Chapter 28Chapter 28Atomic PhysicsAtomic Physics

Conceptual questions: 2,4,8,9,16

Quick quizzes: 1,2,3,4

Problems: 18, 39

Examples: 1,3,4

Rutherford’s Model of the Rutherford’s Model of the AtomAtom

Planetary modelPlanetary model Positive charge is Positive charge is

concentrated in concentrated in the center of the the center of the atom, called the atom, called the nucleusnucleus

Electrons orbit Electrons orbit the nucleus like the nucleus like planets orbit the planets orbit the sunsun

Examples of SpectraExamples of Spectra

a) emission b) absorption

Absorption SpectraAbsorption Spectra

An element can also absorb light at An element can also absorb light at specific wavelengthsspecific wavelengths

The absorption spectrum consists of The absorption spectrum consists of a series of dark lines superimposed a series of dark lines superimposed on the otherwise continuous on the otherwise continuous spectrumspectrum The dark lines of the absorption The dark lines of the absorption

spectrum coincide with the bright lines spectrum coincide with the bright lines of the emission spectrumof the emission spectrum

Spectral Lines of HydrogenSpectral Lines of Hydrogen

The Balmer Series The Balmer Series has lines whose has lines whose wavelengths are wavelengths are given by the given by the preceding equationpreceding equation

Examples of Examples of spectral linesspectral lines n = 3, n = 3, λ = 656.3 nmλ = 656.3 nm n = 4, n = 4, λ = 486.1 nmλ = 486.1 nm

Balmer’s Equation Balmer’s Equation

The wavelengths of hydrogen’s spectral The wavelengths of hydrogen’s spectral lines can be found fromlines can be found from

RRHH is the is the Rydberg constantRydberg constant RRHH = 1.0973732 x 10 = 1.0973732 x 1077 m m-1-1

n is an integer, n = 1, 2, 3, …n is an integer, n = 1, 2, 3, … The spectral lines correspond to different The spectral lines correspond to different

values of nvalues of n

22H n

1

2

1R

1

Bohr’s Assumptions for Bohr’s Assumptions for HydrogenHydrogen

Circular orbits around Circular orbits around the proton under the the proton under the influence of the influence of the Coulomb force of Coulomb force of attractionattraction

Only certain electron Only certain electron orbits are stableorbits are stable

Radiation is emitted by Radiation is emitted by the atom when the the atom when the electron “jumps” from a electron “jumps” from a more energetic initial more energetic initial state to a lower statestate to a lower state

Mathematics of Bohr’s Mathematics of Bohr’s Assumptions and ResultsAssumptions and Results

Electron’s orbital angular momentumElectron’s orbital angular momentum mmee v r = n v r = n ħ where n = 1, 2, 3, …ħ where n = 1, 2, 3, …

The total energy of the atomThe total energy of the atom

Coulomb’s force provides centripetal Coulomb’s force provides centripetal accelerationacceleration

Radiation emitted, ERadiation emitted, Eii – E – Eff = hf = hf

r

ekvm

2

1PEKEE

2

e2

e

2 2

2ek e mv

r r

Radii and Energy of OrbitsRadii and Energy of Orbits The radii of the Bohr The radii of the Bohr

orbits are quantizedorbits are quantized n = 1, the orbit has the n = 1, the orbit has the

smallest radius, called smallest radius, called the the Bohr radiusBohr radius, a, aoo

aaoo = 0.0529 nm = 0.0529 nm

A general expression for A general expression for the radius of any orbit in the radius of any orbit in a hydrogen atom isa hydrogen atom is rrnn = n = n22 a aoo

The energy of any orbit The energy of any orbit isis EEnn = - 13.6 eV/ n = - 13.6 eV/ n22

2 2

21, 2, 3,

n

e e

nr n

m k e

Energy Level DiagramEnergy Level Diagram

The value of RThe value of RHH from Bohr’s from Bohr’s analysis is in analysis is in excellent excellent agreement with the agreement with the experimental valueexperimental value

A more generalized A more generalized equation can be equation can be used to find the used to find the wavelengths of any wavelengths of any spectral linesspectral lines

Generalized EquationGeneralized Equation

For the Balmer series, nFor the Balmer series, nf f = 2 = 2

For the Lyman series, nFor the Lyman series, nff = 1 = 1

Whenever an transition occurs between a Whenever an transition occurs between a state, nstate, nii to another state, n to another state, nff (where n (where nii > > nnff), a photon ), a photon is emittedis emitted The photon has a frequency f = (EThe photon has a frequency f = (Eii – E – Eff)/h )/h

and wavelength and wavelength λ λ

2i

2f

H n

1

n

1R

1

Successes of the Bohr Successes of the Bohr TheoryTheory

Explained several features of the hydrogen Explained several features of the hydrogen spectrumspectrum Accounts for Balmer and other seriesAccounts for Balmer and other series Predicts a value for RPredicts a value for RHH that agrees with the that agrees with the

experimental valueexperimental value Gives an expression for the radius of the atomGives an expression for the radius of the atom Predicts energy levels of hydrogenPredicts energy levels of hydrogen Gives a model of what the atom looks like and how it Gives a model of what the atom looks like and how it

behavesbehaves Can be extended to “hydrogen-like” atomsCan be extended to “hydrogen-like” atoms

Those with one electronThose with one electron ZeZe22 needs to be substituted for e needs to be substituted for e22 in equations in equations

Z is the atomic number of the elementZ is the atomic number of the element

de Broglie Wavesde Broglie Waves

In this example, three In this example, three complete wavelengths complete wavelengths are contained in the are contained in the circumference of the circumference of the orbitorbit

In general, the In general, the circumference must circumference must equal some integer equal some integer number of wavelengthsnumber of wavelengths 2 2 r = n r = n λ n = 1, 2, …λ n = 1, 2, …

Electron CloudsElectron Clouds

Modifications of the Bohr Modifications of the Bohr Theory – Elliptical OrbitsTheory – Elliptical Orbits

Sommerfeld extended the results to Sommerfeld extended the results to include elliptical orbitsinclude elliptical orbits Retained the Retained the principle quantum numberprinciple quantum number, ,

nn Added the Added the orbital quantum numberorbital quantum number, , ℓℓ

ℓ ℓ ranges from 0 to n-1 in integer stepsranges from 0 to n-1 in integer steps All states with the same principle All states with the same principle

quantum number are said to form a quantum number are said to form a shellshell The states with given values of n and ℓ The states with given values of n and ℓ

are said to form a are said to form a subshellsubshell

Modifications of the Bohr Modifications of the Bohr Theory – Zeeman EffectTheory – Zeeman Effect

Another modification was needed to Another modification was needed to account for the account for the Zeeman effectZeeman effect The Zeeman effect is the splitting of The Zeeman effect is the splitting of

spectral lines in a strong magnetic spectral lines in a strong magnetic fieldfield

This indicates that the energy of an This indicates that the energy of an electron is slightly modified when the electron is slightly modified when the atom is immersed in a magnetic fieldatom is immersed in a magnetic field

A new quantum number, mA new quantum number, m ℓℓ, called the , called the orbital magnetic quantum numberorbital magnetic quantum number, , had to be introducedhad to be introduced

m m ℓℓ can vary from - ℓ to + ℓ in integer steps can vary from - ℓ to + ℓ in integer steps

Quantum numbers for the Quantum numbers for the hydrogen atomhydrogen atom

In an analysis relating Bohr's theory to the de Broglie wavelength of electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the wavelength of the electron becomes 9 times as long, or (d) the electron is moving 9 times as fast.

QUICK QUIZ 28.1

Quantum Number Quantum Number SummarySummary

The values of n can range from 1 The values of n can range from 1 to to in integer steps in integer steps

The values of The values of ℓ can range from 0 to ℓ can range from 0 to n-1 in integer stepsn-1 in integer steps

The values of The values of mm ℓℓ can range from -ℓ can range from -ℓ to ℓ in integer stepsto ℓ in integer steps

How many possible orbital states are there for (a) the n = 3 level of hydrogen? (b) the n = 4 level?

QUICK QUIZ 28.2

When the principal quantum number is n = 5, how many different values of (a) and (b) m are possible?

QUICK QUIZ 28.3

Modifications of the Bohr Modifications of the Bohr Theory – Fine StructureTheory – Fine Structure

High resolution spectrometers show High resolution spectrometers show that spectral lines are, in fact, two that spectral lines are, in fact, two very closely spaced lines, even in very closely spaced lines, even in the absence of a magnetic fieldthe absence of a magnetic field This splitting is called This splitting is called fine structurefine structure Another quantum number, mAnother quantum number, mss, called , called

the the spin magnetic quantum number,spin magnetic quantum number, was introduced to explain the fine was introduced to explain the fine structurestructure

Spin Magnetic Quantum Spin Magnetic Quantum NumberNumber

It is convenient to think It is convenient to think of the electron as of the electron as spinning on its axisspinning on its axis The electron is The electron is notnot

physically spinningphysically spinning There are two directions There are two directions

for the spinfor the spin Spin up, mSpin up, mss = = ½½ Spin down, mSpin down, mss = - = -½½

There is a slight energy There is a slight energy difference between the difference between the two spins and this two spins and this accounts for the Zeeman accounts for the Zeeman effecteffect

The Pauli Exclusion The Pauli Exclusion PrinciplePrinciple

No two electrons in an atom can No two electrons in an atom can ever be in the same quantum stateever be in the same quantum state In other words, no two electrons in In other words, no two electrons in

the same atom can have exactly the the same atom can have exactly the same values for n, same values for n, ℓ, ℓ, mm ℓℓ, and m, and mss

This explains the electronic This explains the electronic structure of complex atoms as a structure of complex atoms as a succession of filled energy levels succession of filled energy levels with different quantum numberswith different quantum numbers

The Periodic TableThe Periodic Table

The outermost electrons are primarily The outermost electrons are primarily responsible for the chemical properties of responsible for the chemical properties of the atomthe atom

Mendeleev arranged the elements Mendeleev arranged the elements according to their atomic masses and according to their atomic masses and chemical similaritieschemical similarities

The electronic configuration of the The electronic configuration of the elements explained by quantum numbers elements explained by quantum numbers and Pauli’s Exclusion Principle explains the and Pauli’s Exclusion Principle explains the configurationconfiguration

Krypton (atomic number 36) has how many electrons in its next to outer shell (n = 3)?

(a) 2 (b) 4(c) 8 (d) 18

QUICK QUIZ 28.4

ProblemsProblems18. A particle of charge q and mass m, moving with a constant speed v, perpendicular to a constant magnetic field, B, follows a circular path. If the angular momentum about the center of this circle is quantized so that mvr = nħ , show that the allowed radii for the particle are

where n = 1, 2, 3, . . .

qB

nrn

39. Zirconium (Z = 40) has two electrons in an incomplete d subshell. (a) What are the values of n and l for each electron? (b) What are all possible values of ml and ms ? (c) What is the electron configuration in the ground state of zirconium?

Characteristic X-RaysCharacteristic X-Rays

Explanation of Explanation of Characteristic X-RaysCharacteristic X-Rays

The details of atomic structure can be used The details of atomic structure can be used to explain characteristic x-raysto explain characteristic x-rays A bombarding electron collides with an electron A bombarding electron collides with an electron

in the target metal that is in an inner shellin the target metal that is in an inner shell If there is sufficient energy, the electron is If there is sufficient energy, the electron is

removed from the target atomremoved from the target atom The vacancy created by the lost electron is filled The vacancy created by the lost electron is filled

by an electron falling to the vacancy from a by an electron falling to the vacancy from a higher energy levelhigher energy level

The transition is accompanied by the emission The transition is accompanied by the emission of a photon whose energy is equal to the of a photon whose energy is equal to the difference between the two levelsdifference between the two levels

Atomic Transitions – Atomic Transitions – Energy LevelsEnergy Levels

An atom may have An atom may have many possible energy many possible energy levelslevels

At ordinary At ordinary temperatures, most of temperatures, most of the atoms in a sample the atoms in a sample are in the ground stateare in the ground state

Only photons with Only photons with energies energies corresponding to corresponding to differences between differences between energy levels can be energy levels can be absorbedabsorbed

Atomic Transitions – Atomic Transitions – Stimulated AbsorptionStimulated Absorption

The blue dots The blue dots represent electronsrepresent electrons

When a photon with When a photon with energy energy ΔE is ΔE is absorbed, one absorbed, one electron jumps to a electron jumps to a higher energy levelhigher energy level These higher levels are These higher levels are

called called excited statesexcited states ΔE = hƒ = EΔE = hƒ = E22 – E – E11

In general, ΔE can be In general, ΔE can be the difference between the difference between any two energy levelsany two energy levels

Atomic Transitions – Atomic Transitions – Spontaneous EmissionSpontaneous Emission

Once an atom is in Once an atom is in an excited state, an excited state, there is a constant there is a constant probability that it probability that it will jump back to a will jump back to a lower state by lower state by emitting a photonemitting a photon

This process is This process is called called spontaneous spontaneous emissionemission

Atomic Transitions – Atomic Transitions – Stimulated EmissionStimulated Emission

An atom is in an excited An atom is in an excited stated and a photon is stated and a photon is incident on itincident on it

The incoming photon The incoming photon increases the probability increases the probability that the excited atom that the excited atom will return to the ground will return to the ground statestate

There are two emitted There are two emitted photons, the incident photons, the incident one and the emitted oneone and the emitted one The emitted photon is in The emitted photon is in

exactly in phase with exactly in phase with the incident photonthe incident photon

LasersLasers To achieve laser action, three conditions To achieve laser action, three conditions

must be metmust be met The system must be in a state of population The system must be in a state of population

inversioninversion The excited state of the system must be a The excited state of the system must be a

metastable statemetastable state Its lifetime must be long compared to the normal Its lifetime must be long compared to the normal

lifetime of an excited statelifetime of an excited state The emitted photons must be confined in the The emitted photons must be confined in the

system long enough to allow them to stimulate system long enough to allow them to stimulate further emission from other excited atomsfurther emission from other excited atoms

This is achieved by using reflecting mirrorsThis is achieved by using reflecting mirrors

Production of a Laser Production of a Laser BeamBeam

Laser Beam – He Ne Laser Beam – He Ne ExampleExample

The energy level diagram The energy level diagram for Nefor Ne

The mixture of helium and The mixture of helium and neon is confined to a neon is confined to a glass tube sealed at the glass tube sealed at the ends by mirrorsends by mirrors

A high voltage applied A high voltage applied causes electrons to sweep causes electrons to sweep through the tube, through the tube, producing excited statesproducing excited states

When the electron falls to When the electron falls to EE22 in Ne, a 632.8 nm in Ne, a 632.8 nm photon is emittedphoton is emitted

Conceptual questionsConceptual questions

2.Does a light emitted by a neon sign 2.Does a light emitted by a neon sign constitute a continuous spectrum or only a few constitute a continuous spectrum or only a few colors.colors.

4. Must an atom first be ionized before it can 4. Must an atom first be ionized before it can emit light?emit light?

8. If matter has a wave nature, why is this not 8. If matter has a wave nature, why is this not observable in our daily experiences?observable in our daily experiences?

9. Discuss consequences of the exclusion 9. Discuss consequences of the exclusion principle.principle.

16. A 1 mW laser might damage your eye if 16. A 1 mW laser might damage your eye if you look directly at it, but there is no harm at you look directly at it, but there is no harm at looking directly at a 100 W lightbulb. Why?looking directly at a 100 W lightbulb. Why?

Energy Bands in SolidsEnergy Bands in Solids

Sodium exampleSodium example Blue represents energy bands Blue represents energy bands

occupied by the sodium occupied by the sodium electrons when the atoms are electrons when the atoms are in their ground statesin their ground states

Gold represents energy bands Gold represents energy bands that are emptythat are empty

White represents energy gapsWhite represents energy gaps Electrons can have any energy Electrons can have any energy

within the allowed bandswithin the allowed bands Electrons cannot have Electrons cannot have

energies in the gapsenergies in the gaps

Energy Level DefinitionsEnergy Level Definitions

The The valence bandvalence band is the highest is the highest filled bandfilled band

The The conduction bandconduction band is the next is the next higher empty bandhigher empty band

The energy gap has an energy, EThe energy gap has an energy, Egg, , equal to the difference in energy equal to the difference in energy between the top of the valence between the top of the valence band and the bottom of the band and the bottom of the conduction bandconduction band

ConductorsConductors When a voltage is applied When a voltage is applied

to a conductor, the to a conductor, the electrons accelerate and electrons accelerate and gain energygain energy

In quantum terms, In quantum terms, electron energies electron energies increase if there are a increase if there are a high number of high number of unoccupied energy levels unoccupied energy levels for the electron to jump tofor the electron to jump to

For example, it takes very For example, it takes very little energy for electrons little energy for electrons to jump from the partially to jump from the partially filled to one of the nearby filled to one of the nearby empty statesempty states

InsulatorsInsulators The valence band is The valence band is

completely full of completely full of electronselectrons

A large band gap A large band gap separates the valence separates the valence and conduction bandsand conduction bands

A large amount of A large amount of energy is needed for energy is needed for an electron to be able an electron to be able to jump from the to jump from the valence to the valence to the conduction bandconduction band

SemiconductorsSemiconductors A semiconductor has a small A semiconductor has a small

energy gapenergy gap Thermally excited electrons Thermally excited electrons

have enough energy to cross have enough energy to cross the band gapthe band gap

The resistivity of The resistivity of semiconductors decreases semiconductors decreases with increases in with increases in temperaturetemperature

The white area in the The white area in the valence band represents valence band represents holesholes

Semiconductors, contSemiconductors, cont HolesHoles are empty states in the valence band are empty states in the valence band

created by electrons that have jumped to created by electrons that have jumped to the conduction bandthe conduction band

Some electrons in the valence band move to Some electrons in the valence band move to fill the holes and therefore also carry currentfill the holes and therefore also carry current

The valence electrons that fill the holes The valence electrons that fill the holes leave behind other holesleave behind other holes It is common to view the conduction process in It is common to view the conduction process in

the valence band as a flow of positive holes the valence band as a flow of positive holes toward the negative electrode applied to the toward the negative electrode applied to the semiconductorsemiconductor

Current Process in Current Process in SemiconductorsSemiconductors

An external voltage is An external voltage is suppliedsupplied

Electrons move Electrons move toward the positive toward the positive electrodeelectrode

Holes move toward Holes move toward the negative the negative electrodeelectrode

There is a There is a symmetrical current symmetrical current process in a process in a semiconductorsemiconductor

Doping in SemiconductorsDoping in Semiconductors

DopingDoping is the adding of impurities is the adding of impurities to a semiconductorto a semiconductor Generally about 1 impurity atom per Generally about 1 impurity atom per

101077 semiconductor atoms semiconductor atoms Doping results in both the band Doping results in both the band

structure and the resistivity being structure and the resistivity being changedchanged

A p-n JunctionA p-n Junction

A p-n junction is A p-n junction is formed when a p-formed when a p-type semiconductor type semiconductor is joined to an n-typeis joined to an n-type

Three distinct Three distinct regions existregions exist A p regionA p region An n regionAn n region A depletion regionA depletion region

Diode ActionDiode Action

The p-n junction has the ability The p-n junction has the ability to pass current in only one to pass current in only one directiondirection

When the p-side is connected When the p-side is connected to a positive terminal, the to a positive terminal, the device is device is forward biasedforward biased and and current flowscurrent flows

When the n-side is connected When the n-side is connected to the positive terminal, the to the positive terminal, the device is device is reverse biasedreverse biased and a and a very small reverse current very small reverse current resultsresults

Applications of Applications of Semiconductor DiodesSemiconductor Diodes

RectifiersRectifiers Change AC voltage to DC voltageChange AC voltage to DC voltage A A half-wave rectifierhalf-wave rectifier allows current to flow during allows current to flow during

half the AC cyclehalf the AC cycle A A full-wave rectifierfull-wave rectifier rectifies both halves of the AC rectifies both halves of the AC

cyclecycle TransistorsTransistors

May be used to amplify small signalsMay be used to amplify small signals Integrated circuitIntegrated circuit

A collection of interconnected transistors, diodes, A collection of interconnected transistors, diodes, resistors and capacitors fabricated on a single piece resistors and capacitors fabricated on a single piece of siliconof silicon