Pr. Dr. I. Nasser Phys, -Oct-7 FERMI_DIRAC GASSES Fermins: Are particles hal-integer spin that bey Fermi-Dirac statistics. Fermins bey the Pauli exclusin principle, which prhibits the ccupancy an available quantum state by mre than ne particle. Ideal ermin gas: Cnsisting N nn-interacting and indistinguishable ermins in a cntainer vlume V held at abslute temperature. Fermi-Dirac distributin: Fr an ideal FD gas (nn-interactins between the indistinguishable particles) N mlecules in a vlume V, the mst prbable number particles with energy is: * ni ( i ) ( i ) gi e and the cntinuum is: N ( ) ( ), y ( ) y g( ) e Fermi unctin gives the prbability that a single particle state will be ccupied by a ermin. Clearly, ( ). i i he Fermi unctin at = ( ) at three dierent temperatures in FD statistics Ntes: - n need t be negative, due t the + in the denminatr. may be psitive r negative. - I, then - I ( ) e, and ( ) reduces t the Maxwell-ltzmaan distributin. i- i ii- i e and all expnentially like Maxwell-ltzmaan distributin. iii- i iv- In the limit we have sharp drp and g ( ) d ( ) N g ( ) ( ) d Where V g ( ) g / s m, g s s h / m V h, r electrns
Pr. Dr. I. Nasser Phys, -Oct-7 Exclusin principle implies that a FD gas has a large mean energy even at abslute zer, ( ()). [Nte that: Degenerate here means illed, nt as the case QM], Very lw temperature Cmpletely degenerate Lw temperature degenerate Intermed iate temperature Slightly degenerate High temperature Classical limit At abslute zer, due t exclusin principle, all the states with ( ()) are cmpletely illed and all the states with Cmpletely Degenerate Fermi-Gas tal number particles: are cmpletely empty. / / m 8V m N g( ) ( ) d V d h h / h N m8 V Fr cnvenience, we intrduce a Fermi temperature such that F k. his can be written as: h N k mk 8 V Example: Metallic ptassium has.86 kg/m and atmic weight M 9 kg/kmle. Find,, and v. Slutin: We will cnsider ne ree electrn per atm r mnvalent atms. hus the cncentratin is: 6 N Na (6. atms/kmle)(.86 kg/m ) 8. atms/m, V M 9 kg/kmle / / 7 / hc (. ev.m) 8 atms 6 h N N.. ev 8m V 8mc V 8. ev m hen - 8.67 K /. ev 79 K k ev S, even at rm temperature we have t treat the metallic ptassium quantum mechanically. Use p p m m 8. ev /. ev. m/s c 7. m /s 6 v m mc v 8. m/s
Pr. Dr. I. Nasser Phys, -Oct-7 he speed the electrn in metals is times the speed sund. Internal energy: U g ( ) ( ) d N Other thermdynamic unctins are: S, / / / / d h m 8 V m V h PV U S N N, N 7 P.7 atm r electrns) V V hus at a ermin gas exerts a pressure. I the electrns in a metal were neutral they wuld exert a pressure abut atm. Given this tremendus pressure, we can appreciate the rle the surace ptential barrier in keeping the electrns rm evaprating rm the metal. In ther wrds, the Culmb attractin t the ins cunterbalances the pressure. Fr the value is psitive and large. K H.W. Prve that () 6 K ( ), Ntes: - ( ) is psitive r temperature belw the Fermi temperature and negative r higher temperature. - As the temperature increases abve, mre and mre the ermins are in the excited states and the mean ccupancy the grund state alls belw /. In this regin, () e which implies that k r. Fr bsn gas ( ) is negative at all temperatures and is zer at abslute zer. A high temperature the ermin gas apprximates the classical ideal gas. In the classical limit: / mk V k ln z k ln h N
Pr. Dr. I. Nasser Phys, -Oct-7 Example: r kil-mle the ermin.69 K, s that 9, then classically He gas atms at SP, / z mk V ln ln.7 k N h N And e.. he average ccupancy single particle states is very small, as in the case an ideal gas beying the Maxwell-ltzmann distributin.
Pr. Dr. I. Nasser Phys, -Oct-7 Strngly degenerate gas / d x dx N g( ) ( ) d G V G V ( ) k xx e s s e Where we used x ( ) and x. Our target is the calculatin as a unctin the abslute temperature. ( ) Mathematical Nte: Smerield s integral he standard integral in the abve equatin culd be slved with the help Smerield s integral, i.e. s s x ss ( ) ( x x ) x dx, s 6 x Fr the case the analytic unctin ( x x ) e ( x x ) and, we have e x / / ( x x dx x ) 8x s / ----------------------------------------------------------------------------------- / x dx / k x x x 8 e x k 8 S the ttal number will be / k N GsV 8 / Using N G V, ne inds s / / k ( ) 8 As a irst rder apprximatin, we put in the square bracket t have hen, using k F / / k ( ) 8, we get / / ( ) 8 F Nw, we can calculate ( ) as a unctin as the llwing: k ( ) 8 / / / ( ) 8 F F
Pr. Dr. I. Nasser Phys, -Oct-7 Example: Calculate ( ) ptassium at. ev at F. K. Answer:. K. Nte that r ptassium,. F. ( )...986. ev Which is a slight change with respect t. ev. his is because F is high r ptassium and we can cnsider ( ). H.W. Prve i- the internal energy U N F 6 F ii- Speciic heat U CV, e U V F F tal speciic heat metal culd be written as: U, C C C V ttal V e V A (Debye) ree electrn Deby's he llwing igure is a cmparisn between the speciic heat in the three distributin 6
Pr. Dr. I. Nasser Phys, -Oct-7 Example: Calculate the speciic heat ptassium at. K. Nte that r ptassium F. K. Answer: C. K V, e.9 F F. K 6. Which is very small cmpare t at rm temperature. calculate the values A and in the equatin linear equatin, see the llwing igure. C A, ne can treat it as a V ttal ii- Entrpy CVe, S d F F S, S as. iii- Helmhltz F U S F F 8 F iv- Pressure F F U P V N, V F 8 F V 7