bei48482_FM

Since Fermi energies for metals are usually several electronvolts (Table 9.2), the

average electron energy in them at 0 K will also be of this order of magnitude. The

temperature of an ideal gas whose molecules have an average kinetic energy of 1 eV is

11,600 K. If free electrons behaved classically, a sample of copper would have to be at

a temperature of about 50,000 K for its electrons to have the same average energy they

actually have at 0 K!

The failure of the free electrons in a metal to contribute appreciably to its specific

heat follows directly from their energy distribution. When a metal is heated, only those

electrons near the very top of the energy distribution—those within about kTof the

Fermi energy—are excited to higher energy states. The less energetic electrons cannot

absorb more energy because the states above them are already filled. It is unlikely

that an electron with, say, an energy that is 0.5 eV below Fcan leapfrog the filled

states above it to the nearest vacant state when kTat room temperature is 0.025 eV

and even at 500 K is only 0.043 eV.

A detailed calculation shows that the specific heat of the electron gas in a metal is

given by

cVe R (9.60)

At room temperature, kTFranges from 0.016 for cesium to 0.0021 for aluminum for

the metals listed in Table 9.2, so the coefficient of Ris very much smaller than the clas-

sical figure of ^32 . The dominance of the atomic specific heat cVin a metal over the elec-

tronic specific heat is pronounced over a wide temperature range. However, at very

low temperatures cVebecomes significant because cVis then approximately proportional

to T^3 whereas cVeis proportional to T. At very high temperatures cVhas leveled out at

about 3Rwhile cVehas continued to increase, and the contribution of cVeto the total

specific heat is then detectable.

9.11 DYING STARS

What happens when a star runs out of fuel

Metals are not the only systems that contain degenerate fermion gases—many dead

and dying stars fall into this category also.

kT



F

^2



2

Electron specific

heat

Statistical Mechanics 327

Table 9.2 Some Fermi Energies

Metal Fermi Energy, eV

Lithium Li 4.72

Sodium Na 3.12

Aluminum Al 11.8

Potassium K 2.14

Cesium Cs 1.53

Copper Cu 7.04

Zinc Zn 11.0

Silver Ag 5.51

Gold Au 5.54

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