PHYS-3301

Nov. 20, 2018

Lecture 23

Chapter. 9Statistical Physics

Outline:

n 9.1 Historical Overviewn 9.2 Maxwell Velocity Distributionn 9.3 Equipartition Theoremn 9.4 Maxwell Speed Distributionn 9.5 Classical and Quantum Statisticsn 9.6 Fermi-Dirac Statisticsn 9.7 Bose-Einstein Statistics

Classical and Quantum Distributions 9.6: Fermi-Dirac Statistics

n This shows an important fact about the Fermi energy: q When E = EF, the exponential term is 1. FFD =

n Consider now the T dependence of FFD, q In the limit as T → 0,

n At T = 0, fermions occupy the lowest energy levels available to them. They can’t all be in the lowest level due to Pauli principle.

n Near T = 0, there is little chance that thermal agitation will kick a fermion to an energy greater than EF.

12

FFD =1

exp β E −EF( )#$ %&+1

provide the basic of our understanding of the behavior of a collection of fermions

FFD =1

BFD exp βE( )+1 BFD = exp(-bEF)

BFD : normalization factor

Fermi energy

See Problem 30

Fermi-Dirac Statistics

n As the temperature increases from T = 0, more & more fermions jump to higher energy levels. The Fermi-Dirac factor �smears out� from the sharp step function to a smoother curve.

n Fermi temperature, defined as TF ≡ EF / k (useful!)

n When T >> TF, FFD approaches a simple decaying exponential

T > 0

T >> TFT = TF

T = 0

Fig 9.9 The Fermi-Dirac factor FFD at various temperatures: (a) T = 0, (b) T > 0, (c) T = TF = EF/k, and (d) T >> TF. At T = 0 the Fermi-Dirac factor is a step function. As the temperature increases, the step is gradually rounded. Finally, at very high temperatures, the distribution approaches the simple decaying exponential of the Maxwell-Boltzmann distribution.

Classical Theory of Electrical Conduction

n Paul Drude (1900) showed that the current in a conductor should be linearly proportional to the applied electric field that is consistent with Ohm�s law.

n The numerical prediction of the electrical conductivity:

n Mean free path is

n Then, the electrical conductivity can be expressed as:

n: the number of density of conduction electronsnote: can be measure by Hall effect (Ch.11)

e: the electron charget: the average time between electron-ion collision

note: not as easy to measure, can be estimated using transport theorym: electronic mass

mean speed of electron

Classical Theory of Electrical Conductionn Mean speed is proportional to the square root of the absolute T.

n According to the Drude, the conductivity should be proportional to T−1/2.

n But for most conductors, s is very nearly proportional to T−1 (except @ very low T); so, Drude’s classical model of electrical conduction is not accurate!

n Finally, there’s the problem of calculating the electronic contribution to the heat capacity (Cv). As discussed, Cv of a solid can be accounted for by considering the 6 d.o.f in the lattice vibrations. cv = 6(1/2R) = 3R

n According to equipartition theorem, add another 3 (1/2R) = (3/2)R for the heat capacity of the electron gas, giving a total of is R.

n This is not consistent with experimental results; typically only about 0.02R per mole at room T. Clearly a new theory is needed!!!!!

92

a molar hear capacity

Quantum Theory of Electrical Conductionn New, let’s turn to quantum theoryn It is necessary to understand how the electron energies are distributed on

conductor. Use the Fermi-Dirac distribution.n Q: how to find g(E)?n The allowed energies for electrons (in 3-dim infinite square-well potential) are

n We will solve the problem @ T = 0 first & consider the effect of T later.n Since T << TF, (e.g. TF ~ 80,000 K for copper), rewrite this as E = r2E1

n The parameter r is the �radius� of a sphere in phase space. [note: do not confused with a radius in coordinate sys]

n The # of allowed states up to r is related to spherical volume πr 3.n The exact number of states up to radius r is .

43

Nr = 2( ) 18( ) 4

3πr3( )

The # of allowed states per unit E what E values should we use?

E1 = h2/8mL2

The extra factor 2 is due to spin degeneracy; spin up & down! The factor 1/8 is necessary because we are restricted to one octant of the 3-dim number space

Quantum Theory of Electrical Conductionn Two eq. can be used to express Nr as a function of E:

n At T = 0, the Fermi energy is the energy of the highest occupied level.n If there are a total of N electrons, then

n Solve for EF:

n The density of states can be calculated by differentiating Nr(E) with respect to energy:

n Can be expressed more conveniently in terms of EF rather than E1

n Finally, the distribution of electronic energies is then given by

E1 = h2/8mL2

Useful! Why? The ratio (N/L3) si well known for most conductors

g(E) = dNr/dE = (⇡/2)E�3/21 E1/2

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Quantum Theory of Electrical Conductionn @ T = 0, it’s possible to use the step function form pf the Fermi-Dirac factor,

we have

n With the distribution function n(E), the mean electronic energy:

n Now, proceed to find the electronic contribution to

the heat capacity (CV) of a conductor.

n Internal energy of the system:

n Recall: the general expression for CV at constant volume:

n Important question for determining the CV : how U increase with T

n Because the E levels are filled up to EF, only those electrons within about kTof EF will be able to absorb thermal energy & jump to a higher state.

Therefore the fraction of electrons (capable of participating in this thermal

process) is on the order of kT / EF.

U = NE =35NEF

CV =@U

@T<latexit sha1_base64="ydXB53uqlpwkEysyFKQU5WuoHGU=">AAACEXicbZBPS8MwGMbT+W/Wf1WPXoJD2Gm0IuhFGO7iccK6DbZS0izdwtK0JKkwSr+CF7+KFw+KePXmzW9juhXUzQcCP573fZO8T5AwKpVtfxmVtfWNza3qtrmzu7d/YB0edWWcCkxcHLNY9AMkCaOcuIoqRvqJICgKGOkF01ZR790TIWnMO2qWEC9CY05DipHSlm/VW37WzeE1HIYC4WyYIKEoYtDNf7iTm6bpWzW7Yc8FV8EpoQZKtX3rcziKcRoRrjBDUg4cO1FeVtyJGcnNYSpJgvAUjclAI0cRkV423yiHZ9oZwTAW+nAF5+7viQxFUs6iQHdGSE3kcq0w/6sNUhVeeRnlSaoIx4uHwpRBFcMiHjiigmDFZhoQFlT/FeIJ0tEoHWIRgrO88ip0zxuO5ruLWvOmjKMKTsApqAMHXIImuAVt4AIMHsATeAGvxqPxbLwZ74vWilHOHIM/Mj6+AYmCnCk=</latexit><latexit sha1_base64="ydXB53uqlpwkEysyFKQU5WuoHGU=">AAACEXicbZBPS8MwGMbT+W/Wf1WPXoJD2Gm0IuhFGO7iccK6DbZS0izdwtK0JKkwSr+CF7+KFw+KePXmzW9juhXUzQcCP573fZO8T5AwKpVtfxmVtfWNza3qtrmzu7d/YB0edWWcCkxcHLNY9AMkCaOcuIoqRvqJICgKGOkF01ZR790TIWnMO2qWEC9CY05DipHSlm/VW37WzeE1HIYC4WyYIKEoYtDNf7iTm6bpWzW7Yc8FV8EpoQZKtX3rcziKcRoRrjBDUg4cO1FeVtyJGcnNYSpJgvAUjclAI0cRkV423yiHZ9oZwTAW+nAF5+7viQxFUs6iQHdGSE3kcq0w/6sNUhVeeRnlSaoIx4uHwpRBFcMiHjiigmDFZhoQFlT/FeIJ0tEoHWIRgrO88ip0zxuO5ruLWvOmjKMKTsApqAMHXIImuAVt4AIMHsATeAGvxqPxbLwZ74vWilHOHIM/Mj6+AYmCnCk=</latexit><latexit sha1_base64="ydXB53uqlpwkEysyFKQU5WuoHGU=">AAACEXicbZBPS8MwGMbT+W/Wf1WPXoJD2Gm0IuhFGO7iccK6DbZS0izdwtK0JKkwSr+CF7+KFw+KePXmzW9juhXUzQcCP573fZO8T5AwKpVtfxmVtfWNza3qtrmzu7d/YB0edWWcCkxcHLNY9AMkCaOcuIoqRvqJICgKGOkF01ZR790TIWnMO2qWEC9CY05DipHSlm/VW37WzeE1HIYC4WyYIKEoYtDNf7iTm6bpWzW7Yc8FV8EpoQZKtX3rcziKcRoRrjBDUg4cO1FeVtyJGcnNYSpJgvAUjclAI0cRkV423yiHZ9oZwTAW+nAF5+7viQxFUs6iQHdGSE3kcq0w/6sNUhVeeRnlSaoIx4uHwpRBFcMiHjiigmDFZhoQFlT/FeIJ0tEoHWIRgrO88ip0zxuO5ruLWvOmjKMKTsApqAMHXIImuAVt4AIMHsATeAGvxqPxbLwZ74vWilHOHIM/Mj6+AYmCnCk=</latexit><latexit sha1_base64="ydXB53uqlpwkEysyFKQU5WuoHGU=">AAACEXicbZBPS8MwGMbT+W/Wf1WPXoJD2Gm0IuhFGO7iccK6DbZS0izdwtK0JKkwSr+CF7+KFw+KePXmzW9juhXUzQcCP573fZO8T5AwKpVtfxmVtfWNza3qtrmzu7d/YB0edWWcCkxcHLNY9AMkCaOcuIoqRvqJICgKGOkF01ZR790TIWnMO2qWEC9CY05DipHSlm/VW37WzeE1HIYC4WyYIKEoYtDNf7iTm6bpWzW7Yc8FV8EpoQZKtX3rcziKcRoRrjBDUg4cO1FeVtyJGcnNYSpJgvAUjclAI0cRkV423yiHZ9oZwTAW+nAF5+7viQxFUs6iQHdGSE3kcq0w/6sNUhVeeRnlSaoIx4uHwpRBFcMiHjiigmDFZhoQFlT/FeIJ0tEoHWIRgrO88ip0zxuO5ruLWvOmjKMKTsApqAMHXIImuAVt4AIMHsATeAGvxqPxbLwZ74vWilHOHIM/Mj6+AYmCnCk=</latexit>

Quantum Theory of Electrical Conductionn In general,

Where α is a constant > 1 (due to the shape of the distribution curve)

n The exact number of electrons depends on temperature.n Heat capacity :

n As with idea gas it’s common to express this result as a molar heat capacity cV. For 1 mole, N = NA. Using NAk = R & EF = kTF,

(a) The density of states g(E) = (3N/2)EF

–3/2E1/2

(b) distribution function n(E) = g(E)FFDfor an electron gas.

The function n(E) is shown at T = 0 (dashed line) and T = 300 K (solid line).

Quantum Theory of Electrical Conductionn Arnold Sommerfield used correct distribution n(E) at room temperature and

found a value for α of π2 / 4.

n With the value TF = 80,000 K for copper, we obtain cV ≈ 0.02R, which is consistent with the experimental value! Quantum theory has proved to be a success.

n Back to the electrical conductivity again

n Replace mean speed in Eq (9.37) by Fermi speed uF defined from .

n Conducting electrons in a metal are loosely bound to their atoms

these electrons must be at the high energy level

@ room T, the highest energy level is close to the Fermi energy

n So, we should use for copper!

EF = 12muF

2

Unfortunately, this is an even higher speed than mean speed ?????

Quantum Theory of Electrical Conduction

n Drude thought that the mean free path ( ) ) could be no more than several tenths of a nanometer, but it was longer than his estimation.

n But in quantum theory, the electrons can be thought of as wave. So, the mean free path can be longer than Drude estimated.

n Einstein calculated the value of ℓ to be on the order of 40 nm in copper at room temperature.

n The conductivity :

n Sequence of proportions

The electrical conductivity varies inversely with T. This is another success for quantum theory, because electrical conductivity is observed to vary inversely with T for most pure metals.

9.7: Bose-Einstein Statistics

Blackbody Radiationn Intensity of the emitted radiation :

n Use the Bose-Einstein distribution (to find how the photons are distributed by energy) because photons are bosons with spin 1.

n Key is being able to find the density of state g(E).n For a free particle of mass m the E = p2/2m n Write the eq. in terms of momentum:

n The energy of a photon is pc, so

Like Fermi-Dirac statistics, Bose-Einstein statistics can be used to solve problems that are beyond the scope of classical physics. Two examples: a derivation of the Planks formula for blackbody radiation& an investigation of the properties of liqy=uid helium.

Bose-Einstein Statisticsn Now, proceed to calculate the g(E) = dNr/dEn The number of allowed energy states within �radius� r :

,where 1/8 comes from the restriction to positive values of ni and 2 comes from the fact that there are two possible photon polarizations.

n Energy is proportional to r

n The density of states g(E) :

n The Bose-Einstein factor:

Nr = 2( ) 18( ) 4

3πr3( )

Bose-Einstein Statisticsn Next step: Convert from a number distribution (n(E)) to an energy density

distribution u(E).q Multiply by a factor E/L3 (E per unit volume) :

q For all photons in the range E to E + dE

n Using E = hc/λ and |dE| = (hc/λ2) dλ :

n In the SI system, multiplying by constant factor c/4 is required to change the energy density to a spectral intensity

u(E) =En(E)

L3=

8⇡

h3c3E3 1

eE/kT � 1<latexit sha1_base64="81eDEV/2cPS0An+npFdHMQNyP1A=">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</latexit><latexit sha1_base64="81eDEV/2cPS0An+npFdHMQNyP1A=">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</latexit><latexit sha1_base64="81eDEV/2cPS0An+npFdHMQNyP1A=">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</latexit><latexit sha1_base64="81eDEV/2cPS0An+npFdHMQNyP1A=">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</latexit>

u(E)dE =8⇡

h3c3E3dE

eE/kT � 1<latexit sha1_base64="HxMxSnDz1bssm/fVj98mSnPiB2Y=">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</latexit><latexit sha1_base64="HxMxSnDz1bssm/fVj98mSnPiB2Y=">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</latexit><latexit sha1_base64="HxMxSnDz1bssm/fVj98mSnPiB2Y=">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</latexit><latexit sha1_base64="HxMxSnDz1bssm/fVj98mSnPiB2Y=">AAACJXicbVDLSgMxFM34rPU16tJNsAi6UGdUsAuFogy4rNAXtLVkMpk2NPMgyQglzM+48VfcuLCI4MpfMdPOQlsPJJyccy8397gxo0Ja1pexsLi0vLJaWCuub2xubZs7uw0RJRyTOo5YxFsuEoTRkNQllYy0Yk5Q4DLSdId3md98IlzQKKzJUUy6AeqH1KcYSS31zOvkyDn2HHgDOz5HWJU7MU3V4FFdpDi70lx3sofnpIo8KudsWEtP7LTYM0vWqTUBnCd2TkogR7VnjjtehJOAhBIzJETbtmLZVYhLihlJi51EkBjhIeqTtqYhCojoqsmWKTzUigf9iOsTSjhRf3coFAgxClxdGSA5ELNeJv7ntRPpl7uKhnEiSYing/yEQRnBLDLoUU6wZCNNEOZU/xXiAdKpSB1sFoI9u/I8aZyf2po/XJYqt3kcBbAPDsARsMEVqIB7UAV1gMEzeAXvYGy8GG/Gh/E5LV0w8p498AfG9w/QQ6Q7</latexit>

u(�, T )d� =8⇡hc

�5

1

ehc/�kT � 1<latexit sha1_base64="fE6qaOXuVoodbKBYpC7G5Fua1bo=">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</latexit><latexit sha1_base64="fE6qaOXuVoodbKBYpC7G5Fua1bo=">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</latexit><latexit sha1_base64="fE6qaOXuVoodbKBYpC7G5Fua1bo=">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</latexit><latexit sha1_base64="fE6qaOXuVoodbKBYpC7G5Fua1bo=">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</latexit>

This is identical with eq. (3.23)

Plank didn’t use the Bose-Einstein distribution to derive his radiation law. Nevertheless, it’s an excellent example of the power of the statistical approach to see that such a fundamental law can be derived: first solved in 1924 by Bose before full development of QM!!!!

Liquid Heliumn Helium is an element with a number of remarkable properties.n It has the lowest boiling point of any element (4.2 K @ 1 atmosphere pressure)

and has no solid phase at normal pressure.

n 1st discovered in 1098 by Heike Kamerlingh Onnesn Used in experiment as a cryogenic

(cooling) device. n Many effect can be observed only

@ extremely low T, and liquid He can be used to cool materials to 4.2K and lower.

n The density of liquid helium as a function of T (by Onnes):

Liquid HeliumThe specific heat of liquid helium as a function of T :

n The temperature at about 2.17 K is referred to as the critical temperature(Tc), transition temperature, or lambda point.

n As the temperature is reduced from 4.2 K toward the lambda point, the liquid boils vigorously. But, @ 2.17 K the boiling suddenly stops.

n What happens @ 2.17 K is a transition from the normal phase to the superfluid phase.

Liquid Helium

n The rate of flow increases dramatically as the temperature is reduced (see Fig) because the superfluid has a low viscosity.

n Creeping film – formed when the viscosity is very low

Liquid Helium

n Fritz London claimed (1938) that liquid helium below the lambda point is part superfluid and part normal.q As the temperature approaches absolute zero, the superfluid approaches

100% superfluid.

n The fraction of helium atoms in the superfluid state:

n Superfluid liquid helium is referred to as a Bose-Einstein condensation.not subject to the Pauli exclusion principleall particles are in the same quantum state

Liquid Heliumn Such a condensation process is not possible with fermions because

fermions must �stack up� into their energy states, no more than two per energy state.

n 4He isotope is a fermion and superfluid mechanism is radically different than the Bose-Einstein condensation.

n Use the fermions� density of states function and substituting for the constant EF yields

n Bosons do not obey the Pauli principle, therefore the density of states should be less by a factor of 2.

Liquid Heliumn m is the mass of a helium atom.n The number distribution n(E) is now

n In a collection of N helium atoms the normalization condition is

n Substituting u = E / kT,

Liquid Helium

n Use minimum value of BBE = 1; this result corresponds to the maximum value of N.

n Rearrange this,

n Can be evaluated numerically. The result is T ≥ 3.06 K.n The value 3.06 K is an estimate of Tc.

Bose-Einstein Condensation in Gases

n By the strong Coulomb interactions among gas particles it was difficult to obtain the low temperatures and high densities needed to produce the condensate. Finally success was achieved in 1995.

n First, they used laser cooling to cool their gas of 87Rb atoms to about 1 mK. Then they used a magnetic trap to cool the gas to about 20 nK. In their magnetic trap they drove away atoms with higher speeds and further from the center. What remained was an extremely cold, dense cloud at about 170 nK.