Physics devices function property MATERIALS.pdf

physics devices function property.

1. Physics of Functional Materials & Devices
Prof. Amreesh Chandra
Department of Physics, IIT KHARAGPUR
Module03:Introductiontotheoryofsolids
Lecture14:TheoryofSolidsNPTEL

2. Definitionsan dClassificationsofSolids
Drude- L
ClassicalTheory
Sommerfeld’sQ
uantumTheory
ApplicationsofF
reeElectronGasModelNPTEL

3. 3
Definition of solids:
Asolid isa state of matterwhich hasa fixedshape,mass,andvolume.Itshowsverysmallchangesinvolumeby
changing thetemperature.
Accordingtothe natureofbandoccupationbyelectrons,allsolidscanbe broadlyclassifiedintothreetypes.
Metals Semiconductors Insulators
Solids
The fi completelyfilledvalanceband overlappingwithapartiallyfilled
conduction band
(Metal).
Depending on thewidthof the forbidden band, the second groupcanbe dividedinto:
Insulator (band gap????????????
????????????>????????????eV)
Semiconductor (band gap????????????
????????????≤1.5eV)NPTEL

4. 4
Difference between metal, insulator, and semiconductor in terms of energy band
Ingeneral,metalisanyclassofsubstancescharacterizedby highelectricalandthermal
conductivity aswellasbymalleability, ductility,and highreflectivity oflight.
Thetheory of metalsarepostulatedby Drudein1900.
Thetheory was named asfreeelectrongas model.NPTEL

5. Postulates of free electron gas model :
Theme consistofpositiveioncoreswiththevalanceelectronsmovingfreelyamongthesecores.
Thee
areboundtomovewithinthemetal,dueto electrostaticattractionbetweenthepositiveioncoresand
electrons.
Thepot
fieldoftheseioncores,whichisresponsibleforsuchan attraction,isassumedtobeconstant throughout the
metal.
The m
amongtheelectrons,isneglected.
Theb
offreeelectronsinside themetalsisconsideredsimilarto that of atoms ormoleculesinaperfectgas.
Hence,it is calledfreeelectrongas.
5
Free Electron Gas Model
Definitely, there are some basic differences
between ordinary gas and free electron gas.NPTEL

6. 6
Basic differences between ordinary gas and free electron gas
Thef electrongasis negativelycharged, whereasthemoleculesof an ordinary gasaremostlyneutral.
Theconcentration ofelectrons inanelectrongasis quitelarge comparedtotheconcentration ofmoleculesinan ordinary
gas.
Thevalanceelectronsare responsiblefortheconduction ofelectricitythroughmetals,hencecalledconductionelectrons.
Since theconductionelectronsmoveinauniformelectrostatic fieldof ioncores,their potentialenergyremainsconstant
andis normallytaken aszero,ignoring theexistenceof ioncores.
Asthemovementofelectronsisrestrictedto withinthecrystalonly,thepotentialenergyof a stationaryelectroninsidea
metal islessthan thepotentialenergyof anidenticalelectronjustoutsideit.
Theenergy differenceV
0servesas a potentialbarrierandstopsthe innerelectronsfrom
leavingthesurfaceofthemetal.
Some important points about free electron gas
Thus, themovementof afreeelectron in metal is similar tothemovementof afreeelectron
gasinsidea“potentialenergybox”.NPTEL

7. 7
Fig. Metallic surface bounded by a potential barrier V
0
Free Electron
Gas Model
Successfully
explained
Lorentzin190 9,postulatedthattheelectrons constitutingtheelectrons
constitutingtheelectrongas obey Maxwell-Boltzmanstatistics.
TheDrude-Lorentztheoryoftheelectrongasis calledclassicaltheory.
Themodelissuccessfulinexplaining various properties ofmetals.
Failure of Free
Electron Gas
Model
Can not
explain
Temperature dependence resistivity
Heat capacity
Paramagnetic susceptibility, etc.
Electrical conductivity
T
hermal conductivity
Thermionic emission
Thermoelectric effect
GalvanomagneticeffectNPTEL

8. 8
Toovercomethefailure oftheFreeelectrongas model, Sommerfeld’squantum theorywas adopted.
Sommerfeld treatedtheproblemquantummechanicallyusing theFermi-Diracstatistics.
Free Electron Gas in One- Dimensional box
Potential
Fig. one dimensional potential box
V
0 (for x ≤ 0 and x ≥ L
V(x) =
0 (for 0 ˂ x ˂L)
????????????
????????????
????????????(????????????)
????????????????????????
????????????
+
????????????????????????
????????????
????????????
????????????????????????
????????????=????????????
The Schrödinger takes the following form inside the crystal:
By solving the equation with the proper boundary conditions: (i) ψ????????????= 0 = 0
(ii) ψ????????????=????????????= 0
The general solution to this equations is:
ψ????????????=????????????????????????????????????????????????(????????????????????????) +????????????cos(????????????????????????)
(1)
(2)
The solution of eq. 1 in the region 0 ˂ x ˂a
becomes,
????????????
????????????=
????????????????????????????????????
????????????????????????????????????
????????????
(3)NPTEL

9. 9
Sommerfeld’sQuantumTheory
For each value of ‘n’, there is a corresponding quantum state????????????
????????????; whose energy????????????
????????????can be obtained as:
????????????
????????????=
ћ
????????????
????????????????????????
(
????????????????????????
????????????
)
????????????
The bo haveonly discrete energy values correspondingto????????????=????????????,????????????,????????????,…
The lowest energyoftheparticle is obtainedas????????????
????????????=
????????????
????????????
????????????????????????????????????
????????????
(for????????????=????????????) which is not equalstozero.
The spacing betweentwoconsecutive levels increasesas:(????????????+????????????)
????????????
????????????
????????????− ????????????
????????????
????????????
????????????=
(????????????????????????+????????????)????????????
????????????
Substituting equation 3 in equation 1, the energy value can be calculated.
(4)NPTEi

10. Free Electron Gas in One- Dimensional box
Fig. First three energy levels and wave function of a
free electron
oThequa mechanical energy levelsarediscrete.
oElectronsa
accommodated according toPauli’sexclusionprinciple.
oMoreth
oneorbitalmayhavethesameenergy,and thatnumberiscalled the
degeneracy.
????????????
????????????=
ћ
????????????
????????????????????????
(
????????????
????????????????????????
????????????????????????
)
????????????The fermi energy is given as:
The topmost filled energy level at 0 K is known as the Fermi level and the energy
corresponding to this level is called the Fermi energy E
F.
Where n
Frepresents the principal quantum number of
the Fermi level.
(5)NPTEi

11. Free Electron Gas in One- Dimensional box
According to FD statistics,oneenergy state canbeoccupiedbyamaximumof2 electrons.
Hence to
number ofaccommodated electrons,uptoFermi level is:N = 2n
F
Hence equa
For example: If we accommodate ????????????×????????????????????????
−????????????????????????
electrons on one centimetre length of line the fermi
energy of the topmost electron would be
????????????
????????????=
????????????.????????????×????????????????????????
−????????????????????????
erg = 2.4 eV
Thus the Fermi energy depends on the length of the box and the number of electrons in the box.
????????????
????????????=
ћ
????????????
????????????????????????
(
????????????????????????
????????????????????????
)
????????????NPTEL

12. Free Electron Gas in Three Dimensions
12
The free particle Schrödinger equation in three dimensional cube of edge L is:
Fig. Three dimensional potential box
−
ћ
????????????
????????????????????????
(
????????????
????????????
????????????????????????
????????????
+
????????????
????????????
????????????????????????
????????????
+
????????????
????????????
????????????????????????
????????????
)????????????
????????????(????????????) =????????????
????????????????????????
????????????
Similar to the 1D potential well case, here the wavefunction can be expressed as:
????????????
????????????(????????????) =????????????????????????????????????????????????
????????????????????????
????????????????????????
????????????
????????????????????????
????????????
????????????????????????
????????????????????????
????????????
????????????????????????
????????????
????????????????????????
????????????????????????
????????????
where, ????????????
????????????,????????????
????????????and ????????????
????????????are positive integers
Proceeding further, we have the energy value ????????????
????????????as:
????????????
????????????=
ћ
????????????
????????????????????????
????????????
????????????
=
ћ
????????????
????????????????????????
(????????????
????????????
????????????+????????????
????????????
????????????+????????????
????????????
????????????)
where;????????????
????????????=????????????????????????
????????????
????????????
, ????????????
????????????=
????????????????????????
????????????
????????????
and ????????????
????????????=
????????????????????????
????????????
????????????
Several c canyieldthesamevalueof energy.
Eachcombinationofthequantum numbers iscalleda quantumstate,and
severalstateshavingthesameenergyaretermeddegenerate.NPTEi

13. Filling of Energy Levels
Applications oftheFree ElectronGasModel
oThed ofelectrons followsPauli’sexclusionprinciple.
oAvailable s
of quantum numbers:k
x, k
y, k
zandm
s
�
????????????
????????????
− �
????????????
????????????
or
Hence each level can accommodate two electrons with set of (k
x, k
y, k
z, ±1/2)
Or
Each energy level is doubly degenerate
????????????????????????????????????????????????????????????????????????????????????????????????????????????
???????????????????????????????????? −????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????????????????????????????????????????
It needs ????????????/????????????
energy levels to fill up???????????????????????????????????? ????????????
Energy (E)
Surface
energy (E
s
)
Vacuum
E
F0
????????????/????????????????????????????????????l
????????????????????????????????????????????????????????????level
Unlike classical theory, Sommerfeld’s theory does not a llow condensation of all
electrons into the zero energy level even at 0.
Electrons are distributed among the discrete energy levels having energies ranging f
rom 0 to E
F0.
Fig. Sommerfeld’s free electron model at 0 K.NPTEi

14. 1414
The density of states (D(E)) can be also found by applying Sommerfeld’squantum theory.
The density of states (D(E)) is the total number of available electronic states (or orbitals) per
unit energy range at energy E.NPTEL

15. 1515
Derivation of the Density of States
Linear momentum is represented by quantum mechanical operator P ⇒
Now, ????????????????????????
????????????????????????=−????????????ћ ▽????????????
????????????????????????= ћ????????????????????????
????????????????????????
From eigen value, particle velocity in the orbital ????????????is given by:
−????????????ћ▽
????????????=�ћ????????????
????????????
Fig. Schematic representation of electron in ????????????space
????????????
????????????
????????????
????????????
????????????
????????????
Fermi surface
at energy????????????
????????????
oInth groundstateofasystemofNfreeelectrons,theoccupied orbitalsofthesystemfill a
sphere ofradius????????????
????????????, in the????????????space.
oTheFermienergyatthe surfaceofthe sphereis givenby:
????????????
????????????=
ћ
????????????
????????????????????????
????????????
????????????
????????????
oTheallowed wave vectors, i.e.,????????????
????????????,????????????
????????????and????????????
????????????occupythevolume element(⁄
????????????????????????
????????????
)
????????????
of????????????space.
So,thetotalnumber of orbitalintheFermi sphereofvolume
�
????????????????????????????????????
????????????
????????????
????????????is:
2.
�
4????????????????????????
????????????
3
3
(⁄
2????????????
????????????
)
3
=
????????????
3????????????
2
????????????
????????????
3=???????????? ????????????
????????????= (
????????????????????????
????????????
????????????
????????????
)
�
????????????
????????????
(1)
(2)NPTEL

16. Density of States
1616
From (1) and (2)⇒ ????????????
????????????=
ћ
????????????
????????????????????????
(
????????????????????????
????????????
????????????
????????????
)
�
????????????
????????????
⇒????????????=
????????????
????????????????????????
????????????
(
????????????????????????????????????
ћ
????????????
)
�
????????????
????????????
Density of states D(E) is the total number of available electronic
states (or orbitals) per unit energy range at energy E.
Fig. Density of states as a function of electron
energy
D(E) is given as:
????????????????????????=
????????????????????????
????????????????????????
=
????????????
????????????????????????
????????????
(
????????????????????????
ћ
????????????
)
�
????????????
????????????
.????????????
�
????????????
????????????
=
????????????????????????
????????????????????????NPTEL

17. Theb classical(Drude-Lorentz) andquantum(Sommerfeld)theoriesofmetalswere
discussed.
Thed
ofstatesoftheelectronsinmetalswas derivedfromSommerfeld’squantum
theory.
Somed
oftheclassicaltheorycouldbesolvedbythequantumtheory.NPTEL

18. •Physicsof F unctional MaterialsbyHasse Fredriksson&UllaAkerlind
•IntroductiontoNan
otechnology,CharlesP.Poole,Jr.andFrankJ.Owens,wiley-
interscience.
•ShriverA
AtkinsInorganicChemistrybyPeterAtkinsTina Overton,Jonathon Rourke.NPTEL

19. Thank you…NPTEL

20. Physics of Functional Materials & Devices
Prof. Amreesh Chandra
Department of Physics, IIT KHARAGPUR
Module03:IntroductiontoTheoryofSolids
Lecture15:NearlyfreeelectronmodelNPTEL

21. Nearlyfreeelectronmodel
Bloch’sTheorem
KronigPenneyModel
ExtendedandReducedZoneSchemeNPTEL

22. 3
In continuation of the free electron model of solids, it can be inferred that
Drawbacks of the free electron model:
Thefreeelectronmodelcannotexplainsomephenomena.
1.Itcouldnotaccountforthedifferenceinconductorsand insulators,i.e.whytheconductorshaveplentyof
freeelectronsbutthe insulatorsdonothaveany.
2.Itcannotexplainthevariationofresistivitywithtemperature,fortheinsulators.
3.Thevariouspropertiesofsemiconductorscouldnotbeexplainedwiththebasicmodel.
Toovercomethedeficiencies,theelectronsareconsideredtomovein
aperiodicpotentialwithaperiodequaltothelatticeconstant,which
isknownasthe“Nearlyfreeelectronmodel”.NPTEL

23. 4
Figure: Periodic lattice in one dimension
Accordingtothenearlyfreeelectronmodel,it isassumedthat theioncoresareatrestand thepotentialexperiencedby
theelectronsina crystalisperiodicwith aperiodequaltothelattice constant.
Nearlyfreeelectronmodel:
Theassumption ofthemodelisbasedon the fact thatthe
ioncoresinthecrystalsaredistributedperiodicallyonthe
latticesites.
Thepotential contributionduetoallotherfreeelectronsare
taken as a constant.
This typeofperiodicpotentialextends uptoinfinity inall
directionsexceptatthesurfaceofthecrystal, asthere
remainsinterruptionintheperiodicity.
Importantfactsofnearlyfreeelectronmodel:NPTEL

24. 5
Inthecaseoffreeelectrontheory,thereisno upperlimitofenergy.Accordingto that:
????????????????????????????????????????????????????????????????????????,????????????=
ћ
????????????
????????????
????????????
????????????????????????
k can havediscretevaluesinanyrangewhichmeans thattheenergylevelsarediscreteandmay haveanyspacing
dependingonthe dimensionsofthe box.
Accordingtothefreeelectronmodel,the one-dimensional Schrodingerequation for anelectroninaconstantpotential
V
0is:????????????
????????????
????????????
????????????????????????
????????????
+
????????????????????????
ћ
????????????
(???????????? − ????????????
????????????)????????????=???????????? (1)
????????????
????????????=????????????
±???????????????????????????????????? (2)
Thesolution to equation 1,isatypeofplanewave as given:
????????????????????????????????????????????????????????????,????????????−????????????
????????????=
ћ
????????????
????????????
????????????
????????????????????????
=
????????????
????????????
????????????????????????
= ????????????
????????????????????????????????????
(3)NPTEL

25. 6
Whenweconsider, periodicpotential V(x),the Schrodingerequationiswritten as:
Thepotential V(x)isperiodicwiththelattice constant a. i.e. V (x + a) = V (x)
Thesolution of equation 4isgovernedbya famous theorem,knownas
Bloch’stheorem.
????????????
????????????
????????????
????????????????????????
????????????
+
????????????????????????
ћ
????????????
(???????????? − ????????????
????????????)????????????=???????????? (4)
Bloch’sTheorem:
Where, ????????????
????????????????????????halattice. i.e., ????????????
????????????(????????????+????????????)= ????????????
????????????(????????????)
The Schrödinger equation in a periodic potential possesses the form:????????????????????????=????????????
????????????(????????????)????????????
±????????????????????????????????????
(5)
Thesolution of equation 4 takestheform of equation 5accordingtoBloch’stheorem.
Inthreedimensions,Bloch’stheoremisexpressedas:????????????????????????=????????????
????????????(????????????)????????????
????????????????????????.????????????
Figure: Periodic lattice in one dimension NPTEL

26. 7
Figure: Ideal periodic square well potential suggested by Kronigand Penney
KronigPenney model
Thepotentialenergyof anelectroninalineararrayof positivenucleiisassumedto havetheformof aperiodicarray
ofsquarewellswith aperiodof (a+b).
Themodel illustratesthebehavior ofelectronsinaperiodicpotentialby assuminga relativelysimpleone-dimensional
model of aperiodicpotential.
For,????????????<????????????<????????????(Region -I),atthebottom of awell,V = 0,theelectronisassumedtobeinthevicinityofthe nucleus.
Whereasoutsideawell,i.e., for−????????????<????????????<????????????,t hepotentialenergyisassumedtobeV
0.NPTEL

27. 8
KronigPenney model
Schrödinger equation for two region can be written as:
????????????
????????????
????????????
????????????????????????
????????????
+
????????????????????????
ћ
????????????
????????????????????????=????????????
????????????
????????????
????????????
????????????????????????
????????????
+
????????????????????????
ћ
????????????
(????????????−????????????
????????????)????????????=????????????
For, ????????????<????????????<????????????
For, 0>????????????>−????????????
Region-I
Region-II
Region-I
Region-II
(6)
(7)
Assumingthat theenergy Eoftheelectrons is lessthanV
0,wecan definetworealquantities????????????and????????????,as
????????????
????????????
=
????????????????????????????????????
ћ
????????????
????????????
????????????
=
????????????????????????
ћ
????????????
(????????????
????????????− ????????????)and
So,theequations 6 and 7 become
????????????
????????????
????????????
????????????????????????
????????????
+????????????
????????????
????????????=????????????
????????????
????????????
????????????
????????????????????????
????????????
− ????????????
????????????
????????????=????????????
(8)
(9)
For, ????????????<????????????<????????????
For, 0>????????????>−????????????
Figure: Ideal periodic square well potential suggested by Kronig and
PenneyNPTEL

28. 9
Since the above two Schrödinger equations obey Bloch’s theorem, the wave function takes the following form:
????????????????????????=????????????
????????????(????????????)????????????
????????????????????????????????????
????????????
????????????
????????????
????????????
????????????????????????
????????????
+????????????????????????????????????
????????????????????????
????????????
????????????????????????
−????????????
????????????
+????????????
????????????
????????????
????????????=????????????
????????????
2
=
2????????????????????????
ћ
2
where,
????????????
2
=
2????????????
ћ
2
(????????????
0−????????????)where,
(9)
Where, ????????????
????????????
????????????is the periodic function in x with periodicity of (a+b), i.e., ????????????
????????????(????????????)= ????????????
????????????(????????????+????????????+????????????)
From equation 9, we have ????????????
????????????
????????????????????????????????????
????????????
????????????(????????????)????????????
????????????????????????????????????
and
????????????
= -????????????
????????????????????????????????????
????????????
????????????????????????+????????????????????????????????????????????????
????????????????????????????????????
????????????
????????????????????????????????????
(10)
Substituting equations 9 and 10 in equations 8 and 9, we get two equations for two regions (I) and (II).
(11)
(12)NPTEL

29. 10
Boundary conditions
????????????
1(????????????)
????????????=0=????????????
2(????????????)
????????????=0
????????????????????????
1
????????????????????????
????????????=0
=
????????????????????????
2
????????????????????????
????????????=0
????????????
1(????????????)
????????????=????????????=????????????
2(????????????)
????????????=−????????????
????????????????????????
1
????????????????????????
????????????=????????????
=
????????????????????????
2
????????????????????????
????????????=−????????????
(ii)
(i) (iii)
(iv)
????????????+????????????=????????????+????????????
???????????????????????????????????? −????????????=????????????(???????????? − ????????????)
????????????????????????
????????????????????????????????????
+????????????????????????
−????????????????????????????????????
=????????????????????????
−????????????????????????
+????????????????????????
????????????????????????
????????????
????????????????????????(????????????+????????????)
????????????????????????????????????????????????
????????????????????????????????????
+????????????????????????
−????????????????????????????????????
=????????????????????????????????????
−????????????????????????
− ????????????????????????
????????????????????????
????????????
????????????????????????(????????????+????????????)
(i)
(ii)
(iii)
(iv)
General Solutions of
Equations 11 and 12
????????????
????????????(????????????) =????????????????????????
????????????(????????????−????????????)????????????
+????????????????????????
−????????????(????????????+????????????)????????????
????????????
????????????(????????????) =????????????????????????
????????????(????????????−????????????????????????)????????????
+????????????????????????
−????????????(????????????+????????????????????????)????????????
Obtained modified
equations after applying
all the boundary
conditions
Now, simplifying the determinant of these equations, we can obtain:
????????????
2
+????????????
2
2????????????????????????
sinℎ????????????????????????sin????????????????????????+cosℎ????????????????????????cos????????????????????????=cos????????????(????????????+????????????)
For, ????????????<????????????<????????????
For, 0>????????????>−????????????
(13)NPTEL

30. 11
(i) Potential barrier ????????????
0→ ∞
(ii) Barrier width b→0
Such a way that (????????????
????????????????????????)
remains finite
New modified equation:????????????
2
+????????????
2
2????????????????????????
????????????????????????sin????????????????????????+cos????????????????????????=cos????????????????????????
mV
0b
ħ
2
K
sin????????????????????????+cos????????????????????????=????????????????????????????????????
????????????????????????
[where, ????????????=
????????????????????????
????????????????????????????????????
ħ
????????????
]
As, b → 0, sinℎ????????????????????????→ ????????????????????????And, cosℎ????????????????????????→ 1
Also,
????????????
2
+????????????
2
2????????????????????????
=
????????????????????????0
????????????????????????ħ
2
(14)
????????????
sin????????????????????????
????????????????????????
+cos????????????????????????=????????????????????????????????????????????????????????????
(15)
Assumptions for
simplificationNPTEL

31. 12
Fig. Plot of ⁄
????????????
????????????????????????????????????????????????????????????????????????????????????+????????????????????????????????????????????????vs ????????????????????????
Theabovecondition must besatisfiedforthe
solutions tothewaveequationstoexist.
Onlythosevaluesof????????????????????????areallowedforwhichthe
left-hand sideoftheequationliesbetween+1and-1,
asthevalues of????????????????????????????????????????????????????????????liebetweenthose values.
Theother values of????????????????????????arenotallowedand they
resideinthe forbiddenenergybands.
????????????
????????????????????????????????????????????????????????????
????????????????????????
+????????????????????????????????????????????????????????????=????????????????????????????????????????????????????????????
Theenergyspectrumoftheelectronconsistsofalternate regions ofallowedenergybands
(solidlinesontheabscissa)andforbiddenenergybands(brokenlines).
The widthoftheallowedenergybandincreaseswith????????????????????????ortheenergy.
The widthof a particularenergybanddecreaseswith anincreaseinthevalueofP,i.e.,
withtheincreaseinthe bindingenergyofelectrons.
Important pointsNPTEL

32. 13
Thewidthof allowedandforbidden energybandcanbe changed onbasis of????????????value,andvariousbandstructures canbeformedwithin the lattice
crystal.
If ???????????? → ∞,
sin????????????????????????= 0
????????????????????????=????????????????????????
????????????=
????????????????????????
????????????
????????????=
ћ
2
2????????????
????????????????????????
????????????
2
If ???????????? →0,
cos????????????????????????=cos????????????????????????
????????????
2
=????????????
2
2????????????????????????
ћ
2
=
2????????????
λ
2
????????????=
1
2????????????
ℎ
λ
2
=
????????????
2
2????????????
Fig. Band structurefor ????????????→∞
Fig. Band structurefor ????????????→∞
Fig. Band structurefor ????????????= ????????????????????????NPTEL

33. 14
????????????
????????????????????????????????????????????????????????????
????????????????????????
+????????????????????????????????????????????????????????????=????????????????????????????????????????????????????????????
Therelationinfersthat????????????????????????????????????????????????????????????takes aspecificvalue foreachofthe
allowedenergyvalue of E.
????????????????????????????????????????????????????????????isanev enperiodic function,withperiod
????????????????????????????????????
????????????
,wherenisany
integer.Hence,Eisalsoanevenperiodicfunction off k withperiodof
????????????????????????
????????????
.
Thereexists ageneralconvention toexpresstherelationshipbetweentheenergyEandk.Thegeneralconvenienttwo
schemesare:
(a)Theextendedzonescheme
(b)ThereducedzoneschemeNPTEL

34. 15
Fig. Energy vs wave vector for one dimensional lattice
(extended zone scheme)
Thesolid lines intheadjacent figurerepresenttheE-k
relationshipintheextendedzonescheme.
The correspondingdottedparaboliccurveforfreeelectronsin
theconstant potentialisalsoshownfor comparison.
TheE-kcurveoftheextendedzoneschemeisnot continuous and
has discontinuitiesat
????????????= ±
????????????????????????
????????????
,where n = 1, 2, 3 ….
Theallowedvaluesof kdefine the boundariesoftheBrillouin
zones.
ThefirstBrillouinzoneextendsfrom -
????????????
????????????
to +
????????????
????????????
;thesecondone
extendsfrom
????????????
????????????
to
????????????????????????
????????????
andfrom -
????????????
????????????
to -
????????????????????????
????????????
.
Notablepoints regardingextendedzonescheme:NPTEL

35. 16
TheE-kcurveinthereducedzoneisshownintheadjacent figure,whosex-axis
is limitedfrom -
????????????
????????????
to +
????????????
????????????
.
Theschemeisobtainedbyreducingthe contentsoftheotherzonesas to
correspond,ingeneral,tothefirstzone,i.e. totheregion -
????????????
????????????
to +
????????????
????????????
.
Thewave vector
????????????belongingtothisregionis calledthereducedwave vector.
Notablepoints regarding thereducedzonescheme:
Fig. Reduced zone scheme
Major applications of the Nearly free electron model
and band theory of solids:
(a)Tocalculatethe bandgap ofsolids.
(b)Todifferentiate amongtheinsulators,
semiconductors,andmetals.NPTEL

36. Thenea electronmodelconsidersthattheelectronsarenottotallyfreebutit
encountersthepotentialduetothepresenceof positive ion cores.
Thenea
electronmodelissuccessfulinexplainingthedifferencesinthebandgap
amongmetals,semiconductors,andinsulators.
Extended a
nd reducedzoneschemesofbandtheory havebeen discussed.NPTEL

37. •Physicsof F unctional MaterialsbyHasse Fredriksson&UllaAkerlind
•Solid S
PhysicsbyCharlesKittel.
•ShriverA
AtkinsInorganicChemistrybyPeterAtkinsTina Overton,Jonathon Rourke.NPTEL

38. Thank you…NPTEL

39. Physics of Functional Materials and Devices
Prof. Amreesh Chandra
Department of Physics, IIT KHARAGPUR
Module03:IntroductiontoTheoryofSolids
Lecture16:BondsinMoleculesandSolidsNPTEL

40. Introduction to Bonds in Molecules and Solids
Atomic energies: Binding, Dissociation, Ionization, Electron Affinity,
S
ublimation, Condensation, Cohesive and Lattice.
Bonds in Molecules: Molecular Bonds, Ionic Bonds, and Covalent Bonds.
Metallic BondsNPTEL

41. Bonds in Molecules and Solids
What ar bonds?
Abondistheattractiveforcethatconnectsdistinctconstituents(atoms,ions,etc.)ofvarious
typesof moleculestogether.
Whyd
bondsform?
Atomst
tostaystable,which iswhy they createbondswithoneanother.Atomsattainthis
stabilitybyeither sharing electronsordonatingelectronstootheratoms.
Sharing of electrons Donation of electronsNPTEL

42. Bonds in Molecules and Solids
Gettingfa withgeneralterms:
Potential en
ergy:Forparticles,theconcept ofpotential energyis oftenusedtodescribethe
energy associatedwiththeinteractionsbetween particles.
Particles c
an interactwitheach otherthroughvariousfundamentalforces,suchas the
electromagneticforce,thestrongnuclearforce,and the weak nuclearforce.
PotentialEne
rgyintermsof forcecanbewrittenas:
????????????
????????????????????????????????????=�
????????????
????????????
−????????????????????????????????????
a:Zero levelofthe potential energy
The ab
concept isusedtounderstandtheinteractionbetweentwo
particles.NPTEL

43. Bonds in Molecules and Solids
Fig: Potential Energy of a free molecule as a function of interionic distance r.NPTEL

44. Bonds in Molecules and Solids
Differentt ypesof particleinteractions/energies
Bindingan
DissociationEnergy
IonizationEne
rgy
Electron A

SublimationEne
rgyandCondensation Energy
CohesiveEne
rgy
LatticeEne
rgyNPTEL

45. Bonds in Molecules and Solids: Binding Energy and dissociation Energy
Binding Energy:Itisthe energy that holds thenucleus ofanatomtogether, andis requiredto
overcometheattractiveforcesthat exist betweenprotonsandneutrons.Further,wecan also say
thatthebindingenergyofan atomisameasureofthestability ofthenucleus.Themorestable
thenucleus,the greater thebindingenergy.
Letusu
withan example:
Consideringt
woatomsAandB.
AtomsAan
dBare at aninfinitedistance(∞)suchthatnointeraction ispresent between the
atoms.
Sincet
atomsare apart they are highlyunstable.
So,t
atoms graduallycomes closetoeach other.
The en
twoatomsAandBare movedfrom
infinitytotheirequilibriumdistanceandformastablemoleculeAB.NPTEL

46. Bonds in Molecules and Solids: Binding Energy and dissociation Energy
Binding Energy:
E
B~ Binding Energy
Energy released when the two atoms A and B are moved from infinity to their
equilibrium distanceNPTEL

47. Bonds in Molecules and Solids: Binding Energy and dissociation Energy
DissociationEne rgy:Itistheamount ofenergyrequiredtobreakabondbetweentwoatomsina
moleculeand separatethemcompletely.Inotherwords,it istheminimumenergy neededtobreak
thebondandconvertthemoleculeintoits constituentatoms.
Letusu
withan example:
Consideras
tablediatomic moleculeAB.
AtomsAan
dBare at anequilibriumdistancesuchthat themoleculeisunderitsminimum
energycondition.
Thed
energyisequaltothebindingenergy, D
0,ofthe
moleculeorthe energyrequiredtoseparate theatomsAandBand
movethemtoinfinitedistance(∞)fromeach other.NPTEL

48. Bonds in Molecules and Solids: Binding Energy and dissociation Energy
DissociationEne rgySchematic:
D
0~Dissociation Energy
Energy required to separate the atoms A and BNPTEL

49. Bonds in Molecules and Solids: Ionization Energy
IonizationEne rgy:The energy neededtoremovean electronfromanatomormoleculeisknownas
ionizationenergy.Dependingonthetypeofatomormoleculeandthe electron being extracted,
differentamountsofenergy arerequired.For instance,someatomstakemoreenergy than
otherstoremovean electron.
Na
Na atom
The first ionization
energy of sodium
~5.14eV
NaNPTEL

50. Bonds in Molecules and Solids: Electron Affinity
Electron Affinity:The energyrequiredtotransferanelectronfrominfinityto thelowest
feasibleorbitinan atomor molecule.Inotherwords,it istheenergy change associatedwith
addingan electrontoaneutral atomor molecule.
Atom
Nucleus
Cl atom
For chlorine gaining an electron, energy
is released ~ - 3.62 eV
Atom
NucleusNPTEL

51. Bonds in Molecules and Solids: Sublimation Energy and Condensation Energy
SublimationEne rgy:Theamountofenergyrequiredtochangeasubstancefrom its solidtoits
gaseousstatewithoutfirst goingthroughtheliquidstateisknownassublimationenergy.Toput
itsimply,itisthe energy neededtorelease thebonds bindingtogetherthemolecules ofasolid
andturnitintoagas.Atomic sublimationenergyisdefined as:
????????????
????????????=????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
CondensationEne rgy:Thereverseprocessofsublimationi.e.thereleaseofenergywhenvapour
changesdirectlyintoasolid. The energy released per atomisgiven asfollows:
????????????
????????????????????????????????????????????????=????????????
????????????
Sublimationan dcondensationenergyis frequentlyexpressedin joules
permole (J/mol)or kilojoulespermole(kJ/mol).NPTEL

52. Bonds in Molecules and Solids: Cohesive Energy
CohesiveE nergy:Theamountofenergy needed,startingfromthesubstance'smost stableform,
to completelyseparateasubstanceinto itsindividualatoms ormoleculesisknownascohesive
energy.The energy neededtobreak everyintermolecularbondholdingtheatomsormoleculesin
thesubstancetogether.
Thesefor
cesmayincludehydrogenbonds,otherkindsofchemicalbonds,dipole-dipole
interactions,andLondondispersionforces.
EnergyrequiredNPTEL

53. Bonds in Molecules and Solids: Lattice Energy
LatticeEne rgy:Theamountofenergy neededtoseparateasolid ionic compound,which ismadeup
ofcharged particlesknownasions,intoitsindividualgaseousions isknownaslatticeenergy.Itis
awayto estimatehowstronglytheions inasolid'scrystallatticeattractoneanother.
The en
whichhastobeaddedtoonestoichiometricunit ofacrystaltoseparateits
componentionsintofreeions.NPTEL

54. Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
What ar differenttypesofbonds?
Types of Bonds
Molecular
Ionic
Covalent
MetallicNPTEL

55. Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
MolecularB onds: Dipole interaction inamoleculeacts as thesourceof originofmolecularbonds.
Theunequal distribution ofelectrons among theatomsinapolar covalentconnectionresultsin
theformation ofadipolemomentwithinamolecule.Thisdipolemomentcaninteractwith
adjacentmolecules' dipolestoproduce dipole-dipole interactions.
Dipole I Between Molecules
Dipolem
ofadipole isdefined as:????????????=????????????????????????
whereqistheelectrical charge andr avectordirectedfromthe negativetowardsthepositive
chargeNPTEL

56. Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
IonicBo : The electrostaticattractionbetweenionswithopposingcharges iswhatcreatesanionicbond.
Ions areatomsormoleculeswithone ormoreelectrons addedorremoved,creating anetelectrical charge.
Whentw
formanionicbond, oneofthemgivesone ormoreofitselectronstothe other,resultingin
theformationofa positively charged cationandanegativelycharged anion.
Asanil
andchlorinecanunitetoformsodiumchloride(NaCl),usuallyreferredto astable
salt. Inorderto generatea sodiumcation(Na
+
),oneelectronislost,whereas,in a chlorineanion(Cl
-
),an
electronisgained. The ionic bondinNaClis produced by electrostatic interactionbetweentheionsofNa
+
andCl
-
.
Now,on
canimaginea Sodium ionandachlorineionformingabond. Butonecan
ask:
How
doesthebondattainstability?NPTEL

57. Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
Ast cloudsofthetwosubshells starttooverlap,astrong repulsiveforceappears,
whichcauses thedistancebetween theionstoshrinkuntil itequals theequilibriumdistancefor
theNa
+
Cl
-
molecule.
Att
point,theelectrostaticattraction forcebalances thestrong repulsiveforce.
Owingtot
hePauliexclusionprinciple, noelectrons can havefourequalquantumnumbers.Hence
someelectronsmustbeexcitedtohigher energy levels.
Thispr
requiresalotofenergy andresultsinasteep energycurve.
Theequilibriumcorrespondstothe lowestpossibletotalenergyofthe
systemandresultsinastableionic molecule.NPTEL

58. Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
IONICBO NDSCHEMATICNPTEL

59. Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
CovalentB: Whentwoatomsshare electrons,it formsacovalentbond,atypeofchemical
bond.Whentwoatomswithsomeelectrons intheiroutershellscombine,youcanimaginethis.
Bothoftheseatomscan haveamorestableelectronconfigurationbysharing electronswithone
anothertoformabond.
Covalentb
ondstendtooccurbetween nonmetalatoms,suchashydrogen,oxygen,andcarbon.
They canbefound inmanymolecules, includingwater, methane, andglucose.
Let's t
theexampleofwater(H
2O)toexplain covalentbonding.Waterisamoleculethat
consistsoftwohydrogenatomsandoneoxygen atom.
Inac
bond,thetwohydrogenatomsand the oxygenatom
share electronstoformastablemolecule.Each hydrogen atom has
oneelectronin its outershell, while theoxygen atomhassix
electronsin its outershell.NPTEL

60. Bonds in Molecules and Solids: Bonds in Molecules and Nonmetallic Solids
Toco theirouter shells,thehydrogenatomssharetheirelectronswith theoxygenatom,while
theoxygenatomshares itselectronswiththehydrogenatoms.
Some m examplesofmoleculeshavingcovalentbondsareCl
2, H
2,CH
4,
Graphiteetc.NPTEL

61. Bonds in Molecules and Solids: Covalent Bonds
CovalentBondProperties:
Theb
areverystrongand covalentsolidsare therefore characterizedbyhighmeltingpoints
and high mechanicalstrength.
They ar
poor conductors ofheat andelectricitybecausethereareno non- localizedelectrons
whichcancarryenergyorchargefromoneplacetoanother.
The el
excitationenergiesofcovalentsolidsare high,ofthe
magnitudeofseveral eV.NPTEL

62. Bonds in Molecules and Solids: Metallic Bonds
MetallicBonds: The valence electronsofmetalatomsinametallicbondare delocalized,which
means they arenotboundtoany particularatomandaretherefore freetomoveaboutthe entire
metallattice.
Anatomisnottightlyboundtoanyofthedelocalizedelectronsinametallicbond.
Rather, theymoveunrestrictedlywithinthe metal lattice,givingmetalstheir specialabilityto
conduct electricity.
Metalscan conduct electricityand heateffectively dueto themobilityofthese
delocalized electrons.NPTEL

63. Bonds in Molecules and Solids: Metallic Bonds
Delocalised electrons
Sea of electronsNPTEL

64. •The t maintypesof chemicalbondsareionicbonds,covalentbonds, andmetallicbonds.
•Further,weha
vesixtypesofbonding energies depending ontheparticle-particle
interactionnamely:
•Binding a
DissociationEnergy
•IonizationEner
gy
•ElectronA
ffinity
•SublimationEner
gy and Condensation Energy
•Cohesive Ener
gy
•LatticeEner
gyNPTEL

65. •Physicsof F unctional MaterialsbyHasse Fredriksson&UllaAkerlind
•IntroductiontoNan
otechnology,CharlesP.Poole,Jr.andFrankJ.Owens,wiley-
interscience.
•ShriverA
AtkinsInorganicChemistrybyPeterAtkinsTina Overton,Jonathon Rourke.NPTEL

66. Thank you…NPTEL

67. Physics of Functional Materials and Devices
Prof. Amreesh Chandra
Department of Physics, IIT KHARAGPUR
Module 03: Theory of solids
Lecture 17 : Introduction to transformation kinetics and reaction ratesNPTEL

68. Transformationorr eaction
Differentc
oftransformation
Endothermica
ndExothermicreactions
Homogeneousa
ndHeterogeneousreactions
Reactionr

Factorsi
reactionrateNPTEL

69. Transformation kinetics
In p refersto a processin whichenergy or matter is
changedfromoneformto another.
In c
inchemistryrefersto aprocessin whichone or
more substances,calledreactants,areconverted into one or more differentsubstances,called
products.
Fromm
science aspects, transformation occurs whenchangesinthe structure ofthe
materialsintheformofeither composition change or change in grain sizesina crystal
structure.
The structure changes occur as a result of rearrangements of atoms in the material through:
Chemical reactions
Phase transformations
DiffusionNPTEL

70. Chemical reactions
Temperature: I n general, increment in the temperature activates more number of reactant
molecules to collide and initiate a reaction.
Pressure: I
ncrease in the pressure increase the frequency of collision between reactant
molecules to initiate a reaction.
Factors that influences the chemical reactions?
What are chemical reactions?
Process of interactions of different atoms, molecules or ions to form a new compounds with
d
ifferent chemical and physical properties.
Deformation and reformation of bonds between atoms or molecules occurs to form new
c
ompounds.
Formation of final product marked by release or absorption of energy, change in
t
emperature, color etc.
Surface area: I
ncrease in surface area provides more number of active
sites for reactants to collide and interact to initiate a reaction.
Concentration:H
igher concentrations increase the chances of reactant
molecules to collide with each other.NPTEL

71. Classifications on the basis of energy changes
Basedofthevariationofinternalenergy duringachemicalreaction, theyarebedividedintotwo
categories:
Endothermic reactions Exothermic reactions
Energy r equired to
breakthe bonds of
reactantsishigher
than the energy
released by the
reaction.
Energy isa
cquired
fromitssurroundings
as heat.
Marked byd
ropin
temperatureofoverall
system.
Examples: Vaporization ofwater,
Photosynthesis,Meltingof ice.
Energyr
tobreakthe
bonds of reactants is lower
than the energy releasedby
the reaction.
Energy isr
eleasedin its
surroundingsas heat.
Markedbyri
seintemperature
ofoverallsystem.
Examples:Combustion reactions,
Rusting of iron, Respiration.NPTEL

72. Classifications on the basis of uniformity
Basedoftheuniformityofreactantsorproducts inareactionmixture,chemicalreactionsaredivided
into twocategories:
Homogeneous reactions Heterogeneous reactions
Homogeneous r eactions, by
definition,refersto a processwhere
thereactantsandproductsareinthe
same chemical phase.
Reactantsa
ndproducts that are
uniformly distributed throughout the
reaction mixture.
Heterogeneous r
eactions, by definition,refersto
a processwherethe reactantsandproductsare
inthedifferentchemical phase.
Reactantsa
ndproducts thatareNOTuniformly
distributed throughout the reaction mixture.Examples:DepletionofOzonelayer,
Photochemical smog formation.
Examples:Haber’sprocess, Combustion reactions,
Corrosion reactions
Solid Solid Solid
Liquid Liquid Liquid SolidLiquid Liquid
LiquidGas LiquidNPTEL

73. Classifications on the basis of spontaneity
Gibbsf energyisathermodynamicfunctionthatrepresentsthemaximum amountofenergy
availabletodousefulworkinachemicalsystem.
Changei
free energy(ΔG) of areactioninfluenceitsspontaneity.
Mostt
chemicalreactionsoccursatconstanttemperatureand pressure.
Atc
temperatureandpressureprocessesalwaysoccurspontaneously inthedirectionof
decreasingGibbsfree energy.
Ino
words,systemis in equilibriumwhentheGibbsfree energyofthesystemisaminimum.
Considertheequilibriumreaction.
A
B
Forthis,
ΔG
ΔG =0
reaction is at equilibrium.
ΔG >0
backward reaction is spontaneous.NPTEL

74. Driving force for a chemical reaction
Wed isspontaneouswhenchangeinGibbsfree
energy (ΔG) of a reactionisnegative or minimal.
Tok
conceptofdrivingforce
isused
.
Driving force = −∆????????????=−∫
????????????????????????
????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????
????????????????????????=−(????????????
????????????−????????????
????????????)
The driving force of a sp ontaneous process is always a positive quantity. The larger the driving
force is, the more likely will be the transformation or reaction.
What about the energy barrier for
driving a chemical reaction ?NPTEL

75. Activation energy: the energy barrier
Considertheequilibrium reaction
Activatione
nergy istheminimumkinetic energythat
reactantsmusthavein orderto form products.
In o
words,theheightofthebarrierbetweenthe
reactantsandproductsistheactivation energyofthe
reaction.
Forin
are numerous collisionsof
reactants occurseachsecond,butonly few collision
that haveenergy greaterthan the activationenergy
resultinto finalproduct.
Afteraddi
tionalenergy, allthe reactantmolecules
haveenough kinetic energy topasstheenergy
barrierto form products.
E
a
Heat
A
BNPTEL

76. Activation energy: influencing the reaction rates
Twon conditions:
(i)∆????????????<????????????
(ii)????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????>????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
How activation energy influences the
rate of reaction?
Letusunderstandwhatisrateofreaction…….NPTEL

77. Reaction rate
????????????=
????????????????????????
????????????????????????????????????????????????????????????/????????????????????????????????????????????????????????????
????????????????????????
The reactionrate atthe time t of a transformationisdefinedas:
where,the fractional transformation????????????
????????????????????????????????????????????????????????????/????????????????????????????????????????????????????????????isdef as:
????????????
????????????????????????????????????????????????????????????/????????????????????????????????????????????????????????????=
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
The reaction rate k can also be expressed as the fraction of the total
numbe
r of particles which reach the final state per unit time.
Unit of reaction r
ate is normally a function of time.NPTEL

78. Factors influencing the reaction rate
Three important factors that influence the rate of reaction:
•Directcorrelation from Arrhenius equation
ActivationEnergy
•Directcorrelation from Maxwell Distribution Law
Distributionof particles in energy states
•Arrheniustheory
•Transition state theory
TemperatureNPTEL

79. Activation energy: influencing the reaction rates
Therelationof reactionrate andactivation energywasgivenby:
Arrhenius Equation
????????????=????????????????????????
−????????????????????????/????????????????????????
where,
k =reactionrate
A =pre-exponentialfactor
Ea= Activationenergy(kJmol
-1
)
R =UniversalGasconstant(JKmol
-1
)
T =Temperature(K)
Ifthe a
energy is comparatively low andthetemperatureis
high, many atomshavekineticenergieshigh enough to overcome the
energybarrierandthetransformation/reactions occurs readily.
If t
activationenergy is highcomparedwiththethermalenergiesof
the atoms/molecules, fewofthemhaveenergieshigh enough to
overcomethebarrier. The reactionratebecomesverylowandthe
transformationorreaction willbeprohibitedinpractice.
Reactants
Products
Activation energy (Ea)
Transition state
Potential Energy
Reaction CoordinateNPTEL

80. Distribution of particles: Influencing reaction rates
The d particlesindifferentavailableenergy states influences the reactionrates
of thermallyactivatedreactionsandtransformations.
Thefr
ofparticles withenergiesabove agivenenergyisdescribedbyMaxwell-
Boltzmann DistributionLaw:
TheM
–Boltzmann distribution lawin itsgeneralformcanbewrittenas:
????????????
????????????=
????????????????????????
????????????
????????????
????????????????????????
−????????????????????????/????????????????????????????????????
Zis c the partition functionandisgivenby
????????????=????????????
????????????????????????
−
????????????
????????????
????????????
????????????
????????????
+????????????
????????????????????????
−
????????????
????????????
????????????
????????????
????????????
+????????????
????????????????????????
−????????????????????????/????????????????????????????????????
…..=∑
????????????????????????
????????????????????????
−
????????????
????????????
????????????
????????????
????????????
Where,
????????????
????????????= energy of particle i
????????????
????????????= number of particles which have the energy ????????????
????????????
????????????????????????= particles
????????????
????????????= statistical weight of energy level ????????????
????????????
????????????
????????????= Boltzmann’s constant
????????????= absolute temperature of the system.NPTEL

81. Distribution of particles: Influencing reaction rates
From Maxwell-Boltzmann Law fraction of particles with energies equal to or greater
than a given energy can be calculated as:
where, ????????????
????????????= the fraction of the No particles which have the thermal energy ????????????
????????????per particle.
????????????
????????????=
????????????????????????
????????????????????????
=
????????????
−????????????
????????????/????????????
????????????????????????
∑????????????
−????????????
????????????/????????????
????????????????????????
Fromhere,wecancalculatethe fractionofmoleculeswhich haveenoughthermalenergyto
overcometheenergybarrier,i.e. the activationenergy
????????????
∗=
????????????????????????
????????????????????????
=
????????????
−????????????
????????????????????????????????????/????????????
????????????????????????
????????????
Z = partition function
If ????????????
????????????????????????????????????>> ????????????
????????????????????????tSMALL and the transformation rate
will be very LOW.
If ????????????
????????????????????????????????????<????????????
????????????????????????tHIGH and the transformation rate
will be very HIGH.NPTEL

82. Temperature of reaction: Influencing reaction rates
From Arrhenius theory:
????????????=????????????????????????
−????????????????????????/????????????????????????
According to theA rrheniustheory,therateofachemical reaction is proportionalto thenumber
ofcollisions betweenreactantmolecules,whichinturnis dependentontemperature.
Herein,
Fore
, i.eheatabsorbduring reaction,Tincreases,rateofreaction
increases.
Fore
reactions, i.eheat releasedduringreaction,Tincreases,rateofreaction
decreases.NPTEL

83. Temperature of reaction: Influencing reaction rates
From Transition state theory:
????????????=
????????????
????????????????????????
????????????
????????????
−∆????????????
∗
/????????????????????????
According to the tra nsition statetheory(TST),thereaction
rateis proportionalto theprobabilityofforming theactivated
complex, which is dependent onthe activationenergyand
temperature.
Ast
temperature increases, theprobabilityofformingthe
activatedcomplex increases, leadingtoanincrease inthe
reactionrate.
Where,????????????= reactionrate,????????????
????????????=Boltzmann’sconstant,
????????????=absolutetemperatureofthesystem,h =
Planck constant,∆????????????
∗
= activationenergyorthefree energydifferencebetweenthe reactants and
the activatedcomplex,R=universalgasconstant.NPTEL

84. Transformation oc curs through chemical reactions, phase transformations, anddiffusion.
Atc
andpressure,transformationalwaysoccur spontaneouslyinthe
directionofdecreasingGibbsfreeenergy.
Drivingfo
rce,a positive quantity used for knowing theprobability ofoccurringa
transformation or reaction.
Activation e
nergyisthe minimum kineticenergythat reactants musthaveinorderto
formproducts.
Threei
therateof reactionareactivationenergy,
temperature,andmolecular distribution ofparticles indifferentenergy states.NPTEL

85. ⮚Physics of Functional Materials by HasseFredriksson & Ulla Akerlind
⮚Introduction to Solid State Physics by Charles Kittle
⮚Atkins’s Physical Chemistry by Peter Atkins, and Julio de Paula.
⮚A Textbook of Nanoscience and Nanotechnology, P. I. Varghese and Thalappil, McGraw Hill
Education, 2017.NPTEL

86. Thank you…NPTEL