Rbk Lect 4 Elect Props of Metals
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Transcript of Rbk Lect 4 Elect Props of Metals
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7/31/2019 Rbk Lect 4 Elect Props of Metals
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DRUDES MODEL THERMAL CONDUCTIVITY
WIEDMANN-FRANZ LAW
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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
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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.
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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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Drude theory: electrical conductivity
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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Drude theory: electrical conductivity
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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Drude theory: electrical conductivity
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.
Drude theory: electrical conductivity
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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
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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
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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
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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 ,
1/ 323
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
37
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
38
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
40
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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
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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
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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 )
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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 )
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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
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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
)(
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Fermi-Dirac Distribution Function
At T > 0 K
11
)( /)( T kEE BFeE f
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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
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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