Question

Ideal Bose gas (a) Consider a 2D ideal Bose gas with density of state D (e) = DoL2, show that Bose- Einstein condensation is not possible in such a gas. (b) Consider a 4D ideal Bose gas with density of state D(e) = DOL6, find the Bose- Einstein condensation temperature in terms of Do, n = N/La, and a dimensionless integral FM A = (6) ex 1 12

Answer 1

(a) Ans Cin awestin Civer Tru Jar Bosons 1 N = E fuom The We can Then isolate Ihe tenm Tar E- O Summation N - I 1 3 o23 3 N,Y Whore The a parhedes occupyny uoundStete E-o no. Parheles no. oluplying 6tates all The excited Now ut's cvaluate <Ne 1 -1 E7O heplace The summation b integuatien as We can follows: DCE stales Density 1

Then de DCe) -1 Gidven for DCE)= Do L2 2 D Do L2 1 de D.L de BEL 3 C 1 take BE=x dt dx 2'-L 3 Naw We Knar chemcal potenHal. luchen Basens -auo highest u fwr Boscns u-0, Hence 3 = 2 dx Ne DsL P 21 Ne This seies du'vorges. L But This intguaf diverges b tenm Hence Thee is no U Pper bound cn Hence Base Einstein conden satian w orit oce ur 2 D far

Here given as Similay We can as 1 3epe- 3 far Besans May au) D CE 1-3J2, D,LE dE BE N Snce - dn dn= 12 uam uestn it is gven Thal A <N D,LKT (A) 4 2 hiphest excited ctnte, This is The maximum ar hence T TE CBFC Bae Linstien Candenat Bose Linstein conden satin tewmporatore ku Since valus her, so Thal 2 BEC used 2 A. N N

Kg Belhmann's canstant 2 14 Po A BEC 1 N BEC FR A 2 Where Shis is The Base-Einstein condensatin temperature feer 4-dinen sional Bose Gas