rbk lect 4 elect props of metals

7/31/2019 Rbk Lect 4 Elect Props of Metals

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DRUDES MODEL THERMAL CONDUCTIVITY

WIEDMANN-FRANZ LAW

1


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Resistivity of materials

10 18 10 16 10 14 10 12 10 10 10 8 10 6 10 4 10 2 10 0 10 -2 10 -4 10 -6

1 ohm - cm

Insulators

Organic insulators

Quartz Ceramics

Mica Glass

Semi conductors Metals

Bi

Fe

Ag

Cu2O

TiO2

Pbs

SeSi

Ge

2


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Mean free path - The average distance that electrons can

move without being scattered by other atoms.Temperature Effect - When the temperature of a metalincreases, thermal energy causes the atoms to vibrateEffect of Atomic Level Defects - Imperfections in crystal

structures scatter electrons, reducing the mobility andconductivity of the metalMatthiessens rule - The resistivity of a metallic material isgiven by the addition of a base resistivity that accounts for

the effect of temperature ( T ), and a temperatureindependent term that reflects the effect of atomic leveldefects, including impurities forming solid solutions ( d ).Effect of Processing and Strengthening

Conductivity of Metals and Alloys

3


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The total scattering rate for a slightly imperfect crystal at finitetemperature;

So the total resistivity ,

This is known as Mattheisens rule and illustrated in followingfigure for sodium specimen of different purity.

0

1 1 1

( ) ph T

Due to phonon Due to imperfections

02 2 20

( )( )

e e e I

ph

m m mT

ne ne T ne

Ideal resistivity Residual resistivity


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5

Electrical resistivity with composition (impurity additions) for various copper alloys.


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Current: (Amps)dqdt

i

q t id

i R

V

R L A

Macroscopic Microscopic2Current Density: (A/m )

d dA

J i

d i A J

where resistivityconductivity

E E J

where carrier densitydrift velocityd

d nv

n e J v

2where scattering timem

ne

Classical Theory of Conduction Review

Drift velocity v d is net motion of electrons (0.1 to 10 -7 m/s).

Scattering time is time between electron-lattice collisions. 6


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Drudes classical theory (1900)

Drude developed this theory of conductivity of metalsby considering metals as a classical gas of electrons,governed by Maxwell-Boltzmann statistics

Electrons wandered through a sea of immobile positivecharges

Drude introduced conduction electron density n=N/V,V=volume of metal, N = Avogadros No.

Applied kinetic theory of gas to conduction electronseven though the electron densities are ~1000 timesgreater

Electron gas is free and independent, meaning noelectron-electron or electron-ion interactions occur

If Electric field is applied, electron moves in straightline between collisions (no electron-electron collisions).Collision time constant. Velocity changesinstantaneously.

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Drude theory: electrical conductivity

Drudes theory gives a reasonable picture for thephenomenon of resistance. Drudes theory gives qualitatively Ohms law (linear

relation between electric field and current density).

It also gives reasonable quantitative values, at least atroom temperature. Drudes theory gives an explanation of why metals do

not transmit light and rather reflect it.

8


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Drudes classical theory (1900)

Drude treated the (free) electrons as a

classical ideal gas but the electronsshould collide with the stationary ions,not with each other.

9

average rms speed

so at room temp.


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10

we apply an electric field. The equation of motion is

integration gives

and if is the average time between collisions then the

average drift speed is


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11

we get

number of electrons passing in unit time

and with

current density

current of negatively charged electrons

Ohms law


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12

Ohms law

and theresistivity

and we can define the conductivity

and the mobility


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vm

lne

v

l

2

mkT

v 2

4

In the Drude model, the conductivity should beproportional to T -1/2 . But for most conductors theconductivity is nearly proportional to T -1except at very lowtemperatures, where it no longer follows a simplerelation.


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2

1( )

E

d

F ma E me e J ne v ne a nne

Metal: Resistance increases with Temperature.

Why? Temp t, n same (same # conductionelectrons) r

Semiconductor: Resistance decreases withTemperature.

Why? Temp t, n (free -up carriers to conduct) r

Temperature dependence of resistivity.

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Failure of Drude s theory: According to Drudes model :

1. Conductivity is proportional to the number density of

valence electrons (n)and to the mobility( ); mobility variesinversely with the mean free path which is taken to beindependent of the size of the ions. However, it is foundthat mobility does not vary substantially among different

metals.2. It could not explain the observation of positive Hallcoefficients in many metals

3. As more became known about metals at lowtemperatures, it was obvious that the conductivityincreased sharply, This could not be explained by simpleelectron-ion scattering.

4. This theory fails to explain ferromagnetism,superconductivity, photoelectric effect, Compton effect and

blackbody radiation. 15


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Failure of Drude s theory contd..:

6.It is a macroscopic theory.7.Dual nature is not explained.8.Atomic fine spectra could not be accounted.

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Failures of the Drude model: heat capacity

Experimentally, one finds a value of about at roomtemperature, independent of the number of valenceelectrons (rule of Dulong and Petit), as if the electrons do

not contribute at all.17

consider the classical energy for one mole of solid in a heatbath: each degree of freedom contributes with

energy heat capacity

monovalent

divalent

trivalent

el. transl. ions vib.

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18

Failures of the Drude model: electrical conductivity of analloy

The resistivity of an alloy should be between those of itscomponents, or at least similar to them.

It can be much higher than that of either component.


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19

Conduction electron Density n

calculate as

#valenceelectronsper atom

density atomicmass

#atomspervolume


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20

Electrical conductivity of materials


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Thermal conductivity, K

metals non metalsK K

13 V F

K C v l V C

Due to the heat tranport by the conduction electrons

Electrons coming from a hotter region of the metalcarry more thermal energy than those from a cooler region,resulting in a net flow of heat. The thermal conductivity

where is the specific heat per unitvolume

21


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Wiedemann-Franz law

Wiedemann and Franz found in 1853 that the ratio of thermal and electrical conductivity for ALL METALSis constant at a given temperature (for roomtemperature and above).

Later it was found by L. Lorenz that this constant is

proportional to the temperature.

22

constant

22


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Wiedemann-Franz law

2

e

nem

2 2

3 B

e

nk T K m

228 22.45 10

3K k

x W K T e

B

8 22.23 10K

L x W K T

The ratio of the electrical and thermal conductivities is

independent of the electron gas parameters;

Lorentznumber

For copper at 0 C

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Comparison of the Lorenz number to experimental data

24

L = 2.45 10 -8 Watt K -2

at 273 K

metal 10-8 Watt K -

2

Ag 2.31 Au 2.35

Cd 2.42Cu 2.23Mo 2.61Pb 2.47Pt 2.51Sn 2.52W 3.04Zn 2.31

f h d l


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Assumptions of the FREE ELECTRON Model1. Metals have high electrical conductivity and no apparent activationenergy, so at least some of their electrons are free and not bound toatoms

4. :Therefore model the behavior of the freeelectrons with U = 0 inside the volume of themetal and a finite potential step at the

surface. Assume each atom has n 0 freeelectrons, where n 0 = chemical valence. Assume that resistance comes fromelectrons interacting with lattice throughoccasional collisions

2. Coulomb potential energy of positive ions U 1/r is screened bybound electrons and is weaker at large distances from nucleus

3. Electrons would have lowest U (highest K) near nuclei, so they spendless time near nuclei and more time far from nuclei where U is notchanging rapidly

25

F i E d D i f S


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Fermi Energy and Density of States

Solutions have

1. Wave functions:

2. Energies:

Time-independentSchrdinger Equation:

E U m

22

2

With U = 0: E m

2

2

2 22 2mE 22 k

)( t r k i Ae Traveling waves (plane waves)

mk

E 2

22

Parabolic energy bands

E

kx26

0


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Electron Density per unit Energy

The density of states dN/dE is often written g(E) The number of electrons per unit energy, N(E) also

depends upon the probability that a given state isoccupied, F(E). It is,

N(E) = g(E)F(E)

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We know that the number of allowed k values inside aspherical shell of k-space of radius k of

2

2( ) ,2

Vk g k dk dk

where g(k) is the density of statesper unit magnitude of k.

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The number of allowed statesper unit energy range?

Each k state represents two possible electron states, one forspin up, the other is spin down.

( ) 2 ( )g E dE g k dk ( ) 2 ( )

dk g E g k

dE 2 2

2k

E m

2dE k dk m

2

2 mE k

( )g E 2 ( )g k dk

dE

3/ 2 1/ 22 3 (2 )2

( )V

m E g E

29


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Ground state of the free electron gas

Electrons are fermions (s= 1/2) and obey Pauliexclusion principle; each state can accommodateonly one electron.

The lowest-energy state of N free electrons istherefore obtained by filling the N states of lowest energy.

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Thus all states are filled up to an energy E F, known as Fermienergy , obtained by integrating density of states between 0 andEF, should equal N. Hence

Remember

Solve for E F (Fermi energy);

2/ 32 232F

N E

m V

3/ 2 1/ 22 3 (2 )2

( )V

m E g E

3/ 2 1/ 2 3/ 22 3 2 3

0 0

( ) (2 ) (2 )2 3

F F E E

F

V V N g E dE m E dE mE

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The occupied states are inside the Fermi sphere in k-space shown below;

radius is Fermi wave number k F.

2 2

2F

F e

k E m

kz

ky

kx

Fermi surfaceE=EF

kF

2 / 32 23

2F

N E

m V

From these two equation k Fcan be found as ,

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F N k

V

The surface of the Fermi sphere represent the boundary betweenoccupied and unoccupied k states at absolute zero for the freeelectron gas. 32


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Typical values may be obtained by using monovalent potassiummetal as an example; for potassium the atomic density andhence the valance electron density N/V is 1.402x10 28 m -3 sothat

Fermi (degeneracy) Temperature T F by

193.40 10 2.12F E J eV 10.746F k A

F B F E k T

42.46 10F F B

E T K

k

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It is only at a temperature of this order that the particles in aclassical gas can attain (gain) kinetic energies as high as E F .

Only at temperatures above T F will the free electron gasbehave like a classical gas.

Fermi momentum

These are the momentum and the velocity values of the

electrons at the states on the Fermi surface of the Fermisphere.

So, Fermi Sphere plays important role on the behaviour of metals.

F F P k

F e F P m V

6 10.86 10F F e

PV ms

m

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The Fermi Parameters

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2/ 32 2

3 2.122F

N E eV m V

1/ 3213 0.746F

N k A

V

6 10.86 10F F e

PV ms

m

42.46 10F F B

E T K

k

Typical values of monovalent potassium metal;

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The free electron gas at finite temperature

At a temperature T the probability of occupation of anelectron state of energy E is given by the Fermi distributionfunction

Fermi distribution function determines the probability of

finding an electron at the energy E.

( ) /

11 F BFD E E k T f e

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Reality of the Fermi Energy


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Reality of the Fermi Energy

The valence bandwidth is in reasonableagreement with the FEG prediction of E F = 11.7

eV

There are several spectroscopic techniques that allow the measurementof the distribution of valence electron states in a metal. The simplest issoft x-ray spectroscopy, in which the highest-lying core level in a sample

is ionized. Only higher-lying valence electrons can fall down to occupythe core level, and the spectrum of emitted x-rays can be measured:

EF 13 eV

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Utility of the Density of States


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Utility of the Density of States

We can simplify by using the relation:

With N(E) we can immediatelycalculate the average energy per

electron in the 3-D FEG system:

Why the factor 3/5? A look at thedensity of states curve should give theanswer:

F

F

E

E

dE E N

dE E E N

electrons

energytotal E

0

0

)(

)(

#

2 / 1)( CE E N

F E

E

E

E

E

E

E

dE E C

dE E C

E F

F

F

F

53

32

52

0

2 / 3

0

2 / 5

0

2 / 1

0

2 / 3

N(E)

E

EF39


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3D Cubic Infinite Potential Well

1-D Well

3-D Cubic Well (with sides length L)

22

22

2/1

2

sin2)(

nmL

E

nLxLx

n

n

20232221222

321

2 / 3

2

sinsinsin2

)(

321

321

n E nnnmL

E

n L z

n L y

n L x

L x

nnn

nnn

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3D Infinite Potential Well : Degeneracy

Consider three differentwave functions(quantum states) for aparticle in the 3-D Well:

i, j, and k are integers

Although the states aredifferent, the energy ofthese states are the same,i.e. these are degenerate

iL

z jL

yk

L

x

Lx

kLzi

L y

jLx

Lx

kL

z jL

yi

L

x

Lx

kji

jik

ijk

sinsinsin2

)(

sinsinsin2

)(

sinsinsin2

)(

2/3

2/3

2/3

2

22

0

2220

2220

2220

2

mLE

i jkE

ki jE

k jiEE

41


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Electrons in 3D Infinite Potential Well

Each electron is described by the wavefunction of a

particle in the infinite well, i.e. the electron state isdefined by three quantum number n1 , n2 , n3; however, in addition, we have to include the spinquantum number , ms

The electron states are thus determined by four quantumnumbers: n1 , n2 , n3 , ms The energy, of course, still depends only on n1 , n2 , n3!

It is convenient to use the following notations:

for ms = , we shall call it the spin up ( ) for ms = - , we shall call it the spin down ( )

Example: ),,,()21,,,( 321321 nnnmnnn s42


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Electrons in 3D Infinite Potential Well:Paulis Principle

What is the ground state configuration ofmany electrons in the 3D infinite potentialwell?

Electrons cannot be in the lowest energystate, since this would violate the PauliExclusion Principle. consider case of solid with 34 electrons

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34 Electrons in 3D Infinite Well

(n1, n1, n1) The lowest energy for this system is 3 E0 , which corresponds

to n1 = n2 = n3 = 1 Thus only 2 (two) electrons can have this energy: one with

spin and one with spin

Next energy level (6 E0), for which one of ns is 2 (112) Thus total of 6 (six) electrons can have this energy

Next energy level (9 E0) (122) can also accommodate6 electrons

What are the combinations of n s for this energy level?

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Electrons in 3D Infinite Well

Can demonstrate with

diagram Energy is plotted in terms

of E0

Each arrow represents anelectron with up ordown spin

Numbers in parenthesisshow the set of ns for agiven energy level

0

1

2

3

45

6

7

8

9

1011

12

13

14

15

1617

18

(3,2,2)

(3,2,1)

(3,1,1)

(2,2,2)

(2,2,1)

(2,1,1)

(1,1,1)

E n e r g y

( i n u n i

t s o

f E

0 )

222

0

2

3

2

2

2

10 2 ,

321 mLEwithnnnEE

nnn

45


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Electrons in 3D Infinite Well

In this configuration, What is the probability at

T =0 that a level withenergy 14 E0 or less will beoccupied?

It is 1! What is the probability

that the level with energyabove 14 E

0will be

occupied? It is 0!

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

(3,2,2)

(3,2,1)

(3,1,1)

(2,2,2)

(2,2,1)

(2,1,1)

(1,1,1)

E n e r g y

( i n u n

i t s o

f E

0 )

46


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Fermi Energy

Generally: The highest filled energy iscalled the Fermi Energy

It is often denoted as E F In our case: EF = 14E0 An electron with E = 14E0 is

said to be at the Fermi level

0

1

2

34

5

6

7

8

9

10

11

12

13

14

15

16

17

18

(3,2,2)

(3,2,1)

(3,1,1)

(2,2,2)

(2,2,1)

(2,1,1)

(1,1,1)

E n e r g y

( i n u n

i t s o

f E

0 )

47


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Velocity Distribution

Find the velocity distribution the probability perunit velocity of finding a state with velocity v

222 / 3

22 / 3

2 / 1

2 / 12

2 / 1

2 / 12

2)(

)(2))(2()(

,22

1E

andforsexpressionneedSo

)(,21

vvm

C vg

dvvgdvv

m

C mvdvv

m

C dE CE dE E g

mvdvdE vm

E mv

dE E

dE CE dE E gmv E

48


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The free electron gas at finite temperature

At a temperature T the probability of occupation of an electron state of energy E is given by the Fermidistribution function

Fermi distribution function determines the probability

of finding an electron at the energy E.

( ) / 11 F BFD E E k T f

e

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Fermi-Dirac Distribution Function

We introduce the probability distribution function, F (E ),

which describes the probability that a state with energyE

isoccupied For electrons this function is the

F

F

EEEE

E f for ,1 for ,0

)(

50


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Fermi-Dirac Distribution Function

At T > 0 K

11

)( /)( T kEE BFeE f

51


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EF EEF

0.5

f FD(E,T)

E

( ) /

11 F BFD E E k T

f e

Fermi Function at T=0and at a finite temperature

f FD=? At 0K

i. EEF

( ) / 1 11 F BFD E E k T f

e

( ) / 1 0

1 F BFD E E k T f

e

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T>0

T=0

n(E,T)

E

g(E)

EF

n(E,T) number of freeelectrons per unit energyrange is just the areaunder n(E,T) graph.

( , ) ( ) ( , )FDn E T g E f E T

Number of electrons per unit energy range according to thefree electron model?

The shaded area shows the change in distribution betweenabsolute zero and a finite temperature.

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Fermi-Dirac distribution function is a symmetricfunction; at finite temperatures, the same

number of levels below E F is emptied and samenumber of levels above E F are filled by electrons.

T>0

T=0

n(E,T)

E

g(E)

EF

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Fermi-Dirac distribution function at varioustemperatures

Incg temp

55

F El t M d l


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Free Electron Models Classical Model:

Metal is an array of positiveions with electrons that arefree to roam through theionic array Electrons are treated as an ideal

neutral gas, and their totalenergy depends on thetemperature and applied field

In the absence of an electricalfield, electrons move withrandomly distributed thermalvelocities

When an electric field isapplied, electrons acquire a netdrift velocity in the directionopposite to the field

Quantum MechanicalModel:

Electrons are in a potentialwell with infinite barriers:They do not leave metal, but free to roam inside

Electron energy levels arediscrete (quantized) andwell defined, so averageenergy of electron is notequal to (3/2) kBT

Electrons occupy energylevels according to Paulisexclusion principle

Electrons acquire additionalenergy when electric field isapplied

56

f l h


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Consequences for Metal Theories

At T = 0 K

N is the electron density, i.e. the number of electrons perunit volume of metal

Calculations show that

Thermal energy at room temperature:

k BT ~ 0.025 eV k BT


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Consequences for Metal Theories

Only electrons occupying levels close to the FermiEnergy will participate in the conduction, sinceonly these electrons can be excited into the higherenergy states by the electric field

From QM point of view, energy supplied by theelectric field excites electrons into higher lyingenergy levels

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Heat capacity of the free electron gas

From the diagram of n(E,T) the change in the distribution of electrons can be resembled into triangles of height 1/2g(E F)and a base of 2k BT so 1/2g(E F)kBT electrons increased theirenergy by k BT.

T>0

T=0

n(E,T)

E

g(E)

EF

The difference in thermal energyfrom the value at T=0K

21( ) (0) ( )( )2 F B

E T E g E k T


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Differentiating with respect to T gives the heat capacity atconstant volume,

2( )v F B E C g E k T T

2( )

33 3

( )2 2

F F

F F B F

N E g E

N N g E

E k T

2 23( )2v F B B B F

N C g E k T k T k T

32v B F

T C Nk T

Heat capacity of Free electron gas


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Total heat capacity at low temperatures

where and are constants and they can be found

drawing C v/T as a function of T 2

3

C T T

ElectronicHeat capacity

Lattice HeatCapacity

rbk lect 4 elect props of metals

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