239 0022-2291/03/1100-0239/0 © 2003 Plenum Publishing Corporation Journal of Low Temperature Physics, Vol. 133, Nos. 3/4, November 2003 (© 2003) Transition Temperatures of Bose–Einstein Condensation in Traps G.-X. Hu, 1, 2 X.-X. Dai, 2, 4 J.-X. Dai, 3 and W. E. Evenson 4 1 Department of Physics, Fudan University, Shanghai 200433, People’s Republic of China 2 State Key Laboratory of Applied Surface Physics, Fudan University, Shanghai 200433, People’s Republic of China E-mail: xxdai@fudan.ac.cn 3 New York University, Department of Chemistry, Washington Place, New York, New York 10003, USA 4 Department of Physics & Astronomy, Brigham Young University, Provo, Utah 84602-4645, USA E-mail: evenson@byu.edu (Received November 7, 2002; revised June 3, 2003) A careful study is made of the definitions of the transition temperature T c of an ideal Bose system in traps. We review several physical quantities that are used to define the transition temperature, which corresponds to different kinds of T c . The different definitions give different values for T c and the differences are quite large for finite systems. This makes the comparisons of theoretical predictions of the transition temperature shift with experimental results quite difficult. We also find that the derivative of chemical potential with respect to temperature is nearly discontinuous. This implies that in the thermodynamic limit the system might undergo a first-order phase transition. KEY WORDS : Bose–Einstein condensation; finite size effects; T c . 1. INTRODUCTION The realization of Bose–Einstein condensation ( BEC ) in dilute atomic gases trapped in a magnetic potential 1–3 has triggered intense research all over the world. 4–8 The experimental results have led to a need to determine a clear understanding of the nature of BEC in a finite sample. 9 It is seen that T c plays an important role in both theoretical and experimental studies. 10–17 Because T c can be measured directly by experiment, 17 there are many predictions of the shift of T c of an interacting system from an ideal system. Different predictions result from different theories. Mean-field theory 10 predicts the shift is t= T c -T 0 c T 0 c =-1.33 a a ho N 1/6 , (1)