Fermi-Dirac statistics

Transcription

1 UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch s attrbuted to spn. Partcles that have half odd-ntegral values of spn 2, 3 2, 5 2,...) are called ferons. Electron, proton, neutron, neutrno etc are all ferons. A syste wth two ferons, labelled A and B, located at x and x 2 have a wavefuncton that s antsyetrc under the exchange of partcles. ψx, x 2 ) ψ A x )ψ B x 2 ) ψ B x 2 )ψ A x ) ψx, x 2 ) It s clear fro the above that f we nterchange the locatons of the two ferons, the wavefuncton only changes by a sgn. ψx, x 2 ) 2 does not change and thus the probablty denstes do not change under ths nterchange. he two ferons are sad to be ndstngushable. One portant consequence of the above wavefunctos that t vanshes f we put x x 2. hs eans that two dentcal ndstngushable) ferons cannot be found n the sae poston. More generally, t s true that two dentcal ferons cannot be found n sae state. hs fact s called Paul s excluson prncple. hs property akes t nterestng to fnd the dstrbuton of ferons collected nsde a box held at soe teperature. We wll consder a box of N ferons. Each feros allowed to have one of the dfferent values of energy E, E 2,..., E. Let g, g 2,..., g be the nuber of ways of occupyng each of these energy levels. If n, n 2,..., n be the nuber of partcles n each of these energy levels the total energy E, we have N n + n n ) E n E + n 2 E n E E 2) We wll now derve a forula for the occupaton nuber - the nuber of partcles n the th energy level as a functon of the energy E of that level. Wth ths objectve we wll count the nuber of ways N ferons can be dstrbuted n the anner descrbed above. Startng wth the frst energy level E, the n partcles that occupy ths level has g ways to do t. hese can be thought of as g cells, all correspondng to energy E. We need to fll these cells by requrng Paul s excluson prncple to hold. hs eans that we can put at ost one feron per cell. hus at ost one partcle can go nto one cell. Clearly ths requres g n. We wll frst fnd the nuber of ways n ferons can be dstrbuted nto g cells. akng the frst feron, t has all the g cells open to t. After the frst one s provded a cell, the second feron has g cells to choose fro. here are g g ) of fllng the frst two hose that have ntegral values of spn 0,, 2,...) are bosons.

2 ferons. Contnung ths, we can see that there are g g )... g n + ) ways of fllng n ferons nto g cells. We can wrte ths count as g g )... g n + ) g! g n )! hese n ferons are dentcal and any of ther perutatons ust be ndstngushable. hus the nuber of dstnct ways n whch n ferons can be flled n g cells s g! n!g n )! 3) Slarly the nuber of dstnct ways to fll n 2 ferons n g 2 cells s g 2! n 2!g 2 n 2 )! he nuber of dstnct ways to fll n ferons n g cells, n 2 ferons n g 2 cells,..., n ferons n g cells gves us the therodynac probablty W E) g! n!g n )! g 2! n 2!g 2 n 2 )!... g! n!g n )! 4) akng a logarth of ths we get g! ln W E) ln n!g n )! + ln g 2! n 2!g 2 n 2 )! ln g! n!g n )!! ln n! )! 5) For,, we can use the Strlng s approxaton to splfy eqn??) to obtan ln W E) [ ln ln + ) ln ) + ] [ ln ln ) ln )] 6) We put ths box n contact wth a heat bath and let t reach equlbru. After t reaches equlbru f we observe ths box for a short enough perod of te we see no change n energy, e. E reans constant 2. hs requres E 0 E 0 7) At equlbru, the aount of energy suppled or taken by the heat bath s not suffcent to affect energy values E or allowed for the ndvdual partcles nor ther degeneraces. herefore nether the E nor vares at equlbru. 2 Evef we observe for long enough tes, we wll see fluctuatons of energy E that are too sall copared to the energy E. hus the syste can be consdered to be at equlbru for all tes 2

3 Note that the total nuber of partcles N always rean the sae despte the varaton the occupaton nubers. hus N 0 0 8) Further the therodynac probablty W E) wll be axu at equlbru, whch requres W E) to vansh at the equlbru value of energy. ln W E) also vanshes at the sae values of energy at whch W E) vanshes. ln W E) W E) W E) 0 9) Usng the for of W E) obtaned n eqn6), we can wrte the condton eqn8) as ln W E) 0 [ ln + ln ) + ) n ] 0 ln g ) 0 0) We ust requre the condton eqn0) to hold along wth the condtons n eqn7) and eqn8). It s obvous that f we ultply an arbtrary constant γ that s ndependent of ) to the left hand sde of eqn0) t ust stll vansh. γ ln ) 0 ) Slarly, ultplyng eqn7) and eqn8) wth arbtrary constants β and α β α E 0 2) 0 3) Clearly f su the left hand sdes of eqns,2, 3) t ust vansh too. γ ln g ) + β E + α 0 4) Now, the only way eqn4) can be satsfed for arbtrary varatons s f ts coeffcent vanshes for each. γ ln + β E + α 0 5) Dvdng through out by γ and defnng new arbtrary constants α α γ and β β γ, we can rewrte eqn5) as ln + βe + α 0 ln βe α 6) 3

4 Exponentatng ths eqn6) and wrtng n ters of the rest of the quanttes e α e βe + 7) he quantty e α s called fugacty. α s proportonal to the aount of work requred to nsert one ore feronto the box called the checal potental of the syste. In order to evaluate the arbtrary constant β, we wll supply a sall aount of energy to ths box of ferons at a fxed volue and fnd out by how uch the entropy changes. he rate of change of entropy wth energy at constant volue s the nverse of the teperature of the syste n absolute scale Kelvns). S 8) E V When a sall aount of energy E s suppled t leads to a change n the nuber of ferons n dfferent energy levels E E. he aount of energy suppled wll be nsuffcent to change the nature of the energy levels E of ndvdual ferons or ther degeneraces and hence these rean constant. E 0; 0 he entropy of the syste s found usng eqn6) to be S ln W E) [ ln ln ) ln )] 9) We have used Strlng s approxaton to get the fnal expresson eqn9). he change n entropy due to an excess energy E s obtaned fro eqn9) as [ S ln ) + ln ) + ) n ] n [ ln g ] Substtutng the expresson for fro eqn7) nto eqn20) S ln e α βe ) kb α β α βe ) 20) E 2) β E 22) In eqn2), the frst su s zero as the total nuber of partcle rean unchanged as n eqn8) and the second su gves us the quantty n eqn22). Usng the result of eqn22) n eqn8) we get S E β β 4 23)

5 hs fxes the arbtrary constant β. Substtutng ths nto eqn7) we get the Fer-Drac dstrbuton for the occupaton nuber as e α e E + Now consder the box of N ferons at 0K. At absolute zero all the partcles tend to st at the lowest possble energy level avalable to the. But ferons obey Paul s excluson prncple and only one at a te can occupy any quantu state. If the total nuber of ferons N s uch greater than the degeneracy g of the lowest energy level E, the reanng ferons wll have to occupy the reanng lower energy levels. hus, even at 0K there could be ferons up to a certan energy level E F. hs energy level s called the Fer level of the syste. At 0K, or energy levels wth E E F, wll have all the cells wll be flled up by one feron each. he nuber of ferons n such energy levels wll be thus equal to the nuber cells,. For energy levels wth E > E F we fnd no ferons 0. hus the rato { for E EF ; 0 for E > E F Fro Fer-Drac probablty dstrbuton eqn24) we see that ths rato s 24) e α e E + 25) he expresson eqn25) on the rght hand sde has the desred behavour f α E F as explaned below. At 0K for E < E F, the quantty E E F becoes. hen the expresson on the RHS of eqn25) becoes e hs atches wth the expected value of. At 0K for E > E F, the quantty E E F becoes. hen the expresson on the RHS of eqn25) becoes e Agan the expected value 0 s reproduced correctly. hus havng fxed both the arbtrary constants we can wrte down the Fer-Drac dstrbuton as 26) e E E F + At 0K, ths dstrbutos, strctly speakng, not vald for E E F. Matheatcally ths s because the rato E E F s undeterned n ths case. Physcally, ths ndcates an abguty for the nuber of ferons n F at ths energy level as all the g F cells correspondng to ths energy value need not be flled up. However at any fnte teperature 0K, Fer level s the energy value at whch the rato n F gf. 2 Next we wll use Fer-Drac dstrbuton to obtan the energy dstrbuton of electrons n etals 5