§8. Bose Statistics and Fermi Statistics

1. §8. Bose Statistics and Fermi Statistics §8.1 The Statistic Expression of Thermodynamic Quantities 8.1.1 Bose system The mean total particle number Introduce a new function called grand partition function then

2. For the internal energy we have equation Generalized force is For instance

3. The total differential of lnΞ follows as Then β is aintegral factor of

4. Comparing above two equations we have Insert the formula (8.1.2) into (8.1.9) and then compare with the formula (6.7.4) S=klnΩ (8.1.10) This is well known Maxwell relation.

5. Letcalled grand thermodynamic potential.We have 8.1.2 Fermi system Grand partition functionis All above equations can be applied.

6. §8.2 Weak Degeneracy Bose Gas and Fermi Gas 8.2.1 The basic concepts (1)Non-degeneracy gas:The gas satisfying classical limit condition ,and dealt by Boltzmann distribution is called non-degeneracy gas. (2) Degeneracy gas: The gases which can be dealt by Bose distribution and Fermi distribution, is called degeneracy gas. 8.2.2 Weak degeneracy Bose gas and Fermi gas

7. With , , the possible state number follows that g is the degeneracy for spin. Introduce variable ,

8. This case corresponds to weak degeneracy. We can finally obtain that

9. With two equations we have “-” for Fermi gas; “+” for Bose gas

10. §8.3Photon Gas Radiant field in a cavity can be regarded as photon gas which is Boson. Based on ω=ck, one gets ε=cp (8.3.2) Because the number of photon is not conservational, only one Lagrange factor is introduced. We have Because , and α=0, thus μ=0 in equilibrium state.

11. The number of quantum state in volume V and in the scope from p to p+dp follows as Insert the equation (8.3.1) and (8.3.2) into (8.3.4), it gets The mean number of photon is

12. The internal energy of radiant fieldis This is called Planck expression which is agreement with experiment.

13. Discussion: • , This is Rayleigh-Jones expression. • , This is Wien expression.

14. (3) The radiant field is regarded as a system of more vibrational freedom degree. The energy of one vibrational freedom degree (one plane wave) One plane wave is in a state with quantum number n corresponding to n photons in this state. From the point of view of partucle, is photon number. From the point of view of wave, is the mean value of quantum number. The image of particle and wave unify each other.

15. (4)For the degree of freedom satisfying its energy is regarded as continuous variable. The conclusion having the mean energy kT in one vibrational degree of freedom is applied. When , the degree of freedom is frozen at ground state, which can explain some difficult question in classical statistics. This is Stefan-Boltzmann’s law.

16. Introduce a variable ωm is in direct proportion to T, This conclusion is Wien displacement law. The pressure of radiant field is The density of radiant flux Juis

17. §8.4 Bose- Einstein Condensation Suppose that the spin of the Bose gas is zero. The number of particle at energy level εl is It must be hold al>0, thus , Thechemical potential of an ideal Bose gas must be smaller than the energy of the lowest one-particle level. εl>μ (8.4.2) If ε0=0, thus μ<0 (8.4.3)

18. Thechemical potential has been determined from the equation We have

19. When the temperaturereachesTc, μ will be zero, thus the critical temperature Tc is determined by Let it gets

20. Discussion: • When temperature is lower than Tc, we can not obtain negative chemical potential in the formula (8.4.5), because the term of ε=0 is neglected from the formula (8.4.5). • We correct the formula (8.4.6) from as follows The first term n0 represents the number of particles per unit volume in the state ε=0 ; The second term nε>0 represents the number of particles per unit volume in excited state.

21. Insert this formula into (8.4.9) one gets When T<Tc, the excited state no longer contain all particles, and it becomes favorable for the system to fill the state ε=0 with the excess particles. This phenomenon is called Bose- Einstein condensation. Tc is called condensed temperature.

22. When T<Tc, the energy of the particles which ε>0 is So that In the region of Bose condensation, T<Tc, the specific heat increase like to the maximum value.

23. At T=Tc a spike appears, and for , CV approaches the value of the ideal gas. The specific heat of 4He for example shows a behavior at very low temperatures. At T λ=2.17K there is a phase change called λ phase change. In the region of T > T λ the liquid called He Ⅰis normal; In the region of T< T λthe liquid called He Ⅱhas superfluidity. London already had surmised that Bose- Einstein condensation might be responsible for this peculiar behavior.

24. §8.5 The Free Electronic Gas in Metals The mean number of electron at one level ε is We have 8.5.1 In the region of T=0K From the formula (8.5.1), at T=0K , we have f=1 ε<μ(0) f=0 ε>μ(0) (8.5.5)

25. One obtain that

26. Let , it gets p(0) called Fermi momentum is the maximum momentum at T=0K. The internal energy is The mean energy is . Let kTF=μ(0), TF is called Fermi temperature. TF~7.8×104K for Cu , so that μ(0) is very big. The μ at general temperature is closed to μ(0). So μ usuallycalledFermi energy εF.. μ >>kT , then The free electronic gas in metals is high degeneracy.

27. 8.5.2In the region ofT>0K From the formula (8.5.1), at T>0K , we have f >1/2 ε<μ f =1/2 ε=μ (8.5.9) f <1/2 ε>μ

28. The electronic number in the region of level kT nearμ take part in contribution to heat capacity. The heat capacity follows that In the region of room temperature, T /TF~1/260, so that the heat capacity of electrons can be neglected.

29. The integralterm I is written by here η(ε) is or , Let ε- μ=kTx, it follows that

30. Let,thusit gets Remark! μ/kT>>1, the integral comes from the contribution of small x. The integral function is expanded as power series of x.

31. So The equations (8.5.11) and (8.5.12) are written by From the equation (8.5.15) one gets When , thus

32. Similarly we have The heat capacity of ions is in direct proportion to T3; The heat capacity of electrons is in direct proportion to T.

33. The non-relativity gas pressure is At T=0K, p~3.7 ×1010Pa, which is considerable.

34. §8.6 Examples for Degeneracy Ideal Fermi Gases 8.6.1 Heat-electron emission • Heat-electron emission:The phenomenon emitting electrons in high temperature metal. (2) Work function χ, the depth of a potential through equal the least work to move electron from ε =0 to outside of the metal. The least work to move electron from ε = μ to outside of the metal is W= χ- μ (8.6.1) called work function.

35. The electron number in unit volume with momentum in dpx dpy dpz is In time of unit the electron number colliding unit area with momentum in dpx dpy dpz is here

36. εx> χ (8.6.5) The electrons satisfying the expression (8.6.5) can disengage bound be emitted to outside of the metal. The density of emission current is here

37. In general, θ>>1, thus 8.6.2 Contact potential difference

38. VA>0, VB<0 (8.6.7) The potential energy -eVA<0, -eVB>0 (8.6.8)

39. In the case of equilibrium we have -eVB- (-eVA)=WB-WA namely Remark ! The mechanism of introducing contact potential difference is to compensate the difference of origin chemical potential by using contact potential difference introduced between two contacting conductors so that the electronic gas is in equilibrium.

40. 8.6.3 Pauli paramagnetism Pauli paramagnetism: The free electronic gases in metals show weak paramagnetism. The electronic energy is(ε- μB)forthe spin magnetic momentum in the direction of magnetic field: The electronic energy is(ε+ μB)forthe spin magnetic momentum in the inverse direction of magnetic field.

41. The electronic number with the direction of magnetic momentum changing is .The change of every electron magnetic momentum is 2 μ, so the magnetization of the metal is

42. §8.7 The Electronic Gas of Quasi-two Dimensions and Quantum Hall Effect 8.7.1 The electronic gases of quasi-two dimensions (1) The electrons of quasi-two dimensions : Its movement is free in two directions, but limited in third direction. (2) The characterization of the electronic gases of quasi-two dimensions The density of state per unit area in the plane (x, y) is a constant . Here spin is not counted in this formula.

43. In direction of Z the possible energy is The density of state of i-th sub-band is Magnetic field points z direction.

44. It gets ωc is called cyclotron resonance frequency. Bohr quantum condition reads It introduce thus we obtain Virtual magnetic momentum orbital current orbital area

45. The energy of one electron is Insert the formula (8.7.6) it gets The space between two separate level is called landau level. The rigid quantum result is

46. The degeneracy of landau level is

47. 8.7.2 Quantum hall effect • Hall effect The current pointing direction x is at a magnetic field pointing direction z, which will introduce a transverse potential difference in the direction of y.

48. When Lorentz force equals transverse force, we have The electronic field Ey is Hall resistivity density of current Finally we get that

49. (2) Quantum hall effect 1980, Si-SiO2