SME
Thermal & Statistical Physics · Problem type
Bose–Einstein condensation temperature
Use this for “A cloud of sodium-23 atoms holds 5 × 10¹⁹ atoms per cubic meter. Below what temperature does it form a Bose–Einstein condensate?”
Table
Atom density (n)
(m⁻³)
Critical temperature (Tc)
(nK)
This is an example. Type your own values to start a new problem.
2πħ²
m (u)
k
B
n (m⁻³)
2.612
2
÷
3
=
Tc (
)
λₜₕ
Thermal wavelength at Tc
calculated · nm
Formulas
Tc
=
3.048 ×
10
-18
m
×
(
n
2.612
)
2
3
3.9838 ×
10
-7
=
3.048 ×
10
-18
86.909
×
(
1 ×
10
20
2.612
)
2
3
λₜₕ
=
∛(
2.612
n
)
×
10
9
296.7
=
∛(
2.612
1 ×
10
20
)
×
10
9
Formulas are worked in K. The step-by-step shows the conversions.
Assumptions
•
An ideal gas of bosons in a box: ⁸⁷Rb is 86.909 u, ²³Na is 22.990 u, ⁴He is 4.003 u.
•
Below Tc a large share of the atoms falls into the lowest state: nλₜₕ³ reaches ζ(3/2) = 2.612, so λₜₕ at Tc depends on n alone.
•
With m in u, 2πħ² ÷ (mkB) = 3.048 × 10⁻¹⁸ ÷ m K·m².
•
Liquid ⁴He is not ideal: the formula gives 3.1 K, the real transition is at 2.17 K.
Step-by-step
WE KNOW
m = 86.909 u, n = 1 × 10²⁰ m⁻³
FIND
critical temperature (Tc), thermal wavelength at Tc (λₜₕ)
Step 1 · Find critical temperature (Tc)
Use Tc = 3.048 × 10⁻¹⁸ ÷ m × (n ÷ 2.612)2/3
Condensation starts when the thermal wavelength reaches the atoms’ spacing: nλₜₕ³ = 2.612, solved for T.
Tc
=
3.048 ×
10
-18
86.909
×
(
1 ×
10
20
2.612
)
2
3
Tc
=
3.048 ×
10
-18
86.909
×
(3.828 ×
10
19
)
2
3
Tc
=
3.048 ×
10
-18
86.909
×
(1.136
×
10
13
)
Tc
=
(3.5071
×
10
-20
)
×
(1.136
×
10
13
)
Tc
=
3.9838 ×
10
-7
K
Step 2 · Find thermal wavelength at Tc (λₜₕ)
Use λₜₕ = ∛(2.612 ÷ n) × 10⁹
At
Tc,
nλₜₕ³
=
2.612:
the
wave
packets
just
overlap;
×
10
9
turns
meters
into
nm.
λₜₕ
=
∛(
2.612
10
20
)
×
10
9
λₜₕ
=
∛(2.612
×
10
-20
)
×
10
9
λₜₕ
=
(2.967
×
10
-7
)
×
10
9
λₜₕ = 296.7 nm
Step 3 · Convert the answers
Change each answer back to the unit you chose.
Tc = 3.9838 × 10⁻⁷ K = 398.4 nK (1 K = 10⁹ nK)
Check
Put the numbers back in. Both sides should match.
3.9838 ×
10
-7
=
3.048 ×
10
-18
86.909
×
(
1 ×
10
20
2.612
)
2
3
✓
296.7
=
∛(
2.612
10
20
)
×
10
9
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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