Chapter43Problems
1,2,3=straightforward,intermediate,
challenging
Section43.1MolecularBonds
1.
Reviewproblem.AK+ionandaClion
areseparatedbyadistanceof5.001010m.
Assumingthetwoionsactlikepointcharges,
determine(a)theforceeachionexertsonthe
otherand(b)thepotentialenergyofthetwoion
systeminelectronvolts.
2.
Potassiumchlorideisanionically
bondedmolecule,soldasasaltsubstitutefor
useinalowsodiumdiet.Theelectronaffinity
ofchlorineis3.6eV.Anenergyinputof0.7eV
isrequiredtoformseparateK+andClions
fromseparateKandClatoms.Whatisthe
ionizationenergyofK?
3.
Onedescriptionofthepotentialenergy
ofadiatomicmoleculeisgivenbytheLennard
Jonespotential,
U=
A
B
6
12
r
r
whereAandBareconstants.Find,intermsofA
andB,(a)thevaluer0atwhichtheenergyisa
minimumand(b)theenergyErequiredtobreak
upadiatomicmolecule.(c)Evaluater0in
metersandEinelectronvoltsfortheH2
molecule.Inyourcalculations,takeA=0.124
10120eVm12andB=1.4881060eVm6.
(Note:Althoughthispotentialiswidelyusedfor
modeling,itisknowntohaveseriousdefects.
Forexample,itsbehavioratbothsmalland
largevaluesofrissignificantlyinerror.)
4.
Potassiumiodidecanbetakenasa
medicinetoreduceradiationdosagetothe
thyroidglandbeforeorafterexposureto
radioactiveiodine.Inthepotassiumiodide
molecule,assumethattheKandIatomsbond
ionicallybythetransferofoneelectronfromK
toI.(a)TheionizationenergyofKis4.34eV,
andtheelectronaffinityofIis3.06eV.What
energyisneededtotransferanelectronfromK
toI,toformK+andIionsfromneutralatoms?
Thisissometimescalledtheactivationenergy
Ea.(b)Amodelpotentialenergyfunctionforthe
KImoleculeistheLennardJonespotential:
12 6
U ( r ) = 4 + E a
r
r
whereristheinternuclearseparationdistance,
andandareadjustableparameters.TheEa
termisaddedtoensurethecorrectasymptotic
behavioratlarger.Attheequilibrium
separationdistance,r=r0=0.305nm,U(r)isa
minimum,anddU/dr=0.NowU(r0)isthe
negativeofthedissociationenergy:U(r0)=
3.37eV.Evaluateand.(c)Calculatethe
forceneededtobreakupaKImolecule.(d)
Calculatetheforceconstantforsmall
oscillationsaboutr=r0.(Suggestion:Setr=r0
+swheres/r0<<1,andexpandU(r)inpowers
ofs/r0uptosecondorderterms.)
5.
AvanderWaalsdispersionforce
betweenheliumatomsproducesaveryshallow
potentialwell,withadepthontheorderof1
meV.Ataboutwhattemperaturewouldyou
expectheliumtocondense?
emittedwhenanNaClmoleculeundergoesa
transitionfromtheJ=2statetotheJ=1state.
Section43.2EnergyStatesandSpectraof
Molecules
11.
TherotationalspectrumoftheHCl
moleculecontainslineswithwavelengthsof
0.0604,0.0690,0.0804,0.0964,and0.1204
mm.Whatisthemomentofinertiaofthe
molecule?
6.
Thecesiumiodide(CsI)moleculehasan
atomicseparationof0.127nm.(a)Determine
theenergyofthelowestexcitedrotationalstate
andthefrequencyofthephotonabsorbedinthe
J=0toJ=1transition.(b)WhatIf?What
wouldbethefractionalchangeinthisfrequency
iftheestimateoftheatomicseparationisoffby
10%?
7.
AnHClmoleculeisexcitedtoitsfirst
rotationalenergylevel,correspondingtoJ=1.
Ifthedistancebetweenitsnucleiis0.1275nm,
whatistheangularspeedofthemoleculeabout
itscenterofmass?
8.
TheCOmoleculemakesatransition
fromtheJ=1toJ=2rotationalstatewhenit
absorbsaphotonoffrequency2.301011Hz.
Findthemomentofinertiaofthismolecule
fromthesedata.
9.
Adiatomicmoleculeconsistsoftwo
atomshavingmassesm1andm2andseparated
byadistancer.Showthatthemomentofinertia
aboutanaxisthroughthecenterofmassofthe
moleculeisgivenbyEquation43.3,I=r2.
10.
(a)Calculatethemomentofinertiaofan
NaClmoleculeaboutitscenterofmass.The
atomsareseparatedbyadistancer=0.28nm.
(b)Calculatethewavelengthofradiation
12.
UsethedatainTable43.2tocalculate
theminimumamplitudeofvibrationfor(a)the
HImoleculeand(b)theHFmolecule.Which
hastheweakerbond?
13.
Takingtheeffectiveforceconstantofa
vibratingHClmoleculeask=480N/m,findthe
energydifferencebetweenthegroundstateand
thefirstexcitedvibrationallevel.
14.
ThenucleioftheO2moleculeare
separatedby1.201010m.Themassofeach
oxygenatominthemoleculeis2.661026kg.
(a)Determinetherotationalenergiesofan
oxygenmoleculeinelectronvoltsforthelevels
correspondingtoJ=0,1,and2.(b)The
effectiveforceconstantkbetweentheatomsin
theoxygenmoleculeis1177N/m.Determine
thevibrationalenergies(inelectronvolts)
correspondingtov=0,1,and2.
15.
FigureP43.15isamodelofabenzene
molecule.Allatomslieinaplane,andthe
carbonatomsformaregularhexagon,asdothe
hydrogenatoms.Thecarbonatomsare0.110
nmapartcentertocenter.Determinethe
allowedenergiesofrotationaboutanaxis
perpendiculartotheplaneofthepaperthrough
thecenterpointO.Hydrogenandcarbonatoms
havemassesof1.671027kgand1.991026
kg,respectively.
17.
IftheCOmoleculewererigid,the
rotationaltransitionintowhatJstatewould
absorbthesamewavelengthphotonasthe0to1
vibrationaltransition?(Useinformationgivenin
Table43.2.)
18.
Calculatethemomentofinertiaofan
HClmoleculefromitsinfraredabsorption
spectrumshowninFigure43.9.
FigureP43.15
16.
Calculatethelongestwavelengthinthe
rotationalspectrumofHCl.TaketheClatomto
betheisotope35Cl.Theequilibriumseparation
oftheHandClatomsis0.12746nm.The
atomicmassoftheHatomis1.007825u,and
thatofthe35Clatomis34.968853u.(b)What
If?Repeatthecalculationin(a),buttaketheCl
atomtobetheisotope37Cl,whichhasatomic
mass36.965903u.Theequilibriumseparation
distanceisthesameasin(a).(c)Naturally
occurringchlorinecontainsapproximatelythree
partsof35Cltoonepartof37Cl.Becauseofthe
twodifferentClmasses,eachlineinthe
microwaverotationalspectrumofHClissplit
intoadoublet.Calculatethedoubletseparation
forthelongestwavelength.(Figure43.9shows
thedoubletsintheinfraredvibrational
spectrum).
19.
AnH2moleculeisinitsvibrationaland
rotationalgroundstates.Itabsorbsaphotonof
wavelength2.2112mandjumpstothev=1,
J=1energylevel.Itthendropstothev=0,J=
2energylevel,whileemittingaphotonof
wavelength2.4054m.Calculate(a)the
momentofinertiaoftheH2moleculeaboutan
axisthroughitscenterofmassand
perpendiculartotheHHbond(b)the
vibrationalfrequencyoftheH2molecule(c)the
equilibriumseparationdistanceforthis
molecule.
20.
Photonsofwhatfrequenciescanbe
spontaneouslyemittedbyCOmoleculesinthe
statewithv=1andJ=0?
21.
Mostofthemassofanatomisinits
nucleus.Modelthemassdistributionina
diatomicmoleculeastwospheres,eachof
radius2.001015mandmass1.001026kg,
locatedatpointsalongthexaxisinFigure
43.5a,andseparatedby2.001010m.Rotation
abouttheaxisjoiningthenucleiinthediatomic
moleculeisordinarilyignoredbecausethefirst
excitedstatewouldhaveanenergythatistoo
hightoaccess.Toseewhy,calculatetheratioof
theenergyofthefirstexcitedstateforrotation
aboutthexaxistotheenergyofthefirstexited
stateforrotationabouttheyaxis.
Section43.3BondinginSolids
22.
Useamagnifyingglasstolookatthe
tablesaltthatcomesoutofasaltshaker.
ComparewhatyouseetoFigure43.10(a).The
distancebetweenasodiumionandanearest
neighborchlorineionis0.261nm.(a)Makean
orderofmagnitudeestimateofthenumberNof
atomsinatypicalgrainofsalt.(b)WhatIf?
Supposethatyouhadanumberofgrainsofsalt
equaltothisnumberN.Whatwouldbethe
volumeofthisquantityofsalt?
23.
UseEquation43.18tocalculatetheionic
cohesiveenergyforNaCl.Take=1.7476,r0
=0.281nm,andm=8.
24.
ThedistancebetweentheK+andClions
inaKClcrystalis0.314nm.Calculatethe
distancesfromoneK+iontoitsnearest
neighborK+ions,toitssecondnearestneighbor
K+ions,andtoitsthirdnearestneighborK+
ions.
25.
Consideraonedimensionalchainof
alternatingpositiveandnegativeions.Showthat
thepotentialenergyassociatedwithoneofthe
ionsanditsinteractionswiththerestofthis
hypotheticalcrystalis
U ( r ) = k e
e2
r
wheretheMadelungconstantis=2ln2andr
istheinterionicspacing.[Suggestion:Usethe
seriesexpansionforln(1+x).]
Section43.4FreeElectronTheoryofMetals
Section43.5BandTheoryofSolids
26.
ShowthatEquation43.25canbe
expressedasEF=(3.651019) ne
2/3
eVwhere
EFisinelectronvoltswhenneisinelectronsper
cubicmeter.
27.
TheFermienergyforsilveris5.48eV.
Silverhasadensityof10.6103kg/m3andan
atomicmassof108.Usethisinformationto
showthatsilverhasonefreeelectronperatom.
28.
(a)Findthetypicalspeedofa
conductionelectronincopper,takingitskinetic
energyasequaltotheFermienergy,7.05eV.
(b)Howdoesthiscomparewithadriftspeedof
0.1mm/s?
29.
Sodiumisamonovalentmetalhavinga
densityof0.971g/cm3andamolarmassof23.0
g/mol.Usethisinformationtocalculate(a)the
densityofchargecarriersand(b)theFermi
energy.
30.
Whensolidsilverstartstomelt,whatis
theapproximatefractionoftheconduction
electronsthatarethermallyexcitedabovethe
Fermilevel?
31.
Calculatetheenergyofaconduction
electroninsilverat800K,assumingthe
probabilityoffindinganelectroninthatstateis
0.950.TheFermienergyis5.48eVatthis
temperature.
Toconfinetheelectroninsidethebox,takeU=
0insideandU=outside.)
32.
Consideracubeofgold1.00mmonan
edge.Calculatetheapproximatenumberof
conductionelectronsinthiscubewhose
energieslieintherange4.000to4.025eV.
35.
(a)Considerasystemofelectrons
confinedtoathreedimensionalbox.Calculate
theratioofthenumberofallowedenergylevels
at8.50eVtothenumberat7.00eV.(b)What
If?CopperhasaFermienergyof7.0eVat300
K.Calculatetheratioofthenumberofoccupied
levelsatanenergyof8.50eVtothenumberat
theFermienergy.Compareyouranswerwith
thatobtainedinpart(a).
33.
Showthattheaveragekineticenergyof
aconductionelectroninametalat0KisEav=
3/5EF.(Suggestion:Ingeneral,theaverage
kineticenergyis
Section43.6ElectricalConductioninMetals,
Insulators,andSemiconductors
whereneisthedensityofparticles,N(E)dEis
givenbyEquation43.22,andtheintegralisover
allpossiblevaluesoftheenergy.)
34.
Reviewproblem.Anelectronmovesin
athreedimensionalboxofedgelengthLand
volumeL3.Thewavefunctionoftheparticleis
=Asin(kxx)sin(kyy)sin(kzz).Showthatits
energyisgivenbyEquation43.20,
E=
(
2 2
2
2
2
nx + n y + nz
2
2 me L
)
wherethequantumnumbers(nx,ny,nz)are
integers1.(Suggestions:TheSchrdinger
equationinthreedimensionsmaybewritten
2 2 2 2
+
+
2m x 2 y 2 z 2
= (U E )
36.
Theenergygapforsiliconat300Kis
1.14eV.(a)Findthelowestfrequencyphoton
thatwillpromoteanelectronfromthevalence
bandtotheconductionband.(b)Whatisthe
wavelengthofthisphoton?
37.
Lightfromahydrogendischargetubeis
incidentonaCdScrystal.Whichspectrallines
fromtheBalmerseriesareabsorbedandwhich
aretransmitted?
38.
Alightemittingdiode(LED)madeof
thesemiconductorGaAsPemitsredlight(=
650nm).DeterminetheenergybandgapEgin
thesemiconductor.
39.
Mostsolarradiationhasawavelengthof
1morless.Whatenergygapshouldthe
materialinasolarcellhaveinordertoabsorb
thisradiation?Issiliconappropriate(seeTable
43.5)?
40.
Assumeyouaretobuildascientific
instrumentthatisthermallyisolatedfromits
surroundings,butsuchthatyoucanusean
externallasertoraisethetemperatureofatarget
insideit.(Itmightbeacalorimeter,butthese
designcriteriacouldapplytootherdevicesas
well.)Sinceyouknowthatdiamondis
transparentandagoodthermalinsulator,you
decidetouseadiamondwindowinthe
apparatus.Diamondhasanenergygapof5.5eV
betweenitsvalenceandconductionbands.What
istheshortestlaserwavelengthyoucanuseto
warmthesampleinside?
41.
Reviewproblem.Siliconisa
semiconductorwidelyusedincomputerchips
andotherelectronicdevices.Itsmostimportant
propertiesresultfromdopingitwithimpurities
inordertocontrolitselectricalconductivity.
Phosphorus,whichisadjacenttosiliconinthe
periodictable,hasfiveoutervalenceelectrons
ascomparedtofourforsilicon.Whena
phosphorusatomissubstitutedforasilicon
atominacrystal,fourofthephosphorus
valenceelectronsformbondswithneighboring
atomsandtheremainingelectronismuchmore
looselybound.Youcanmodeltheelectronas
freetomovethroughthecrystallattice.The
phosphorusnucleushasonemorepositive
chargethandoesthesiliconnucleus,however,
sotheextraelectronprovidedbythephosphorus
atomisattractedtothissinglenuclearcharge
+e.Theenergylevelsoftheextraelectronare
similartothoseoftheelectronintheBohr
hydrogenatomwithtwoimportantexceptions.
First,theCoulombattractionbetweenthe
electronandthepositivechargeonthe
phosphorusnucleusisreducedbyafactorof
1/fromwhatitwouldbeinfreespace(seeEq.
26.21),sotheorbitradiiaregreatlyincreased.
Hereisthedielectricconstantofthecrystal,
withavalueof11.7insilicon.Second,the
influenceoftheperiodicelectricpotentialofthe
latticecausestheelectrontomoveasifithadan
effectivemassm*,quitedifferentfromthemass
meofafreeelectron.OnecanusetheBohr
modelofhydrogentoobtainfairlyaccurate
valuesfortheallowedenergylevelsoftheextra
electron.Theseenergylevels,calleddonor
states,playanimportantroleinsemiconductor
devices.Assumethatm*=0.220me.Calculate
theenergyandtheradiusforanextraelectronin
thefirstBohrorbitaroundadonoratomin
silicon.
Section43.7SemiconductorDevices
Note:Problem74inChapter27canbe
assignedwiththissection.
42.
ForwhatvalueofthebiasvoltageVin
Equation43.27does(a)I=9.00I0?(b)(What
If?)I=0.900I0?AssumeT=300K.
43.
ThediodeshowninFigure43.28is
connectedinserieswithabatteryanda150
resistor.Whatbatteryemfisrequiredfora
currentof25.0mA?
44.
Youputadiodeinamicroelectronic
circuittoprotectthesystemincaseanuntrained
personinstallsthebatterybackward.Inthe
correctforwardbiassituation,thecurrentis200
mAwithapotentialdifferenceof100mV
acrossthediodeatroomtemperature(300K).If
thebatterywerereversed,whatwouldbethe
magnitudeofthecurrentthroughthediode?
indiameterwhena0.0200Tmagneticfield
directedperpendiculartotheringissuddenly
decreasedtozero.Theinductanceoftheringis
3.10108H.
45.
Adiode,aresistor,andabatteryare
connectedinaseriescircuit.Thediodeisata
temperatureforwhichkBT=25.0meV,andthe
saturationvalueI0ofthecurrentis1.00A.The
resistanceoftheresistoris745,andthe
batterymaintainsaconstantpotentialdifference
betweenitsterminalsof2.42V.(a)Find
graphicallythecurrentintheloop.Proceedas
follows.Onthesameaxes,drawgraphsofthe
diodecurrentIDandthecurrentinthewireIW
versusthevoltageacrossthediodeV.Choose
valuesofVrangingfrom0to0.250Vinsteps
of0.025V.DeterminethevalueofVatthe
intersectionofthetwographlines,andcalculate
thecorrespondingcurrentsIDandIW.Dothey
agree?(b)Findtheohmicresistanceofthe
diode,whichisdefinedastheratioV/ID.(c)
Findthedynamicresistanceofthediode,which
isdefinedasthederivatived(V)/dID.
48.
Aconvincingdemonstrationofzero
resistance.Adirectandrelativelysimple
demonstrationofzeroDCresistancecanbe
carriedoutusingthefourpointprobemethod.
TheprobeshowninFigureP43.48consistsofa
diskofYBa2Cu3O7(ahighTcsuperconductor)
towhichfourwiresareattachedbyindium
solderorsomeothersuitablecontactmaterial.
Currentismaintainedthroughthesampleby
applyingaDCvoltagebetweenpointsaandb,
anditismeasuredwithaDCammeter.The
currentcanbevariedwiththevariable
resistanceR.ThepotentialdifferenceVcd
betweencanddismeasuredwithadigital
voltmeter.Whentheprobeisimmersedinliquid
nitrogen,thesamplequicklycoolsto77K,
belowthecriticaltemperatureofthematerial,
92K.Thecurrentremainsapproximately
constant,butVcddropsabruptlytozero.(a)
Explainthisobservationonthebasisofwhat
youknowaboutsuperconductors.(b)Thedata
inTableP43.48representactualvaluesofVcd
fordifferentvaluesofItakenonthesampleat
roomtemperature.A6Vbatteryinserieswitha
variableresistorRsuppliedthecurrent.The
valuesofRrangedfrom10to100.The
dataarefromoneauthorslaboratory.Makean
IVplotofthedata,anddeterminewhetherthe
samplebehavesinalinearmanner.Fromthe
dataobtainavaluefortheDCresistanceofthe
sampleatroomtemperature.(c)Atroom
temperatureitisfoundthatVcd=2.234mVfor
I=100.3mA,butafterthesampleiscooledto
Section43.8Superconductivity
46.Athinrodofsuperconductingmaterial2.50
cmlongisplacedintoa0.540Tmagneticfield
withitscylindricalaxisalongthemagneticfield
lines.(a)Sketchthedirectionsoftheapplied
fieldandtheinducedsurfacecurrent.(b)Find
themagnitudeofthesurfacecurrentonthe
curvedsurfaceoftherod.
47.
Determinethecurrentgeneratedina
superconductingringofniobiummetal2.00cm
77K,Vcd=0andI=98.1mA.Whatdoyou
thinkmightcausetheslightdecreaseincurrent?
Figure P43.48: Circuit diagram used in the
fourpoint probe measurement of the DC
resistance of a sample. A DC digital
ammeterisusedtomeasurethecurrent,and
thepotentialdifferencebetween c and d is
measuredwithaDCdigitalvoltmeter.Note
thatthereisnovoltagesourceintheinner
loopcircuitwhereVcdismeasured.
TableP43.48
Current Versus Potential Difference Vcd
Measured in a Bulk Ceramic Sample of
YBa2Cu3O7atRoomTemperature
I(mA)
Vcd(mV)
57.8
61.5
68.3
76.8
87.5
102.2
123.7
155
1.356
1.441
1.602
1.802
2.053
2.398
2.904
3.61
AdditionalProblems
49.
AsyouwilllearninChapter44,carbon
14
14( C)isanisotopeofcarbon.Ithasthesame
chemicalpropertiesandelectronicstructureas
themuchmoreabundantisotopecarbon12
(12C)buthasdifferentnuclearproperties.Its
massis14u,greaterbecauseithastwoextra
neutronsinitsnucleus.AssumethattheCO
molecularpotentialisthesameforbothisotopes
ofcarbon,andthatthetablesandexamplesin
Section43.2refertocarbonmonoxidewith
carbon12atoms.(a)Whatisthevibrational
frequencyof14CO?(b)Whatisthemomentof
inertiaof14CO?(c)Whatwavelengthsoflight
canbeabsorbedby14COinthe(v=0,J=10)
statethatwillcauseittoendupinthev=1
level?
50.
Theeffectivespringconstantassociated
withbondingintheN2moleculeis2297N/m.
Thenitrogenatomseachhaveamassof2.32
1026kg,andtheirnucleiare0.120nmapart.
Assumethatthemoleculeisrigidandinthe
groundvibrationalstate.CalculatetheJvalueof
therotationalstatethathasthesameenergyas
thefirstexcitedvibrationalstate.
51.
Thehydrogenmoleculecomesapart
(dissociates)whenitisexcitedinternallyby4.5
eV.Assumingthatthismoleculebehaveslikea
harmonicoscillatorhavingclassicalangular
frequency=8.281014rad/s,findthehighest
vibrationalquantumnumberforastatebelow
the4.5eVdissociationenergy.
52.
Underpressure,liquidheliumcan
solidifyaseachatombondswithfourothers,
andeachbondhasanaverageenergyof1.74
1023J.Findthelatentheatoffusionforhelium
injoulespergram.(ThemolarmassofHeis
4.00g/mol.)
53.
Showthattheioniccohesiveenergyof
anionicallybondedsolidisgivenbyEquation
43.18.(Suggestion:StartwithEquation43.17,
andnotethatdU/dr=0atr=r0.)
54.
Thedissociationenergyofgroundstate
molecularhydrogenis4.48eV,whileitonly
takes3.96eVtodissociateitwhenitstartsin
thefirstexcitedvibrationalstatewithJ=0.
Usingthisinformation,determinethedepthof
theH2molecularpotentialenergyfunction.
55.
Aparticlemovesinonedimensional
motionthroughafieldforwhichthepotential
energyoftheparticlefieldsystemis
U(x)=(A/x3)(B/x)
whereA=0.150eVnm3andB=3.68eVnm.
Thegeneralshapeofthisfunctionisshownin
Figure43.11,wherexreplacesr.(a)Findthe
staticequilibriumpositionx0oftheparticle.(b)
DeterminethedepthU0ofthispotentialwell.
(c)Inmovingalongthexaxis,whatmaximum
forcetowardthenegativexdirectiondoesthe
particleexperience?
56.
Aparticleofmassmmovesinone
dimensionalmotionthroughafieldforwhich
thepotentialenergyoftheparticlefieldsystem
is
U(x)=(A/x )(B/x)
3
whereAandBareconstantswithappropriate
units.Thegeneralshapeofthisfunctionis
showninFigure43.11,wherexreplacesr.(a)
Findthestaticequilibriumpositionx0ofthe
particleintermsofm,A,andB.(b)Determine
thedepthU0ofthispotentialwell.(c)Inmoving
alongthexaxis,whatmaximumforcetoward
thenegativexdirectiondoestheparticle
experience?
57.
AsanalternativetoEquation43.1,
anotherusefulmodelforthepotentialenergyof
adiatomicmoleculeistheMorsepotential
[
]
U ( r ) = B e a ( r r0 ) 1
2
whereB,a,andr0areparametersusedtoadjust
theshapeofthepotentialanditsdepth.(a)What
istheequilibriumseparationofthenuclei?(b)
Whatisthedepthofthepotentialwell,i.e.,the
differenceinenergybetweenthepotentials
minimumvalueanditsasymptoteasr
approachesinfinity?(c)Ifisthereducedmass
ofthesystemoftwonuclei,whatisthe
vibrationalfrequencyofthediatomicmolecule
initsgroundstate?(Assumethatthepotentialis
nearlyparabolicaboutthewellminimum).(d)
Whatamountofenergyneedstobesuppliedto
thegroundstatemoleculetoseparatethetwo
nucleitoinfinity?
58.
TheFermiDiracdistributionfunction
canbewrittenas
f ( E) =
1
e
( E E F ) / k B T
+1
=
1
e
( E / E F 1 ) T F / T
+1
whereTFistheFermitemperature,defined
accordingto
k BTF E F
Writeaspreadsheettocalculateandplotf(E)
versusE/EFatafixedtemperatureT.Examine
thecurvesobtainedforT=0.1TF,0.2TF,and
0.5TF.
59.
TheMadelungconstantforsodium
chloridemaybefoundbysumminganinfinite
Copyright2004Thomson.Allrightsreserved.
alternatingseriesoftermsgivingtheelectric
potentialenergybetweenaNa+ionanditssix
nearestClneighbors,itstwelvenextnearest
Na+neighbors,andsoon(Fig.43.10a).(a)From
thisexpression,showthatthefirstthreetermsof
theseriesyield=2.13fortheNaClstructure.
(b)WhatIf?Doesthisseriesconvergerapidly?
Calculatethefourthtermasacheck.