
  B l
2m
3.16
where B is called the Bohr magneton and its value is given by
e
2m
It will be observed in Eq.3.16 that l is directed antiparallel to the orbital angular momentum.
B 
The ratio of the magnetic moment to the orbital angular momentum is called the classical
gyromagnetic ratio,
l 
l
l


e
 B
2m
3.17
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.
s 
s
s

e
m
3.18
59
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
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.
Since
s
1
2
the magnetic moment due to the spin of the electron is
s   s s 
e
.  B
m 2
Thus, the smallest unit of magnetic moment for the electron is the Bohr magneton, whether
one combines orbital or spin angular momentum.
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  with respect to the z - axis, as shown in Fig.3.4 below.
60
Fig.3.4 The Precession of The Angular Momentum Vector in A Magnetic Field

The torque on l is given by



 l  l  B
3.19
this is directed into the plane of the page, in the -direction.
Now, the torque also equals the rate of change of the angular momentum, so we have
l 
But
dl
 l  B   l l  B
dt
3.20

dl  l sin  d
so that the scalar form of Eq.3.20 becomes
l sin .
d
  l lB sin
dt
3.21
We define the precessional velocity by
L 
d
dt
So that Eq.3.21 becomes
L   l B 
e
B
2m
3.22
61
The angular velocity L 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
 E   L  
e B
  B B
2m
3.23
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
 E    .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 B is E0 in the absence of an applied field, then it can take on
one of the energies
E0   B B
in a magnetic field B.
Thus, in a collection of identical atomic particles of the type discussed, a magnetic field
produces a triplet of levels, called a Lorentz triplet whose energies are E0, and E0   B 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 behaviour, the more general
Zeeman effect, including spin was historically misnamed as the anomalous Zeeman effect.
62
(a) Single Transition without an
applied Magnetic Field
(b) Five transitions with an
applied external magnetic field
Fig.3.5 Transitions With and without a Magnetic Field
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
total  nlm(r , ,  ). ( spin ).e iEnt / 
 total  Rnl .e  iEnt /  l , ml s, ms
3.24
The assumption made above implies that there is no interaction between L and S,
ie.
 
 L, S   0


In this case, total is an eigenfunction of both Lz and Sz and so ml and ms are good quantum


numbers; in other words, the projections of L and S are constants of the motion.


But in reality there is an interaction by L and S called the Spin-Orbit interaction, expressed in
 
terms of the quantity L . S .
 

Since L . S does not commute with either L or
ms cease to be good quantum numbers.

S , Eq.3.24 is no longer correct and ml and
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
63

 
Ze 
r
r2
(cgs)
and a magnetic field
 

j r
He  2
r
(cgs)

where r is directed from the nucleus toward the electron.

Assuming that v is the velocity of the electron in the rest frame of the nucleus, the current
produced by the nuclear motion is

Ze 
j  v
c
in the rest frame of the electron.
Then
 

Ze v  r
1 
He  
 v  
2
c r
c
The spin moment of the electron precesses in this field at the Larmor frequency


e  
v
m0 c 2
3.25
 
 
Ee   s .H e   e .S
3.26
 e  H e  
with the potential energy
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

L  
e  
v
2m 0 c 2
3.27
64
and by an additional energy given by
1  
2
E    e .S
3.28
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
 V


F  r
 e
r
and so
  1  V   1 1 V  
1 1 V 
v 
vr 
vr 
L
e r
e r r
em0 r  r
Eq. 3.27 then becomes

L  
1 1 V 
L
2m02 c 2 r  r
and the additional energy
E  
1 1 V  
L.S
2m 02 c 2 r  r
3.29
The scalar product

L.S  ml s
For spin = ½,

1
1
L.S  ml .   ml  2
2
2
The energy splitting then becomes
E 
 2 ml 1  V
4m02 c 2 r  r
For the Coulomb potential the energy splitting can be approximated by
65
E 
 2c ml Ze 2
r3
3.30
where
c 
h
m0 c
c 

m0 c
is the Compton wavelength and
c
.
2
or
A useful result in computation is quoted without proof. The average value of 1/r 3
ie.
1
Z2
 2 2
r3
a0 n l (l  1 / 2)( l  1)
3.31
for l  0.
So that the energy splitting becomes
E 
2
3
c ml Z e
a02 n2 l( l  1 / 2 )( l  1 )
for l  0.
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

 
J 1  L1  S1
and



J 2  L2  S 2


The total angular momentum is obtained by combining J 1 and J 2 :
66
  
J  J1  J 2
and
j  j1  j2 , j1  j2  1, ......., j1  j2
We illustrate j-j coupling by applying it to two inequivalent p-electrons.
For each electron
j1  j 2 
1 3
or
2 2
Then the possible ways of combining these are shown in the Table 3.1.
Table 3.1: j-j Coupling of Two Inequivalent p - Electrons
j1
j2
j
Spectral Terms
3/2
3/2
3,2,1,0
3 3
 , 
 2 2 3, 2,1, 0
3/2
1/2
1/2
1/2
1/2
3/2
2,1
1,0
2,1
Number of States in
a Magnetic Field
16
3 1
 , 
 2 2  2,1
8
1 1
 , 
 2 2 1, 0
4
1 3
 , 
 2 2  2,1
8
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
and
  
L  L1  L2
  
S  S1  S 2
67
The total angular momentum is given, as before, by
  
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
l
s
j
2
1
3,2,1
2
0
2
1
1
1
2,1,0
3
1
0
1
1
0
1
1
3
0
0
0
1
Spectral
Terms
3
D1, 2,3
Number of States in
a Magnetic Field
15
5
D2
P0,1, 2
9
P1
3
S1
3
S0
1
36 states
The Lande g-Factor and The Zeeman Effect
The orbital and spin contributions to the magnetic moment are given by
l  
and
s  

gl e
L   gl  B l  l  1  l
2m0

gs e
S   g s  B s  s  1 s
2m0
where gl = 1 and gs = 2.004  2.
68


Now, when L and S are coupled, we have
  
J  LS
and




B 
  l   s


L  2S 

3.32


It is evident from the expressions for J and  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





Since L and S precess about J , it is apparent that  also precesses about J . However,


the effective magnetic moment, that is the component of  along J , maintains the constant
value,

  

  
 
 B L.J  2S .J
 B L.( L  S )  2S .( L  S )
j    





J
J
J
 .J


 B L2  2S 2  3L.S

J


B

L2  2S 2 

3 2
J  L2  S 2
2
J

69

 
J2  LS
(where

j  
j  

2

 L2  2 L.S  S 2 )
 B ( L2  4S 2  3J 2  3L2  3S 2

2J

 B   3J 2  S 2  L2 
J 


j  
B  
2J 2
J 1 
 



J 2  S 2  L2 

2J 2

B

j ( j  1)  s( s  1)  l (l  1) 
 j ( j  1) .1 


2 j ( j  1)


 j ( j  1)  s( s  1)  l (l  1) 
 j    B j ( j  1) .1 

2 j ( j  1)


We define the Lande g factor as
g 1
j ( j  1)  s( s  1)  l (l  1)
2 j ( j  1)
3.33
and the effective magnetic moment becomes
 j  g B j ( j  1)
3.34
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.


In a weak magnetic field, the angular momentum J will precess about B such that the

projection of J along the field direction will be one of the allowed values of m j  .
The corresponding magnetic moment along the field direction, taken to be the z-direction, will
then be
z   g B m j
having a magnetic dipolar energy of
70
E  gm j  B B
3.35
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 sstate.
Table 3.3: g-Factor Calculations
l
1
1
0
orbital state
p
p
s
j
3/2
1/2
1/2
g
4/3
2/3
2
In a magnetic field B, such that BB 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
p
3/2
mj
3/2
g
E  gm j
4/3
in units of BB
2
71
p
p
p
p
p
s
s
3/2
3/2
3/2
1/2
1/2
1/2
1/2
1/2
-1/2
-3/2
1/2
-1/2
1/2
-1/2
4/3
4/3
4/3
2/3
2/3
2
2
2/3
-2/3
-2
1/3
-1/3
1
-1
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 
 R 2  2 
 n f ni 



1
where R  1.0967758  103 Å-1
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.
72
Fig. 3.8 Hydrogen Spectrum
Source:http://www2.kutl.kyushu.ac.jp/seminar/MicroWorld1_E/Part4_E/P43_E/Bohr_theory_
E.htm
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.
73
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.
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
_ _ _ 2p
74
_ 1s
_ 2s
_ 2s
_ 1s
_ 2s
_ 2s
_ 1s
_ 1s
_ 1s
_ 1s
Source: http://www.science.uwaterloo.ca/~cchieh/cact/c120/eleconfg.html
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 sub-shell 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
75
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.
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 )  A exp(   r ) where A and  are constants and r the
distance between the electron and the nucleus. Show the following:
Z
2
 2 4  0
(a) A 2 
(b)  
where a0 
a0
me e 2

m
(c) the energy E   Z E 0 , where E 0  e2
2
2
 e2

 4  0



2
(d) the expectation values of the potential and kinetic energies are 2E and -E
respectively,
(e) the expectation value r 
3a0
a
and (f) the most probable value of r is 0
2Z
Z
2. For the state 210 of hydrogen atom, calculate the expectation values < r >, <1/r> and
<p2> 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. max = 1215 Å and min = 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 )
76
6. Show that L.S 
1
 J  J  1  L  L  1  S  S  1 
2
2
.
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 perpendicular to the
e
field. ( Hint  E  B ms )
m
77
ACTIVITY 4: X-Rays
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 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.
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
78
Reading 2: The Origin of Characteristic X-Rays
Complete reference : http://www4.nau.edu/microanalysis/Microprobe/XrayCharacteristic.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#SECTION0001100000
0000000000
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:ecollection.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.
List of Relevant MM Resources
1. 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.
79
2. 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 alternately the
x-ray wavelength , the Bragg angle  and the interplanner distance d and for each
variation of a parameter study the effect thereof.
3. 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
Resource #2:
Title:- Notes on the X-Ray Tube
URL http://en.wikipedia.org/wiki/X-ray_tube
Screen Capture:
80
Coolidge side-window tube (scheme)
Abstract:- This is an encylopediac presentation of x-ray tubes and x-ray generation.
Rationale: The material including the links there in are most relevant to this Learning
Activity.
Date consulted: April 2007
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
analyse the x-ray spectrum at hand.
Rationale: The material is very good and relevant to this learning activity.
Date consulted: April 2007
81
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
82
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.
83
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.
If K is the kinetic energy of the incident electron, then
K  h 
hc
min
. The cut off
wavelength depends solely on the accelerating voltage.
84
hvmax 
hc
min
 eV
where V is the accelerating voltage.
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 subsequent
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 K , K , ... 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.
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,.
85
Fig.4.4 X – Ray Transitions without Fine Structure
Fig.4.5 X – Ray Transitions with Fine Structure
Not all transitions are allowed. Only those transitions which fulfil the following
selection rule are allowed: l = 1.
86
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
 1 / 2  A  Z  Z0 
4.1
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 K line can be calculated approximately using Bohr atomic
theory. The wavelength of lines emitted by hydrogenic atoms is given by the Rydberg
formula.

1 
 RZ 2  2

 n  n2 

u
 l
1
4.2
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.
For the K line the effective nuclear charge is (Z-1), nl = 1 and nu = 2, so that Eq.4.2
becomes
 K 
 K 
1 
2 1
 cR  Z  1  2  2 

2 
1
c
3cR
 Z  1 2
4
4.3
A plot of  1K/2 versus Z yields a straight line. Eqn 4.3 is another way of expressing
Moseleys relation.
87
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
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 = 2dsin so that by Bragg’s law, we have
2dsin = n
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.
88
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 selected 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.
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 incomplete; there is an intense and yet
unidentified reflection at 2  30.
The chemistry of the mineral phases mentioned above is as follows: quartz is SiO 2, 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
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Fig.4.10 A Diffractogram of an Iron Sulphide Sample from Mwamela, Nzega, Tanzania
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 min for the continuous spectrum of xrays emitted when 35 keV electrons fall on a molybdenum target.
3. Determine the wavelength of the K 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  = 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 K line for a molybdenum Z = 42 target.
(Ans.  = 0.721 Å)
4. An experiment measuring the K 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  = 1.54 Å and n = 1, calculate the dvalues for the planes responsible for the reflection of the most intense ( 100%) lines of
pyrrhotite and quartz.
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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.
XI
COMPILED LIST OF ALL KEY CONCEPTS (GLOSSARY)
1. 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
2.
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.
3. 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)
4. Planetary orbit - the path that a planet makes around the sun under the influence
of gravitational force.
Source http://en.wikipedia.org/wiki/Orbit.
5. Atomic shell – an arrangement of electrons in an atom, in compliance with appropriate
physical laws.
6. 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
7. 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
8. Rydberg formula : is an impirical relation that gives all wavelengths of atomic Hydrogen.
9. 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
10. Quantisation of Angular Momentum- The Angular momentum quantum number can
take only certain values in multiples of . This phenomenon is also referred to as space
quantisation
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11. Angular Momentum Coupling:- The orbital and spin angular momentum of a particle
can interact through spin-orbit interaction. The procedure of constructing 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
12. 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 viscous fluid:
F  6 r v where: F is the frictional force, r is the radius of the particle, η is the fluid
viscosity, and v is the particle's velocity.
Source: http://en.wikipedia.org/wiki/Stokes'_law
13. Bremsstrahlung radiation – Radiation ( X-rays) produced by slowing down energetic
electrons (or any charged particles) upon impact on a target (or an absorber). Source:
http://www.ionactive.co.uk/glossary/Bremsstrahlung.html
14. Bragg’s Law - the result of experiments into the diffraction of X-rays or
crystal surfaces at certain angles.
Source: http://en.wikipedia.org/wiki/Bragg's_law
neutrons off
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XII
COMPILED LIST OF COMPULSORY 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.
Reading 2:
Complete reference : Cathode Ray Tube
From Wikipedia the free encyclopedia
URL : http://en.wikipedia.org/wiki/cathoderay
Accessed on the 20th April 2007
tube
Abstract :
This reading is compiled from wikipedia page indicated above and the links available in the
page. The pages provide a good supplementary material for activity two of this module
Rationale:
This article provides additonal material and links are available for futher reading..
Reading 3: Zeeman Effect.
Complete reference : Zeeman Effect
From from wikipedia
URL : http://en.wikipedia.org/w/index.php?title=Zeeman_effect&printable=yes
Accessed on the 24th April 2007
Abstract :
The Zeeman effect is presented from a theroretical stand point covering the weak field effect
with the Lyman alpha transition in hydrogen as an example and the strong field effect.
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Rationale:
This article provides additonal material and links are available for futher reading.
Reading 4: X-Ray Crystallography
Complete reference
From Wikipedia.
URL : http://en.wikipedia.org/wiki/X-Ray_Crystallography#Definition
Accessed on the 20th April 2007
Abstract :
The article provides reviews on theory of scattering and diffraction and on the methodolgy of
x-ray crystallography.
Rationale:
This article has a well illustrated content on x-rays. It is a good supplement for Activity 4 and
the links there are useful reading materials.
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XIII
COMPILED LIST OF (OPTIONAL) MM RESOURCES
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
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
e the resources provided. The rationale for the resource provided should also be explained (maximum length : 50-75 words). An electronic version of each
source is required.
1.
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.
2.
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.
3.
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.
4.
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
5.
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.
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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
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 .
Rationale:
.
Date Consulted:- May 2007
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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
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
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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 .
Rationale:
.
Date Consulted: May 19, 2007
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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 sub- atomic 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.
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 and
q
E
 2 where the symbols have the satandard meaning.
derive the exprsession
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. (b) Consider the
two ways in which L and S vectors add to form the vector J when l = 1 and s = ½ .
j( j  1 )  l( l  1 )  s( s  1 )
If the angle between L and S is , show that cos  
2 l( l  1 )s( s  1 )
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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? (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
Orbital
1s
Binding Energy , keV
20.000
Calculate the wavelengths of the
LI
LII
2s
2p
2.866
2.625
K X-rays emitted.
LIII
2p
2.520
Answer Key:
1.(a) Bohr postulates: Postulate 1: Coulomb force balanced by centripetal force, postulate 2:
me qe4 1
L = n ; postulate 3: E n   2 2 2 ;
8h  0 n
postulate 4:  E  Ei  E f .
(b) Bohr radius rn 
 0 nh2
. (c) In the ground state n = 1, so that the Bohr
 me Ze 2
radius is r1.
2. In the Thompson tube the electric force is balanced by the magnetic force.
3. (a) Orbital angular momentum L is due to rotation of electron in its orbit. Spin angular
momentum S has no classical analogue. Total angular momentum is the vector sum of L
and S .
(b) Vector sum of L and S
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Applying cosine rule to the triangle made by vectors J, L, and S we have
J 2  L2  S 2
cos  
2 LS
But L  l( l  1 ) ; S  s( s  1 )
Upon substitution we have cos  
and J 
j( j  1 )
j( j  1 )  l( l  1 )  s( s  1 )
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
number
n
l
ml
ms
Electron 1
1
0
0
+1/2
Electron 2
1
0
0
-1/2
(c ) No two electrons can occupy an energy state definied by the same quantum
numbers.
Outermost electrons
Quantum
number
n
l
ml
ms
Electron 1
3
0
0
+1/2
Electron 2
3
0
0
-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.
101
(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  = hc/E = 2.82 x10-7 m.
6 (b) Using cot  / 2  
=> b 
3.795
0
4 0T
2 Ze 2
cot  / 2 
b where T = 5.3 MeV and Z = 79.
this expression is adequate since  is unknown.
7. Transitions are subject to the selection rule l = 1.

c


12.4 keV . A
hc
hc
where Ei is the initial energy and Ef is the final


h Ei  E f
Ei  E f
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 Evaluation should
consistitute 60%.
102
XVII. References:
1. Foot C.J.(2005) Atomic Physics, Oxford University Press, Chapters 1 and 2..
2. Willmont, J.C. (1975), Atomic Physics, Wiley.
3. Beiser A., (2004) Applied Physics, 4th ed., Tata McGraw_Hill edition, New Delhi, India.
4. Bernstein, J.Fishbane, P.M. and Gasiorowicz, Modern Physics, Prentice Hall.
5. Anderson, E.E. 1971, Modern Physics and Quantum Mechanics, W.B.Saunders Co.
Philadelphia.
6. Cohen-Tannoudji,C., Diu, B., Laloe, F. 1977, Quantum Mechanics, John Wiley and
Sons Inc., Paris.
7. Gasiorowicz, S. 1974, Quantum Physics, John Wiley and Sons Inc., New York:
8. Liboff, R.L. 1980, Introductory Quantum Mechanics, Addison-Wesley Publishing
Co. Inc., New York.
9. Landau, L.D. and Lifshiftz, E.M., 1958, Quantum Mechanics Non-Relativistic
Theory, Addison-Wesley Publishing Co. Inc., London.
10. Merzbacher,E., 1961, Quantum Mechanics, John Wiley and Sons Inc., New York
11. Rae, A.I.M., 1986, Quantum Mechanics, Adam Hilger/English Language Book
Society, Bristol.
XVIII. Main Author of the Module
About the author of this module:
Name:Christopher Amelye KIWANGA
Title:
Associte Professor of Physics
Address:
The Open Universty of Tanzania
P.O.Box 23409
DAR ES SALAAM
TANZANIA.,
E-mail: ckiwanga@yahoo.com , kiwanga5@hotmail.com .
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 -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
University of Tanzania for six years todate.
103
You are always welcome to communicate with the author regarding any
question, opinion, suggestions, etc in respect of this module.
XX. 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. .
104