The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number
w
of dimensions has not been restricted to three. The energy levels are taken to be
ε
i
≡
ε
s
1
,
.
.
.
.
,
s
10
=
constant
h
2
m
s
1
α
−
1
a
1
2
+
.
.
.
+
s
w
α
a
w
2
(
1
≤
α
≤
2
)
the quantum numbers being
s
1
,
w
= 1, 2, ..., and a
1
, ..., a
w
being certain characteristic lengths. (For α = 2, the potential field is that of the
w
-dimensional rectangular box; for α = 1, we obtain the
w
-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q =
w
/α determines the appearance of an Einstein transition temperature T
0
·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ
-
per particle and the specific heat dϵ
-
/dT are continuous at T = T
0
· For q > 2, the specific heat is discontinuous at T = T
0
, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T
0
is finite only if the quantity
v
= N/(a
1
.... a
w
)2/ α
¯
is finite. For a rectangular box,
v
is equal to the mean density of the gas. If
v
tends to zero or infinity as N→ ∞, then T
0
likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T
0
' there is a finite fraction N
0
/N of the particles, given by N
0
/N = 1-(T/T
0
)
q
, in the lowest state. London’s formula (1938
b
) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.
Bose–Einstein Condensation of Magnons in TlCuCl3: Phase Diagram and Specific Heat from a Self-consistent Hartree–Fock Calculation with a Realistic Dispersion Relation
