Statistical Physics

170 10 Dense Gases – Ideal Gases at Low Temperaturegas is hard to liquefy. At 1 atm, it liquefies at 4.2 K. This is the lowest gas–liquid phase transition temperature of any substance except for helium-3. Atatmospheric pressure, all other substances solidify at much higher temperatures.However, 4.2 K is still not low enough for a Bose gas to show its mostpeculiar characteristics. On the other hand, liquid helium-4 remains a liquiddown to T → 0 below about 25 atm. Even though it is a liquid and the interactionsbetween the atoms are not negligible, some aspects of the liquid, suchas the superfluid phase transition, can be understood as a property of a Bosegas. Other, much better examples of Bose gases have recently been realized:these are provided by vapors of alkali metals, cooled down to temperaturesin the microkelvin range. We are now going to study the properties of sucha Bose gas at low temperature.Since the Bose distribution function diverges if the chemical potential coincideswith a single-particle energy of the Bose gas, µ must be lower thanthe lowest single-particle energy, which is zero for gas of atoms. Therefore,the chemical potential must always be negative at finite temperature for thesecond kind of Bose gas.10.5.2 Properties at Low TemperatureBose–Einstein CondensationThe energy spectrum of an atomic Bose particle is the same as that for a Fermigas, and so the density of single-particle states has the same form:D(E)dE =V2π 2 3 m√ 2mE dE.A Bose particle has an integer spin. Here we consider the simplest case, namelythat of particles with spin zero. In this case the number of particles is givenby the following equation:N = ∑ i1e β(Ei−µ) − 1 . (10.80)At high temperature, we can calculate this summation as an integral withrespect to the energy:N =∫ ∞01dED(E)e β(E−µ) − 1 . (10.81)However, at low temperature, this replacement is not permissible. This isa special property of a Bose gas. We shall see in the following why this replacementis not correct.The Fermi distribution function is finite for any energy, and is less than orequal to one. On the other hand, the Bose distribution function can diverge as