tF. London, Superfluids, Vol. II (Wiley, New York, 1954), Appendix. in Java

Implementation QR Code in Java tF. London, Superfluids, Vol. II (Wiley, New York, 1954), Appendix.
tF. London, Superfluids, Vol. II (Wiley, New York, 1954), Appendix.
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O:$;z:$;l For comparison we recall that 0 :$; Z < 00 in the case of Fermi statistics. For small z, the power series (12.42) furnishes a practical way to calculate g3/2(Z): Z2 Z3 g3/2(Z) = Z + 2 3/ 2 + 33/ 2 + ... (12.43)
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t(~) = 2.612... (12.44) I where t(x) is the Riemann zeta function of x. Thus for all z between 0 and 1,
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A graph of g3/2(Z) is shown in Fig. 12.3. Let us rewrite (12.41) in the form
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(12.45)
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(12.46)
This implies that (no)/V > 0 when the temperature and the specific volume are such that (12.47) This means that a finite fraction of the particles occupies the level with p = O. This phenomenon is known as the Bose-Einstein condensation. The condition (12.47) defines a subspace of the thermodynamic P-v-T space of the ideal Bose gas, which corresponds to the transition region of the Bose-Einstein condensation. As we see later, in this region the system can be considered to be a mixture of two thermodynamic phases, one phase being composed of particles with p = 0 and the other with p O. We refer to the region (12.47) as the condensation region. It is separated from the rest of the P-v-T space by the two-dimensional surface
(12.48)
STATISTICAL MECHANICS
____L
o(ir)
Fig. 12.4 (a)
0'-----------'-1---2.612
Graphical solution of (12.41); Bose gas contained in a finite volume V.
the fugacity for an ideal
For a given specific volume v, (12.48) defines a critical temperature Tc :
A.Jc = vg 3 / 2 (1)
(12.49)
(12.50)
As indicated by (12.49), Tc is the temperature at which the thermal wavelength is of the same order of magnitude as the average interparticle separation. For a given temperature T, (12.48) defines a critical volume vc:
>-..3
(12.51)
In terms of Tc and Vc the region of condensation is the region in which T < ~. or v < vc. To find z as a function of T and v we solve (12.41) graphically. For a large but finite value of the total volume V the graphical construction in Fig. 12.4a yields the curve for z shown in Fig. 12.4b. In the limit as V ~ 00 we obtain
(~ ~ g3/2(l))
(~ ~ g3/2(1))
(12.52)
For (>-..3/V) ~ g3/2(1), the value of z must be found by numerical methods. A graph of z is given in Fig. 12.5. To make these considerations more rigorous the following point must be noted. It is recalled that (12.41) is derived from the condition
1 = -
>+ V
(no>
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' - - - - - - 1 ' - - - - - - ~3
Fig. 12.5 The fugacity for an ideal Bose gas of infinite
volume.
by replacing the sum on the right side by an integral. It is clear that this integral is unchanged if we subtract from the sum any finite number of terms. More generally, (12.41) should be replaced by the equation
~g (z) + (no) + (n 1 ) + (n 2 ) + ... ) >-..3 3/2 V V V
where, in the parentheses, there appear any finite number of terms. Every term in the parentheses, however, approaches zero as V ~ 00. For example, (n 1 ) 1 1 1 1 --=- l <-----,--f3 <l-l V V Z- e f3<l -1 - V e where
2m 1
(21Th) V 2/ 3 sum of the squares of three integers not all zero (n 1) 1 2m{3V 2/ 3 --<-0 V - V (21Th)2{32/ 1 V-'" 00
Hence (12.53)
This shows that (12.41) is valid. By (12.52) and the fact that (no)
z/(l - z) we can write
1_(~)3/2=1_~
(~ s g3/2(l)) (~ ~ g3/2(l))
(12.54)
A plot of (n 0) / N is shown in Fig. 12.6. It is seen that when T < Te , a finite fraction of the particles in the system occupy the single level with p = O. On the other hand (12.53) shows that (np)/N is always zero for p =1= O. Therefore we have the following situation: For T> Te no single level is occupied by a finite fraction of all the particles. The particles "spread thinly" over all levels. For T < Te a finite fraction 1 - (T/TJ3/2 occupies the level with p = 0 while the rest of the particles" spread thinly" over the levels with p =1= O. At absolute zero all particles occupy the level with p = O.