compulsory - cdlteachingcommons.cdl.edu/avu.old/physics/documents/atomic physics... · atomic...

CCOOMMPPUULLSSOORRYY

RREEAADDIINNGGSS11

1 According to the author of the module, the compulsory readings do not infringe known copyright.

Reading 1Complete reference : Atomic Models From: wikipediaURL : http://en.wikipedia.org/wiki/Atomic_physicsAccessed 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’splum pudding model, Rutherford’s alpha scattering experiment that led to the pla-netary 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 DateConsulted: June 2007Abstract :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 linkto go to the derivation of the Bohr radius and click next again for calculation of energylevels.Rationale: The material is presented in a manner that it is easy to follow.

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

Reading 4: A Look Inside the Atom

Complete Reference: http://www.aip.org/history/electron/jjhome.htm

Date Consulted: June 2007

Abstract: This is an account of the work by J.J.Thomson on Cathode rays that culminated in the discovery of the electron as a fundamental part of atom. Follow the links by clicking next.

Rationale: The article is qualitative but very informative and relevant to this course.

Reading 5: Nobel Prize Lecture on Cathode Rays

Complete Reference: http://nobelprize.org/nobel_prizes/physics/laureates/1905/le-nard-lecture.html


Abstract: In the context of what you already know now, this is a light reading but informative article on cathode rays and misconceptions at the time.

Rationale: The presentation is by a Physics Nobel Prize winner, Philipp Lenard, 1905. This is good motivational material for you.

Reading 6: The Millikan Oil Drop Experiment

Complete reference: http://hep.wisc.edu/~prepost/407/millikan/millikan.pdf


Abstract: This is a good quantitative article on the practical aspects of the Millikan Oil Drop Experiment.

Rationale: The material is good and relevant to the course.

Reading 7

Complete reference : URL: http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.htmlDate consulted: June 2007Abstract : 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 8: Emission Spectrum of Hydrogen

Complete reference : URL : http://chemed.chem.purdue.edu/genchem/topicre-view/bp/ch6/bohr.htmlDate 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 pro-blems makes this reading very important.

Reading 9: Hydrogen Atom

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

Reading 10: Mathematical Solution of the Hydrogen Atom

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

Reading 11: Fine Structure of Hydrogen Atom

Complete Reference: http://farside.ph.utexas.edu/teaching/qmech/lectures/node107.htmlAbstract: This article is part of a series of lecture notes in non relativistic quan-tum mechanics. Rationale: The material is good but requires a strong link with knowledge in quantum mechanics.

Reading 12: X-Ray Production

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

Reading 13: The Origin of Characteristic X-Rays

Complete reference : http://www4.nau.edu/microanalysis/Microprobe/Xray-Cha-racteristic.htmlAbstract : 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 14:X-Ray Di�raction .

Complete reference : http://www.physics.upenn.edu/~heiney/talks/hires/whatis.html#SECTION00011000000000000000

Date Consulted: Junel 2007Abstract : In this article, x-ray is concisely presented. Rationale: The article covers the contents of this activity

Reading 15: X-Ray Di�raction

Reference link: http://e-collection.ethbib.ethz.ch/ecol-pool/lehr/lehr_54_folie2.pdfComplete reference: http://www.google.com/search?q=cache:qLs7iI81agwJ:e-collec-tion.ethbib.ethz.ch/ecol-pool/lehr/lehr_54_folie2.pdf+X-RAY+MOSLEY’S+LAWAbstract: This article contains Power Point Slides on practical aspects of X-Ray

elcitra eht ssecca oT .waL s’yelsoM dna murtcepS yaR-X ,ebuT yaR-X ,noitcarffiDstart with the complete reference and then click on reference link.Rationale: The material is relavant to this activity. Please read it.

RREEAADDIINNGG ##11

Atomic physics From Wikipedia, the free encyclopedia

Atomic physics (or atom physics) is the field of physics that studies atoms as isolated systems comprised of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and the processes by which these arrangements change. This clearly includes ions as well as neutral atoms and, unless otherwise stated, for the purposes of this discussion it should be assumed that the term atom includes ions.

The term atomic physics is often associated with nuclear power and nuclear bombs, due to the synonymous use of atomic and nuclear in standard English. However, physicists distinguish between atomic physics - which deals with the atom as a system of electron(s) and a nucleus - and nuclear physics - which considers atomic nuclei alone.

As with many scientific fields, strict delineation can be highly contrived and atomic physics is often considered in the wider context of atomic, molecular, and optical physics. Physics research groups are usually so classified.

Isolated atoms Atomic physics always considers atoms in isolation - i.e. a model will consist of a single nucleus which may be surrounded by one or more bound electrons. It is not concerned with the formation of molecules (although much of the physics is identical) nor does it examine atoms in a solid state as condensed matter. It is concerned with processes such as ionization and excitation by photons or collisions with atomic particles.

While modelling atoms in isolation may not seem realistic, if one considers atoms in a gas or plasma then the time-scales for atom-atom interactions are huge in comparison to the atomic processes that we are concerned with. This means that the individual atoms can be treated as if each were in isolation because for the vast majority of the time they are. By this consideration atomic physics provides the underlying theory in plasma physics and atmospheric physics even though both deal with huge numbers of atoms.

Contents 1 Isolated atoms 2 Electronic configuration 3 History and developments 4 Significant atomic physicists 5 See also

5.1 Fundamental atomic physics5.2 Common units 5.3 Applications 5.4 Related fields

6 References 7 External links

Page 1 of 4Atomic physics - Wikipedia, the free encyclopedia

5/19/2007http://en.wikipedia.org/wiki/Atomic_physics

Additionally, the properties of an atom in isolation is different from that of individual atoms in relatively close proximity to each other. This is because medium and long range forces come into play with proximity.

Electronic configuration Electrons form notional shells around the nucleus. These are naturally in a ground state but can be excited by the absorption of energy from light (photons), magnetic fields, or interaction with a colliding particle (typically other electrons). The excited electron may still be bound to the nucleus and should, after a certain period of time, decay back to the original ground state. The energy is released as a photon. There are strict selection rules as to the electronic configurations that can be reached by excitation by light - however there are no such rules for excitation by collision processes.

An electron may be sufficiently excited so that it breaks free of the nucleus and is no longer part of the atom. The remaining system is an ion and the atom is said to have been ionized having been left in a charged state.

History and developments The majority of fields in physics can be divided between theoretical work and experimental work and atomic physics is no exception. It is usually the case, but not always, that progress goes in alternate cycles from an experimental observation, through to a theoretical explanation followed by some predictions which may or may not be confirmed by experiment, and so on. Of course, the current state of technology at any given time can put limitations on what can be achieved experimentally and theoretically so it may take considerable time for theory to be refined.

Clearly the earliest steps towards atomic physics was the recognition that matter was composed of atoms, in the modern sense of the basic unit of a chemical element. This theory was developed by the British chemist and physicist John Dalton in the 18th century. At this stage, it wasn't clear what atoms were although they could be described and classified by their properties (in bulk) in a periodic table.

The true beginning of atomic physics is marked by the discovery of spectral lines and attempts to describe the phenomenon, most notably by Joseph von Fraunhofer. The study of these lines led to the Bohr atom model and to the birth of quantum mechanics itself. In seeking to explain atomic spectra an entirely new mathematical model of matter was revealed. As far as atoms and their electron shells were concerned, not only did this yield a better overall description, i.e. the atomic orbital model, but it also provided a new theoretical basis for chemistry (quantum chemistry) and spectroscopy.

Since the Second World War, both theoretical and experimental fields have advanced at a great pace. This can be attributed to progress in computing technology which has allowed bigger and more sophisticated models of atomic structure and associated collision processes. Similar technological advances in accelerators, detectors, magnetic field generation and lasers have greatly assisted experimental work.

Significant atomic physicists



Pre quantum mechanics

John Dalton Joseph von Fraunhofer Johannes Rydberg J.J. Thomson

Post quantum mechanics

David Bates Niels Bohr Max Born Clinton Joseph Davisson Charlotte Froese Fischer Vladimir Fock Douglas Hartree Harrie S. Massey Nevill Mott Mike Seaton John C. Slater George Paget Thomson

See also

Fundamental atomic physics

Energy level Wavefunction Atomic orbital Electron configuration

Common units

SI units Electron volt Hartree Rydberg wavenumber

Applications

Plasma physics Stellar atmosphere Atmospheric physics Atomic clock

Related fields

Quantum optics

Physics Portal



This page was last modified 17:43, 16 May 2007. All text is available under the terms of the GNU Free Documentation License. (See Copyrights for details.) Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a US-registered 501(c)(3) tax-deductible nonprofit charity.

Molecular physics

References Bransden, BH; Joachain, CJ (2002). Physics of Atoms and Molecules, 2nd Edition, Prentice Hall. ISBN 0-582-35692-X. Foot, CJ (2004). Atomic Physics. Oxford University Press. ISBN 0-19-850696-1.

External links Atomic Physics on the Internet Atomic Physics Links JILA (Atomic Physics)

Retrieved from "http://en.wikipedia.org/wiki/Atomic_physics"

Categories: Atomic physics | Atomic, molecular, and optical physics



RREEAADDIINNGG ##22

BELIEVE ME NOT! - - A SKEPTICs GUIDE

Next: Relativistic Energy Up: Fudging The Bohr Atom Previous: The Bohr Radius

Bohr's Energy Levels

Going on, we can calculate the net energy (kinetic plus potential) of an electron in the Bohr orbital of the H atom:

or [again using Eq. (8) to substitute for pn]

Now we replace rn with our expression (13) to get

which simplifies to

where J = 13.6055 eV (where 1 eV = J). We have thus reproduced

Bohr's explanation for the empirical formulae of Balmer and Rydberg! Note that whereas the energy of confinement of a particle in a box increases as n2 (where n-1 is the number of nodes inside the box), the Bohr energy levels of an atom increase as -1/n2 (they get less negative and closer together as n increases). Of course, so far all these calculations have been done in the classical (nonrelativistic) limit. If the momenta get big enough (p comparable to or greater than mc) we have to do our calculations differently . . . .

Next: Relativistic Energy Up: Fudging The Bohr Atom Previous: The Bohr Radius Jess H. Brewer 2000-01-17

(24.15)

Page 1 of 1Bohr's Energy Levels

4/16/2008http://musr.physics.ubc.ca/~jess/hr/skept/QM1D/node4.html

BELIEVE ME NOT! - - A SKEPTICs GUIDE

Next: Black Holes Up: Particle in a Box Previous: Bohr's Energy Levels

Relativistic Energy Let's generalize our formula for kinetic energy so that it is relativistically correct. For a massless particle (like a photon) the expression (4) doesn't make any sense and is in fact wrong. Without stopping now to explain where it comes from, I will just give you the relativistically correct and completely general formula for the total energy of a particle:

Note that this TOTAL RELATIVISTIC ENERGY has the irreducible value E0 = m c2 when the particle is at rest (momentum = zero). This should ring a bell. To separate the KINETIC ENERGY K from the total relativistic energy we just subtract off E0.

It turns out [Don't you love that phrase?] that de Broglie's relation (1) is relativistically correct! Thus we can still use it to calculate the total energy even if the confined particle is ultrarelativistic or massless. In fact, any particle acts pretty much like a photon at high enough momentum, where we can ignore m2 c4 in comparison with p2 c2, in which case the formula simplifies to E = pc or (for our ultrarelativistic particle in a box)

Black Holes The Planck Length

Next: Black Holes Up: Particle in a Box Previous: Bohr's Energy Levels Jess H. Brewer 2000-01-17

(24.16)

(24.17)

Page 1 of 1Relativistic Energy

4/16/2008http://musr.physics.ubc.ca/~jess/hr/skept/QM1D/node5.html

RREEAADDIINNGG ##33

Rutherford Scattering Alpha particles from a radioactive source were allowed to strike a thin gold foil. Alpha particles produce a tiny, but visible flash of light when they strike a fluorescent screen. Surprisingly, alpha particles were found at large deflection angles and some were even found to be back-scattered.

This experiment showed that the positive matter in atoms was concentrated in an incredibly small volume and gave birth to the idea of the nuclear atom. In so doing, it represented one of the great turning points in our understanding of nature.

If the gold foil were 1 micrometer thick, then using the diameter of the gold atom from the periodic table suggests that the foil is about 2800 atoms thick.

Some history Questions raised Geiger-Marsden data

Index

Rutherford concepts

Scattering concepts

Great

experiments of physics

HyperPhysics***** Mechanics ***** Nuclear R Nave

Go Back

Rutherford Scattering Formula The scattering of alpha particles from nuclei can be modeled from the Coulomb force and treated as an orbit. The scattering process can be treated statistically in terms of the cross-section for interaction with a nucleus which is considered to be a point charge Ze. 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:

Page 1 of 3Rutherford Scattering

4/16/2008http://hyperphysics.phy-astr.gsu.edu/hbase/rutsca.html

The predicted variation of detected alphas with angle is followed closely by the Geiger-Marsden data. The above form includes the cross-section for scattering for a given nucleus and the nature of the scattering film to get the scattered fraction. Another common form for the Rutherford equation is just the differential cross section for scattering from a given nucleus.

For this equation, some of the constants have been combined to express the cross section in terms of the fine-structure constant, α .

The departure from the point-particle form of scattering has been an indicator of nuclear structure and then at higher energies, the structure of the proton.

This form of the scattering formula serves as a signature for scattering off a point target in which no structure is evident. The point of departure from Rutherford scattering in the case of the nucleus was the basis for the earliest evaluations of the nuclear radius.

Index

Rutherford concepts

Scattering concepts

HyperPhysics***** Mechanics ***** Nuclear R Nave Go Back

Alpha Scattering Geometry



The scattering of the alpha particle by the central repulsive Coulomb force leads to a hyperbolic trajectory. From the scattering angle and momentum, one can calculate the impact parameter and closest approach to the target nucleus.

Calculation of impact parameter and closest approach

Cross section for scattering

Add annotation showing relationships for calculation

Index

Rutherford concepts

Scattering concepts

Beiser

reference

HyperPhysics***** Mechanics ***** Nuclear R Nave Go Back



RREEAADDIINNGG ##44

[an error occurred while processing this directive]

SITE MAP

Click on any image for a big picture and more information.

ne hundred years ago, amidst glowing glass tubes and the hum of electricity, the

British physicist J.J. Thomson was venturing into the interior of the atom. At the Cavendish Laboratory at Cambridge University, Thomson was experimenting with currents of electricity inside empty glass tubes. He was investigating a long-standing puzzle known as "cathode rays." His experiments prompted him to make a bold proposal: these mysterious rays are streams of particles much smaller than atoms, they are in fact minuscule pieces of atoms. He called these particles "corpuscles," and suggested that they might make up all of the matter in atoms. It was startling to imagine a particle residing inside the atom--most people thought that the atom was indivisible, the most fundamental unit of matter.

A simple cathode ray tube.

Thomson in his office.

homson's speculation was not unambiguously supported by his experiments. It took more

experimental work by Thomson and others to sort out the confusion. The atom is now known to contain other particles as well. Yet Thomson's bold suggestion that cathode rays were material constituents of atoms turned out to be correct. The rays are made up of electrons: very small, negatively charged particles that are indeed fundamental parts of every atom.

"Could anything at first sight seem more impractical than a

odern ideas and technologies based on the

electron, leading to television and the computer and much else, evolved through many difficult steps.

Page 1 of 2A Look Inside the Atom

4/16/2008http://www.aip.org/history/electron/jjhome.htm

body which is so small that its mass is an insignificant fraction of the mass of an atom of hydrogen?" -- J.J. Thomson.

Thomson's careful experiments and adventurous hypotheses were followed by crucial experimental and theoretical work by many others in the United Kingdom, Germany, France and elsewhere. These physicists opened for us a new perspective--a view from inside the atom.

Table of Contents:

Exhibit Home J.J. Thomson Mysterious Rays 1897 Experiments Corpuscles to Electrons Legacy for Today Exhibit Info

Mysterious Rays

Click here for information about this exhibit and

suggested readings and links.

Sign up for our e-mail list:

find out when we put more exhibits online.

The Discovery of the Electron is brought to you by the

All material on this site copyright ©1997- 2008 American Institute of Physics.

This site created 3/97

Center for History of Physics

a division of the American Institute of

Physics

Page 2 of 2A Look Inside the Atom

4/16/2008http://www.aip.org/history/electron/jjhome.htm

RREEAADDIINNGG ##55

P H I L I P P E. A . V O N L E N A R D

On cathode rays

Nobel Lecture, May 28, 1906

I am pleased to fulfil my obligation as a Nobel Prize winner to talk to youhere on cathode rays. I assume that you would prefer me to tell you whatothers could not tell you. I shall describe to you the development of the sub-ject - which also embraces recent theories concerning electricity and matter- as it has appeared to me, on the basis of my own experience.* This willgive me a welcome opportunity of showing on the one hand how my workhas depended on that of others, and on the other how in one or two pointssubsequent, or more or less contemporary, work by other investigators isrelated to mine. Thus - using the simile which you, my esteemed colleaguesof the Academy of Sciences, have used at the head of your member’s diplo-ma** - I shall now speak not only of the fruits but also of the trees whichhave borne them, and of those who planted these trees. This approach is themore suitable in my case, as I have by no means always been numberedamong those who pluck the fruit; I have been repeatedly only one of thosewho planted or cared for the trees, or who helped to do this.

In the time at my disposal I can deal at length with only a few aspects ofmy work in the field under discussion.

The start takes me back 26 years to Crookes. I had read his lecture on"radiating matter" (5)*** - his term for cathode rays**** - and was greatlyimpressed by it. You are all acquainted with the tests he made. Here Fig. 1 isone as a reminder: the glass tubes with highly rarefied air; the negativelycharged plate or cathode (a) on which the rays are produced; a cross (b) inthe path of the rays, and here the shadow of the cross (d) thrown by the rays

* In this paper I have tried hard to put into their historical perspective all the publica-tions which in my opinion have made basic contributions to knowledge, even if theycame to my notice too late for them to influence my work.** Coat of arms: gardener planting young trees, with the motto "For our descendants".*** The numbers in brackets refer to the bibliography at the end; here "p." gives thepage number in the case of voluminous publications.**** After Faraday, Hittorf (2) and then Goldstein (4) had already produced and pro-gressively studied "glow rays" or "cathode rays". But Crookes made more progressthan these workers, because he carried out experiments at a higher vacuum.

106

Fig. 1.

onto the phosphorescent glass wall. The shadow moves when a magnet isbrought near; this is a sign that the cathode rays - unlike light rays - are bentin a magnetic field.

I have always attached great importance in my work to the problem ofisolating the phenomenon being studied from interference sources, irrespec-tive of the difficulties that this entails, an approach already adhered to byCrookes in his work. For it was he who produced these cathode rays in a pureform as never reached before, and showed that the phenomena concernedare of a very special type differing from other discharge phenomena throughtheir attractive simplicity. The real nature of "radiating matter" or "thefourth state of aggregation", as he called it, was then beyond my compre-hension, just as it must, we may now be sure, have been beyond his. But Ireadily shared his enthusiasm when he said: "Here, I believe, are the ultimaterealities." And we were right: that is why I stand here today!

My interest in these matters found no direct expression during my studentdays. Electrical gas discharges were not considered a suitable object of studyfor beginners, and rightly so. But even mature investigators achieved noth-ing really significant in this field in the years following Crookes’ work. Theydid not obtain any results that in themselves opened new vistas, and so far aspurity of experimental conditions was concerned they hardly progressed be-yond Crookes’ work.

It was only later, when I was assistant to Quincke in Heidelberg, that I hadthe opportunity and the facilities for building a mercury air pump capableof giving very high rarefaction - then by no means a standard item of equip-ment in physics institutes - and for carrying out tests myself on cathode rays.I wanted to advance as directly as possible, and thought how fine it wouldbe, in particular, to bring these rays from the tube out into the open air; itwould then be possible to carry out direct experiments with them. To dothis it was necessary to fit into the wall of the tube an airtight seal that would

O N C A TH O D E RA Y S 107

allow the rays to pass through. Now radiating matter would not readily passthrough airtight seals, but might not Eilhard Wiedemann be right in as-suming that cathode rays were a form of ultra-ultraviolet light? Finally,quartz appeared to me to be the most promising material, since it best trans-mitted all the radiations that were then known. Here (Fig. 2) is the tube Ibuilt at the time with the cathode plate, and at the top of it you will see theopening sealed by a quartz plate 2.4 mm thick. The test was unsuccessful,however; outside the quartz I found no phosphorescence nor even any elec-trical* effects that could be ascribed with certainty to the light issuing fromthe tube.

It was four years later, in 1892, that another opportunity arose. Hertz,whose assistant I then was, had found that thin metal leaf transmits cathoderays (15). He used quite thin, very soft and porous gold, silver and aluminiumleaf used in bookbinding, but showed that the cathode rays not only passthrough the holes but through the material itself, the metal of the leaf. Oneday he called me over - an event which to my great regret at the time didnot occur often - and showed me what he had just found: uranium glasscovered with aluminium leaf inside a discharge tube, glowed under the leafwhen irradiated from above. He said to me: "We ought - and I might sim-ply do this for he was prevented - to separate two chambers with aluminiumleaf, and produce the rays as usual in one of the chambers. It should then bepossible to observe the rays in the other chamber more purely than has beendone so far and even though the difference in air pressure between the twochambers is low because of the softness of the leaf, it might be possible tocompletely evacuate the observation chamber and see whether this impedesthe spread of the cathode rays - in other words find out whether the rays are* Hertz’ discovery of such effects of ultraviolet light had at that time just been made (8).

108 1905 P.E.A .V O N LEN A RD

phenomena in matter or phenomena in ether." He appeared to consider thislast question to be the most important one. I did carry out the test later;but I was primarily interested in my earlier question, that of cathode rays inthe open air. I was not put off by the softness of the leaf used by Hertz. I laidmore and more of such leaves on top of each other in a suitable tube andfound that 10 and 15 layers still transmitted the rays fairly well. I then pro-cured some pieces of aluminium foil of comparable thickness, to see whetherthey would withstand the air pressure. Such was the case, provided that asufficiently small area of foil was used. Then, taking the old tube again, Ireplaced the quartz by a metal plate containing a small hole sealed withaluminium foil, spread a few small grains of alkaline-earth phosphor onthis small aluminium window, excited the tube and, lo and behold, thegrains glowed brightly! I then fixed them slightly above the aluminiumwindow and they glowed brightly there as well! Thus not only had the ca-thode rays passed out of the interior of the discharge tube to which they hadbeen hitherto confined, in addition - and nobody could have predicted this -they could pass through air of normal density. It thus became clear that avast new field of investigation had opened up in front of me, a field that notonly embraced hitherto unseen phenomena but also gave promise of a break-through into the unknown. Cathode rays, which had hitherto stubbornlyeluded explanation, had yielded their secret and, more important, now forthe first time tests of maximum purity could be carried out. Let us comparethe position with that of another type of radiation, light: hitherto it was as ifit had been impossible to study light except in the interior of furnaces andflames where it is produced, like the cathode rays in the tube. Where thenwould the great and detailed science of optics have stopped? ! A window hadnow been built in the furnace through which pure light only could emerge,freed from the complex and still unexplained processes of its formation. Suchprocesses remain confined to the interior of the discharge tube and, as hassince been found, could not be understood until a sufficient study had beenmade of the cathode rays themselves. As we shall see in our historical survey,this study has also provided a great deal of other information, some of whichis now general knowledge, on X-rays and radioactivity, as well as a deeperunderstanding of electricity and matter.

It was now necessary above all to widen the inroad already made into thenew field of knowledge. It was important to increase the intensity of the rayscoming out of the window, and to improve the conditions of their produc-tion compared with the first tube. This led to the construction of the tube

ON CATHODE RAYS 109

S

Fig. 3.

illustrated in Fig3, which was used in a large number of experiments (18).Here will be seen the production chamber with the anode (A) and the cath-ode (C), the seal ( m m) with the window, and beyond it the observationchamber into which the rays emerge. The number of phenomena possiblehere is such that although the most obvious ones and also the slightly lessobvious ones have probably now all been discovered, so far the consequencesof all the phenomena have not yet been studied sufficiently.

It must be noted that the rays are not directly visible; it would be uselessto put one’s eye to the window, as this organ is not receptive to cathoderays. On the other hand, materials that are capable of becoming luminouswithout heat, phosphorescent materials as they are called, are suitable formaking the rays visible. It is best to use sheets of paper coated with suchmaterials, e.g. a certain ketone, platinum cyanide, or an alkaline-earth phos-phor and to hold them as a screen against the rays. If the screen glows, itindicates that it has been hit by the rays. The rays can also be photographeddirectly. These are the same methods that are used to make visible ultra-violet light, at that time the only known example of such demonstrable in-visible radiation.

When we use the phosphorescent screen, we find it glowing brightly closeto the window; as the distance from the window increases, the intensity ofthe rays progressively diminishes until at a distance of about 8 cm the screenremains quite dark. Apparently air at full atmospheric pressure is not verypermeable to cathode rays, certainly far less permeable than it is to light. Butit was far more interesting to find that air is even a turbid medium for theserays, just as milk is for light. If we place an impermeable wall with a hole

110 1905 P.E.A . VON LENA RD

in it a suitable distance from the window and put the edge of the screenagainst it, we then get this view (Fig. 4). Here the dotted lines indicate thenarrow pencil of rays that we should expect in the case of rectilinear prop-agation; but it is the broad bent bunch of rays that we really see on thescreen in the open air, just as if we had passed light through the same holeinto a tank containing slightly diluted milk. What clouds the air? In milk it

Fig. 4.

is numerous small suspended fat particles that make it turbid to light. Pureair on the other hand contains nothing except molecules of the gases con-tained in it, suspended in the ether. These molecules are extremely small,10,000 times smaller than the fat particles, far too small to act individually onlight. But, as we see, the cathode rays are hindered by each of these mole-cules. Thus these rays must be extremely fine, so fine that the molecular struc-ture of matter, which is minute compared with the very fine light waves,becomes pronounced in comparison with them. It may then be possible toobtain data by means of these rays concerning the nature of molecules andatoms.

It is therefore particularly interesting to study the behaviour of a widevariety of materials relative to cathode rays. The first point to be studied wasthe permeability. Some idea of this can be obtained by holding a thin layer ofthe material being studied, between the window and the screen. It is abun-dantly clear that the permeability or impermeability of a material to light isis not even slightly related to its behaviour in relation to cathode rays. Hereis an example (Fig. 5), a print of a direct photograph taken at the aluminiumwindow. In the top half will be seen the deep shadow of a completely light-permeable ½ mm thick rectangular quartz plate, and in the left half, as avery mat veil, the picture of an ordinary aluminium leaf, impermeable tolight and with somewhat irregular boundaries, laid over the whole of this

ON CATHODE RAYS 1 1 1

Fig. 5.

half. Great care must be taken in selecting the thickness of the layer throwingthe shadow. Thus, for example, the quartz plate used in this experiment isimpermeable simply because it is too thick, and the reason why metal leafwas found to be the only example of permeable layers in Hertz’ tests lay inthe thinness at which metal leaf is available. We shall soon see that most othermaterials, when of the same thickness, are even more permeable than goldand silver. It is soon evident that the absorption of cathode rays in any sub-stance is a very gradual process, just as in the case of light, where as we know,gold is permeable when it is made sufficiently thin. Here (Fig. 6) we see theshadow of leaves of aluminium laid stepwise on top of each other; the num-bers on the left give the number of leaves, those on the right their totalthickness. Each increase in the number of leaves, and also any unevenness in

Fig. 6.

112 1905 P.E.A . V O N LEN A RD

the thickness of the individual leaves, can be noted, and it will be seen howalmost total permeability changes into almost total impermeability. Thus,with each material it was not a question of simply deciding between "per-meable" and "impermeable", but of finding a numerical measure of the de-gree of absorption of cathode rays therein, of measuring its absorptive power,

and I did this for a large number of solid and gaseous materials.The result was astounding. All the great multiplicity of properties that we

associate with the different materials around us, disappeared. The sole deter-mining characteristic was the weight of the materials (21). Everything of equalweight absorbed equally, anything heavier absorbed more, anything lighterabsorbed less, and always in proportion to the weights or the masses. At a firstapproximation, the chemical composition of the materials, their state of ag-gregation and other properties, did not count at all - a quite unprecedentedfinding that was not valid for any radiation known at the time. * At a secondapproximation, on closer inspection it is seen that the chemical compositionhas a slight effect: thus, e.g. hydrogen, and anything containing hydrogen,absorbs slightly more than one would expect in proportion to its weight. Imust forbear to discuss these deviations and their significance in detail here.**As an illustration of the law of proportionality between mass and cathoderay absorption, that is valid at a first approximation, let us see the directphotographs of the shadows of layers of aluminium, silver and gold of equalthickness (Fig. 7). It will be seen that the heavy silver absorbs more than thelighter aluminium, and that gold, which is the heaviest, absorbs the most. Ifon the other hand we take layers of equal weights of the three metals (Fig.8), we also get equal shadows and equal absorption, and the result would

AIuminium

Silver

Gold

Fig. 7. Fig. 8.

* Later X-rays were found to be a second example of radiation that is absorbed moreor less in proportion to the mass.** Compare (21, 47, 52).

ON CATHODE RAYS 113

be the same if we had taken any other materials in layers of equal weight.In fact, in relation to cathode rays, not only the absorption, but also the

turbidity, which I also studied in relation to a number of different materials

(18b p. 257; 21, p. 265) was found to be related solely to the weight, themass of the material in question, the quantity of matter - as Newton put it -and not the quality of the material.* If we now recall that cathode rays as theyspread in matter are simply affected by the individual molecules of the ma-terial, we can conclude that the molecules of the most varied materials, andthus also the atoms of the different chemical elements vary, not qualitativelybut only quantitatively, from each other, i.e. they all consist of the same basicmaterial but contain different amounts of it. This old but because of the lackof valid data almost forgotten hypothesis of the alchemists was brought backvividly to our mind, this time however not to disappear again but to beproved; as evidence of this we can quote the recent results of Ramsay (54)and Rutherford (51) concerning the amazing transformation of radium**into other elements. But in order to use the law of proportionality betweenthe mass and cathode ray absorption as a basis for drawing more detailedconclusions on the constitution of matter, it was first necessary to knowsomething about the nature of cathode rays themselves. Let us now turn tothis problem, which I have also borne in mind throughout my work.

Straightway we can decide whether cathode rays are phenomena that takeplace in matter or in the ether. When we completely evacuate a chamber bymeans of an air pump, it then does not contain any matter, only the ether, aspresent in the heavens. Now it has long been known for instance that soundcannot pass through such evacuated chambers, while light, and electrical andmagnetic forces can. Thus there is no doubt that sound is a phenomenon inmatter while light and electrical and magnetic forces are phenomena in theether. We had been unable to carry out the corresponding test in relation tocathode rays in the ordinary discharge tubes, because once all the air is re-moved the production of the rays in such a tube ceases. But, without inter-fering in the least with production, we were able to completely evacuate ourobservation chamber on the other side of the window and see whether de-

* The diffuse reflection of cathode rays, which can be considered to be pronouncedbackwards-directed scatter, is also determined by the mass, as can be clearly seen fromthe measurements of A. Becker (52, p. 448).** As evidence of the elementary nature of radium its spectrum and atomic weightare given (38, 39).

114 1905 P.E.A. VON LENARD

spite this the cathode rays spread in this chamber. We found that the propaga-tion of the rays is particularly good in an extreme vacuum; all absorptionand turbidity due to the gas molecules disappear, the rays attain lengths ofseveral meters and are of such rectilinear sharpness as we are accustomed tofind only in light rays (18). Thus cathode rays are phenomena in the ether. In

particular, on the basis of the hypotheses which we have mentioned, it couldbe stated that cathode rays were not radiating matter, nor emitted gas mole-cules, as they had come to be regarded, especially in England. * We were stillnot clear as to what type of phenomena in the ether they were. Many of myreaders believed, very wrongly, that I had concluded beforehand that ca-thode rays were "waves in the ether"; I had no desire to say this or in factanything unless it was shown to be so in my experiments and appeared toprovide an explanation. I had the means available to discover new thingsdaily from Nature herself, in further experiments. I hoped so or thought so,at least. I greatly regretted, therefore, that at this stage my experimentswere interrupted for considerable time, first by a far-from-simple task thatdevolved unexpectedly on me through the untimely death of HeinrichHertz - the publication of his Prinzipien der Mechanik (Principles of Mechan-ics) and then when I was appointed to a theoretical professorship.

It is barely worth mentioning, but not unimportant for the further devel-opment of our subject, that even before this interruption I had designed anew and far more convenient type of discharge tube. I had tested it as far aspossible, and had recommended its use and made it generally available (18b,

p. 228). Here (Fig. 9) the window seal is fitted to a platinum tube, which inturn is fused into the glass; this means the large amount of puttying whichoften made the tube very difficult to use, is avoided. This type of tube hadhowever a special advantage that could not be foreseen at the time. In it the

* Even after the experiments with the aluminiumwindow had been reported, thistheory continued to be held for some time, and it was suggested that molecular col-lisions actuated, via the window, the molecules of the outside air or molecules remain-ing in the vacuum.

ON CATHODE RAYS 115

intensive cathode rays impinge on the large area of platinum - the metalwhich, as we now know, is most effective in turning the rays into the - thenundiscovered - X-rays. Thus X-rays are produced here in very large quan-tities, and they are also able to pass through the window, either mixed withor separate from the cathode rays, into the observation chamber. This wasnot possible in the earlier tube because of the large thick metal cover locatedin the path of the rays (27). The discovery soon after this of X-rays by Rönt-gen (22), the first investigator to use the type of tube described above, is gen-erally considered to be a good example of a lucky discovery. But, given thetube, the fact that the attention of the observer was already turned from theinterior to the outside of the tube, and the presence of phosphorescent screensoutside the tube because of the purpose of the tube, it appeared to me thatthis discovery had of necessity to be made at this stage of development.

On resuming my experiments I soon occupied myself exclusively with anidea already put forward by Hertz (7b, p. 275) and Schuster (13), which inrelation to the nature of the cathode rays had appeared to me to be very impor-tant right from the start, and which I had already begun to pursue in the firstperiod of my experiments. It had been known since Hittorf’s days that cath-ode rays are deflected by magnets (2) ; similarly the deflection of cathode raysnoted by Goldstein (4) could be interpreted as being an influence of electrical

forces on the rays. Now both the magnetic and the electrical deflection ofthe rays suggest that the cathode rays consist of emitted negatively chargedmasses. Moreover, from experiments made to measure the magnitude of themagnetic and electrical effects on a ray it is even possible to calculate the ve-locity of the supposed masses and also the electric charge per unit of mass(the charge/ mass ratio). This is what Hertz and Schuster had done, but theirresults had contradicted each other. Hertz found that his observations refutedthe theory of ejected gas molecules, while Schuster found that his fitted inwith it, and thus he took them as support for this theory.

This contradiction did not surprise me. For both investigators had observedthe interior of the discharge tube, and they might have been confused by thecomplications of the production process and the presence of the gas, as infact they both admitted with reserve. It was now time to carry out theseimportant tests under clearly-defined experimental conditions, i.e. outsidethe discharge tube and in a very high vacuum, and some excitement as tothe result was permissible. For if we know already that the rays are ether,not material phenomena, it should be astonishing that their behaviour stillresembled so deceptively that of ejected negatively charged gas molecules.


Nothing known so far had made solving this dilemma of the streaming mole-cules and the ether phenomena possible; these experiments would do this,and in any case therefore they would reveal something quite new.

While I was still doing the preparatory work for the experiments* I heardthat others were already convinced of their importance. J.J. Thomson wasthe first to publish a detailed publication on the subject (25). His experiments,like those of Hertz and Schuster, were carried out in the discharge tube. Hesought to avoid the danger of confusion as a result of interaction of the pro-duction discharge, by means of shielding devices, and to compensate the lackof reliability due to the presence of the gas by greatly varying the test condi-tions. It was clear to me that all that followed would have to rest on thispillar which I had learned to distrust from the discrepancy between Hertz andSchuster. It seemed to me that before it was made a part of the structure ofscience it should be tested as directly and rigorously as was possible with themeans at our disposal. I therefore concluded my experiments, the results ofwhich were as follows (28).

The velocity of the supposed masses was about one third the velocity oflight, and the ratio of the charge to the mass was about 1,000 times that of ahydrogen atom in electrolysis, which atom is the lightest material carrier ofelectricity known to us. So, if the rays were streaming hydrogen atoms,their charge would have to be taken as being 1,000 times that in electrolysis.This possibility had however been excluded by my previous tests, which hadshown that the rays are not material bodies. It seemed evident that I haddiscovered hitherto unknown parts of the ether, representing electric chargesand moving like inert masses. The smallness of the inertia determined -1/ 1,000 of the inertia of the hydrogen ion, at an equal charge - and the otherbehaviour (30) of these parts of the ether, made it easier to identify them withwhat had long been known as the "electrical fluidum". The solution to thedilemma was therefore as follows: The rays are not emitted electrically-charged molecules but simply streaming electricity. Thus, in cathode rays wehave found under our very noses what we never believed we should see:electricity without material, electrical charges without charged bodies. Wehave, in a sense, discovered electricity itself, a thing whose existence or non-existence and whose properties have puzzled investigators since Gilbert andFranklin. Earlier workers, even Coulomb, had referred naively to electric-

* These were resumed at Aix under the direction of and with a grant from Wüllner’sInstitute (1896), and they had again been interrupted when I was appointed to anotherprofessorship elsewhere.

ON CATHODE RAYS 117

ities as something that existed and could almost be grasped. But the num-ber of electrical phenomena known grew steadily without anybody beingable to state that they had seen anything of the supposed electricity itself.Thus it was that - about a generation after Coulomb - Faraday (1) and thenMaxwell (3) turned their attention completely away from the electricities toconcentrate on the electrical forces that could be observed. These forces -thought of as states in the ether - appeared in fact in the famous experimentsof Hertz (9) to be so likely to exist independently that from then on one feltincreasingly inclined to forget their centres, the electricities, that had former-ly been regarded as indispensible. Now - again about a generation after Fara-day and Maxwell - the picture has changed somewhat: it has become morecomplete. We have found in cathode rays just as good a way of studyingelectricity as we found earlier for the electrical forces alone; we can followthe motion of electricity to and fro in these rays over distances stretchingseveral meters, at will and directly with our senses - without any intermediatetheoretical conclusions; we can see how electricity behaves under differentconditions, and what its properties are; we are now in a position to give tothe old term "electricity" a new content based on experience.

This new content, of which we now know a great deal, appears quite dif-ferent now in many ways from what could have been supposed earlier.

Here it should be noted that all our remarks concerning electricity applyonly to negative electricity, not positive electricity, about which even todaylittle can be said that is concrete. We cannot claim to be acquainted with it;we can only recognize positively-charged material, whether it be atoms,molecules or groups of molecules.* We thus use the unitary means of expres-sion, and say that a piece of material is positively charged when it has lostnegative electricity.

Let us now therefore consider negative electricity, as it appears in our tests.Here we are amazed by the freedom of its motion, which we hitherto be-lieved was only present inside metallic conductors. Already in the dischargetube, in the centre of the gas, we set this electricity in accelerated motionthrough the voltage applied to the electrodes, and immediately its velocitybecomes one third that of light, 100,000km/ set, and it represents a cathoderay. Now it impinges on the aluminium window. "It will adhere to it andflow to earth" one would have said on the basis of previous knowledge. Far

* Thus, for instance, canal rays and, so far as they have been studied, also the α-raysof radium, have been found to be emitted positively charged molecules.


from it: it passes through the metal plate (28, p. 28) and, as I was able tocheck, its velocity does not diminish appreciably (19; 46, p. 479). Beyondthe window it can enter a more or less total vacuum, in which it continuesits course linearly, representing an electric current in the empty ether, a phe-nomenon which we had earlier also thought to be impossible. When itfinally hits a piece of metal of sufficient thickness, it penetrates it and sticksthere; finally, after following such unusual courses it appears as an ordinarycharge on the surface of the metal (28).

The problem of whether electricity fills space continuously or not, ofwhether it has a structure, is of particular interest. I have seen two cathoderays pass through the same chamber in opposite directions, and found in aquantitative investigation of the phenomena that the two rays did not inter-fere with each other in the slightest (44, p. 165). This indicates that the elec-tricity of these rays consists of discrete and very small parts separated by alarge volume of free space. We can represent the parts themselves as beingmore or less impenetrable to each other, because according to Coulomb’s law,as soon as two of the parts come very close to each other they must exertenormous repulsive forces on each other. But the best indication of the struc-ture of electricity comes from quite a different source, and is much older.

Here we come to the connexion between our findings and earlier knowl-edge. Such knowledge was very scanty, and was related to phenomena takingplace in and on individual atoms, i.e. phenomena that could not be studieddirectly, but the connexion was a very good and fruitful one.

Years earlier, Helmholtz in his lecture in memory of Faraday had notedthat electrolysis phenomena would suggest that electricity is split up intopieces of constant size, just as matter is split up in atoms (6). This was the in-dication already available concerning the structure of electricity, the exist-ence of electrical atoms, electrical elementary quanta , as Helmholtz called them.*

In the field of optics moreover, the theory already firmly supported byHertz’ famous experiments (9), that each luminous atom be regarded as anelectrical oscillator, had suddenly been given a tangible form by the discoveryof Zeeman, who in conjunction with Lorentz concluded from his observa-tions that it is negative** - not positive - electrical mass that oscillates in the lu-

* I recall hearing him use this expression many times in his demonstration lecture inthe summer term of 1885.** It is interesting that in Zeeman’s first publications the word "positive" - not "neg-ative" - was printed (24, p. 18), so that there was some delay in recognizing therelationship between his discovery and cathode rays.

ON CATHODE RAYS 119

minous atoms of a sodium or other metal flame, and that there is a definiteratio between the charge and mass of the oscillating material (24). The ratio wasof the same magnitude as that found shortly afterwards, in the way described,for cathode rays.

It seemed likely that in all these cases, in the ions in electrolysis, in the lu-minous metal atoms and in the cathode rays, and perhaps everywhere whereelectricity plays a part, we might be concerned with the same electrical el-ementary quanta, the existence of which had first been indicated by Fara-day’s electrolysis law and which might be further elucidated by means ofcathode rays. This theory has been proved, so much so that it has engendereda new branch of physics, so fruitful and already so vast that in this paper,which is devoted primarily to my own work, I cannot say any more in gen-eral on the subject. I would simply like to mention three points.

First, as an important initial quantitative check on our conclusions, the di-rect experimental measurement of the velocity of cathode rays carried out byWiechert, in which the figure obtained was the same as that obtained fromthe electrical and magnetic deflections (see above) - about one third of thevelocity of light (29).

Second, Kaufmann’s experimental result, obtained on the basis of thework of J.J. Thomson and Heaviside (10), and concerning electrical elementaryquanta, namely that their mass and inertia are of a purely electromagneticnature (55), a result that we can interpret as follows : We have no evidence that(negative) electricity is a special material with inertia; it appears to be simply astate, the state of the ether which we were accustomed, after Faraday (1), Max-well (3), and Hertz (9), to denote as the electric force field in the environ-ment of electrified bodies, a state which according to Hertz (20) and Bjerknes(33) might consist of latent motion of the ether. Thus, even with the pureelementary quanta of electricity nothing else has been discovered except thisether state in their area. These elementary quanta themselves appear to us inMaxwell’s sense to be the probably empty and only purely geometric centresof the electrical forces, except that we can now claim to be able successfullyto observe these centres individually, follow their courses and study the ge-ometric proportions of their size and shape. According to this finding, cath-ode rays, the streaming centres of state, appear to us, more than ever, to bewhat they seemed to us to be from the beginning, pure ether phenomena.

Thirdly, we must list the names given to these parts of electricity,or centres of state: I have called them elementary quanta of electricity orfor short quanta, after Helmholtz; J.J. Thomson speaks of corpuscles, Lord


Kelvin of electrions; but the name preferred by Lorentz and Zeeman electrons

has become the everyday term.

So far we have spoken of cathode rays as such; we shall now discuss theirmodes of formation, their generation.

The oldest, and for a long time the only known method of generationand the one which we have hitherto used to the exclusion of all others, is thedischarge tube. Here, as their name suggests, the rays originate at the cathode.The gas molecules which are under the influence of the prevailing electricalforces have an effect - the proximity effect as I call it (53) - on the electrodemetal, whereby quanta are withdrawn from the latter. Immediately they arefree they are subjected to the accelerating forces of the field between theelectrodes and thus move with increasing velocity away from the cathode;the ray is complete. The ultimate velocity at which we allow it to leave thetube through, say, a window is given by the size of the voltage used; and thevery fact that in effect this whole voltage and not just a fraction thereof isdeterminative for the ultimate velocity proves that the origin of the ray mustbe sought at the cathode surface and not, say, in the centre of the gas.* Bythe magnitude of the voltage we can thus produce faster or slower cathoderays and when we previously spoke of 1/ 3 the speed of light that applied onlyto one particular voltage about 30,000 V, which I used generally throughoutmy experiments.

How would faster or slower rays behave? Some predictions could bemade on the basis of my first experiments in which the voltage, and hencethe velocity, was slightly varied (18b, p. 266; 19; 21, p. 261). Very fast rayscould be expected to have extremely slight absorbability (high penetratingpower) ** ; the slow rays on the other hand appeared best suited to yield in-formation on the forces of atoms, the constitution of matter. For a long time,however, it seemed impossible to carry out pure tests over a sufficiently wide

* To start with this was an arbitrary assumption made in many previous studies oncathode rays; the proof of its correctness was supplied with increasing accuracy astime went on, with most accuracy probably by A. Becker (52, p. 404).* * When X-rays were discovered this expectation of them appeared to be fulfilled;their first properties to become known agreed with those to be anticipated from mytests for the fastest cathode rays (27). It was just at first Righi’s convincing observationthat the X-rays do not carry with them a negative charge (23) that had shown the untena-bility of the theory that they were extremely fast cathode rays. They are nowadays consid-ered to be short, transverse impulses in the ether, a kind of ultra-ultraviolet light. The fact

O N C A TH O D E RA Y S 121

range of velocities since the glass of the discharge tube could not withstandthe heavy voltages needed for the very fast rays, and the slow rays, althougheasy to produce in the tube, failed to emerge through the window; theywere too absorbable. Other arrangements failed, too.*

Both problems, that of the slowest and that of the fastest rays, were finallysolved in quite novel ways.

A discovery by Hertz as early as 1887(8) completed shortly afterwards byHallwachs (11), had shown that by mere exposure to ultraviolet light metalplates give off negative electricity to the air. This remarkable fact - nowadaysusually referred to as the photo-electric effect - immediately captured myinterest at that time and has also continued to do so since. Experiments car-ried out in collaboration with the astronomer Wolf showed me first of allthat ultraviolet light roughens substances or pulverizes them (12; 46, p.490). Subsequent experiments, however, caused me to think it unlikely thatmetal particles carried the negative charge off the plate. At the time I con-ducted my first experiments on cathode rays, when I had discovered that theair in front of the aluminium window becomes conductive (18) I formed theidea that cathode rays could be driven from the plate into the air by ultra-violet light. Both then and later I made repeated vain attempts to detect pos-sible rays in the vacuum on fluorescent screens. Only my decision - based onRighi’s work (14) - to use the electrometer instead of the fluorescent screenrevealed the existence of the rays. The apparatus used is illustrated in Fig. 10.U is the plate to be irradiated and is in a complete vacuum; the quartz seal atB admits the ultraviolet light. The cathode rays start from U and a narrowbeam is separated out by the hole in the counterplate E. This beam impingeson the small plate α which collects the negative charge brought by the beamand thereby indicates the existence of the radiation on the electrometer. Webring a magnet or the coil indicated by a broken line close to the tube in asuitable manner and then find the charge on the plate β instead of on α, in-

that such impulses can actually occur side by side with heavy absorption of cathoderays, such as e.g. in heavy platinum, follows, as first put forward by Heaviside, fromMaxwell’s theory (10). The fact that extremely short-wave ultraviolet light will passunrefracted through prisms was foreseen by Helmholtz in his dispersion theory (17c,p. 517). The low-absorbability cathode rays forecast first became a reality in the raysemitted by radium.* Amongst other measures I had tried inserting, instead of the window, narrow chan-nels between generating chamber and completely evacuated observation chamber, butthey allowed too much gas through.


dicating that the invisible ray is actually deflected by the magnet and inthe appropriate direction for cathode rays. When carried out quantitatively,the experiment showed that the deflection is also of the correct degree, andthat the same ratio obtains between charge and mass of the quanta as in thecase of the rays generated in discharge tubes (32; 44, p. 150; 46).

Immediately it had been established beyond doubt in this way that ca-thode rays are produced by ultraviolet light and that their behaviour hadbecome suffciently well known, I was soon able to detect them on fluores-cent screens (44), then follow them further and use them. I shall refer to thoseaspects later. The following should be noted as regards the actual generation.

Firstly - an important point for pure experiments - it also occurs in a com-plete vacuum where the usual method fails. A gas need not be present but itdoes not interfere with the generation of the rays. What is involved is thedirect action of the light on the metal of the plate. The initial velocities withwhich the quanta leave the plate are so slight that a negative charge of onlya few volts on the counterplate is sufficient to compel the rays to reverse be-fore reaching it. They then return to the irradiated plate in the same way asa stone thrown upwards falls black to the ground (32; 44).*

* My first detailed communication on the subject (32) appeared in the Sitzungsberichte der

Kaiserl. Akademie der Wiss. zu Wien for 19th October 1899. In the December issue of ThePhilosophical Magazine of the same year J. J. Thomson published studies "On the mass ofthe ions in gases at low pressures" in which the photo-electric effect is involved al-though its centre is still sought in the gas adjacent to the irradiated plate, as the remarkson p. 552 indicate. In the same author’s book Conduction of Electricity through Gases, 2nd.ed., 1903, p. 109, my publication is dated one year later than that just mentioned sincea later reprint (Ann. Physik, 2 (1900) 359, where it is expressly marked as a reprint), andnot the original is cited.

ON CATHODE RAYS 123

Here, therefore, we obtain extremely slow cathode rays; faster ones can beproduced merely by charging the counterplate positively. The velocity of therays can be controlled freely by the level of the voltage of the counterplate.

Secondly, considering the effect of the ultraviolet light on the plate, wemust imagine that the light waves cause the interior of the metal atoms in theplate to vibrate. We have previously mentioned that Zeeman’s discovery hasproved atoms to contain negative electricity capable of vibration. If the co-vibration of a negative quantum in the atom with the light waves becomestoo violent, the quantum escapes from the atom* and so from the plate; wehave a cathode ray.

The velocity at escape we have already mentioned as very low. I have alsofound that the velocity is independent of the ultraviolet light intensity (M), andthus concluded that the energy at escape does not come from the light at all,but from the interior of the particular atom. The light only has an initiatingaction, rather like that of the fuse in firing a loaded gun. I find this conclusionimportant since from it we learn that not only the atoms of radium - theproperties of which were just beginning to be discerned in more detail at thattime - contain reserves of energy, but also the atoms of the other elements;these too are capable of emitting radiation and in doing so perhaps complete-ly break down, corresponding to the disintegration and roughening of thesubstances in ultraviolet light. This view has quite recently been corroboratedat the Kiel Institute by special experiments which also showed that the photo-electric effect occurs with unchanged initial velocities even at the temper-ature of liquid air.

We cannot regard the action of the light as restricted only to the solid stateof aggregation. The molecules, or atoms of gases undergo a completely anal-ogous effect under the action of ultraviolet light (35; 40); it is reasonable toassume that quanta escape from them (49, p. 486); the gas thus becomes elec-trically conductive in a manner which we shall discuss in detail later. If thegas contains oxygen like the air, ozone is formed as a by-product (35).**

* This is a process which was earlier anticipated in Helmholtz’s comprehensive disper-sion theory (17c, p. 518).** In the light of subsequent studies by Warburg it can be assumed that the mostproductive of current methods of producing ozone, i.e.those using what are termed"silent electrical discharge" are wholly or largely effective owing to the ultravioletlight of these discharges (48). The rich sources of ultraviolet light obtainable nowadays,e.g.electrical mercury quartz lamps, propagate such a noticeable odour of ozone intheir environment that this effect of ultraviolet light has now became a commonplace.


This same action of light, namely the production of cathode rays, the vibra-tion of atoms and the releasing of quanta therefrom, is also involved in phos-phorescence (50, p. 671) and hence probably also in fluorescence, perhaps,too, in all photochemical effects. Bearing in mind that we have detectedtransformation of energies from the interior of the atoms associated with thephotoelectric effect, we should not be surprised if in future perhaps we en-counter phenomena of the same type acting as sources of energy not intro-duced from outside.

It should also be mentioned that the research carried out by Curie andSagnac (37) as well as that by Dom (42) indicates that in common with ultra-violet light, X-rays too have the effect of generating cathode rays. This is con-sistent with their ability to make gases electrically conductive and inducephosphorescent and photochemical effects.

Scarcely had ultraviolet light been shown suited for the generationof the slowest rays when the solution was found to the problem of how thefastest rays originate. The rays emitted by uranium and radium were al-ready known; Becquerel, P. and M. Curie were engaged in pursuing furtherthese discoveries of theirs. By applying to these new rays the methods devel-oped as described for the cathode rays from discharge tubes, it was shownthat the new rays are partly cathode rays (34; 36; 41)*, and amazingly ca-thode rays of almost or entirely the speed of light (43). What no discharge tubecould withstand is thus achieved by the radium atom - and quite sponta-neously - although admittedly not without being completely broken downin the process (51; 54).

Once the entire range of velocities from rest to the speed of light was thusavailable it was worthwhile re-examining in more detail the behaviour ofmatter to irradiation.

From the turbidity of all substances, including e.g. air, to cathode rays, wehad concluded that each molecule or atom acts as a separate obstacle to therays, an obstacle which deflects them from their path to a greater or lesser

Nevertheless, the meteorological significance of the action of ultraviolet sunlight in theupper layers of the atmosphere (35, p. 504) still does not appear to have been sufficientlyappreciated. Whether the ionization by cathode rays first found at the aluminiumwindow (18) is also an effect of the light which occurs there, or else the direct effect ofthe cathode rays, is still obscure.* This is the part of the uranium or radium radiation normally designated as β.

ON CATHODE RAYS 125

extent. How should we visualize this deflection? Let us first examine whetherperhaps the quanta of the rays are reflected from the molecules of the sub-stance in the same way as the molecules of a gas are reflected on one anotherwhen they collide. Were that so a cathode ray in the gas would be restrictedto that length which can readily and accurately be calculated as the mean freepath of very small particles between the gas molecules from the data ofthe kinetic gas theory. These path lengths are, however, very small; in hy-drogen at 40 mm pressure, for example, about two hundredths of a milli-meter. Beyond this short length, a ray would not be able to develop at all inthis gas, that is to say almost instantaneous diffusion would ensue. However,gases are by no means so turbid as my observations published in diagram-matic form had earlier shown (18b). Even the air at full atmospheric pressureproved much clearer, as we have already seen (Fig. 4), and the lighter hy-drogen rarified to the specified pressure of 40 mm is much clearer still; Fig.11 illustrates the path of the ray in hydrogen as observed on the fluorescent

Fig. 11.

screen. The broken lines show the extent to which rectilinear light wouldpropagate under the same conditions. It will be seen that over a length of 10cm the cathode ray still scarcely deviates from this rectilinear propagation,and it becomes distinctly broader only beyond that length. The length of 10cm is, however, 5,000-fold that of the free path length of 0.02 mm. It thusfollows that here the radiant quanta must have traversed 5,000 hydrogenmolecules before undergoing the first noticeable change of direction. We areamazed to see that we have transcended the old impermeability of matter.Each atom in the substance occupies a space which is impermeable to the


other atoms*; but vis-à-vis the fine quanta of electricity all types of atomsare highly permeable structures as if built up of fine constituents with a greatmany interstices.

What are these fine constituents of atoms? That in all atoms they are thesame, only present in varying numbers, we have already concluded from thelaw of proportionality between mass and absorption. We can now learn fur-ther details. We can use the quanta of the cathode rays as small test particleswhich we allow to traverse the interior of the atoms and thus provide us withinformation thereon.

The first and most noticeable thing to happen to them during this traverse,i.e. deflection from the rectilinear path, we have just discussed as diffusion of

the rays. As far as we know, cathode rays experience such a deflection owingonly to electrical and magnetic forces. To assume magnetic forces within theatoms would imply the assumption of mobile electricity in the atoms, thusagain electrical forces. We must therefore regard the diffusion of cathode raysin matter as proof for the existence of electrical forces in the interior of the atoms.

The magnitude of these forces can be estimated by considering the extent ofthe deflection together with the transit time, which latter depends on thevelocity of the quanta and, of course, on the size of the atoms. If we takeprogressively slower rays, the transit times will become longer and corre-spondingly the diffusions occurring will be stronger (19, p. 30; 46, p. 480).In this way we find for the interior of the atoms electrical field intensities ofunusual magnitude such that we can never produce by any means known tous owing to lack of sufficient resistance in even the best insulators: field in-tensities compared with which those occurring during the most violentstorms are insignificantly small (47). The force effects of the radium atom thencease to seem so surprising but we should be more amazed that most of theatoms around us behave so placidly, only revealing something of the forcestored within them when subjected to the photoelectric action or throughother similar causes.

The further quantitative study of cathode-ray diffusion in the various mate-rials promises to yield valuable information on the precise nature of theelectrical fields of atoms. For the present we must turn to a second phenom-

* In any case at the normal velocities of the molecules. For very high velocities suchas occur with the α-particles of radium, mutual penetrability of even whole atoms, al-though probably accompanied by their destruction, would not be impossible in thelight of the concepts which we have arrived at regarding the constitution of atoms.Recent studies by Bragg and Kleemann promise to throw light on this subject.

ON CATHODE RA YS 127

enon, somewhat more easy to determine numerically, which is apt to occurin the course of such atom traverses. It can easily happen that the quantum,after successfully passing thousands of atoms, finally stops in an atom anddoes not emerge at once. This is the absorption of cathode rays. I have deter-mined this effect quantitatively for the entire scale of available ray velocitiesand found the following (47).

The absorption increases with decreasing ray velocity, in common withthe diffusion. This is also to be expected if the absorption, like the diffusion,is an effect of the electrical fields of force within the atoms and if these fieldsof force concentrate about certain centres in the atoms in the vicinity ofwhich their intensity is greater than at longer range, in the same way as theintensity of a magnetic field of force concentrates about the two poles. Aradiation quantum which traverses such fields with mobile centres is onlystopped when it enters sufficiently intense parts of these fields along its partic-ular path; otherwise it will pass through and be deflected to a greater or les-ser extent. The entire cross-section of the atom, the area which the atompresents to a ray, thus consists of two parts, an absorbing and a transmittingpart, and the former - which I refer to briefly as the absorbing cross-section- is known in square centimetres from my measurements. It affords a meas-ure for the size of those parts of the force fields of the atom, the intensity ofwhich is greater than the relevant level which is just sufficient to arrest theparticular velocity of the quantum. The slower the quantum, the larger theparts of the atom’s force field which act as absorbing cross-sections. For theslowest rays I have found that the absorbing cross-section not only becomesequal to the entire cross-section of the atom or molecule - which entire cross-section was known from the kinetic theory of gases - but even slightlylarger. This is tantamount to direct proof for the existence of electrical fieldsof forces both within the atoms and molecules and also for a certain distancearound them. It is probably correct to identify these external electrical forcesof the molecules with the forces of strength, elasticity, cohesion and adhe-sion, in short with the molecular forces in general which have long been knownalthough not immediately regarded as of an electrical nature. Berzelius’ viewthat the chemical forces of atoms are of an electrical nature will now be heldwith all the more reason and as the research in progress continues it is to behoped that concepts of the electrical force fields of the atoms will emergewhich give a better and more complete picture of their chemical behaviourthan the simple concept of a number of fixed valency positions equippedwith electrical charges.

128 1905 P.E.A . VON LENA RD

Of equal interest to the transition to the lowest velocities was that to thehighest. As the ray velocity increases, the absorbing cross-section contracts;ultimately only those quanta are stopped whose path lies through the highestintensity parts of the internal force fields close to their centres. For that veryreason the fastest rays are also capable of supplying the answer to the ques-tion whether perhaps these centres have a special, impenetrable proper vol-ume, or in more general terms: whether apart from the force fields there issomething else in the atoms which holds back our test particles. What hap-pens when the test is carried out with the fastest rays can best be illustrated bymeans of an example. Let us imagine a cubic metre block of the most solidand heavy substance known to us, say, platinum. In this block we find al-together not more impenetrable proper volume than at most one cubic mil-limetre. Apart from this pinhead-sized portion, we find the remainder of ourblock as empty as the sky. We ought to be astounded at the insignificantdegree to which the space in matter is actually filled! What we have foundin the space occupied by matter have only been fields of force such as canalso form in the free ether. What are then the basic constituents of all atomsto which we have been led by the mass dependence of cathode-ray absorp-tion? Clearly they too are in the main only fields of force in common withthe whole atoms. I have therefore termed these basic constituents of all mat-ter "dynamids".

As constituents of electrically neutral atoms the dynamids will also be re-garded as electrically neutral, and hence possess the same amounts of neg-ative and positive electricity as the centres of their fields. We may then state :matter - all the tangible, ponderable substances around us - consists ul-timately of equal quantities of negative and positive electricity. The pre-viously mentioned findings derived from the Zeeman phenomenon, thephoto-electric effect and the secondary cathode radiation, which will pre-sently be discussed, show that the negative electricity is contained in theatoms as precisely the same quanta which we found in the cathode rays, andwhich the research worker has since variously encountered in their own right,separate from matter. The positive electricity, on the other hand, appears tobe something much more specifically proper to the atoms of matter; as hasbeen previously stressed they have not been found with certainty other thanin atoms. From our findings on the packing of space it follows that for nega-tive quanta the proper volume, impenetrable for things of the same kind,must then be extraordinarily small. This is consistent with the previouslymentioned experiments by Kaufmann. The probable proper volume of the

ON CATHODE RAYS 129

positive electricity, provided it too were not extremely small, should be re-garded as completely penetrable for negative quanta.

With this constitution of matter, the third phenomenon occurring duringatom traverses, and one which has still to be referred to, can readily beunderstood. Owing to the repelling force which it exerts on the other neg-ative quanta proper to the atom the traversing ray quantum will be capableof setting up a tremendous disturbance within the atom and as a result ofthis disturbance a quantum belonging to the atom can be flung out.*The process is termed secondary cathode radiation. We have allowed onecathode ray - the primary - to penetrate into the atom, against which twoemerge, the primary and the secondary** (46, p. 481).

The velocity of the secondary rays - in common with that of the photo-electrically generated cathode rays - is very low, even when that of the pri-mary rays is high. The amount of the secondary radiation, i.e. the probabilityof quanta emission from atoms during traverse, is largest at a given opti-mum of the primary velocity; both faster and slower primary rays are lesseffective and at quite a low primary velocity - below 1/ 200th of the speedof light - the secondary radiation is absent altogether (44, p. 188 et seq.; 46,p. 474 et seq.; 49). This is quite understandable for if the primary quantumapproaches too slowly, it has too little energy to cause adequate disturbanceof the interior of the atom, and if it approaches too quickly it will generallyremain for too short a time in the atom to have that effect.

At its low velocity the secondary radiation must succumb to strong ab-sorption in the surrounding molecules of the material. In gases where themolecules are free, a molecule which has absorbed a secondary quantum willact as a mobile carrier of negative electricity, while the molecule from which thesecondary quantum has escaped has an excess of positive electricity and isthus a po sitive electricity carrier. The migrations of such carriers, however -for which knowledge we are indebted to the unremitting efforts of Arrhe-nius and after him to J. J. Thomson in particular - constitute the electricalconductivity in gases*** and in the secondary cathode radiation we have thus

* Two and more quanta can also escape from the atom (46, p. 485).** It seems often to be assumed that the secondary radiation is an exclusive result ofthe absorption of the primary radiation; but this is not the conception which I haveformed from observation (46, p. 474 et seq.).*** Three consecutive steps seem to me to substantiate the belief that molecules orgroups thereof carry the electrical conductivity in gases, notably: (i) the thorough studyof one of the first cases of gas electrification in which dust - which at first was regarded


found and, as I believe, adequately ascertained by thorough observations, themechanism whereby cathode rays cause a gas to become electrically con-ductive (46, p. 474).* This case of conductivity induced and maintained bycathode rays must also be a factor in all gas discharges where sufficiently fastcathode rays occur; hence also in the normal discharge tube which we tookat the outset as our first generator of cathode rays.

In other cases as well where gases become electrically conductive themechanism appears to be the same: escape of quanta from molecules, and re-absorption by other molecules; this occurs under the influence of ultravioletlight on gases, as aforementioned, and also in flames (45) but the cause un-derlying the escape of quanta is apt to vary from case to case (53, p. 242).

In conclusion, I wish to thank you for your attention.

as the sole or at any rate the main carrier of electrical discharges in gases - was founddefinitely not to be involved. This was the case of waterfall electricity (16); (ii) the firstmeasurements of the migration speeds of gas carriers in various cases of conductinggases, performed in rapid sequence by a number of different observers, first of all byRutherford (26) ; (iii) J. J. Thomson’s essentially irreproachable first measurements of theabsolute electrical charge of the individual gas carriers in different cases (31 ). - In thelight of simple considerations of gas kinetics the size of the gas carriers was calculatedfrom (ii) and (iii) as equal to that of molecules or groups of molecules.* This mechanism differs from that of ion formation in liquid electrolytes where itconsists in the splitting of electrically neutral molecules into two oppositely chargedatoms or groups of atoms. In other respects, too, the analogy between the conductionof electricity in gases and in liquids breaks down precisely in the most characteristicpoints. Magnitude and sign of the charge on a gas carrier are - quite unlike the case withthe ions in liquid electrolytes - not determined by the chemical nature of the carrier,and electrolysis proper - that so typically chemical mode of decomposition - hencedoes not occur at all in gases (53, p. 236). I therefore considered it better to emphasizethis analogy no longer, notwithstanding its heuristic usefulness at the outset, and forthat reason I have always avoided applying the name ions to the gas carriers, and callingthe conduction of electricity in gases electrolytic.

ON CATHODE RAYS 131

Chronological list of publications*

(1860) 1. M. Faraday, Experimental Researches in Electricity (from 1820 onwards).(1869) 2. W. Hittorf, "Über die Elektrizitätsleitung der Gase" (The electrical con-

ductivity of gases), Pogg. Ann. Physik, Vol. 136.(1873) 3. J. C. Maxwell, A Treatise on Electricity and Magnetism.(1876) 4. E. Goldstein, "Über elektrische Entladungen in verdünnten Gasen" (Elec-

trical discharges in rarified gases), Monatsber. Berl. Akad. Also continued in the vo-lumes of the same journal for 1880 and 1881.

(1879) 5. W. Crookes, Strahlende Materie oder der vierte Aggregutzustand (Radiatingmatter or the fourth state of aggregation). (German translation published in Leip-zig.) Also Phil. Trans. Roy. Soc., Vol. 170.

(1881) 6. H. Helmholtz, "On the modern development of Faraday’s conceptions ofelectricity" J. Chem. Soc., Vol. 39; Wiss. Abhandl., Vol. 3, p. 52.

(1883) 7. H. Hertz, "Versuche über die Glimmentladung". (Experiments on the glowdischarge). (a) Wied. Ann. Physik, Vol. 19. Also (b) Ges. Werke (Collected works),Vol. 1, p. 242.

(1887) 8. H. Hertz, "Über einen Einfluss des ultravioletten Lichtes auf die elektrischeEntladung" (An effect of ultraviolet light on electrical discharge), Sitz.-ber. Berl.

Akad., 9th June. In more detail in Wied. Ann. Physik, Vol. 31.(1887-1888) 9. H. Hertz, Untersuchungen über die Ausbreitung der elektrischen Kraft (Re-

searches on the propagation of electrical force).(1888) 10. O. Heaviside, "On electromagnetic waves", Phil. Mag., Ser. 5, Vols. 25 and

26. Further: "On the electromagnetic effects due to the motion of electrificationthrough a dielectric", Phil. Mag., Ser. 5, Vol. 27 (1889); "A charge suddenly jerkedinto motion", "Sudden stoppage of charge", The Electrician, May 1, 1891. AlsoElectrical Papers, Vol. 2, pp. 375, 504, and Electromagnetic Theory, Vol. 1, p.54.- 11. W. Hallwachs, "Über den Einfluss des Lichtes auf elektrostatisch geladene Kör-per" (The effect of light on electrostatically charged substances), Wied. Ann. Physik,

Vol. 33.(1889) 12. P. Lenard and M. Wolf, "Zersduben der Körper durch das ultraviolette

Licht" (Pulverization of substances under the action of ultraviolet light), Wied. Ann.

Physik, Vol. 37.(1890) 13. A. Schuster, "The discharge of electricity through gases", Proc. Roy. SO c,

Vol. 47. (Received 20th March.)- 14. A Righi, "Sulle traiettorie percorse nella convezione foto-electrica, ..." (Thetrajectories described in photoelectric convection), Atti Accad. Naz. Lincei, Vol. 6[2]. (Received 3rd August.)

(1892) 15. H. Hertz, "Über den Durchgang von Kathodenstrahlen durch dünne Me-

* The position of a publication in this sequence has been determined by its date ofsubmission to a learned societyor by its date of receipt by a journal editor. Where nosuch dates were available, the date of publication, as far as can be determined, has beentaken.


tallschichten" (The passage of cathode rays through thin metal layers), Wied. Ann.

Phys, Vol. 45. Also Ges. Werke (Collected works), Vol. 1, p. 355.- 16. P. Lenard, "Über die Elektricität der Wasserfälle" (Waterfall electricity), Wied.

Ann. Physik, Vol. 46.- 17. H. Helmholtz, "Elektromagnetische Theorie der Farbenzerstreuung" (Electro-

magnetic theory of colour dispersion). (a) Sitz.-ber. Berl. Akad., 15th December- (b) Wied. Ann. Physik, Vol. 48. - (c) Wiss. Abhandl., Vol. 3, p. 505.

(1894) 18. P. Lenard, "Uber Kathodenstrahlen in Gasen von atmosphärischem Druckund im äussersten Vakuum" (Cathode rays in gases at atmospheric pressure and inthe highest vacuum). (a) Sitz.-ber. Berl. Akad. for 12th January 1893. - (b) In moredetail, Wied. Ann. Physik, Vol. 51.

- 19. P. Lenard, "Über die magnetische Ablenkung der Kathodenstrahlen" (Themagnetic deflection of cathode rays), Wied. Ann. Physik, Vol.52.

- 20. H. Hertz, Die Principien der Mechanik in neuem Zusammenhange dargestellt (Theprinciples of mechanics displayed in a new context.)

(1895) 21. P. Lenard, "Über die Absorption der Kathodenstrahlen" (The absorptionof cathode rays), Wied. Ann. Physik, Vol. 56. (Published in October issue.)

- 22. W. C. Röntgen, "Über eine neue Art von Strahlen" (A new type of ray),Sitz.-ber. Würzburger Phys. Med. Ges., for December. Reprinted in Wied. Ann. Physik,

Vol. 64.(1896) 23. A. Righi, "Sulla propagazione dell’elettricità nei gas attraversati dai raggi

di Röntgens" (The propagation of electricity in gases traversed by X-rays), Mem.

Accad. Sci. Bologna, Ser. 5, Vol. 6 (Submitted 31st May.)- 24. P. Zeeman, "On the influence of magnetism on the nature of the light emitted

by a substance", Comm. Phys. Lab. Leiden, transl. from: Verslag. Kongl. Ned. Akad.

Wetenschap., 31st October.(1897) 25. J. J. Thomson, "Cathode rays", Phil. Mag., Ser. 5, Vol. 44. (Published in

October issue.)- 26. E. Rutherford, "The velocity and rate of recombination of the ions of gases

exposed to Röntgen radiation"., Phil. Mag., Ser. 5, Vol. 44. (Published in Novemberissue.)

- 27. P. Lenard, "Über die elektrische Wirkung der Kathodenstrahlen auf atmos-phärische Luft" (The electrical effect of cathode rays on the air of the atmosphere),Wied. Ann. Physik, Vol. 63. (Published 11th December.)

(1898) 28. P. Lenard, "Über die elektrostatischen Eigenschafen der Kathodenstrah-len" (The electrostatic properties of cathode rays), Wied. Ann., Vol. 64. (Received 2nd January.)

- 29. E. Wiechert, "Experimentelle Untersuchungen über die Geschwindigkeit unddie magnetische Ablenlbarkeit der Kathodenstrahlen" (Experimental studies on thevelocity and magnetic deflectability of cathode rays), Nachr. Kgl. Ges. Wiss. Göttin-

gen (Submitted on 19th March.)- 30. P. Lenard, "Über das Verhalten von Kathodenstrahlen parallel zu elektrischer

Kraft" (The behaviour of cathode rays parallel to electrical force), Wied. Ann. Physik,

Vol. 65. (Received 1st May.) Also published in Math. Naturw. Ber. Ungarn, Vol. 16.- 31. J. J. Thomson, "On the charge of electricity carried by the ions produced by

ON CATHODE RAYS I33

Röntgen rays" Phil. Mag., Ser. 5, Vol. 46. (Published in December issue.) Continuedand subsequently corrected in Phil. Mag., December issue 1899, and March issue1903.

(1899) 32. P. Lenard, "Erzeugung von Kathodenstrahlen durch ultraviolettes Licht"(Production of cathode rays by ultraviolet light), Sitz.-ber. Kaiserl. Akad. Wiss. Wien

for 19th October. Reprinted in Ann. Physik, Vol. 2.(1900) 33. V. Bjerknes, Vorlesungen über hydrodynamische Fernkräfte nach C. A. Bjerknes’

Theorie (Lectures on the hydrodynamic distant forces in accordance with C. A.Bjerknes’ theory).

- 34. H. Becquerel, "Contribution à l’étude du rayonnement du radium" (Contribu-tion to the study of the radiation from radium), and "Déviation du rayonnement duradium dans un champ électrique" (Deflection of radium radiation in an electricalfield), Compt. Rend. (Paris) for 29th January and 26th March, respectively.

- 35. P. Lenard, "Über Wirkungen des ultravioletten Lichtes auf gasförmige Körper"(Effects of ultraviolet light on gaseous substances), Ann. Physik, Vol. 1. (Received6th February.)

- 36. P. and M. Curie, "Sur la charge électrique des rayons déviables du radium" (Theelectrical charge of deflectable radium rays), Compt. Rend. (Paris) for 5th March.

- 37. P. Curie and G. Sagnac, "Électrisation negative des rayons secondaires produitsau moyen des rayons Röntgen" (Negative electrification of secondary rays producedby means of X-rays), Compt. Rend. (Paris) 130, for 9th April.

- 38. E. Demarcay, "Sur le spectre du radium" (The radium spectrum), Compt. Rend.

(Paris) 131, for 25th July.- 39. M. Curie, "Sur le poids atomique du baryum radifère" (The atomic weight of

radiferous barium), Compt. Rend. (Paris) 131, for 6th August. Also "Sur le poids atom-ique du radium" (The atomic weight of radium), same journal 21st July, 1902.

- 40. P. Lenard, "Über die Elektrizitätszerstreuung in ultraviolet durchstrahler Luft"(Scattering of electricity in air irradiated with ultraviolet light), Ann. Physik, Vol.3.(Received 17th August.)

- 41. E. Dorn, "Elektrostatische Ablenkung der Radiumstrahlen" (Electrostatic deflec-tion of radium rays), Abhandl. Naturforsch. Ges. Halle, Vol.22.

- 42. E. Dorn, "Versuche über Sekundärstrahlen" (Experiments on secondary rays),Arch. Néerl. Sci. (Haarlem), Ser. 2, Vol. 5.

(1901) 43. W. Kaufmann, "Die magnetische und elektrische Ablenkbarkeit der Bec-querelstrahlen und die scheinbare Masse der Elektronen" (The magnetic and electricaldeflectability of Becquerel rays and the apparent mass of electrons), Nachr. Kgl. Ges.

Wiss. Göttingen, Vol. 8, for 8th November.(1902) 44. P. Lenard, "Über die lichtelektrische Wirkung" (The photoelectric effect),

Ann. Physik, Vol. 8. (Received 17th March.)- 45. P. Lenard, "Über die Elektrizitätsleitung in Flammen" (Electrical conductivity

in flames), Ann. Physik, Vol. 9. (Received 18th August.)(1903) 46. P. Lenard, "Über die Beobachtung langsamer Kathodenstrahlen mit Hilfe

der Phosphoreszenz und über Sekundärentstehung von Kathodenstrahlen" (The ob-servation of slow cathode rays by means of phosphorescence, and the secondaryoccurrence of cathode rays), Ann. Physik, Vol. 12. (Received 28th June.)

I34 1905 P.E. V O N LEN A RD

- 47. P. Lenard, "Über die Absorption von Kathodenstrahlen verschiedener Ge-schwindigkeit" (The absorption of cathode rays of various velocities), Ann. Physik,

Vol. 12. (Received 30th July.)- 48. E. Warburg, "Über die Ozonisierung des Sauerstoffs durch stille elektrische Ent-

ladungen" (The ozonization of oxygen by silent electrical discharges), Sitz.-ber.

Berl. Akad. for 11th November.(1904) 49. P. Lenard, "Über sekundäre Kathodenstrahlung in gasförmigen und festen

Köpern" (Secondary cathode radiation in gaseous and solid substances), Ann. Physik,

Vol. 15. (Received 22nd August.)- 50. P. Lenard and V. Klatt, "Über die Erdalkaliphosphore" (The alkaline-earth

phosphors), Ann. Physik, Vol. 15. (Received 22nd August.) Extracts also publishedin the Mat. Naturw. Ber. Ungarn, Vol. 22.

- 51. E. Rutherford, "Slow transformation products of radium", Phil. Mag., Ser. 6,Vol. 8. (Published in November issue.)

(1905) 52. A. Becker, "Messungen an Kathodensstrahlen" (Measurements on cathoderays), Ann. Physik, Vol. 17. (Inaugural dissertation submitted to the Kiel Philosoph-ical Faculty on 1st March.)

- 53. P. Lenard, "Über die Lichtemissionen der Alkalimetalldämpfe und Salze undüber die Zentren dieser Emissionen" (The photo-emissions of alkaline metal vapoursand salts, and the centres of these emissions), Ann. Physik, Vol. 17. (Received 13thApril.)

- 54. F. Himstedt and G. Meyer, "Über die Bildung von Helium aus der Radium-emanation" (The formation of helium from the emanation of radium), Ber. Natur-forsch. Ges. Freiburg i. B., for May. Also Ann. Physik, Vol. 17.

(1906) 55. W. Kaufmann, "Über die Konstitution des Elektrons" (The constitutionof the electron), Ann. Physik, Vol. 19. (Received 3rd January.)

It is not without interest to see from this list how, during the years 1887-1894, the

subject under discussion has suddenly, as it were, become the field of more abundant

and more successful activity. The years prior to that period, with the exception of

the fundamental work by Faraday and Maxwell, are marked by only sporadic and

isolated symptoms of that activity, the years thereafter and down to the present by

its increasingly fruitful pursuit.

RREEAADDIINNGG ##66

(revised 1/23/02)

MILLIKAN OIL-DROP EXPERIMENT

Advanced Laboratory, Physics 407University of Wisconsin

Madison, Wisconsin 53706

Abstract

The charge of the electron is measured using the classic technique of Millikan. Mea-surements are made of the rise and fall times of oil drops illuminated by light froma Helium–Neon laser. A radioactive source is used to enhance the probability that agiven drop will change its charge during observation. The emphasis of the experimentis to make an accurate measurement with a full analysis of statistical and systematicerrors.

Introduction

Robert A. Millikan performed a set of experiments which gave two important results:

(1) Electric charge is quantized. All electric charges are integral multiples of aunique elementary charge e.

(2) The elementary charge was measured and found to have the value e = 1.60 ×10−19 Coulombs.

Of these two results, the first is the most significant since it makes an absoluteassertion about the nature of matter. We now recognize e as the elementary chargecarried by the electron and other elementary particles. More precise measurementshave given the value

e = (1.60217733± 0.00000049)× 10−19 Coulombs

The electric charge carried by a particle may be calculated by measuring the forceexperienced by the particle in an electric field of known strength. Although it isrelatively easy to produce a known electric field, the force exerted by such a field ona particle carrying only one or several excess electrons is very small. For example,a field of 1000 volts per cm would exert a force of only 1.6 × 10−14 N on a particlebearing one excess electron. This is a force comparable to the weight of 10−12 grams.

The success of the Millikan Oil-Drop experiment depends on the ability to measuresmall forces. The behavior of small charged droplets of oil, weighing only 10−12 gramor less, is observed in a gravitational and electric field. Measuring the velocity of fallof the drop in air enables, with the use of Stokes’ Law, the calculation of the mass ofthe drop. The observation of the velocity of the drop rising in an electric field thenpermits a calculation of the force on, and hence the charge, carried by the oil drop.

Although this experiment will allow one to measure the total charge q on a drop, itis only through an analysis of the data obtained and a certain degree of experimentalskill that the charges can be shown to be quantized. By selecting droplets which riseand fall slowly, one can be certain that the drop has a fairly small charge. A numberof such drops should be observed and their respective charges q calculated. If thecharges q on these drops are integral multiples of a certain smallest charge e, thenthis is an observation that charge is quantized.

2

Question

Some students measure a medium size charge q and then divide it by whatever largeinteger, n, which will give

q

n= 1.60× 10−19 coulombs!

What is wrong with this?Since a different droplet has been used for measuring each charge, there remains

the question as to the effect of the drop itself on the charge. This uncertainty can beeliminated by changing the charge on a single drop while the drop is under observation.An ionization source placed near the drop will accomplish this. In fact, it is possibleto change the charge on the same drop several times. If the results of measurementson the same drop then yield charges q which are integral multiples of some smallestcharge e, then this strongly suggests that charge is quantized.

The measurement of the charge of the electron also permits the calculation ofAvogadro’s number. The charge F required to electrodeposit one gram equivalentof an element on an electrode (the Faraday) is equal to the charge of the electronmultiplied by the number of molecules in a mole. Through electrolysis experiments,the Faraday has been found to be F = 9.625× 107 coulombs per kilogram equivalentweight.

Hence Avogadro’s Number N = F/e =9.625× 107 coulombs/kg equiv. wt.

1.60× 10−19 coulombs/molecule

= 6.02× 1026 molecules/kg equiv. wt.

Equations for calculating the charge on a drop

An analysis of the forces acting on the oil drop let will yield the equations for thedetermination of the charge carried by the droplet.

Figure 1a shows the forces acting on the drop when it is falling in air and hasreached its terminal velocity (terminal velocity is reached in a few milliseconds forthe droplets used in this experiment). In Fig. 1a, vf is the velocity of fall, k is thecoefficient of friction between the air and the drop, m1 is the mass of the drop, m2 isthe mass of air displaced by the drop and g is the acceleration due to gravity.

The downward force due to gravity is m1g −m2g. The viscous retarding force iskvf . Then

m1g −m2g = kvf .

3

Figure 1: Fig. 1a and Fig. 1b

Let m = (m1 −m2), thenmg = kvf . (1)

Figure 1b shows the forces acting on the drop when it is rising under the influenceof an electric field. In Fig. 1b, E is the electric intensity, q is the charge carried bythe drop and vr is the velocity of rise. Adding the forces vectorially yields:

Eq = mg + kvr . (2)

Eliminating k from equations (1) and (2) and solving for q yields:

q =mg(vf + vr)

Evf. (3)

To eliminate m from equation (3), one uses the expression for the volume of asphere:

m = (m1 −m2)

m = (4/3)πa3(σ1 − σ2) (4)

4

where a is the radius of the droplet, σ1 is the density of the oil and σ2 is the densityof the air. Equations (3) and (4) are combined:

q =4π

3

g

E(σ1 − σ2)(1 +

vrvf

)a3 . (5)

To calculate a, one employs Stokes’ Law, relating the radius a of any sphericalbody to its velocity of fall in a viscous medium (with the coefficient of viscosity, η):

velocity falling =2

9

ga2

η(σ1 − σ2)

In the theoretical derivation of Stokes’ Law the following five assumptions aremade:

(1) that the inhomogeneities in the medium are small in comparison with the sizeof the sphere;

(2) that the sphere falls as it would in a medium of unlimited extent;

(3) that the sphere is smooth and rigid;

(4) that there is no slipping of the medium over the surface of the sphere;

(5) that the velocity with which the sphere is moving is so small that the resistanceto the motion is all due to the viscosity of the medium and not at all due to theinertia of such portion of the medium as is being pushed forward by the motionof the sphere.

In the case of our small drops, the assumptions (2), (3), (4) and (5) are valid.However, the assumption (1) is not completely valid since the drop radii are about 1or 2 microns and not much greater than the mean free path of the air molecules.

The drop will tend to fall more quickly in the “holes” between the air molecules.Kinetic theory indicates that a correction must be made to the formula for the fallingvelocity

vf =2

9

ga2

η(σ1 − σ2)

(1 + A

`

a

),

where ` is the mean free path of the air molecules and A is a dimensionless correctionfactor.

5

The mean free path ` is dependent upon the air pressure P and so we use a moreconvenient form

vf =2

9

ga2

η(σ1 − σ2)

(1 +

b

Pa

)(6)

where b is a correction factor, and P is the pressure.To calculate the radius a we must solve this equation. First rearrange.

9ηvf2g(σ1 − σ2)

= a2 +b

Pa .

Letθ = b/2P , (7)

and

φ =9ηvf

2g(σ1 − σ2). (8)

Then a2 + 2θa− φ = 0. The solution must be the positive root

a = −θ +√θ2 + φ . (9)

The electric intensity is given by E = V/d, where V is the potential differenceacross the parallel plates separated by a distance d. E, V and d are all expressed inthe mks system of units and so E is measured in volts/meter.

The charge q may be obtained by calculating θ and φ with equations (7) and (8)then calculating the radius a with equation (9) and finally q with equation (5).

6

Calculations

It is suggested that you lay out the results of your calculations for each drop inthe form of a table so that errors may be found more easily. Use MKS units and thefollowing columns:

averagefall

timetf

averagerisetimetr

(1 +

tftr

)=

(1 +

vrvf

) fallvelocityvf

φ (θ2 + φ)√θ2 + φ a a3 q

(1) The fall velocity vf = s/tf where s is distance between the images of thegraticule lines. You will have to determine s experimentally.

(2) From vf , calculate φ by multiplying by a factor 9η/2g(σ1 − σ2) where:

η = ? N sec m2 (Use graph in Fig. 2: 1 N sec m2 = 1 kg m−1 sec−1).

g = 9.81 meters/sec2.

σ1 = ? kg/meters3 (density of oil to be measured).

σ2 = 1.192 kg/meters3 (density of air at 1 atmosphere and 22◦C).

(3) Calculate θ from the barometric pressure P measured in cm of mercury and useb = 6.17× 10−6 (cm of Hg-meter).

(4) Add θ2 + φ. The θ2 will be small.

(5) Take the square root√θ2 + φ.

(6) Subtract θ to obtain a.

(7) At this stage, check that the radius a is reasonable.

(8) Obtain a3.

(9) Compute a multiplier 4π3gd(σ1−σ2)

V. (You will have to measure d.)

(10) Combine the multiplier, (1 + vrvf

) and a3 to obtain q.

Apparatus

7

1.8840

1.8800

1.8760

1.87201.86801.86401.86001.85601.8520

1.8480

1.8440

1.84001.83601.83201.8280

1.82401.82001.8160

1.81201.80801.80401.8000

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Figure 2: Viscosity(η) of dry air as a function of temperature (◦C). Vertical scale is

in units of 10−5 kg m−1 sec−1.

8

The experiment uses:

1. A PASCO SCIENTIFIC CORP apparatus.

2. A 4 milliwatt He-Ne laser for illumination of the drops. The laser is fastenedto the supporting plate so that its light cannot enter your eyes accidentally.However, be careful, and do not thoughtlessly unfasten the laser or introduceshiny or reflecting objects into the laser beam.

3. The high voltage is generated and stabilized within the PASCO apparatus and ismeasured precisely with a Data Precision Model 1450 multimeter. A correctionfor the finite input impedance of the meter may be required.

4. The time is measured by a small counter controlled by a microswitch and a 100kHz crystal oscillator. The count is visible on LEDs and is precise to the leastcount of 0.01 seconds. The counter is zeroed when the microswitch is pressedand the count starts. The count stops when the switch is released.

5. The apparatus and laser are mounted on one plate (for safety) which can beadjusted so that the electric field E is vertical.

6. An atomizer contains a non-volatile oil of known density.

The apparatus controls are:PLATE CHARGING CONTROL SWITCH - When the three way lever plate

control switch is in the OFF position the condenser plates are disconnected fromthe high voltage supply and grounded. When the switch is in the TOP PLATE +,position the top plate is positive with respect to the bottom plate. When the switchis in the TOP PLATE –, position the top plate is negative with respect to the bottomplate.

RADIATION SOURCE - when the lever is at the OUT position the radiationsource is shielded on all sides by plastic, so that virtually no radiation enters the areaof the drops. At the IN position the plastic shielding is removed and the drop areais exposed to the radiation source. NOTE: move the radiation source lever gentlyto avoid jarring the condenser assembly and knocking the droplet from the viewingarea.

The radiation source, initial strength and date of initial strength are all specifiedon the radiation tag to the left of the radiation source lever. The power Supply

specifications are:

Power Input: 110/130 VAC, 50/60 cps.

9

Range: 300–400 VDC, continuously variable.

Regulation: 1% for 10% line variation.

Ripple: less than 0.1 volt.

Stability: within 1% after warm up.

10

Procedure

Two people are advisable since one person must follow the drops while the otherrecords the times.

1. Darken the room. A small desk lamp can be used to illuminate your data book.

2. Turn the POWER switch ON, the PLATE CONTROL switch to OFF, theRADIATION SOURCE LEVER to IN.

3. The RETICULE ILLUMINATION control should be set so that the reticulelines are just bright enough to be easily visible. Excessive illumination of theselines may make it difficult to observe very small droplets.

4. Introduce some oil drops into the condenser by placing the nozzle of the atomizerinto the hole of the condenser housing cover. A few quick “squirts” of oil will fillthe upper chamber of the condenser with drops and begin to force some dropsinto the viewing area. If no drops are seen, squeeze the atomizer bulb gentlyuntil drops appear in the viewing area. If repeated “squirts” of the atomizer failto produce any drops in the viewing area, but rather a cloudy brightening ofthe field, the hole in the top plate is probably clogged, and should be cleaned.

The exact technique of introducing drops will have to be developed by theexperimenter. The object is to get a small number of drops, not a large, brightcloud, from which a single drop can be chosen. It is important to rememberthat the drops are being forced into the viewing area by the pressure of theatomizer. Therefore, excessive use of the atomizer can cause too many dropsto be forced into the viewing area and, more important, into the area betweenthe condenser wall and the focal point of the scope. Drops in this area preventobservation of drops at the focal point of the scope.

NOTE: If the entire viewing area becomes filled with drops, so that no one dropcan be isolated, either wait three or four minutes until the drops settle out ofview, or disassemble the condenser, thus removing the drops. When the amountof oil on the condenser parts becomes excessive, clean the assembly.

5. Select drops which have a mass which can be accurately measured. If the massis too large the drop will fall quickly and your percentage error in timing willbe poor.

If the mass is too small, the drop will bounce around due to random collisionswith air molecules (Brownian motion) and it will be difficult to estimate when

11

it crosses a line. A very small drop may cross a line 10 or 20 times! The lasershows the Brownian motion quite clearly.

We recommend using drops which fall with times between 10 and 40 seconds.

6. Now move the RADIATION SOURCE lever to the OUT position (so that thecharges will be unlikely to change.)

7. Immediately select those drops with measurable charges. If the drop risesquickly then it has a large charge q and so will not be much use in findingthe elementary charge.

Look for drops which take 10 to 40 seconds to rise.

8. If you still have many drops in sight, repeat 5 and 7 to concentrate on a usefuldrop.

9. Take about 10 measurements of the fall time and of the rise time of a particulardrop. Do not jar the apparatus or you may lose your drop.

10. Plot the fall time and rise time of this drop before measuring the next drop.Write the drop number beside its point.

Falltime

sec

Rise time

sec

40

30

20

10

10 20 30 40

12

11. Calculate the q for the first drop before you make any further measurements.If the drop is outside the range of 1 to 10 × 10−19 coulombs then find yourprocedural or arithmetic error before continuing.

12. Repeat 4 through 10 for about 30 drops. Choose the fall and rise times so thatyou have a fairly uniform scatter within the 10–40 sec square on your fall time,rise time plot. This will prevent you from selecting too many identical charges.Include a few points outside the square if you wish.

13. Try a variation on one or two drops by changing the charge. Drops are againintroduced into the viewing area and a new drop is selected. After about 20measurements on this drop have been made, the drop is brought to the top ofthe field of view and allowed to fall with the RADIATION SOURCE lever atthe IN position. A few seconds later the plates should be charged, and, if therising velocity has changed, the RADIATION SOURCE is moved to the OUTposition and a new series of measurements taken. If, however, the charge hasnot changed, then turn the Plate Control Switch to OFF and allow the dropto continue falling. After a few seconds, again check for a change in the risingvelocity. Continue this procedure until the drop has captured an ion.

If the drop captures an ion such that the drop moves rapidly downward, thenreverse the polarity of the plates so that the drop can be made to rise.

Make about 20 measurements of the rising and falling velocity of the drop, and,if possible, change the charge again and repeat the measurement procedure.

14. Record the barometric pressure and the voltage on the condenser for each mea-surement set.

15. At this time (before doing the calculations) decide which of your observationsmay be in error. It is very important that you not reject any data after yousee the result of the calculations. Why?

16. Now calculate q for each of your drops.

17. Make a histogram of your charges q. Use a bin width of 0.2× 10−19 coulombs.Label each square in the histogram with the drop number.

18. Does your evidence indicate that all charges are integral multiples of an elemen-tary charge?

13

19. Choose a charge Q such that all charges less than Q, fall in groups which areobvious multiples of an elementary charge. (The errors of the charges largerthan Q are too large for clean grouping).

20. Tabulate all charges q less than Q (it is now too late to reject a charge), divideeach q by the integer appropriate for its group. The final step is an averagecharge determined from the average charge of each of the integer subgroups.The final result should include an error resulting from correct error propagationof individual error contributions. Be sure to separate systematic and statisticalerrors.

14

RREEAADDIINNGG ##77

Electron Transitions The Bohr model for an electron transition in hydrogen between quantized energy levels with different quantum numbers n yields a photon by emission with quantum energy:

This is often expressed in terms of the inverse wavelength or "wave number" as follows:

Index

Atomic structure concepts

HyperPhysics***** Quantum Physics R Nave

Go Back

Hydrogen Energy Levels The basic hydrogen energy level structure is in agreement with the Bohr model. Common pictures are those of a shell structure with each main shell associated with a value of the principal quantum number n.

Index

Hydrogen concepts

Page 1 of 6Hydrogen energies and spectrum

4/20/2008http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html

This Bohr model picture of the orbits has some usefulness for visualization so long as it is realized that the "orbits" and the "orbit radius" just represent the most probable values of a considerable range of values. If the radial probabilities for the states are used to make sure you understand the distributions of the probability, then the Bohr picture can be superimposed on that as a kind of conceptual skeleton.

Energy level plot Energies in eV Hydrogen spectrum

Electron energy level diagrams


HyperPhysics***** Quantum Physics R Nave Go Back

Hydrogen Energy Level Plot The basic structure of the hydrogen energy levels can be calculated from the Schrodinger equation. The energy levels agree with the earlier Bohr model, and agree with experiment within a small fraction of an electron volt.



If you look at the hydrogen energy levels at extremely high resolution, you do find evidence of some other small effects on the energy. The 2p level is split into a pair of lines by the spin-orbit effect. The 2s and 2p states are found to differ a small amount in what is called the Lamb shift. And even the 1s ground state is split by the interaction of electron spin and nuclear spin in what is called hyperfine structure.

Electron level calculation Energies in eV

Index

Hydrogen concepts





Hydrogen Spectrum

This spectrum was produced by exciting a glass tube of hydrogen gas with about 5000 volts from a transformer. It was viewed through a diffraction grating with 600 lines/mm. The colors cannot be expected to be accurate because of differences in display devices.

For atomic number Z = ,

a transition from n2 = to n1 =

will have wavelength λ = nm

and quantum energy hν = eV

At left is a hydrogen spectral tube excited by a 5000 volt transformer. The three

Index

Great experiments of physics

Hydrogen concepts




Radiation of all the types in the electromagnetic spectrum can come from the atoms of different elements. A rough classification of some of the types of radiation by wavelength is:

Infrared > 750 nm Visible 400 - 750 nm Ultraviolet 10-400 nm Xrays < 10 nm

prominent hydrogen lines are shown at the right of the image through a 600 lines/mm diffraction grating.

An approximate classification of spectral colors:

Violet (380-435nm) Blue(435-500 nm) Cyan (500-520 nm) Green (520-565 nm) Yellow (565- 590 nm) Orange (590-625 nm) Red (625-740 nm)

Bohr model Measured hydrogen spectrum Other spectra


Measured Hydrogen Spectrum The measured lines of the Balmer series of hydrogen in the nominal visible region are:



The red line of deuterium is measurably different at 656.1065 ( .1787 nm difference).

Wavelength (nm)

Relative

Intensity Transition Color

383.5384 5 9 -> 2 Violet388.9049 6 8 -> 2 Violet397.0072 8 7 -> 2 Violet410.174 15 6 -> 2 Violet434.047 30 5 -> 2 Violet486.133 80 4 -> 2 Bluegreen (cyan)656.272 120 3 -> 2 Red656.2852 180 3 -> 2 Red

Illustration of transitions

Hydrogen fine structure (3->2 transition)

More extensive table of spectral lines

Index

Hydrogen concepts

Atomic spectra

HyperPhysics***** Quantum Physics R Nave

Go Back



RREEAADDIINNGG ##88

Development of Current Atomic Theory

Emission Spectrum of Hydrogen

When an electric current is passed through a glass tube that contains hydrogen gas at low pressure the tube gives off blue light. When this light is passed through a prism (as shown in the figure below), four narrow bands of bright light are observed against a black background.

These narrow bands have the characteristic wavelengths and colors shown in the table below.

Four more series of lines were discovered in the emission spectrum of hydrogen by searching the infrared spectrum at longer wave-lengths and the ultraviolet spectrum at shorter wavelengths. Each of these lines fits the same general equation, where n1 and n2 are integers and RH is 1.09678 x 10-2

nm-1.

Emission Spectrum of Hydrogen

Explanation of the Emission Spectrum

Bohr Model of the Atom

Wave-Particle Duality Bohr Model vs. Reality Wave Functions and Orbitals

Diagram

Wavelength Color656.2 red486.1 blue-green434.0 blue-violet410.1 violet

Page 1 of 7Emission Spectrum of Hydrogen

4/20/2008http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/bohr.html

Explanation of the Emission Spectrum

Max Planck presented a theoretical explanation of the spectrum of radiation emitted by an object that glows when heated. He argued that the walls of a glowing solid could be imagined to contain a series of resonators that oscillated at different frequencies. These resonators gain energy in the form of heat from the walls of the object and lose energy in the form of electromagnetic radiation. The energy of these resonators at any moment is proportional to the frequency with which they oscillate.

To fit the observed spectrum, Planck had to assume that the energy of these oscillators could take on only a limited number of values. In other words, the spectrum of energies for these oscillators was no longer continuous. Because the number of values of the energy of these oscillators is limited, they are theoretically "countable." The energy of the oscillators in this system is therefore said to be quantized. Planck introduced the notion of quantization to explain how light was emitted.

Albert Einstein extended Planck's work to the light that had been emitted. At a time when everyone agreed that light was a wave (and therefore continuous), Einstein suggested that it behaved as if it was a stream of small bundles, or packets, of energy. In other words, light was also quantized. Einstein's model was based on two assumptions. First, he assumed that light was composed of photons, which are small, discrete bundles of energy. Second, he assumed that the energy of a photon is proportional to its frequency.

E = hv

In this equation, h is a constant known as Planck's constant, which is equal to 6.626 x 10-34 J-s.

Example: Let's calculate the energy of a single photon of red light with a wavelength of 700.0 nm and the energy of a mole of these photons.

Red light with a wavelength of 700.0 nm has a frequency of 4.283 x 1014 s-1. Substituting this frequency into the Planck-Einstein equation gives the following result.

A single photon of red light carries an insignificant amount of energy. But a mole of these photons carries about 171,000 joules of energy, or 171 kJ/mol.

Absorption of a mole of photons of red light would therefore provide enough energy to raise the temperature of a liter of water by more than 40oC.

The fact that hydrogen atoms emit or absorb radiation at a limited number of frequencies implies that these atoms can only absorb radiation with certain energies. This suggests that there are only a limited number of energy levels within the hydrogen atom. These energy levels are countable. The energy levels of the hydrogen atom are quantized.

Learning Activity



The Bohr Model of the Atom

Niels Bohr proposed a model for the hydrogen atom that explained the spectrum of the hydrogen atom. The Bohr model was based on the following assumptions.

The electron in a hydrogen atom travels around the nucleus in a circular orbit. The energy of the electron in an orbit is proportional to its distance from the nucleus. The further the electron is from the nucleus, the more energy it has. Only a limited number of orbits with certain energies are allowed. In other words, the orbits are quantized. The only orbits that are allowed are those for which the angular momentum of the electron is an integral multiple of Planck's constant divided by 2p. Light is absorbed when an electron jumps to a higher energy orbit and emitted when an electron falls into a lower energy orbit. The energy of the light emitted or absorbed is exactly equal to the difference between the energies of the orbits.

Some of the key elements of this hypothesis are illustrated in the figure below.

Three points deserve particular attention. First, Bohr recognized that his first assumption violates the principles of classical mechanics. But he knew that it was impossible to explain the spectrum of the hydrogen atom within the limits of classical physics. He was therefore willing to assume that one or more of the principles from classical physics might not be valid on the atomic scale.

Second, he assumed there are only a limited number of orbits in which the electron can reside. He

Practice Problem 5:

Calculate the energy of a single photon of red light with a wavelength of 700.0 nm and the energy of a mole of these photons.

Click here to check your answer to Practice Problem 5

Click here to see a solution to Practice Problem 5



based this assumption on the fact that there are only a limited number of lines in the spectrum of the hydrogen atom and his belief that these lines were the result of light being emitted or absorbed as an electron moved from one orbit to another in the atom.

Finally, Bohr restricted the number of orbits on the hydrogen atom by limiting the allowed values of the angular momentum of the electron. Any object moving along a straight line has a momentum equal to the product of its mass (m) times the velocity (v) with which it moves. An object moving in a circular orbit has an angular momentum equal to its mass (m) times the velocity (v) times the radius of the orbit (r). Bohr assumed that the angular momentum of the electron can take on only certain values, equal to an integer times Planck's constant divided by 2p.

Bohr then used classical physics to show that the energy of an electron in any one of these orbits is inversely proportional to the square of the integer n.

The difference between the energies of any two orbits is therefore given by the following equation.

In this equation, n1 and n2 are both integers and RH is the proportionality constant known as the Rydberg constant.

Planck's equation states that the energy of a photon is proportional to its frequency.

E = hv

Substituting the relationship between the frequency, wavelength, and the speed of light into this equation suggests that the energy of a photon is inversely proportional to its wavelength. The inverse of the wavelength of electromagnetic radiation is therefore directly proportional to the energy of this radiation.

By properly defining the units of the constant, RH, Bohr was able to show that the wavelengths of the light given off or absorbed by a hydrogen atom should be given by the following equation.

Thus, once he introduced his basic assumptions, Bohr was able to derive an equation that matched the relationship obtained from the analysis of the spectrum of the hydrogen atom.

Practice Problem 6:

Calculate the wavelength of the light given off by a hydrogen atom when an electron falls from the n = 4 to the n = 2 orbit in the Bohr model.



According to the Bohr model, the wavelength of the light emitted by a hydrogen atom when the electron falls from a high energy (n = 4) orbit into a lower energy (n = 2) orbit.

Substituting the appropriate values of RH, n1, and n2 into the equation shown above gives the following result.

Solving for the wavelength of this light gives a value of 486.3 nm, which agrees with the experimental value of 486.1 nm for the blue line in the visible spectrum of the hydrogen atom.

Wave-Particle Duality

The theory of wave-particle duality developed by Louis-Victor de Broglie eventually explained why the Bohr model was successful with atoms or ions that contained one electron. It also provided a basis for understanding why this model failed for more complex systems. Light acts as both a particle and a wave. In many ways light acts as a wave, with a characteristic frequency, wavelength, and amplitude. Light carries energy as if it contains discrete photons or packets of energy.

When an object behaves as a particle in motion, it has an energy proportional to its mass (m) and the speed with which it moves through space (s).

E = ms2

When it behaves as a wave, however, it has an energy that is proportional to its frequency:

By simultaneously assuming that an object can be both a particle and a wave, de Broglie set up the following equation.

Click here to check your answer to Practice Problem 6

Click here to see a solution to Practice Problem 6



By rearranging this equation, he derived a relationship between one of the wave-like properties of matter and one of its properties as a particle.

As noted in the previous section, the product of the mass of an object times the speed with which it moves is the momentum (p) of the particle. Thus, the de Broglie equation suggests that the wavelength (l) of any object in motion is inversely proportional to its momentum.

De Broglie concluded that most particles are too heavy to observe their wave properties. When the mass of an object is very small, however, the wave properties can be detected experimentally. De Broglie predicted that the mass of an electron was small enough to exhibit the properties of both particles and waves. In 1927 this prediction was confirmed when the diffraction of electrons was observed experimentally by C. J. Davisson.

De Broglie applied his theory of wave-particle duality to the Bohr model to explain why only certain orbits are allowed for the electron. He argued that only certain orbits allow the electron to satisfy both its particle and wave properties at the same time because only certain orbits have a circumference that is an integral multiple of the wavelength of the electron, as shown in the figure below.

The Bohr Model vs. Reality

At first glance, the Bohr model looks like a two-dimensional model of the atom because it restricts the motion of the electron to a circular orbit in a two-dimensional plane. In reality the Bohr model is a one-dimensional model, because a circle can be defined by specifying only one dimension: its radius, r. As a result, only one coordinate (n) is needed to describe the orbits in the Bohr model.

Unfortunately, electrons aren't particles that can be restricted to a one-dimensional circular orbit. They act to some extent as waves and therefore exist in three-dimensional space. The Bohr model

Diagram



works for one-electron atoms or ions only because certain factors present in more complex atoms are not present in these atoms or ions. To construct a model that describes the distribution of electrons in atoms that contain more than one electron we have to allow the electrons to occupy three-dimensional space. We therefore need a model that uses three coordinates to describe the distribution of electrons in these atoms.

Wave Functions and Orbitals

We still talk about the Bohr model of the atom even if the only thing this model can do is explain the spectrum of the hydrogen atom because it was the last model of the atom for which a simple physical picture can be constructed. It is easy to imagine an atom that consists of solid electrons revolving around the nucleus in circular orbits.

Erwin Schrödinger combined the equations for the behavior of waves with the de Broglie equation to generate a mathematical model for the distribution of electrons in an atom. The advantage of this model is that it consists of mathematical equations known as wave functions that satisfy the requirements placed on the behavior of electrons. The disadvantage is that it is difficult to imagine a physical model of electrons as waves.

The Schrödinger model assumes that the electron is a wave and tries to describe the regions in space, or orbitals, where electrons are most likely to be found. Instead of trying to tell us where the electron is at any time, the Schrödinger model describes the probability that an electron can be found in a given region of space at a given time. This model no longer tells us where the electron is; it only tells us where it might be.



RREEAADDIINNGG ##99

The Hydrogen Atom

The study of the hydrogen atom is more complicated than our previous example of an electron confined to move on a line. Not only does the motion of the electron occur in three dimensions but there is also a force acting on the electron. This force, the electrostatic force of attraction, is responsible for holding the atom together. The magnitude of this force is given by the product of the nuclear and electronic charges divided by the square of the distance between them. In the previous example of an electron confined to move on a line, the total energy was entirely kinetic in origin since there were no forces acting on the electron. In the hydrogen atom however, the energy of the electron, because of the force exerted on it by the nucleus, will consist of a potential energy (one which depends on the position of the electron relative to the nucleus) as well as a kinetic energy. The potential energy arising from the force of attraction between the nucleus and the electron is:

Let us imagine for the moment that the proton and the electron behave classically. Then, if the nucleus is held fixed at the origin and the electron allowed to move relative to it, the potential energy would vary in the manner indicated in Fig. 3-1. The potential energy is independent of the direction in space and depends only on the distance r between the electron and the nucleus. Thus Fig. 3-1 refers to any line directed from the nucleus to the electron. The r-axis in the figure may be taken

An Introduction to the Electronic Structure of Atoms and Molecules

Dr. Richard F.W. Bader Professor of Chemistry / McMaster University / Hamilton, Ontario

Preface1. The Nature of the Problem2. The New Physics3. The Hydrogen Atom

• Introduction • The Quantization of Energy • The Probability Distribution of the

Hydrogen Atom • Angular Momentum of an Electron in

an H Atom • Some Useful Expressions • Problems

4. Many-Electron Atoms5. Electronic Basis for the Properties of the

Elements6. The Chemical Bond7. Ionic and Covalent Binding8. Molecular Orbitals

Table of Contour Values

Page 1 of 2The Hydrogen Atom - Introduction

4/20/2008http://www.chemistry.mcmaster.ca/esam/Chapter_3/intro.html

literally as a line through the nucleus. Whether the electron moves to the right or to the left the potential energy varies in the same manner.

Fig. 3-1. The potential energy of interaction between a nucleus (at the origin) and an electron as a function of the

distance r between them.

The potential energy is zero when the two particles are very far apart (r = ∞ ), and equals minus infinity when r equals zero. We shall take the energy for r = ∞ as our zero of energy. Every energy will be measured relative to this value. When a stable atom is formed, the electron is attracted to the nucleus, r is less than infinity, and the energy will be negative. A negative value for the energy implies that energy must be supplied to the system if the electron is to overcome the attractive force of the nucleus and escape from the atom. The electron has again "fallen into a potential well." However, the shape of the well is no longer a simple square one as previously considered for an electron confined to move on a line, but has the shape shown in Fig. 3-1. This shape is a consequence of there being a force acting on the electron and hence a potential energy contribution which depends on the distance between the two particles. This is the nature of the problem. Now let us see what quantum mechanics predicts for the motion of the electron in such a situation.

Page 2 of 2The Hydrogen Atom - Introduction

4/20/2008http://www.chemistry.mcmaster.ca/esam/Chapter_3/intro.html

The Quantization of Energy

The motion of the electron is not free. The electron is bound to the atom by the attractive force of the nucleus and consequently quantum mechanics predicts that the total energy of the electron is quantized. The expression for the energy is:

where m is the mass of the electron, e is the magnitude of the electronic charge, n is a quantum number, h is Planck's constant and Z is the atomic number (the number of positive charges in the nucleus).

This formula applies to any one-electron atom or ion. For example, He+ is a one-electron system for which Z = 2. We can again construct an energy level diagram listing the allowed energy values (Fig. 3-2).

Fig. 3-2. The energy level diagram for the H atom. Each line dentoes an allowed energy for the atom.

These are obtained by substituting all possible values of n into equation (1). As in our previous example, we shall represent all the constants which appear in the expression for En by a constant K and we shall set Z = 1.

(1)

(2)

Page 1 of 4The Hydrogen Atom -The Quantization of Energy

4/20/2008http://www.chemistry.mcmaster.ca/esam/Chapter_3/section_1.html

Since the motion of the electron occurs in three dimensions we might correctly anticipate three quantum numbers for the hydrogen atom. But the energy depends only on the quantum number n and for this reason it is called the principal quantum number. In this case, the energy is inversly dependent upon n2, and as n is increased the energy becomes less negative with the spacings between the energy levels decreasing in size. When n = ∞ , E = 0 and the electron is free of the attractive force of the nucleus. The average distance between the nucleus and the electron (the average value of r) increases as the energy or the value of n increases. Thus energy must be supplied to pull the electron away from the nucleus.

The parallelism between increasing energy and increasing average value of r is a useful one. In fact, when an electron loses energy, we refer to it as "falling" from one energy level to a lower one on the energy level diagram. Since the average distance between the nucleus and the electron also decreases with a decrease in n, then the electron literally does fall in closer to the nucleus when it "falls" from level to level on the energy level diagram.

The energy difference between E∞ and E

1:

is called the ionization energy. It is the energy required to pull the electron completely away from the nucleus and is, therefore, the energy of the reaction:

This amount of energy is sufficient to separate the electron from the attractive influence of the nucleus and leave both particles at rest. If an amount of energy greater than K is supplied to the electron, it will not only escape from the atom but the energy in excess of K will appear as kinetic energy of the electron. Once the electron is free it may have any energy because all velocities are then possible. This is indicated in the energy level diagram by the shading above the E

∞ = 0 line. An

electron which possesses and energy in this region of the diagram is a free electron and has kinetic energy of motion only.

The Hydrogen Atom Spectrum

As mentioned earlier, hydrogen gas emits coloured light when a high voltage is applied across a sample of the gas contained in a glass tube fitted with electrodes. The electrical energy transmitted to the gas causes many of the hydrogen molecules to dissociate into atoms:

The electrons in the molecules and in the atoms absorb energy and are excited to high energy levels. lonization of the gas also occurs. When the electron is in a quantum level other than the lowest level (with n = 1) the electron is said to be excited, or to be in an excited level. The lifetime of such an excited level is very brief, being of the order of magnitude of only 10-8 sec. The electron loses the energy of excitation by falling to a lower energy level and at the same time emitting a photon to carry off the excess energy. We can easily calculate the frequencies which should appear in the emitted light by calculating the difference in energy between the two levels and making use of



Bohr's frequency condition:

Suppose we consider all those frequencies which appear when the electron falls to the lowest level, n = 1,

Every value of n substituted into this equation gives a distinct value for v. In Fig. 3-3 we illustrate the changes in energy which result when the electron emits a photon by an arrow connecting the excited level (of energy En) with the ground level (of energy E

1). The frequency resulting from each

drop in energy will be directly proportional to the length of the arrow. Just as the arrows increase in length as n is increased, so v increases. However, the spacings between the lines decrease as n is increased, and the spectrum will appear as shown directly below the energy level diagram in Fig. 3-3.

Fig. 3-3. The energy changes and corresponding frequencies which give rise to the Lyman series in the spectrum of

the H atom. The line spectrum degenerates into a continuous spectrum at the high frequency end.

(3)



Each line in the spectrum is placed beneath the arrow which represents the change in energy giving rise to that particular line. Free electrons with varying amounts of kinetic energy (½mυ2) can also fall to the n = 1 level.

The energy released in the reversed ionization reaction:

will equal K, the difference between E∞ and E

1, plus ½mυ2, the kinetic energy originally possessed by

the electron. Since this latter energy is not quantized, every energy value greater than K should be possible and every frequency greater than that corresponding to

should be observed. The line spectrum should, therefore, collapse into a continuous spectrum at its high frequency end. Thus the energy continuum above E

∞ gives rise to a continuum of frequencies

in the emission spectrum. The beginning of the continuum should be the frequency corresponding to the jump from E

∞ to E

1, and thus we can determine K, the ionization energy of the hydrogen atom,

from the observation of this frequency. Indeed, the spectroscopic method is one of the most accurate methods of determining ionization energies.

The hydrogen atom does possess a spectrum identical to that predicted by equation (3), and the observed value for K agrees with the theoretical value. This particular series of lines, called the Lyman series, falls in the ultraviolet region of the spectrum because of the large energy changes involved in the transitions from the excited levels to the lowest level. The first few members of a second series of lines, a second line spectrum, falls in the visible portion of the spectrum. It is called the Balmer series and arises from electrons in excited levels falling to the second quantum level. Since E

2 equals only one quarter of E

1, the energy jumps are smaller and the frequencies are

correspondingly lower than those observed in the Lyman series. Four lines can be readily seen in this series: red, green, blue, and violet. Each colour results from the electrons falling from a specific level, to the n = 2 level: red E3 → E

2; green, E

4→ E

2; blue, E

5→ E

2; and violet E

6 →E

2. Other series,

arising from electrons falling to the n = 3 and n = 4 levels, can be found in the infrared (frequencies preceding the red end or long wavelength end of the visible spectrum).

The fact that the hydrogen atom exhibits a line spectrum is visible proof of the quantization of energy on the atomic level.



The Probability Distributions for the Hydrogen Atom

To what extent will quantum mechanics permit us to pinpoint the position of an electron when it is bound to an atom? We can obtain an order of magnitude answer to this question by applying the uncertainty principle

to estimate Δx. The value of Δx will represent the minimum uncertainty in our knowledge of the position of the electron. The momentum of an electron in an atom is of the order of magnitude of 9 × 10-19 g cm/sec. The uncertainty in the momentum Δp must necessarily be of the same order of magnitude. Thus

The uncertainty in the position of the electron is of the same order of magnitude as the diameter of the atom itself. As long as the electron is bound to the atom, we will not be able to say much more about its position than that it is in the atom. Certainly all models of the atom which describe the electron as a particle following a definite trajectory or orbit must be discarded.

We can obtain an energy and one or more wave functions for every value of n, the principal quantum number, by solving Schrödinger's equation for the hydrogen atom. A knowledge of the wave functions, or probability amplitudes ψ

n, allows us to calculate the probability distributions for the

electron in any given quantum level. When n = 1, the wave function and the derived probability function are independent of direction and depend only on the distance r between the electron and the nucleus. In Fig. 3-4, we plot both ψ

1 and P

1 versus r, showing the variation in these functions as the

electron is moved further and further from the nucleus in any one direction. (These and all succeeding graphs are plotted in terms of the atomic unit of length, a

0 = 0.529 × 10-8 cm.)

Fig. 3-4. The wave function and probability distribution as functions of r for the n = 1 level of the H atom. The functions

and the radius r are in atomic units in this and succeeding figures.

Two interpretations can again be given to the P1 curve. An experiment designed to detect the position

Page 1 of 9The Hydrogen Atom - The Probability Distribution of the Hydrogen Atom


of the electron with an uncertainty much less than the diameter of the atom itself (using light of short wavelength) will, if repeated a large number of times, result in Fig. 3-4 for P

1. That is, the electron

will be detected close to the nucleus most frequently and the probability of observing it at some distance from the nucleus will decrease rapidly with increasing r. The atom will be ionized in making each of these observations because the energy of the photons with a wavelength much less than 10-8 cm will be greater than K, the amount of energy required to ionize the hydrogen atom. If light with a wavelength comparable to the diameter of the atom is employed in the experiment, then the electron will not be excited but our knowledge of its position will be correspondingly less precise. In these experiments, in which the electron's energy is not changed, the electron will appear to be "smeared out" and we may interpret P

1 as giving the fraction of the total electronic charge to be found in every

small volume element of space. (Recall that the addition of the value of Pn for every small volume

element over all space adds up to unity, i.e., one electron and one electronic charge.)

When the electron is in a definite energy level we shall refer to the Pn distributions as electron

density distributions, since they describe the manner in which the total electronic charge is distributed in space. The electron density is expressed in terms of the number of electronic charges per unit volume of space, e-/V. The volume V is usually expressed in atomic units of length cubed, and one atomic unit of electron density is then e-/a0

3. To give an idea of the order of magnitude of an atomic density unit, 1 au of charge density e-/a0

3 = 6.7 electronic charges per cubic Ångstrom. That is, a cube with a length of 0.52917 ×10-8 cm, if uniformly filled with an electronic charge density of 1 au, would contain 6.7 electronic charges.

P1 may be represented in another manner. Rather than considering the amount of electronic charge

in one particular small element of space, we may determine the total amount of charge lying within a thin spherical shell of space. Since the distribution is independent of direction, consider adding up all the charge density which lies within a volume of space bounded by an inner sphere of radius r and an outer concentric sphere with a radius only infinitesimally greater, say r + Δr. The area of the inner sphere is 4πr2 and the thickness of the shell is Δr. Thus the volume of the shell is 4πr2 Δr (Click here for note.) and the product of this volume and the charge density P

1(r), which is the charge or number of

electrons per unit volume, is therefore the total amount of electronic charge lying between the spheres of radius r and r + Δr. The product 4πr2P

n is given a special name, the radial distribution function,

which we shall label Qn(r).

The radial distribution function is plotted in Fig. 3-5 for the ground state of the hydrogen atom.

Fig. 3-5. The radial distribution function Q1(r) for an H atom. The value of this function at some value of r when multiplied by Δr gives the number of electronic charges within the thin shell of space lying between spheres of radius r and r + Δr.



The curve passes through zero at r = 0 since the surface area of a sphere of zero radius is zero. As the radius of the sphere is increased, the volume of space defined by 4πr2Δr increases. However, as shown in Fig 3-4, the absolute value of the electron density at a given point decreases with r and the resulting curve must pass through a maximum. This maximum occurs at r

max = a

0. Thus more of the

electronic charge is present at a distance a0, out from the nucleus than at any other value of r. Since

the curve is unsymmetrical, the average value of r, denoted by , is not equal to rmax

. The average value of r is indicated on the figure by a dashed line. A "picture" of the electron density distribution for the electron in the n = 1 level of the hydrogen atom would be a spherical ball of charge, dense around the nucleus and becoming increasingly diffuse as the value of r is increased.

We could also represent the distribution of negative charge in the hydrogen atom in the manner used previously for the electron confined to move on a plane, Fig. 2-4, by displaying the charge density in a plane by means of a contour map. Imagine a plane through the atom including the nucleus. The density is calculated at every point in this plane. All points having the same value for the electron density in this plane are joined by a contour line (Fig. 3-6). Since the electron density depends only on r, the distance from the nucleus, and not on the direction in space, the contours will be circular. A contour map is useful as it indicates the "shape" of the density distribution.



This completes the description of the most stable state of the hydrogen atom, the state for which n = 1. Before proceeding with a discussion of the excited states of the hydrogen atom we must introduce a new term. When the energy of the electron is increased to another of the allowed values, corresponding to a new value for n, ψn and Pn change as well. The wave functions ψn for the hydrogen atom are given a special name, atomic orbitals, because they play such an important role in all of our future discussions of the electronic structure of atoms. In general the word orbital is the name given to a wave function which determines the motion of a single electron. If the one-electron wave function is for an atomic system, it is called an atomic orbital. (Click here for note.)

For every value of the energy En, for the hydrogen atom, there is a degeneracy equal to n2.

Therefore, for n = 1, there is but one atomic orbital and one electron density distribution. However, for n = 2, there are four different atomic orbitals and four different electron density distributions, all of which possess the same value for the energy, E2. Thus for all values of the principal quantum number n there are n2 different ways in which the electronic charge may be distributed in three-dimensional space and still possess the same value for the energy. For every value of the principal

Fig. 3-6. (a) A contour map of the electron density distribution in a plane containing the nucleus for the n = 1 level of the H atom. The distance between adjacent contours is 1 au. The numbers on the left-hand side on each contour give the electron densityin au. The numbers on the right-hand side give the fraction of the total electronic charge which lies within a sphere of that radius. Thus 99% of the single electronic charge of the H atom lies within a sphere of radius 4 au (or diameter = 4.2 ×10-8 cm). (b) This is a profile of the contour map along a line through the nucleus. It is, of course, the same as that given previously in Fig. 3-4 for P

1, but now

plotted from the nucleus in both directions.



quantum number, one of the possible atomic orbitals is independent of direction and gives a spherical electron density distribution which can be represented by circular contours as has been exemplified above for the case of n = 1. The other atomic orbitals for a given value of n exhibit a directional dependence and predict density distributions which are not spherical but are concentrated in planes or along certain axes. The angular dependence of the atomic orbitals for the hydrogen atom and the shapes of the contours of the corresponding electron density distributions are intimately connected with the angular momentum possessed by the electron.

The physical quantity known as angular momentum plays a dominant role in the understanding of the electronic structure of atoms. To gain a physical picture and feeling for the angular momentum it is necessary to consider a model system from the classical point of view. The simplest classical model of the hydrogen atom is one in which the electron moves in a circular orbit with a constant speed or angular velocity (Fig. 3-7). Just as the ordinary momentum mv plays a dominant role in the analysis of straight line or linear motion, so angular momentum plays the central role in the analysis of a system with circular motion as found in the model of the hydrogen atom.

Fig. 3-7. The angular momentum vector for a classical model of the atom.

In Fig. 3-7, m is the mass of the electron, v is the linear velocity (the velocity the electron would possess if it continued moving at a tangent to the orbit as indicated in the figure) and r is the radius of the orbit. The linear velocity v is a vector since it possesses at any instant both a magnitude and a direction in space. Obviously, as the electron rotates in the orbit the direction of v is constantly changing, and thus the linear momentum mv is not constant for the circular motion. This is so even though the speed of the electron (the magnitude of v which is denoted by υ) remains unchanged. According to Newton's second law, a force must be acting on the electron if its momentum changes with time. This is the force which prevents the electron from flying on tangent to its orbit. In an atom the attractive force which contains the electron is the electrostatic force of attraction between the nucleus and the electron, directed along the radius r at right angles to the direction of the electron's motion.

The angular momentum, like the linear momentum, is a vector and is defined as follows:



The angular momentum vector M is directed along the axis of rotation. From the definition it is evident that the angular momentum vector will remain constant as long as the speed of the electron in the orbit is constant (υ remains unchanged) and the plane and radius of the orbit remain unchanged. Thus for a given orbit, the angular momentum is constant as long as the angular velocity of the particle in the orbit is constant. In an atom the only force on the electron in the orbit is directed along r; it has no component in the direction of the motion. The force acts in such a way as to change only the linear momentum. Therefore, while the linear momentum is not constant during the circular motion, the angular momentum is. A force exerted on the particle in the direction of the vector v would change the angular velocity and the angular momentum. When a force is applied which does change M, a torque is said to be acting on the system. Thus angular momentum and torque are related in the same way as are linear momentum and force.

The important point of the above discussion is that both the angular momentum and the energy of an atom remain constant if the atom is left undisturbed. Any physical quantity which is constant in a classical system is both conserved and quantized in a quantum mechanical system. Thus both the energy and the angular momentum are quantized for an atom.

There is a quantum number, denoted by l, which governs the magnitude of the angular momentum, just as the quantum number n determines the energy. The magnitude of the angular momentum may assume only those values given by:

Furthermore, the value of n limits the maximum value of the angular momentum as the value of l cannot be greater than n - 1. For the state n = 1 discussed above, l may have the value of zero only. When n = 2, l may equal 0 or 1, and for n = 3, l = 0 or 1 or 2, etc. When l = 0, it is evident from equation (4) that the angular momentum of the electron is zero. The atomic orbitals which describe these states of zero angular momentum are called s orbitals. The s orbitals are distinguished from one another by stating the value of n, the principal quantum number. They are referred to as the 1s, 2s, 3s, etc., atomic orbitals.

The preceding discussion referred to the 1s orbital since for the ground state of the hydrogen atom n = 1 and l = 0. This orbital, and all s orbitals in general, predict spherical density distributions for the electron as exemplified by Fig. 3-5 for the 1s density. Figure 3-8 shows the radial distribution functions Q(r) which apply when the electron is in a 2s or 3s orbital to illustrate how the character of the density distributions change as the value of n is increased. (Click here for note.)

(4)



Fig. 3-8. Radial distribution functions for the 2s and 3s density distributions.

Comparing these results with those for the 1s orbital in Fig. 3-5 we see that as n increases the average value of r increases. This agrees with the fact that the energy of the electron also increases as n increases. The increased energy results in the electron being on the average pulled further away from the attractive force of the nucleus. As in the simple example of an electron moving on a line, nodes (values of r for which the electron density is zero) appear in the probability distributions. The number of nodes increases with increasing energy and equals n - 1.

When the electron possesses angular momentum the density distributions are no longer spherical. In fact for each value of l, the electron density distribution assumes a characteristic shape (Fig. 3-9).



Fig. 3-9. Contour maps of the 2s, 2p, 3d and 4f atomic orbitals and their charge density distributions for the H atom. The zero contours shown in the maps for the orbitals define the positions of the nodes. Negative values for the contours of the

orbitals are indicated by dashed lines, positive values by solid lines.

When l = 1, the orbitals are called p orbitals. In this case the orbital and its electron density are concentrated along a line (axis) in space. The 2p orbital or wave function is positive in value on one side and negative in value on the other side of a plane which is perpendicular to the axis of the orbital and passes through the nucleus. The orbital has a node in this plane, and consequently an electron in a 2p orbital does not place any electronic charge density at the nucleus. The electron density of a 1s orbital, on the other hand, is a maximum at the nucleus. The same diagram for the 2p density distribution is obtained for any plane which contains this axis. Thus in three dimensions the electron density would appear to be concentrated in two lobes, one on each side of the nucleus, each lobe being circular in cross section (Fig. 3-10).



Fig. 3-10. The appearance of the 2p electron density distribution in three-dimensional space.

When l = 2, the orbitals are called d orbitals and Fig. 3-9 shows the contours in a plane for a 3d orbital and its density distribution. Notice that the density is again zero at the nucleus and that there are now two nodes in the orbital and in its density distribution. As a final example, Fig. 3-9 shows the contours of the orbital and electron density distribution obtained for a 4f atomic orbital which occurs when n = 4 and l = 3. (Click here for note.) The point to notice is that as the angular momentum of the electron increases, the density distribution becomes increasingly concentrated along an axis or in a plane in space. Only electrons in s orbitals with zero angular momentum give spherical density distributions and in addition place charge density at the position of the nucleus.

We have not as yet accounted for the full degeneracy of the hydrogen atom orbitals which we stated earlier to be n2 for every value of n. For example, when n = 2, there are four distinct atomic orbitals. The remaining degeneracy is again determined by the angular momentum of the system. Since angular momentum like linear momentum is a vector quantity, we may refer to the component of the angular momentum vector which lies along some chosen axis. For reasons we shall investigate, the number of values a particular component can assume for a given value of l is (2l + 1). Thus when l = 0, there is no angular momentum and there is but a single orbital, an s orbital. When l = 1, there are three possible values for the component (2× 1 + 1) of the total angular momentum which are physically distinguishable from one another. There are, therefore, three p orbitals. Similarly there are five d orbitals, (2 × 2+1), seven f orbitals, (2 × 3 +1), etc. All of the orbitals with the same value of n and l, the three 2p orbitals for example, are similar but differ in their spatial orientations.

To gain a better understanding of this final element of degeneracy, we must consider in more detail what quantum mechanics predicts concerning the angular momentum of an electron in an atom.



Angular Momentum of an Electron in an H Atom

The simplest classical model of the hydrogen atom is one in which the electron moves in a circular planar orbit about the nucleus as previously discussed and as illustrated in Fig. 3-7. The angular momentum vector M in this figure is shown at an angle θ with respect to some arbitrary axis in space. Assuming for the moment that we can somehow physically define such an axis, then in the classical model of the atom there should be an infinite number of values possible for the component of the angular momentum vector along this axis. As the angle between the axis and the vector M varies continuously from 0°, through 90° to 180°, the component of M along the axis would vary correspondingly from M to zero to -M. Thus the quantum mechanical statements regarding the angular momentum of an electron in an atom differ from the classical predictions in two startling ways. First, the magnitude of the angular momentum (the length of the vector M) is restricted to only certain values given by:

The magnitude of the angular momentum is quantized. Secondly, quantum mechanics states that the component of M along a given axis can assume only (2l + 1) values, rather than the infinite number allowed in the classical model. In terms of the classical model this would imply that when the

magnitude of M is (the value when l = 1), there are only three allowed values for θ, the angle of inclination of M with respect to a chosen axis.

The angle θ is another example of a physical quantity which in a classical system may assume any value, but which in a quantum system may take on only certain discrete values. You need not accept this result on faith. There is a simple, elegant experiment which illustrates the "quantization" of θ, just as a line spectrum illustrates the quantization of the energy.

If we wish to measure the number of possible values which the component of the angular momentum may exhibit with respect to some axis we must first find some way in which we can physically define a direction or axis in space. To do this we make use of the magnetism exhibited by an electron in an atom. The flow of electrons through a loop of wire (an electric current) produces a magnetic field (Fig. 3-11). At a distance from the ring of wire, large compared to the diameter of the ring, the magnetic field produced by the current appears to be the same as that obtained from a small bar magnet with a north pole and a south pole. Such a small magnet is called a magnetic dipole, i.e., two poles separated by a small distance.

Page 1 of 11The Hydrogen Atom - Angular Momentum of an Electron in an H Atom


Fig. 3-11. The magnetic field produced by a current in a loop of wire.

The electron is charged and the motion of the electron in an atom could be thought of as generating a small electric current. Associated with this current there should be a small magnetic field. The magnitude of this magnetic field is related to the angular momentum of the electron's motion in roughly the same way that the magnetic field produced by a current in a loop of wire is proportional to the strength of the current flowing in the wire.

The strength of the atomic magnetic dipole is given by μ where:

Just as there is a fundamental unit of negative charge denoted by e- so there is a fundamental unit of magnetism at the atomic level denoted by βm and called the Bohr magneton. From equation (5) we can see that the strength of the magnetic dipole will increase as the angular momentum of the electron increases. This is analogous to increasing the magnetic field by increasing the strength of the current through a circular loop of wire The magnetic dipole, since it has a north and a south pole, will define some direction in space (the magnetic dipole is a vector quantity). The axis of the magnetic dipole in fact coincides with the direction of the angular momentum vector. Experimentally, a collection of atoms behave as though they were a collection of small bar magnets if the electrons in these atoms possess angular momentum. In addition, the axis of the magnet lies along the axis of rotation, i.e., along the angular momentum vector. Thus the magnetism exhibited by the atoms provides an experimental means by which we may study the direction of the angular momentum vector.

If we place the atoms in a magnetic field they will be attracted or repelled by this field, depending on whether or not the atomic magnets are aligned against or with the applied field. The applied magnetic field will determine a direction in space. By measuring the deflection of the atoms in this field we can determine the directions of their magnetic moments and hence of their angular momentum vectors with respect to this applied field. Consider an evacuated tube with a tiny opening at one end through which a stream of atoms may enter (Fig. 3-12). By placing a second small hole in front of the first, inside the tube, we will obtain a narrow beam of atoms which will pass the length of the tube and strike the opposite end. If the atoms possess magnetic moments the

(5)



path of the beam can be deflected by placing a magnetic field across the tube, perpendicular to the path of the atoms. The magnetic field must be one in which the lines of force diverge thereby exerting an unbalanced force on any magnetic material lying inside the field. This inhomogeneous magnetic field could be obtained through the use of N and S poles of the kind illustrated in Fig. 3-12. The direction of the magnetic field will be taken as the direction of the z-axis.

Fig. 3-12. The atomic beam apparatus.

Let us suppose the beam consists of neutral atoms which possess units of electronic angular momentum (the angular momentum quantum number l = 1). When no magnetic field is present, the beam of atoms strikes the end wall at a single point in the middle of the detector. What happens when the magnetic field is present? We must assume that before the beam enters the magnetic field, the axes of the atomic magnets are randomly oriented with respect to the z-axis. According to the concepts of classical mechanics, the beam should spread out along the direction of the magnetic field and produce a line rather than a point at the end of the tube (Fig. 3-13a). Actually, the beam is split into three distinct component beams each of equal intensity producing three spots at the end of the tube (Fig. 3-13b).

Fig. 3-13. (a) The result of the atomic beam experiment as predicted by classical mechanics,

(b) The observed result of the atomic beam experiment.

The startling results of this experiment can be explained only if we assume that while in the magnetic field each atomic magnet could assume only one of three possible orientations with respect



to the applied magnetic field (Fig. 3-14).

Fig. 3-14. The three possible orientations for the total magnetic moment with respect to an external magnetic field for an

atom with l =1.

The atomic magnets which are aligned perpendicular to the direction of the field are not deflected and will follow a straight path through the tube. The atoms which are attracted upwards must have their magnetic moments oriented as shown. From the known strength of the applied inhomogeneous magnetic field and the measured distance through which the beam has been deflected upwards, we can determine that the component of the magnetic moment lying along the z-axis is only β

m in

magnitude rather than the value of This latter value would result if the axis of the atomic magnet was parallel to the z-axis, i.e., the angle θ = 0°. Instead θ assumes a value such that the component of the total moment lying along the z-axis is just lβm. Similarly the beam which is deflected downwards possesses a magnetic moment along the z-axis of -βm or -lβm. The classical prediction for this experiment assumes that θ may equal all values from 0° to 180°, and thus

all values (from a maximum of (θ = 0°) to 0 (θ =90°) to (θ = 180°)) for the component of the atomic moment along the z-axis would be possible. Instead, θ is found to equal only those values such that the magnetic moment along the z-axis equals +βm, 0 and -βm.

The angular momentum of the electron determines the magnitude and the direction of the magnetic dipole. (Recall that the vectors for both these quantities lie along the same axis.) Thus the number of possible values which the component of the angular momentum vector may assume along a given axis must equal the number of values observed for the component of the magnetic dipole along the same axis. In the present example the values of the angular momentum component are +1



(h/2π), 0 and -1(h/2π), or since l = 1 in this case, + l(h/2π), 0 and -l(h/2π). In general, it is found that the number of observed values is always (2l + 1) the values being:

for the angular momentum and:

for the magnetic dipole. The number governing the magnitude of the component of M and , ranges from a maximum value of l and decreases in steps of unity to a minimum value of -l. This number is the third and final quantum number which determines the motion of an electron in a hydrogen atom. It is given the symbol m and is called the magnetic quantum number.

In summary, the angular momentum of an electron in the hydrogen atom is quantized and may assume only those values given by:

Furthermore, it is an experimental fact that the component of the angular momentum vector along a given axis is limited to (21 + 1) different values, and that the magnitude of this component is quantized and governed by the quantum number m which may assume the values l, l-1, . . .,0, . . .,-l. These facts are illustrated in Fig. 3-15 for an electron in a d orbital in which l = 2.



(a)

(b)

Fig. 3-15. Pictorial representation of the quantum mechanical properties of the angular momentum of a d electron for which l = 2. The z-axis can be along any arbitrary direction in space. Figure (a) shows the possible components which the

angular momentum vector (of length ) may exhibit along an arbitrary axis in space. A d electron may possess any one of these components. There are therefore five states for a d electron, all of which are physically different. Notice that the maximum magnitude allowed for the component is less then the magnitude of the total angular momentum. Therefore, the angular momentum vector can never coincide with the axis with respect to which the observations are made. Thus the x and y components of the angular momentum are not zero. This is illustrated in Fig. (b) which shows how the angular momentum vector may be oriented with respect to the z-axis for the case m = l = 2. When the atom is in a magnetic field, the field exerts a torque on the magnetic dipole of the atom. This torque causes the magnetic dipole and hence the angular momentum vector to precess or rotate about the direction of the magnetic field. This effect is analogous to the precession of a child's top which is spinning with its axis (and hence its angular momentum vector) at an angle to the earth's gravitational field. In this case the gravitational field exerts the torque and the axis of the top slowly revolves around the perpendicular direction as indicated in the figure. The angle of inclination of M with respect to the field direction remains constant during the precession. The z-component of M is therefore constant but the x and y components are continuously changing. Because of the precession, only one component of the electronic angular momentum of an atom an be determined in a given experiment.



The quantum number m determines the magnitude of the component of the angular momentum along a given axis in space. Therefore, it is not surprising that this same quantum number determines the axis along which the electron density is concentrated. When m = 0 for a p electron (regardless of the n value, 2p, 3p, 4p, etc.) the electron density distribution is concentrated along the z-axis (see Fig. 3-10) implying that the classical axis of rotation must lie in the x-y plane. Thus a p electron with m = 0 is most likely to be found along one axis and has a zero probability of being on the remaining two axes. The effect of the angular momentum possessed by the electron is to concentrate density along one axis. When m = 1 or -1 the density distribution of a p electron is concentrated in the x-y plane with doughnut-shaped circular contours. The m = 1 and -1 density distributions are identical in appearance. Classically they differ only in the direction of rotation of the electron around the z-axis; counter-clockwise for m = +1 and clockwise for m = -1. This explains why they have magnetic moments with their north poles in opposite directions.

We can obtain density diagrams for the m = +1 and -1 cases similar to the m = 0 case by removing the resultant angular momentum component along the z-axis. We can take combinations of the m = +1 and -1 functions such that one combination is concentrated along the x-axis and the other along the y-axis, and both are identical to the m = 0 function in their appearance. Thus these functions are often labelled as px, py and pz functions rather than by their m values. The m value is, however, the true quantum number and we are cheating physically by labelling them px, py and pz . This would correspond to applying the field first in the z direction, then in the x direction and finally in the y direction and trying to save up the information each time. In reality when the direction of the field is changed, all the information regarding the previous direction is lost and every atom will again align itself with one chance out of three of being in one of the possible component states with respect to the new direction.

We should note that the r dependence of the orbitals changes with changes in n or l, but the directional component changes with l and m only. Thus all s orbitals possess spherical charge distributions and all p orbitals possess dumb-bell shaped charge distributions regardless of the value of n.

Table 3-1. The Atomic Orbitals for the Hydrogen Atom

En nl m Symbol for orbital-K 10 0 1s

20 0 2s21 1 2p+1 ⎞21 0 2p0 ⎬px, py, pz21 -1 2p-1 ⎭

30 0 3s31 1 3p+1 ⎞31 0 3p0 ⎬px, py, pz31 -1 3p-1 ⎭32 2 3d+2 ⎞32 1 3d+1 |32 0 3d0 ⎬32 -1 3d-1 |32 -2 3d-2 ⎭



Table 3-1 summarizes the allowed combinations of quantum numbers for an electron in a hydrogen atom for the first few values of n; the corresponding name (symbol) is given for each orbital. Notice that there are n2 orbitals for each value of n, all of which belong to the same quantum level and have the same energy. There are n - 1 values of l for each value of n and there are (2l + 1) values of m for each value of l. Notice also that for every increase in the value of n, orbitals of the same l value (same directional dependence) as found for the preceding value of n are repeated. In addition, a new value of l and a new shape are introduced. Thus there is a repetition in the shapes of the density distributions along with an increase in their number. We can see evidence of a periodicity in these functions (a periodic re-occurrence of a given density distribution) which we might hope to relate to the periodicity observed in the chemical and physical properties of the elements. We might store this idea in the back of our minds until later.

We can summarize what we have found so far regarding the energy and distribution of an electron in a hydrogen atom thus:

Some words of caution about energies and angular momentum should be added. In passing from the domain of classical mechanics to that of quantum mechanics we retain as many of the familiar words as possible. Examples are kinetic and potential energies, momentum, and angular momentum. We must, however, be on guard when we use these familiar concepts in the atomic domain. All have an altered meaning. Let us make this clear by considering these concepts for the hydrogen atom.

Perhaps the most surprising point about the quantum mechanical expression for the energy is that it does not involve r, the distance between the nucleus and the electron. If the system were a classical one, then we would expect to be able to write the total energy En as:

Both the KE and PE would be functions of r, i.e., both would change in value as r was changed (corresponding to the motion of the electron). Furthermore, the sum of the PE and KE must always yield the same value of En which is to remain constant.

(i) The energy increases as n increases, and depends only on n, the principal quantum number.(ii) The average value of the distance between the electron and the nucleus increases as n increases.(iii) The number of nodes in the probability distribution increases as n increases.(iv) The electron density becomes concentrated along certain lines (or in planes) as l is increased.

(6)



Fig.3-16. The potential energy diagram for an H atom with one of the allowed energy values superimposed on it.

Fig 3-16 is the potential energy diagram for the hydrogen atom and we have superimposed on it one of the possible energy levels for the atom, En. Consider a classical value for r at the point A". Classically, when the electron is at the point A", its PE is given by the value of the PE curve at A'. The KE is thus equal to the length of the line A - A' in energy units. Thus the sum of PE + KE adds up to En.

When the electron is at the point B", its PE would equal En and its KE would be zero. The electron would be motionless. Classically, for this value of En the electron could not increase its value of r beyond the point represented by B". If it did, it would be inside the "potential wall." For example, consider the point C". At this value of r, the PE is given by the value at C' which is now greater than En and hence the KE must be equal to the length of the line C - C'. But the KE must now be negative in sign so that the sum of PE and KE will still add up to En. What does a negative KE mean? It doesn't mean anything as it never occurs in a classical system. Nor does it occur in a quantum mechanical system. It is true that quantum mechanics does predict a finite probability for the electron being inside the potential curve and indeed for all values of r out to infinity. However, the quantum mechanical expression for En does not allow us to determine the instantaneous values for the PE and KE. Instead, we can determine only their average values. Thus quantum mechanics does not give equation (6) but instead states only that the average potential and kinetic energies may be known:

A bar denotes the fact that the energy quantity has been averaged over the complete motion (all values of r) of the electron.

Why can r not appear in the quantum mechanical expression for En, and why can we obtain only

(7)



average values for the KE and PE? When the electron is in a given energy level its energy is precisely known; it is En. The uncertainty in the value of the momentum of the electron is thus at a minimum. Under these conditions we have seen that our knowledge of the position of the electron is very uncertain and for an electron in a given energy level we can say no more about its position than that it is bound to the atom. Thus if the energy is to remain fixed and known with certainty, we cannot, because of the uncertainty principle, refer to (or measure) the electron as being at some particular distance r from the nucleus with some instantaneous values for its PE and KE. Instead, we may have knowledge of these quantities only when they are averaged over all possible positions of the electron. This discussion again illustrates the pitfalls (e.g., a negative kinetic energy) which arise when a classical picture of an electron as a particle with a definite instantaneous position is taken literally. (Click here for note.)

It is important to point out that the classical expressions which we write for the dependence of the potential energy on distance, -e2/r for the hydrogen atom for example, are the expressions employed in the quantum mechanical calculation. However, only the average value of the PE may be calculated and this is done by calculating the value of -e2/r at every point in space, taking into account the fraction of the total electronic charge at each point in space. The amount of charge at a given point in three-dimensional space is, of course, determined by the electron density distribution. Thus the value of for the ground state of the hydrogen atom is the electrostatic energy of interaction between a nucleus of charge +1e with the surrounding spherical distribution of negative charge.

We can say more about the and for an electron in an atom. Not only are these values constant for a given value of n, but also for any value of n,

Thus the is always positive and equal to minus one half of the . Since the total energy En is negative when the electron is bound to the atom, we can interpret the stability of atoms as being due to the decrease in the when the electron is attracted by the nucleus.

The question now arises as to why the electron doesn't "fall all the way" and sit right on the nucleus. When r = 0, the would be equal to minus infinity, and the , which is positive and thus destabilizing, would be zero. Classically this would certainly be the situation of lowest energy and thus the most stable one. The reason for the electron not collapsing onto the nucleus is a quantum mechanical one. If the electron was bound directly to the nucleus with no kinetic energy, its position and momentum would be known with certainty. This would violate Heisenberg's uncertainty principle. The uncertainty principle always operates through the kinetic energy causing it to become large and positive as the electron is confined to a smaller region of space. (Recall that in the example of an electron moving on a line, the increased as the length of the line decreased.) The smaller the region to which the electron is confined, the smaller is the uncertainty in its position. There must be a corresponding increase in the uncertainty of its momentum. This is brought about by the increase in the kinetic energy which increases the magnitude of the momentum and thus the uncertainty in its value. In other words the bound electron must always possess kinetic energy as a consequence of quantum mechanics.

The and have opposite dependences on . The decreases (becomes more negative) as decreases but the increases (making the atom less stable) as decreases. A compromise is



reached to make the energy as negative as possible (the atom as stable as possible) and the

compromise always occurs when . A further decrease in would decrease the but only at the expense of a larger increase in the . The reverse is true for an increase in . Thus the reason the electron doesn't fall onto the nucleus may be summed up by stating that "the electron obeys quantum mechanics, and not classical mechanics."



Some Useful expressions

Listed below are a number of equations which give the dependence of , and on the quantum numbers n, l and m. They refer not only to the hydrogen atom but also to any one-electron ion in general with a nuclear charge of Z. Thus He+ is a one-electron ion with Z = 2, Li+2 another example with Z = 3.

The average distance between the electron and the nucleus expressed in atomic units of length is:

Note that is proportional to n2 for l = 0 orbitals, and deviates only slightly from this for l ≠ 0.

The value of decreases as Z increases because the nuclear attractive force is greater. Thus

for He+ would be only one half as large as for H.

Page 1 of 1The Hydrogen Atom - Some Usesful Expression


Problems 1. In 1913 Niels Bohr proposed a model for the hydrogen atom which gives the correct

expression for the energy levels En. His model was based on an awkward marriage of classical

mechanics and, at that time, the new idea of quantization. In the Bohr model of the hydrogen atom the electron is assumed to move in a circular orbit around the nucleus, as illustrated in Fig. 3-7. The energy of the electron in such an orbit is:

(1)

where υ is the tangential velocity of the electron in the orbit. Since the circular orbit is to be a stable one the attractive coulomb force exerted on the electron by the nucleus must be balanced by a centrifugal force, or:

(2)

where ω is the circular velocity of the electron. Up until this point the model is completely classical in concept. However, Bohr now postulated that only those orbits are allowed for which the angular momentum is an integral multiple of (h/2π). In other words, Bohr postulated that the angular momentum of the electron in the hydrogen atom is quantized. This postulate gives a further equation:

(3)

(a)Show that by eliminating r and υ from these three equations you can obtain the correct expression for E

n.

(b) Show that Bohr's model correctly predicts that the KE = -½PE.(c) Show that the radius of the first Bohr orbit is identical to the maximum value of r for the n

= 1 level of the hydrogen atom, , as calculated by quantum mechanics.(d) Criticize the Bohr model in the light of the quantum mechanical results for the hydrogen

atom.2. The part of the hydrogen atom spectrum which occurs in the visible region arises from

electrons in excited levels falling to the n = 2 level. The quantum mechanical expression for the frequencies in this case, corresponding to equation (3) of the text for the Lyman series, is:

(4)

The energy of an emitted photon for a jump from level n to level 2 is:

(5)

Equation (5) predicts that a plot of the photon energies versus (1/n2) should be a straight line. Furthermore, it predicts that the intercept of this line with the energy axis, corresponding to the value of 1/n2 = 0, i.e., n = ∞, should equal (¼)K where:

(6)

The point of this problem is to test these theoretical predictions against the experimental results.

Page 1 of 2The Hydrogen Atom - Problems

4/20/2008http://www.chemistry.mcmaster.ca/esam/Chapter_3/problems.html

Experimentally we measure the wavelength of the emitted light by means of a diffraction grating. A grating for the diffraction of visible light may be made by marking a glass plate with parallel, equally spaced lines. There are about 10,000 lines per cm. The spacings between the lines in the grating d is thus about 1 ×10-4 cm which is the order of magnitude of the wavelength of visible light. The diffraction equation is:(7)as previously discussed in Problem I-1. We measure the angle θ for different orders n = 1, 2, 3, ... of the diffracted light beam. Since d is known, λ may be calculated. The experimentally measured values for the first four lines in the Balmer series are given below.

Balmer Series

The value of the principal quantum number n which appears in equation (5) is given for each value of λ. (This n is totally unrelated to the n of equation (7) for the experimental determination of λ.) Calculate the energy of each photon from the value of its wavelength.

Plot the photon energies versus the appropriate value of 1/n2. Let the 1/n2 axis run from 0 to 0.25 and the energy axis run from 0 to 3.6 ev. Include as a point on your graph ε = 0 for 1/n2 = 0.25, i.e., when n = 2, the excited level and the level to which the electron falls coincide.

λ(Å) n 6563 3 4861 4 4341 5 4102 6

(a) Do the points fall on a straight line as predicted?(b) Determine the value of K by extending the line to intercept the energy axis. This intercept

should equal K/4. Read off this value from your graph.(c) Compare the experimental value for K with that predicted theoretically by equation (5).

Use e = 4.803 × 10-10 esu, express m and h in cgs units and the value of K will be in ergs (1 erg = 6.2420 × 1011 ev). Recall that K is the ionization potential for the hydrogen atom. An electron falling from the n = ∞ level to the n = 2 level will fall only (¼)K in energy as is evident from the energy level diagram shown in Fig. 3-2.

3. A beam of atoms with l = 1 is passed through an atomic beam apparatus with the magnetic field directed along an axis perpendicular to the direc tion of the beam. The undeflected beam from this experiment enters a second beam apparatus in which the magnetic field is directed along an axis which is perpendicular to both the path of the beam and the direction of the field in the first experiment. Will this one component of the original beam be split in the second applied magnetic field? Explain why you think it will be, if this is indeed your answer.

Page 2 of 2The Hydrogen Atom - Problems

4/20/2008http://www.chemistry.mcmaster.ca/esam/Chapter_3/problems.html

RREEAADDIINNGG ##1100

Chapter 2

The Hydrogen atom

In the previous chapter we gave a quick overview of the Bohr model, which is only really valid in thesemiclassical limit. (cf. section 1.7.) We now begin our task in earnest by applying quantum mechanicsto the simplest atom we know, namely the hydrogen atom.

It is well known from classical physics that planetary orbits are characterized by their energy andangular momentum. In this chapter we apply the Schrodinger equation to the hydrogen atom to find theallowed energies and angular momenta of the nucleus-electron system. In classical systems we are alsoable to calculate the precise trajectory of the orbit. This is not possible in quantum systems. The bestwe shall be able to do is to find the wave functions. These will then give us the probability amplitudesthat allow us to calculate all the measurable properties of the system.

2.1 The Schrodinger Equation

The time-independent Schrodinger equation for hydrogen is given by:(− h2

2m∇2 − Ze2

4πε0r

)Ψ(r, θ, φ) = E Ψ(r, θ, φ) . (2.1)

This is written in terms of the spherical polar co-ordinates (r, θ, φ) because atoms are spheres, and theuse of spherical polar co-ordinates simplifies the solutions. Note that we are considering the motion ofthe electron relative to a stationary nucleus here. As with all two-body problems, this means that themass that enters into the equation is the reduced mass defined previously in eqn 1.9:

1m

=1

me+

1mN

. (2.2)

For hydrogen where mN = mp, the reduced mass is very close to me, and has a value of 0.9995me.Written out explicitly, we have

− h2

2m

[1r2

∂

∂r

(r2 ∂Ψ

∂r

)+

1r2 sin θ

∂

∂θ

(sin θ

∂Ψ∂θ

)+

1r2 sin2 θ

∂2Ψ∂φ2

]− Ze2

4πε0rΨ = E Ψ (2.3)

Our task is to find the wave functions Ψ(r, θ, φ) that satisfy this equation, and hence to find the allowedquantized energies E.

2.2 Angular momentum

The classical definition of angular momentum is:

L = r × p . (2.4)

For circular orbits this simplifies to L = mvr, and in Bohr’s model, L was quantized in integer units of h.(See eqn 1.7.) However, the full quantum treatment is more complicated, and requires the introductionof two other quantum numbers l and ml, as we shall now see.

The components of L are given by

Lx

Ly

Lz

=

xyz

×

px

py

pz

=

ypz − zpy

zpx − xpz

xpy − ypx

. (2.5)

9

10 CHAPTER 2. THE HYDROGEN ATOM

In quantum mechanics we represent the linear momentum by differential operators of the type

px = −ih∂

∂x. (2.6)

Therefore, the quantum mechanical operators for the angular momentum are given by:

Lx =h

i

(y

∂

∂z− z

∂

∂y

)(2.7)

Ly =h

i

(z

∂

∂x− x

∂

∂z

)(2.8)

Lz =h

i

(x

∂

∂y− y

∂

∂x

). (2.9)

Note that the “hat” symbol indicates that we are representing an operator and not just a number.The magnitude of the angular momentum is given by:

L2 = L2x + L2

y + L2z .

We therefore define the quantum mechanical operator L2

by

L2

= L2x + L2

y + L2z . (2.10)

Note that operators like L2x should be understood in terms of repeated operations:

L2xψ = −h2

(y

∂

∂z− z

∂

∂y

)(y∂ψ

∂z− z

∂ψ

∂y

)

= −h2

(y2 ∂2ψ

∂z2− y

∂ψ

∂y− z

∂ψ

∂z− 2yz

∂2ψ

∂y∂z+ z2 ∂2ψ

∂y2

).

It can be shown that the components of the angular momentum operator do not commute, that is

LxLy 6= LyLx .

In fact we can show that:[Lx, Ly] = ihLz , (2.11)

where the “commutator bracket” [Lx, Ly] is defined by

[Lx, Ly] = LxLy − LyLx . (2.12)

The other commutators of the angular momentum operators, namely [Ly, Lz] and [Lz, Lx] are obtainedby cyclic permutation of the indices in Eq. 2.11: x → y, y → z, z → x.

This rather esoteric point has deep significance. If two quantum mechanical operators do not commute,then it is not possible to know their values simultaneously. Consider, for example, the operators forposition and momentum in a one-dimensional system:

[x, p]ψ = (xp− px)ψ = −ih x

(dψ

dx

)+ ih

d(xψ)dx

= ihψ .

Thus we have:[x, p] = ih 6= 0 . (2.13)

The fact that [x, p] 6= 0 means that the operators do not commute. This is intrinsically linked to the factthat we cannot measure precise values for the position and momentum simultaneously, which we knowfrom the Heisenberg uncertainty principle. The argument based on commutators is thus a more formalway of understanding uncertainty products.

In the case of the angular momentum operators, the fact that Lx, Ly and Lz do not commute meansthat we can only know one of the components of L at any time. If we know the value of Lz, we cannotknow Lx and Ly as well. However, we can know the length of the angular momentum vector, because wecan show that L2 and Lz commute. In summary:

2.2. ANGULAR MOMENTUM 11

z

Lz = mlh

h)1(|| += llL

x,y

z

Lz = mlh

h)1(|| += llL

x,y

Figure 2.1: Vector model of the angular momentum in an atom. The angular momentum is representedby a vector of length

√l(l + 1)h precessing around the z-axis so that the z-component is equal to mlh.

• We can know the length of the angular momentum vector L and one of its components.

• For mathematical convenience, we usually take the component we know to be the z component, ieLz.

• We cannot know the values of all three components of the angular momentum simultaneously.

This is represented pictorially in the vector model of the atom shown in figure 2.1. In this model theangular momentum is represented as a vector of length

√l(l + 1)h precessing around the z axis so that

the component along that axis is equal to mlh. The x and y components of the angular momentum arenot known.

In spherical polar co-ordinates, the two key angular momentum operators are given by:

Lz =h

i

∂

∂φ(2.14)

and

L2

= −h2

[1

sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1sin2 θ

∂2

∂φ2

]. (2.15)

It is easy to see that the Schrodinger equation given in Eq. 2.3 can be re-written as follows:

− h2

2m

1r2

∂

∂r

(r2 ∂Ψ

∂r

)+

L2

2mr2Ψ− Ze2

4πε0rΨ = E Ψ . (2.16)

The eigenfunctions of the angular momentum operator are found by solving the equation:

L2F (θ, φ) ≡ −h2

[1

sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1sin2 θ

∂2

∂φ2

]F (θ, φ) = CF (θ, φ) . (2.17)

For reasons that will become clearer later, the constant C is usually written in the form:

C = l(l + 1)h2 . (2.18)

At this stage, l can take any value, real or complex. We can separate the variables by writing:

F (θ, φ) = Θ(θ)Φ(φ) . (2.19)

On substitution into eqn 2.17 and cancelling the common factor of h2, we find:

− 1sin θ

ddθ

(sin θ

dΘdθ

)Φ− 1

sin2 θΘ

d2Φdφ2

= l(l + 1)ΘΦ . (2.20)

Multiply by − sin2 θ/ΘΦ and re-arrange to obtain:

sin θ

Θddθ

(sin θ

dΘdθ

)+ sin2 θ l(l + 1) = − 1

Φd2Φdφ2

. (2.21)


The left hand side is a function of θ only, while the right hand side is a function of φ only. The equationmust hold for all values of the θ and φ and hence both sides must be equal to a constant. On writingthis arbitrary separation constant m2, we then find:1

sin θddθ

(sin θ

dΘdθ

)+ l(l + 1) sin2 θ Θ = m2Θ , (2.22)

andd2Φdφ2

= −m2Φ . (2.23)

The equation in φ is easily solved to obtain:

Φ(φ) = Aeimφ . (2.24)

The wave function must have a single value for each value of φ, and hence we require:

Φ(φ + 2π) = Φ(φ) , (2.25)

which requires that the separation constant m must be an integer. Using this fact in eqn 2.22, we thenhave to solve

sin θddθ

(sin θ

dΘdθ

)+ [l(l + 1) sin2 θ −m2] Θ = 0 , (2.26)

with the constraint that m must be an integer. On making the substitution u = cos θ and writingΘ(θ) = P (u), eqn 2.26 becomes:

ddu

((1− u2)

dP

du

)+

[l(l + 1)− m2

1− u2

]P = 0 . (2.27)

Equation 2.27 is known as either the Legendre equation or the associated Legendre equation, dependingon whether m is zero or not. Solutions only exist if l is an integer ≥ |m| and P (u) is a polynomial functionof u. This means that the solutions to eqn 2.26 are of the form:

Θ(θ) = Pml (cos θ) , (2.28)

where Pml (cos θ) is a polynomial function in cos θ called the (associated) Legendre polynomial function.

Putting this all together, we then find:

F (θ, φ) = normalization constant× Pml (cos θ) eimφ , (2.29)

where m and l are integers, and m can have values from −l to +l. The correctly normalized functionsare called the spherical harmonic functions Yl,m(θ, φ).

It is apparent from eqns 2.17 and 2.18 that the spherical harmonics satisfy:

L2Yl,m(θ, φ) = l(l + 1)h2Yl,m(θ, φ) . (2.30)

Furthermore, on substituting from eqn 2.14, it is also apparent that

LzYl,m(θ, φ) = mhYl,m(θ, φ) . (2.31)

The integers l and m that appear here are called the orbital and magnetic quantum numbers respectively.Some of the spherical harmonic functions are listed in Table 2.1. Equations 2.30–2.31 show that themagnitude of the angular momentum and its z-component are equal to

√l(l + 1)h and mh respectively,

as consistent with Fig. 2.1.The spherical harmonics have the property that:

∫ π

θ=0

∫ 2π

φ=0

Y ∗l,m(θ, φ)Yl′,m′(θ, φ) sin θ dθdφ = δl,l′δm,m′ . (2.32)

The symbol δk,k′ is called the Kronecker delta function. It has the value of 1 if k = k′ and 0 if k 6= k′.The sin θ factor in Eq. 2.32 comes from the volume increment in spherical polar co-ordinates: see Eq. 2.47below.

1Be careful not to confuse the magnetic quantum number m with the reduced mass that has the same symbol! Notealso that a subscript l is often added (i.e. ml) to distinguish it from the quantum number for the z-component of the spin(ms).

2.3. SEPARATION OF VARIABLES IN THE SCHRODINGER EQUATION 13

l m Yl,m(θ, φ)

0 0√

14π

1 0√

34π cos θ

1 ±1 ∓√

38π sin θe±iφ

2 0√

516π (3 cos2 θ − 1)

2 ±1 ∓√

158π sin θ cos θe±iφ

2 ±2√

1532π sin2 θe±2iφ

Table 2.1: Spherical harmonic functions.

2.3 Separation of variables in the Schrodinger equation

The Coulomb potential is an example of a central field. This means that the force only lies along theradial direction. This allows us separate the motion into the radial and angular parts. Hence we write:

Ψ(r, θ, φ) = R(r)Y (θ, φ) . (2.33)

On substituting this into Eq. 2.16, we find

− h2

2m

1r2

ddr

(r2 dR

dr

)Y + R

L2Y

2mr2− Ze2

4πε0rRY = E RY . (2.34)

Multiply by r2/RY and re-arrange to obtain:

− h2

2m

1R

ddr

(r2 dR

dr

)− Ze2r

4πε0− Er2 = − 1

Y

L2Y

2m. (2.35)

The left hand side is a function of r only, while the right hand side is only a function of the angularco-ordinates θ and φ. The only way this can be true is if both sides are equal to a constant. Let’s callthis constant −h2`(` + 1)/2m, where ` is an arbitrary number at this stage. This gives us, after a bit ofre-arrangement:

− h2

2m

1r2

ddr

(r2 dR(r)

dr

)+

h2`(` + 1)2mr2

R(r)− Ze2

4πε0rR(r) = ER(r) , (2.36)

andL

2Y (θ, φ) = h2`(` + 1)Y (θ, φ) . (2.37)

On comparing Eqs. 2.30 and 2.37 we can now identify the arbitrary separation constant ` with the angularmomentum quantum number l, and we can see that the function Y (θ, φ) that enters Eq. 2.37 must beone of the spherical harmonics.

We can tidy up the radial equation Eq. 2.36 by writing:

R(r) =P (r)

r.

This gives: [− h2

2m

d2

dr2+

h2l(l + 1)2mr2

− Ze2

4πε0r

]P (r) = EP (r) . (2.38)

This now makes physical sense. It is a Schrodinger equation of the form:

HP (r) = EP (r) , (2.39)


0 2 4 6 8 100

2

4

6

R10

(r)

(Å–3

/2)

radius (Å)

n = 1l = 0

0 2 4 6 8 10

0

1

2

R2l

(r)

(Å–3

/2)

radius (Å )

n = 2

l = 1

l = 0

0 2 4 6 8 10 12 14

0

1

R3l

(r)

(Å–3

/2)

radius (Å )

n = 3l = 0

l = 1

l = 2

0 2 4 6 8 100

2

4

6

R10

(r)

(Å–3

/2)

radius (Å)

n = 1l = 0

0 2 4 6 8 10

0

1

2

R2l

(r)

(Å–3

/2)

radius (Å )

n = 2

l = 1

l = 0

0 2 4 6 8 10 12 14

0

1

R3l

(r)

(Å–3

/2)

radius (Å )

n = 3l = 0

l = 1

l = 2

Figure 2.2: The radial wave functions Rnl(r) for the hydrogen atom with Z = 1. Note that the axes forthe three graphs are not the same.

where the energy operator H (i.e. the Hamiltonian) is given by:

H = − h2

2m

d2

dr2+ Veffective(r) . (2.40)

The first term in eqn 2.40 is the radial kinetic energy given by

K.E.radial =p2

r

2m= − h2

2m

d2

dr2.

The second term is the effective potential energy:

Veffective(r) =h2l(l + 1)

2mr2− Ze2

4πε0r, (2.41)

which has two components. The first of these is the orbital kinetic energy given by:

K.E.orbital =L2

2I=

h2l(l + 1)2mr2

,

where I ≡ mr2 is the moment of inertia. The second is the usual potential energy due to the Coulombenergy.

This analysis shows that the quantized orbital motion adds quantized kinetic energy to the radialmotion. For l > 0 the orbital kinetic energy will always be larger than the Coulomb energy at small r,and so the effective potential energy will be positive. This has the effect of keeping the electron awayfrom the nucleus, and explains why states with l > 0 have nodes at the origin (see below).

2.4 The wave functions and energies

The wave function we require is given by Eq. 2.33. We have seen above that the Y (θ, φ) function thatappears in Eq. 2.33 must be one of the spherical harmonics, some of which are listed in Table 2.1. Theradial wave function R(r) can be found by solving the radial differential equation given in Eq. 2.36. Themathematics is somewhat complicated and is considered in Section 2.5. Here we just quote the mainresults.

Solutions are only found if we introduce an integer quantum number n. The energy depends onlyon n, but the functional form of R(r) depends on both n and l, and so we must write the radial wavefunction as Rnl(r). A list of some of the radial functions is given in Table 2.2. Representative wavefunctions are plotted in Fig. 2.2.

We can now write the full wave function as:

Ψnlm(r, θ, φ) = Rnl(r)Ylm(θ, φ) . (2.42)

The quantum numbers must obey the following rules:

• n can have any integer value ≥ 1.

• l can have integer values up to (n− 1).

2.4. THE WAVE FUNCTIONS AND ENERGIES 15

n l Rnl(r)

1 0 (Z/a0)32 2 exp(−Zr/a0)

2 0 (Z/2a0)32 2

(1− Zr

2a0

)exp(−Zr/2a0)

2 1 (Z/2a0)32 2√

3

(Zr2a0

)exp(−Zr/2a0)

3 0 (Z/3a0)32 2

[1− (2Zr/3a0) + 2

3

(Zr3a0

)2]

exp(−Zr/3a0)

3 1 (Z/3a0)32 (4√

2/3)(

Zr3a0

)(1− 1

2Zr3a0

)exp(−Zr/3a0)

3 2 (Z/3a0)32 (2√

2/3√

5)(

Zr3a0

)2

exp(−Zr/3a0)

Table 2.2: Radial wave functions of the hydrogen atom. a0 is the Bohr radius (5.29 × 10−11 m). Thewave functions are normalized so that

∫∞r=0

R∗nlRnlr2dr = 1.

z

l = 0m = 0

z

l = 1m = 0

m = ±1

z

l = 2m = 0

m = ±1

m = ±2

z

l = 0m = 0

z

l = 0m = 0

z

l = 1m = 0

m = ±1

z

l = 1m = 0

m = ±1

z

l = 2m = 0

m = ±1

m = ±2

z

l = 2m = 0

m = ±1

m = ±2

z

l = 2m = 0

m = ±1

m = ±2

Figure 2.3: Polar plots of the spherical harmonics with l ≤ 2. The plots are to be imagined withspherical symmetry about the z axis. In these polar plots, the value of the function for a given an-gle θ is plotted as the distance from the origin. Prettier pictures may be found, for example, at:http://mathworld.wolfram.com/SphericalHarmonic.html.

• m can have integer values from −l to +l.

These rules drop out of the mathematical solutions. Functions that do not obey these rules will notsatisfy the Schrodinger equation for the hydrogen atom.

The radial wave functions listed in Table 2.2 are of the form:

Rnl(r) = Cnl · (polynomial in r) · e−r/a , (2.43)

where a = naH/Z, aH being the Bohr radius of Hydrogen, namely 5.29× 10−11 m. Cnl is a normalizationconstant. The polynomial functions that drop out of the equations are polynomials of order n − 1, andhave n− 1 nodes. If l = 0, all the nodes occur at finite r, but if l > 0, one of the nodes is at r = 0.

The angular part of the wave function is of the form (see eqn 2.29 and Table 2.1):

Yl,m(θ, φ) = C ′lm · Pml (cos θ) · eimφ , (2.44)

where Pml (cos θ) is a Legendre polynomial, e.g. P 1

1 (cos θ) = constant, P 01 (cos θ) = cos θ, etc. C ′lm is

another normalization constant. Representative polar wave functions are shown in figure 2.3.The energy of the system is found to be:

En = −mZ2e4

8ε20h2

1n2

, (2.45)

which is the same as the Bohr formula given in Eq. 1.10. Note that this depends only on the principalquantum number n: all the l states for a given value of n are degenerate (i.e. have the same energy),


even though the radial wave functions depend on both n and l. This degeneracy with respect to l is called“accidental”, and is a consequence of the fact that the electrostatic energy has a precise 1/r dependencein hydrogen. In more complex atoms, the electrostatic energy will depart from a pure 1/r dependencedue to the shielding effect of inner electrons. In this case, the gross energy depends on l as well as n,even before we start thinking of higher order fine structure effects. We shall see how this works in moredetail when we consider the alkali atoms later.

The wave functions are nomalized so that∫ ∞

r=0

∫ π

θ=0

∫ 2π

φ=0

Ψ∗n,l,mΨn′,l′m′ dV = δn,n′δl,l′δm,m′ (2.46)

where dV is the incremental volume element in spherical polar co-ordinates:

dV = r2 sin θ drdθdφ . (2.47)

The radial probability function Pnl(r) is the probability that the electron is found between r and r + dr:

Pnl(r) dr =∫ π

θ=0

∫ 2π

φ=0

Ψ∗Ψ r2 sin θdrdθdφ

= |Rnl(r)|2 r2 dr . (2.48)

The factor of r2 that appears here is just related to the surface area of the radial shell of radius r (i.e.4πr2.) Some representative radial probability functions are sketched in Fig. 2.4. 3-D plots of the shapesof the atomic orbitals are available at: http://www.shef.ac.uk/chemistry/orbitron/.

Expectation values of measurable quantities are calculated as follows:

〈A〉 =∫ ∫ ∫

Ψ∗AΨdV . (2.49)

Thus, for example, the expectation value of the radius is given by:

〈r〉 =∫ ∫ ∫

Ψ∗rΨdV

=∫ ∞

r=0

R∗nlrRnlr2dr

∫ π

θ=0

sin θdθ

∫ 2π

φ=0

dφ

=∫ ∞

r=0

R∗nlrRnlr2dr . (2.50)

This gives:

〈r〉 =n2aH

Z

(32− l(l + 1)

2n2

). (2.51)

Note that this only approaches the Bohr value, namely n2aH/Z (see eqn 1.15), for the states with l = n−1at large n.

Reading

Demtroder, W., Atoms, Molecules and Photons, §4.3 – §5.1.Haken, H. and Wolf, H.C., The Physics of Atoms and Quanta, chapter 10.Phillips, A.C., Introduction to Quantum Mechanics, chapters 8 & 9.Beisser, A., Concepts of Modern Physics, chapter 6.Eisberg, R. and Resnick, R., Quantum Physics, chapter 7.

2.4. THE WAVE FUNCTIONS AND ENERGIES 17

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

1.2

[rR

10(r

)]2

(Å-

1)

radius (Å)

n = 1l = 0

0 5 10 150.0

0.1

0.2

0.3

0.4

[rR

2l(r

)]2

(Å-

1)

radius (Å )

n = 2

l = 0

l = 1

0 5 10 150.0

0.1

0.2

[rR

3l(r

)]2

(Å-

1)

radius (Å )

l = 0

l = 1l = 2

n = 3

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

1.2

[rR

10(r

)]2

(Å-

1)

radius (Å)

n = 1l = 0

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

1.2

[rR

10(r

)]2

(Å-

1)

radius (Å)

n = 1l = 0

0 5 10 150.0

0.1

0.2

0.3

0.4

[rR

2l(r

)]2

(Å-

1)

radius (Å )

n = 2

l = 0

l = 1

0 5 10 150.0

0.1

0.2

0.3

0.4

[rR

2l(r

)]2

(Å-

1)

radius (Å )

n = 2

l = 0

l = 1

0 5 10 150.0

0.1

0.2

[rR

3l(r

)]2

(Å-

1)

radius (Å )

l = 0

l = 1l = 2

n = 3

0 5 10 150.0

0.1

0.2

[rR

3l(r

)]2

(Å-

1)

radius (Å )

l = 0

l = 1l = 2

n = 3

Figure 2.4: Radial probability functions for the first three n states of the hydrogen atom with Z = 1.Note that the radial probability is equal to r2|Rnl(r)|2, not just to |Rnl(r)|2. Note also that the horizontalaxes are the same for all three graphs, but not the vertical axes.


2.5 Appendix: Mathematical solution of the radial equation

The radial wave equation for hydrogen is given from eqn 2.36 as:

− h2

2m

1r2

ddr

(r2 dR(r)

dr

)+

h2l(l + 1)2mr2

R(r)− Ze2

4πε0rR(r) = ER(r) , (2.52)

where l is an integer ≥ 0. We first put this in a more user-friendly form by introducing the dimensionlessradius ρ according to:

ρ =(

8m|E|h2

)1/2

r . (2.53)

The modulus sign around E is important here because we are seeking bound solutions where E is negative.The radial equation now becomes:

d2R

dρ2+

2ρ

dR

dρ−

(l(l + 1)

ρ2+

λ

ρ− 1

4

)R = 0 , (2.54)

where

λ =1

4πε0

Ze2

h

(m

2|E|)1/2

. (2.55)

We first consider the behaviour at ρ →∞, where eqn 2.54 reduces to:

d2R

dρ2− 1

4R = 0 . (2.56)

This has solutions of e±ρ/2. The e+ρ/2 solution cannot be normalized and is thus excluded, which impliesthat R(ρ) ∼ e−ρ/2.

Now consider the bahaviour for ρ → 0, where the dominant terms in eqn 2.54 are:

d2R

dρ2+

2ρ

dR

dρ− l(l + 1)

ρ2R = 0 , (2.57)

with solutions R(ρ) = ρl or R(ρ) = ρ−(l+1). The latter diverges at the origin and is thus unacceptable.The consideration of the asymptotic behaviours suggests that we should look for general solutions of

the radial equation with R(ρ) in the form:

R(ρ) = L(ρ) ρl e−ρ/2 . (2.58)

On substituting into eqn 2.54 we find:

d2L

dρ2+

(2l + 2

ρ− 1

)dL

dρ+

λ− l − 1ρ

L = 0 . (2.59)

We now look for a series solution of the form:

L(ρ) =∞∑

k=0

akρk . (2.60)

Substitution into eqn 2.59 yields:

∞∑

k=0

[k(k − 1)akρk−2 +

(2l + 2

ρ− 1

)kakρk−1 +

λ− l − 1ρ

akρk

]= 0 , (2.61)

which can be re-written:∞∑

k=0

[(k(k − 1) + 2k(l + 1))akρk−2 + (λ− l − 1− k)akρk−1

]= 0 , (2.62)

or alternatively:

∞∑

k=0

[((k + 1)k + 2(k + 1)(l + 1))ak+1ρ

k−1 + (λ− l − 1− k)akρk−1]

= 0 . (2.63)

2.5. APPENDIX: MATHEMATICAL SOLUTION OF THE RADIAL EQUATION 19

This will be satisfied if

((k + 1)k + 2(k + 1)(l + 1))ak+1 + (λ− l − 1− k)ak = 0 , (2.64)

which implies:ak+1

ak=

−λ + l + 1 + k

(k + 1)(k + 2l + 2). (2.65)

At large k we have:ak+1

ak∼ 1

k. (2.66)

Now the series expansion of eρ is

eρ = 1 + ρ +ρ2

2!+ · · · ρ

k

k!+ · · · , (2.67)

which has the same limit for ak+1/ak. With R(ρ) given by eqn 2.58, we would then have a dependenceof e+ρ · e−ρ/2 = e+ρ/2, which is unacceptable. We therefore conclude that the series expansion mustterminate for some value of k. Let nr be the value of k for which the series terminates. It then followsthat anr+1 = 0, which implies:

−λ + l + 1 + nr = 0 , nr ≥ 0 , (2.68)

orλ = l + 1 + nr . (2.69)

We now introduce the principal quantum number n according to:

n = nr + l + 1 . (2.70)

It follows that:

1. n is an integer,

2. n ≥ l + 1,

3. λ = n .

The first two points establish the general rules for the quantum numbers n and l. The third one fixes theenergy. On inserting λ = n into eqn 2.55 and remembering that E is negative, we find:

En = − me4

(4πε0)22h2

Z2

n2. (2.71)

This is the usual Bohr result. The wave functions are of the form given in eqn 2.58:

R(ρ) = ρl L(ρ) e−ρ/2 . (2.72)

The polynomial series L(ρ) that satisfies eqn 2.59 is known as an associated Laguerre function. Onsubstituting for ρ from eqn 2.53 with |E| given by eqn 2.71, we then obtain:

R(r) = normalization constant× Laguerre polynomial in r × rle−r/a (2.73)

as before (cf. eqn 2.43), with

a =(

h2

2m|E|)1/2

=4πε0h

2

me2

n

Z≡ n

ZaH , (2.74)

where aH is the Bohr radius of hydrogen.

Next: The Zeeman effect Up: Time-independent perturbation theory Previous: The linear Stark effect Contents The fine structure of hydrogen

According to special relativity, the kinetic energy (i.e., the difference between the total energy and the rest mass energy) of a particle of rest mass and momentum is

In the non-relativistic limit , we can expand the square-root in the above

expression to give

Hence,

Of course, we recognize the first term on the right-hand side of this equation as the standard non-relativistic expression for the kinetic energy. The second term is the lowest-order relativistic correction to this energy. Let us consider the effect of this type of correction on the energy levels of a hydrogen atom. So, the unperturbed Hamiltonian is given by Eq. (890), and the perturbing Hamiltonian takes the form

Now, according to standard first-order perturbation theory (see Sect. 12.4), the lowest-order relativistic correction to the energy of a hydrogen atom state characterized by the standard quantum numbers , , and is given by

However, Schrödinger's equation for a unperturbed hydrogen atom can be written

where . Since is an Hermitian operator, it follows that

(945)

(946)

(947)

(948)

(949)

(950)

Page 1 of 6The fine structure of hydrogen

4/20/2008http://farside.ph.utexas.edu/teaching/qmech/lectures/node107.html

It follows from Eqs. (674) and (675) that

Finally, making use of Eqs. (655), (657), and (658), the above expression reduces to

where

is the dimensionless fine structure constant.

Note that the above derivation implicitly assumes that is an Hermitian operator. It turns

out that this is not the case for states. However, somewhat fortuitously, out calculation still gives the correct answer when . Note, also, that we are able to use non-degenerate perturbation theory in the above calculation, using the eigenstates,

because the perturbing Hamiltonian commutes with both and . It follows that there

is no coupling between states with different and quantum numbers. Hence, all coupled states have different quantum numbers, and therefore have different energies.

Now, an electron in a hydrogen atom experiences an electric field

due to the charge on the nucleus. However, according to electromagnetic theory, a non-relativistic particle moving in a electric field with velocity also experiences an effective magnetic field

(951)

(952)

(953)

(954)

(955)

(956)



Recall, that an electron possesses a magnetic moment [see Eqs. (738) and (739)]

due to its spin angular momentum, . We, therefore, expect an additional contribution to the Hamiltonian of a hydrogen atom of the form [see Eq. (740)]

where is the electron's orbital angular momentum. This effect is known as

spin-orbit coupling. It turns out that the above expression is too large, by a factor 2, due to an obscure relativistic effect known as Thomas precession. Hence, the true spin-orbit correction to the Hamiltonian is

Let us now apply perturbation theory to the hydrogen atom, using the above expression as the perturbing Hamiltonian.

Now

is the total angular momentum of the system. Hence,

giving

Recall, from Sect. 11.2, that whilst commutes with both and , it does not commute with either or . It follows that the perturbing Hamiltonian (959) also

commutes with both and , but does not commute with either or . Hence, the

simultaneous eigenstates of the unperturbed Hamiltonian (890) and the perturbing

(957)

(958)

(959)

(960)

(961)

(962)



Hamiltonian (959) are the same as the simultaneous eigenstates of , , and discussed in Sect. 11.3. It is important to know this since, according to Sect. 12.6, we can only safely apply perturbation theory to the simultaneous eigenstates of the unperturbed and perturbing Hamiltonians.

Adopting the notation introduced in Sect. 11.3, let be a simultaneous eigenstate of

, , , and corresponding to the eigenvalues

According to standard first-order perturbation theory, the energy-shift induced in such a state by spin-orbit coupling is given by

Here, we have made use of the fact that for an electron. It follows from Eq. (676)

that

where is the radial quantum number. Finally, making use of Eqs. (655), (657), and (658), the above expression reduces to

where is the fine structure constant. A comparison of this expression with Eq. (953) reveals that the energy-shift due to spin-orbit coupling is of the same order of magnitude as that due to the lowest-order relativistic correction to the Hamiltonian. We can add these two corrections together (making use of the fact that for a hydrogen atom--

(963)

(964)

(965)

(966)

(967)

(968)

(969)



see Sect. 11.3) to obtain a net energy-shift of

This modification of the energy levels of a hydrogen atom due to a combination of relativity and spin-orbit coupling is known as fine structure.

Now, it is conventional to refer to the energy eigenstates of a hydrogen atom which are also simultaneous eigenstates of as states, where is the radial quantum number,

as , and is the total angular momentum

quantum number. Let us examine the effect of the fine structure energy-shift (970) on these eigenstates for and 3.

For , in the absence of fine structure, there are two degenerate states.

According to Eq. (970), the fine structure induced energy-shifts of these two states are the same. Hence, fine structure does not break the degeneracy of the two states of

hydrogen.

For , in the absence of fine structure, there are two states, two states,

and four states, all of which are degenerate. According to Eq. (970), the fine

structure induced energy-shifts of the and states are the same as one another,

but are different from the induced energy-shift of the states. Hence, fine structure

does not break the degeneracy of the and states of hydrogen, but does break

the degeneracy of these states relative to the states.

For , in the absence of fine structure, there are two states, two states,

four states, four states, and six states, all of which are degenerate.

According to Eq. (970), fine structure breaks these states into three groups: the and

states, the and states, and the states.

The effect of the fine structure energy-shift on the , 2, and 3 energy states of a hydrogen atom is illustrated in Fig. 20.

(970)



Note, finally, that although expression (969) does not have a well-defined value for , when added to expression (953) it, somewhat fortuitously, gives rise to an expression (970) which is both well-defined and correct when .

Next: The Zeeman effect Up: Time-independent perturbation theory Previous: The linear Stark effect Contents Richard Fitzpatrick 2006-12-12

Figure 20: Effect of the fine structure energy-shift on the and 3

states of a hydrogen atom. Not to scale.



X-ray Tube

X-rays for medical diagnostic procedures or for research purposes are produced in a standard way: by accelerating electrons with a high voltage and allowing them to collide with a metal target. X-rays are produced when the electrons are suddenly decelerated upon collision with the metal target; these x-rays are commonly called brehmsstrahlung or "braking radiation". If the bombarding electrons have sufficient energy, they can knock an electron out of an inner shell of the target metal atoms. Then electrons from higher states drop down to fill the vacancy, emitting x-ray photons with precise energies determined by the electron energy levels. These x-rays are called characteristic x-rays.

Characteristic x-rays Brehmsstrahlung

Index

HyperPhysics***** Quantum Physics Go Back

Page 1 of 1X-ray Production

4/20/2008http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xtube.html

2.5.3.1. The Origin of Characteristic X-rays

When a sample is bombarded by an electron beam, some electrons are knocked out of their shells in a process called inner-shell ionization. About 0.1% of the electrons produce K-shell vacancies; most produce heat. Outer-shell electrons fall in to fill a vacancy in a process of self-neutralization (Figure 2.5.3.1). The energy required to produce inner-shell ionization is termed the excitation potential or critical ionization potential (Ec).

C.G. Barkla (1877-1944)

The production of "characteristic" X-rays by electron bombardment of pure elements was first observed in 1909 by Charles G. Barkla and C.A. Sadler. However, the physical origin of X-rays was not clear. Barkla received the Nobel Prize in 1917.

Figure 2.5.3.1. Classical models showing the production of bremsstrahlung, characteristic X-rays, and Auger electrons. (left) Electrons are scattered elastically and inelastically by the positively charged nucleus. The inelastically scattered electron loses energy, which appears as bremsstrahlung. Elastically scattered electrons (which include backscattered electrons) are generally scattered through larger angles than are inelastically scattered electrons. (right) An incident electron ionizes the sample atom by ejecting an electron from an inner-shell (the K shell, in this case). De-excitation, in turn, produces characteristic X-radiation (above) or

Page 1 of 2X-rays: Characteristic Radiation

4/20/2008http://www4.nau.edu/microanalysis/Microprobe/Xray-Characteristic.html

When outer-shell electrons drop into inner shells, they emit a quantized photon "characteristic" of the element. The energies of the characteristic X-rays produced are only very weakly dependent on the chemical structure in which the atom is bound, indicating that the non-bonding shells of atoms are the X-ray source. The resulting characteristic spectrum is superimposed on the continuum. An atom remains ionized for a very short time (about 10-14 second) and thus the incident electrons that arrive about every 10-12 second can repeatedly ionize an atom. However, not all outer-shell electrons can fall in to produce X-rays. To more fully understand the production of characteristic X-rays, we must first consider the electronic structure of atoms.

Back: 2.5.2. Continuum X-Rays | Next: 2.5.3.2. Electron Shells | Home: Course Overview

Copyright 1997-2003, James H. Wittke Last update: 01/18/2006 01:47 PM.

an Auger electron (below). Secondary electrons are ejected with low energy from outer loosely bound electron shells, a process not shown.

Page 2 of 2X-rays: Characteristic Radiation

4/20/2008http://www4.nau.edu/microanalysis/Microprobe/Xray-Characteristic.html

What is X-ray diffraction (XRD)

On This Page: Bragg's Law Features of Electron, X-ray, or Neutron Diffraction The ``Ultimate'' (Technically Challenging) Experiment Powder vs. Single Crystal X-ray Diffraction

Bragg's Law

or

A real 3-dimensional crystal contains many sets of planes. For diffraction, crystal must have the correct orientation with respect to the incoming beam.

Page 1 of 4What is X-ray diffraction (XRD)

4/20/2008http://www.physics.upenn.edu/~heiney/talks/hires/whatis.html

Perfect, infinite crystal and perfectly collimated beam: diffraction condition must be satisfied ``exactly.''

Strains, defects, finite size effects, instrumental resolution: diffraction peaks are broadened.

More formally, the scattered intensity is proportional to the square of the Fourier transform of the charge density:

where is the charge density.

For perfect crystals, I(q) consists of delta functions (perfectly sharp scattering). For imperfect crystals, the peaks are broadened. For liquids and glasses, it is a continuous, slowly varying function.

Features of Electron, X-ray, or Neutron Diffraction For a known structure, pattern can be calculated exactly.



Symmetry of the diffraction pattern given by symmetry of the lattice. Intensities of spots determined by basis of atoms at each lattice point. Sharpness and shape of spots determined by perfection of crystal. Liquids, glasses, and other disordered materials produce broad fuzzy rings instead of sharp spots. Defects and disorder in crystals also result in diffuse scattering.

The ``Ultimate'' (Technically Challenging) Experiment Sample is tiny (micron-sized). The effect is weak (light elements, small modulations, subtle modifications of the long-range order). Instrumental resolution (angle and energy) is ``perfect'' allowing detailed measurements of structural disorder. Measurement is time-resolved (nanosecond time scale).

To achieve all of the above, will need lots of intensity in the primary beam together with sensitive detection systems.

Powder vs. Single Crystal X-ray Diffraction

SINGLE CRYSTAL

Put a crystal in the beam, observe what reflections come out at what angles for what orientations of the crystal with what intensities.

Advantages

In principle you can learn everything there is to know about the structure.

Disadvantages

You may not have a single crystal. It is time-consuming and difficult to orient the crystal. If more than one phase is present, you will not necessarily realize that there is more than one set of reflections.

POWDER

Samples consists of a collection of many small crystallites with random orientations. Average over crystal orientations and measure the scattered intensity as a function of outgoing angle.

Disadvantage

Inversion of the measured intensities to find the structure is more difficult and less reliable.

Advantages

It is usually much easier to prepare a powder sample. You are guaranteed to see all reflections. The best way to follow phase changes as a function of temperature, pressure, or some other variable.



Next: X-ray Sources Up: High Resolution X-ray Diffraction Previous: High Resolution X-ray Diffraction

Copyright 1995, 1996, Paul A. Heiney. Individuals should feel free to make links to this document or any images contained in it, or to make a copy for their own personal use. However, you may not further disseminate copies in electronic, printed, or any other form without the express permission of the author, and this copyright notice must appear on any copy.

Last updated December 30, 1996

Paul A. Heiney, [email protected]



(X-ray) Diffraction(X-ray) Diffraction

Some practical aspects of one ofSome practical aspects of one ofthe most important tools in solidthe most important tools in solidstate sciencesstate sciences

Bragg’s Law of DiffractionBragg’s Law of Diffraction

n⋅λ=2d⋅sinθ

constructive interference only, when:∆ = n⋅λ (∆= AB+BC)with:sinθ = (∆/2)/d

Diffraction from Lattice PlanesDiffraction from Lattice Planes

•Each set of planescorresponds to

one structure factor Shkl

Diffraction from Single CrystalsDiffraction from Single Crystals

Diffraction from Powder CrystalsDiffraction from Powder Crystals

QuadraticQuadraticBraggBragg formulas formulas

•• Tungsten wire at 1200-1800Tungsten wire at 1200-1800ooCC(about 35mA heating current)(about 35mA heating current)

•• High Voltage 20-60 kVHigh Voltage 20-60 kV

•• max. Power 2.2-3 kWmax. Power 2.2-3 kW

Working Principle of the X-rayWorking Principle of the X-raytubetube

•Typical operating values for Cu: 40 kV, 35 mAMo: 45 kV, 35 mA

Spectrum of the X-ray tube Spectrum of the X-ray tube

Bremstrahlung (white radiation)Emax.= E0 = e⋅V0 and with E = (h⋅c)/λ:λmin/Å = (h⋅c)/e⋅V0 = 12.34/( V0/kV)

Characteristic radiation

•n=1,2,3 (principal quantum number), corresponds to K, L, M... shells•l=0, 1, ..., n-1 (orbital quantum number)•j=|l±s|; s=1/2 (spin-orbit coupling)•mj=j, j-1, j-2, ..., -j•Rules: Transition only, when ∆l¹0

Kβ1 Kα2 Kα1

•AllowedTransitions

Mosley’s Law (for multipleMosley’s Law (for multipleelectron atoms):electron atoms):

1/λ = c⋅(Z-σ)2⋅(1/n12 - 1/n2

2)•Z = atom number•σ = shielding constant•n = quantum number

⇒ Decreasing wavelength with increasing Z

Characteristic WavelengthsCharacteristic Wavelengths in in Angstroems Angstroems (100pm)(100pm)

Element Symbol Kα2 Kα1 Kβ K abs. edgeCu 1.54433 1.54051 1.39217,

1.381021.380

Mo 0.713543 0.70926 0.62099 0.61977Ag 0.563775 0.559363 0.49701,

0.487010.4858

W 0.213813 0.208992 0.17950 0.17837

µµ vs. vs. λλ

At the absorption edge, the incidentX-ray quantum is energetic enough toknock an electron out of the orbital

Absorption edge

Monochromatisation Monochromatisation of X-raysof X-rays

•Filters•Crystal Monochromators

Different GeometriesDifferent Geometries

•• Debye-ScherrerDebye-Scherrer

•• Bragg-BrentanoBragg-Brentano

•• GuinierGuinier

Debye ScherrerDebye Scherrer

Detection of X-raysDetection of X-rays

•Film (Guinier camera, Debye-Scherrer Camera, precession camera)•Si(Li) solid state detector (powder diffractometers)•Szintillation counter (4-circle diffractometer,Stoe powderdiffractometer)•Position Sensitive Detectors (Stoe powder diffractometer)•Image Plate Detectors(Stoe IPDS)•CCD Detectors (Bruker SMART system)

Resolution:Resolution:

Image plate detectorsImage plate detectors•Metal plate with about 18cm diameter, coated with Eu2+

doped BaFBr•X-rays ionize Eu2+ to Eu3+ and the electrons are trapped incolor centers•Read out process with red laser leads to emission of bluelight, when electrons return to ground state•The blue light is amplified by a photomultiplier andrecorded as a pixel image

Setup for a PowderSetup for a PowderDiffractometerDiffractometer

X-ray tube

Ge-monochromators

shutters

Goniometer

High TemperatureAttachment

Different Sample HoldersDifferent Sample Holders

Capillary

Transmission

Reflection

Preparing a samplePreparing a sample

Capillary:For air sensitive samplesDiameter between 0.1 an 1mm, Standard is 0.3 mmFor samples with high absorption 0.1 mm is better suitedDifficulties with soft samples which are not easy to fill in

Transmission sample holderGood for samples which are not or only moderately air sensitive.Sample is placed on a Scotch (Tesa) strip and covered with a second strip.Be sure, that the sample is only on one(!) side and the second is only for protection.

Reflection sample holderOnly for moderately air sensitive samplesGood for or strongly absorbing samples like for example electrodes or thin films on asubstrateIs used at the moment for in situ electrochemical cell experimentsCannot be used in connection with the large PSD

What Information Can WeWhat Information Can WeExtract from DiffractionExtract from DiffractionExperiments?Experiments?

•• Determination of known phasesDetermination of known phases

•• CrystallinityCrystallinity

•• Determination of lattice constantsDetermination of lattice constants

•• Structure solutionStructure solution

Crystalline and AmorphousCrystalline and AmorphousPhase together:Phase together:

Effect of a Change of theEffect of a Change of theLattice ConstantsLattice Constants

Effect of CenteringEffect of Centering

P

I

F

NumberNumberof linesof lineschangeschangeswithwithsymmetrysymmetry

Overlapping of Reflections:Overlapping of Reflections:

Databases:Databases:

•• ICSD ICSD (Inorganics, Single Crystal Data, on PC‘s)(Inorganics, Single Crystal Data, on PC‘s)

•• CSD CSD (Organics, on Wawona)(Organics, on Wawona)

•• METALS METALS (at vsibm1.mpi-stuttgart.mpg.de,(at vsibm1.mpi-stuttgart.mpg.de,username guest, password guest, metals)username guest, password guest, metals)

Interaction of Electrons withInteraction of Electrons withMatterMatterEmission of electromagnetic radiation: Characteristic radiation, discrete energies, EC<E0

Bremsstrahlung, continuous energie distribution, Eb£E0

Luminescence, in the UV or visible Region

Electron emission: Backscattered electrons (BSE) Auger electrons Secondary electron emission (SE)

Effects in the Target: Electron Absorption (ABS) Heat

X-ray Tube

X-rays for medical diagnostic procedures or for research purposes are produced in a standard way: by accelerating electrons with a high voltage and allowing them to collide with a metal target. X-rays are produced when the electrons are suddenly decelerated upon collision with the metal target; these x-rays are commonly called brehmsstrahlung or "braking radiation". If the bombarding electrons have sufficient energy, they can knock an electron out of an inner shell of the target metal atoms. Then electrons from higher states drop down to fill the vacancy, emitting x-ray photons with precise energies determined by the electron energy levels. These x-rays are called characteristic x-rays.

Characteristic x-rays Brehmsstrahlung

Index

HyperPhysics***** Quantum Physics Go Back

Page 1 of 1X-ray Production

4/20/2008http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xtube.html

compulsory - cdlteachingcommons.cdl.edu/avu.old/physics/documents/atomic physics... · atomic...

Documents