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254 CHAPTER16. LIVEWIRESANDDEADSTARS

16.1 The Kronig-Penny Model

A crystalline solid isa regular array of atoms, and, at first sight, conductionof
electricityisamystery: ifelectronsareboundtoatoms,howispossibleforthemto
movethroughthesolidundertheinfluenceofasmallelectricfield? Theansweris
thatinacrystal, notallof theelectronsareactuallyboundtotheatoms;infact,
someoftheelectronsinthemetalbehavemorelikeagasoffreeparticles,albeitwith
somepeculiarcharacteristicswhichareduetotheexclusionprinciple.
Tounderstandhowelectronsinacrystalcanactasagas,itisusefultosolvefor
theelectronenergyeigenstatesinahighlyidealizedmodelofasolid,knownasthe
Kronig-Pennymodel,whichmakesthefollowingsimplifications:

S1.Thesolidisone-dimensional,ratherthanthree-dimensional. TheNatomsare

spacedadistanceafromoneanother.Inorderthattherearenospecialeffects

attheboundaries,weconsiderasolidhasnoboundaryatall,byarrangingthe

atomsinacircleasshowninFig.[16.1].

S2. InsteadofaCoulombpotential,thepotentialofthen-th atomisrepresented

byadelta-functionattractivepotentialwell

Vn(x)=−gδ(x−xn) (16.2)

wherexnisthepositionofthen-thatom.

S3.Interactionsbetweenelectronsinthe1-dimensionalsolidareignored.

Obviously, these areprettydrastic simplifications. Theimportantfeature of this
model, whichitshares withrealisticsolids, isthat the electronsaremoving ina
periodicpotential. Forpurposesof understandingtheexistenceofconductivity, it
istheperiodicityofthepotential,notitspreciseshape(orevenitsdimensionality)
whichisthecrucialfeature.
Arrangingtheatomsinacircle,asinFig.[16.1],meansthatthepositionvariable
isperiodic,likeanangle.Justasθ+ 2 πisthesameangleasθ,sothepositionx+L
isthesamepositionasx,where
L=Na (16.3)

isthelengthofthesolid. Letthepositionof then-thparticlebexn =na, n=
0 , 1 ,...,N−1,thepotentialthenhastheform

V(x)=−g

N∑− 1

n=0

δ(x−na) (16.4)

Itsclearthatthepotentialsatisfies

V(x+a)=V(x) (16.5)