Energy Levels in One Dimension

7/23/2019 Energy Levels in One Dimension

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Free electron Fermi gas

Figure 1 Schematic model of a crystal of sodium metal. The atomic cores are Na' ions: they are

immersed in a sea of conduction electrons. The conduction electrons are derived from the 3s

valence electrons of the free atoms. The atomic cores contain 10 electrons in the configuration

l s 2 2 s 2 2 pI~n an alkali metal the atomic cores occupy a relatively small part (1! percent" of

the total volume of the crystal# hut in a nohle metal ($u# Ag, %u" the atomic cores are relatively

larger and may he in contact &ith each other. The common crystal structure at room temperature

is hcc for the alkali metals and fcc for the nohle metals.

In a theory which has given results like these, there must

certainly be a great deal of truth,

.%. orent)

*e can understand many physical properties of metals# and not only of the simple metals#

in terms of the free electron model. %ccording to this model# the valence electrons of the

constituent atoms +ecome conduction electrons and move a+out freely through the volume of the

metal. ,ven in metals for &hich the free electron model &orks +est# the charge distri+ution of the

conduction electrons reflects the strong electrostatic potential of the ion cores. The utility of the

free electron model is greatest for properties that depend essentially on the kinectic properties of

the conduction electrons. The interaction of the conduction electrons &ith ions of the lattice is

treated in the ne-t chapter.

The simplest metals are the alkali metalslithium# sodium# potassium# cesium# and

ru+idium. In a free atom of sodium the valence electron is in a3 s

state in the metal this

electron +ecomes a conduction electron in the3 s

conduction +and.


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% monovalent crystal &hich contains N atoms &ill have N conduction electrons and N

positive ion cores. The+

Na

ion cores contains 10 electrons that occupy the

1 s ,2 s ,2p shells of the free ion# &ith a spatial distri+ution that is essentially the same &hen

in the metal as in the free ion. The ion cores fill only a+out15

percent of the volume of a

sodium crystal# as in /ig.1. the radius of the free+

Na

ion is0.98A

# &hereas onehalf of

the nearestneigh+or distance of the metal is 1.83A .

The interpretation of metallic properties in terms of the motion of free electrons &as

developed long +efore the invention of uantum mechanics. The classical theory had several

conspicuous successes# nota+ly the derivation of the form of hm2s la& and the relation +et&een

the electrical and thermal conductivity. The classical theory fails to e-plain the heat capacity and

the magnetic suscepti+ility of the conduction electrons. (these are not failures of the free electron

model# +ut failures of the classical 3a-&ell distri+ution function".

There is a further difficulty &ith the classical model. /rom many types of e-periments it

is clcar that a condition electron in a metal can move freely in a straight path over many atomic

distances# undeflected +y collisions &ith other conduction electrons or +y collisions &ith the

atom cores. In a very pure specimen at lo& temperatures# the mean free path may +e as long as

108

interatomic spacings (more than 1 cm".

*hy is condensed matter so transparent to conduction electrons4 The ans&er to the

uestion contains t&o parts: (a" % conduction electron is not deflected +y ion cores arranged on a

periodic lattice +ecause matter &aves can propagate freely in a periodic structure# as a

conseuence of the mathematics treated in the follo&ing chapter. (+" % conduction electron is

scattered only infreuently +y other conduction electrons. This property is a conseuence of the

pauli e-clusion principle. 5y a free electron /ermi gas# &e shall mean a gas of free electrons

su+6ect to the pauli principle.


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Energy levels in one dimension

$onsider a free electron gas in one dimension# taking account of uantum theory and of

the principle. %n electron of mass m is confined to a length L +y infinite +arriers (fig.7". the

&avefunctionn(x) of the electron is a solution of the schrodinger euation H=

&ith the neglect of potensial energy &e have H=2/2m # &here is the momentum. In

uantum theory may +e represented +y the operator id /dx # so that

Hn=2

2m

d2

n

dx2=nn


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*heren is the energy of the electron in the or+ital.

*e use the term or+ital to denote a solution of the &ave euation for a system of only one

electron. The term allo&s us to distinguish +et&een an e-act uantum state of the &ave euation

of a system of N interacting electrons and an appro-imate uantum state &hich &e construct

+y assigning theN

electrons toN

different or+itals# &here each or+ital is a solution of a

&ave euation for one electron. The or+ital model is e-act only if there are no interactions

+et&een electrons.

The +oundary conditions aren (0 )0 n (L )=0 as imposed +y the infinite

potential energy +arriers. They are satisfied if the &avefunction is sinelike &ith an integral

num+er n of half&avelengths +et&een 0L :

n=A sin( 2 n x) 1

2n n=L #

*here % is a constant. *e see that (7" is a solution of (1"# +ecause

d n

dx=A( nL)cos ( nL x )

d2

n

dx2=A ( nL)

2

sin( nL x )

*hence the energyn is given +y

n=

2

2m ( nL)2

*e &ant to accommodate N electrons on the line. %ccording to the pauli e-clusion

principle# no t&o electrons can have all their uantum num+ers


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/igure 7 first three energy levels and &ave functions of a free electron of mass m confined to a

line of length . The energy levels arc la+eled according to the uantum num+er n &hich gives

the num+er of half&avelengths in the &avefunction. The &avelengths are indicated on the &ave

function. The &avelengths are indicated on the &avefunctions. The energyn of the level of

uantum num+ern

is eual to ( 2

2m )( n2L )2

.

Identical. That is# each or+ital can +e occupied +y at most one electron. This applies to electrons

in atoms# molecules# or solids.

In a linear solid the uantum num+ers of a conduction electron or+ital are n and m# &here

n is any positive interger and the magnetic uantum num+erms=

1

2 # according to spin

orientation. % pair of or+itals la+eled +y the uantum num+er n can accommodate t&o electrons#

one &ith spin up and one &ith spin do&n.

If there are si- electrons# then in the ground state of the system the filled or+itals are

those given in the ta+le:


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3ore than one or+ital may have the same energy. The num+er of or+itals &ith the same energy is

called the degeneracy.

etn f denote the topmost filled energy level# &here &e start filling the levels from the

+ottom (n81" and continue filling higher levels &ith electrons until all N electrons are

accommodated. It is convenient to suppose that N is an even num+er. The condition2nf=N

determinesn f # the value of n for the uppermost filled level.

The /ermi energyf is defined as the energy of the topmost filled level in the ground

state of the N electron system. 5y (9" &ithn=nf &e have in one dimension:

f=

2

2m(nf

L)2

=

2

2m (N2L )2


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Effect of temperature on the Fermi-dirac distribution

The ground state is the state of the N electron system at a+solute )ero. *hat happens as

the temperatuere is increased4 This is a standard pro+lem in elementary statistical mechanics#

and the solution is given +y the /ermidirac distri+ution gives the pro+a+ility that an or+ital at

energy &ill +e occupied in an ideal electron gas in thermal euili+rium:

f()= 1

exp [ kB T]+1

The uantity is a function of the temperature is to +e chosen for the particular

pro+lem in such a &ay that the total num+er of particles in the system comes out correctly that

is# eual to N. at a+solute )eroF # +ecause in the limit T0 the function f( )

$hanges discontinuously from the value 1 (filled" to the value 0 (empty" atF= . %t all

temperatures f() is eual to ; &hen = # for then the denominator of (!" has the value

7.


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/igure 9 /ermidirac distri+ution function (!" at the various la+eled temperatures# for

Tf= f

kb=50000K

. The results apply to a gas in three dimensions. The total num+er of

particles is constant# independent of temperature. The chemical potential at each

temperature may +e read off the graph as the energy at &hichf=0.5 .

The uantity is the chemical potential (T


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Free electron gas in three dimensions

The freeparticle schrodinger euation euation in three dimensions is

2

2m(

2

x2+

2

y2+

2

2

)k(! )=kk(! )

If the electrons are confined to a cu+e of cu+e of edge # the &avefunction is the standing &ave

n (! )=A sin( nxxL )sin ( nyy

L )sin( n

L )

*herenx , ny , n are positive integers. The origin is at one corner of the cu+e.

It is convenient to introduce &avefunctions that satisfy periodic +oundary conditions# as

&e did for phonons in $hapter !. *e no& reuire the &avefunctions to +e periodic in -# y# )#

&ith period . Thus

%nd similarly for the y and ) coordinates. *avefunctions satisfying the free particle schrodinger

euation and the periodicity condition are of the form of a traveling plane &ave:


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exp [ ikx (x+L )]=exp i 2n(x+L )

L

exp(i2nx

L

)exp (i2n)=exp(

i2nx

L

)=exp (i kxx ) .


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n su+stituting (>" in (?" &e have the energyk of the or+ital &ith &avevector k:

k=

2

2mk

2=

2

2m(kx

2+ky2+k

2)

The magniture k of the &avevector is related to the &avelength +y k=2

The linear momentude p may +e represented in uantum mechanics +y the operator

p=i # &hence for the or+ital (>"

pk(! )=i k(! )=k k(! ) #

So that the plane &avek is an eigenfunction of the linear momentum &ith the eigenvalue

k The particle velocity in the or+ital k is given +y "=k/m .

In the ground state of a system of N free electrons# the occupied or+itals may +e

represented as points inside a sphere in k space. The energy at the surface of the sphere is the

/ermi energy the &avevectors at the /ermi surface have a manigtudekF

such that (fig.@":

F=

2

2mkF

2

/rom (10" &e see that there is one allo&ed &avevector Athat is# one distinct triplet of

uantum num+ers k-# ky# k)# for the volume element (2

L )

3

of k space. Thus in the sphere

of volume 4 kF3 /3 the total num+er of or+itals is


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2.4 kF

3 /3

(2

L )

3 =

#

3 2kF

3 =N

*here the factor 7 on the left comes from the t&o allo&ed values of the spin uantum num+er

for each allo&ed value of k. then (1!" gives

kF=( 32N

# )1 /3

*hich depends only on the particle concentration.

/igure @ in the ground state of a system of N free electrons the occupied or+itals of the system

fill a sphere of radiuskF # &here F=

2kF2 /2m is the energy of an electron having a

&avevectorkF .


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Energy Levels in One Dimension