Calculating Critical Temperature (T_c) for BEC

In our last lesson, we established the fundamental criterion for Bose-Einstein condensation in a 3D ideal gas: the transition occurs when the system is cooled to a point where the excited states can no longer accommodate all the particles. We identified this point as the critical temperature, $T_{c}$ , where the chemical potential reaches its upper bound of $μ = 0$ . Today, we will build directly on that foundation.

Our objective is to calculate the critical temperature (T_c) for BEC. This involves translating the physical condition for condensation into a concrete mathematical problem. We will start with the integral expression for the particle number, incorporate the density of states for a 3D free particle system, and perform the necessary integration and algebraic manipulation to arrive at the explicit formula for $T_{c}$ . This calculation is a cornerstone of the theory of ideal Bose gases.

1. The Defining Equation for the Critical Temperature

As we established previously, the total number of particles $N$ in a Bose gas is distributed between the ground state ( $N_{0}$ ) and the excited states ( $N_{e x}$ ):

N = N_{0} + N_{e x} = N_{0} + \int_{0}^{\infty} \frac{g ( ε )}{e ^{β (ε - μ)} - 1} d ε

Here, $g (ε)$ is the density of states, which tells us the number of available states per unit energy interval.

The critical temperature $T_{c}$ is precisely the temperature at which the excited states are just saturated, meaning they hold all $N$ particles with the chemical potential at its maximum value, $μ = 0$ . At this boundary point, the macroscopic occupation of the ground state is just beginning, so we can set $N_{0} \approx 0$ . Therefore, the equation defining $T_{c}$ is:

N = \int_{0}^{\infty} \frac{g ( ε )}{e ^{ε / k_{B} T_{c}} - 1} d ε

To solve this for $T_{c}$ , our first task is to determine the explicit form of $g (ε)$ for a gas of free, non-relativistic bosons in three dimensions.

The following video lecture provides a clear, step-by-step derivation of both the density of states and the subsequent calculation of $T_{c}$ . It's an excellent visual guide for the calculation we are about to undertake.

Bose-Einstein Condensates | Prof B. Prasanna Venkatesh | Lecture 01

This segment of the lecture by Prof. B. Prasanna Venkatesh walks through the entire calculation of the critical temperature, from first principles.

Please watch the portion from this timestamp. Pay close attention to two key parts of the derivation: The Density of States: Observe how the problem of counting quantum states in a box is transformed from a sum over discrete states to an integral over a continuous energy spectrum, leading to the result g(\varepsilon) \propto \sqrt{\varepsilon}. The Integral Evaluation: Follow the change of variables used to transform the integral into a standard form related to the Riemann zeta function, leading to the final expression for T_c.

Let's now work through the key steps of this calculation in detail, solidifying the arguments presented in the video.

2. Calculating the Density of States

For a spin-0 particle in a cubic box of volume $V = L^{3}$ , the allowed wavevectors are quantized, and each state occupies a volume of $(2 π / L)^{3}$ in k-space. To find the total number of states, we replace the sum over discrete states with an integral over the continuous k-space, divided by the volume per state:

k \sum \to \frac{V}{( 2 π ) ^{3}} \int d^{3} k

Since the energy $ε = ℏ^{2} k^{2} / (2 m)$ depends only on the magnitude $k = ∣ k ∣$ , we can integrate over spherical shells in k-space:

\frac{V}{( 2 π ) ^{3}} \int_{0}^{\infty} 4 π k^{2} d k

The density of states $g (ε)$ is defined such that the number of states in an energy interval $d ε$ is $g (ε) d ε$ . To find it, we perform a change of variables from $k$ to $ε$ :

k = \frac{2 m ε}{ℏ} ⟹ d k = \frac{1}{ℏ} \frac{m}{2 ε} d ε

Substituting these into the integral expression gives the number of states:

\int_{0}^{\infty} [\frac{V}{( 2 π ) ^{3}} 4 π (\frac{2 m ε}{ℏ ^{2}})] (\frac{1}{ℏ} \frac{m}{2 ε} d ε) = \int_{0}^{\infty} g (ε) d ε

Identifying the term inside the integral that multiplies $d ε$ gives us the density of states:

g (ε) = \frac{V}{2 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} ε^{1/2}

This $ε$ dependence is characteristic of non-relativistic particles in 3D and is the crucial ingredient for our calculation.

3. Evaluating the Integral for $T_{c}$

We now substitute this density of states into our defining equation for $T_{c}$ :

N = \int_{0}^{\infty} \frac{1}{e ^{ε / k_{B} T_{c}} - 1} [\frac{V}{2 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} ε^{1/2}] d ε

Let's collect the constants and focus on the integral.

N = \frac{V}{2 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} \int_{0}^{\infty} \frac{ε ^{1/2}}{e ^{ε / k_{B} T_{c}} - 1} d ε

To solve this, we make the standard substitution $x = ε / k_{B} T_{c}$ , which implies $ε = x k_{B} T_{c}$ and $d ε = k_{B} T_{c} d x$ .

N = \frac{V}{2 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} \int_{0}^{\infty} \frac{( x k _{B} T _{c} ) ^{1/2}}{e ^{x} - 1} (k_{B} T_{c} d x)

Pulling the temperature-dependent terms out of the integral:

N = \frac{V}{2 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} (k_{B} T_{c})^{3/2} \int_{0}^{\infty} \frac{x ^{1/2}}{e ^{x} - 1} d x

The remaining integral is a standard form known as a Bose-Einstein integral, which can be expressed in terms of the Gamma function $Γ (s)$ and the Riemann zeta function $ζ (s)$ :

\int_{0}^{\infty} \frac{x ^{s - 1}}{e ^{x} - 1} d x = Γ (s) ζ (s)

In our case, $s - 1 = 1/2$ , so $s = 3/2$ . The value of the Gamma function is $Γ (3/2) = π /2$ . Therefore, the integral evaluates to:

\int_{0}^{\infty} \frac{x ^{1/2}}{e ^{x} - 1} d x = Γ (3/2) ζ (3/2) = \frac{π}{2} ζ (3/2)

where $ζ (3/2) \approx 2.612$ is a numerical constant.

Substituting this result back gives:

N = \frac{V}{2 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} (k_{B} T_{c})^{3/2} (\frac{π}{2} ζ (3/2))

This expression connects the total particle number $N$ to the critical temperature $T_{c}$ . The following resource provides a concise summary of this calculation, presenting it in terms of the thermal de Broglie wavelength.

[PDF] 8.08 Statistical Physics II — Spring 2019 Recitation Note 8

These MIT recitation notes offer a compact and rigorous derivation of the critical temperature. They connect the integral we just evaluated to the Bose-Einstein function g_{3/2}(z) that we encountered in the previous lesson.

Start by reading the section "Chemical Potential and Bose-Einstein Condensation". Follow the steps from Eq. (3) to Eq. (6), where the particle number integral is expressed in terms of the thermal wavelength \lambda_T and the Bose-Einstein function g3/2(z). This provides an alternative, equivalent path to our result. Note how the text identifies the saturation point: the right-hand side has a maximum value when z=1, which corresponds to \mu=0. The value is g_{3/2}(1) = \zeta(3/2). Finally, observe the resulting equation for the critical temperature, kBTc, which is precisely what we will now derive.

4. The Final Expression for $T_{c}$

Our final step is to rearrange the equation for $N$ to solve for $T_{c}$ . Let's define the particle density $n = N / V$ and simplify the constants.

n = \frac{1}{2 π ^{2}} (\frac{2 m k _{B} T _{c}}{ℏ ^{2}})^{3/2} (\frac{π}{2} ζ (3/2)) = \frac{( 2 m k _{B} T _{c} ) ^{3/2}}{4 π ^{3/2} ℏ ^{3}} ζ (3/2)

Recalling that $h = 2 π ℏ$ , we have $ℏ^{3} = h^{3} / (8 π^{3})$ .

n = \frac{( 2 m k _{B} T _{c} ) ^{3/2}}{4 π ^{3/2}} \frac{8 π ^{3}}{h ^{3}} ζ (3/2) = (\frac{2 π m k _{B} T _{c}}{h ^{2}})^{3/2} ζ (3/2)

Solving for $k_{B} T_{c}$ :

(k_{B} T_{c})^{3/2} = \frac{n}{ζ ( 3/2 )} (\frac{h ^{2}}{2 π m})^{3/2}

k_{B} T_{c} = (\frac{n}{ζ ( 3/2 )})^{2/3} \frac{h ^{2}}{2 π m}

This is the celebrated formula for the critical temperature of Bose-Einstein condensation for an ideal gas in three dimensions.

It's insightful to see how this calculation can be generalized. The existence of a finite $T_{c}$ depends critically on the behavior of the density of states $g (ε) \propto ε^{α - 1}$ at low energy. For condensation to occur, the integral for the number of excited particles must converge when $μ = 0$ , which requires $α > 1$ . As we've seen, for a 3D free gas, $α = 3/2$ , satisfying the condition. This framework also allows for calculating $T_{c}$ in other systems, such as atoms in a harmonic trap, which is highly relevant to experimental realizations of BEC.

Bose statistics and Bose-Einstein condensation.

These lecture notes by Nobel laureate Anthony Leggett provide a powerful, generalized perspective on the condition for BEC.

Please read the section that starts just after the initial setup. Focus on the argument about the convergence of the integral based on the low-energy form of the density of states, \rho(\epsilon). Notice how the condition (5) depends on \rho(\epsilon) as \epsilon \to 0. This argument elegantly explains why BEC occurs in some dimensions and not others. Then, look at the final results for T_c for both free space and a harmonic trap.

Conclusion

In this lesson, we have performed the explicit calculation of the critical temperature for Bose-Einstein condensation. By combining the fundamental principles of quantum statistical mechanics with the specific properties of a 3D free particle system, we have moved from a qualitative criterion to a quantitative prediction.

Key Takeaways:

The calculation of $T_{c}$ hinges on evaluating the integral for the total number of particles in excited states at the point where the chemical potential $μ = 0$ .
The crucial physical input for this calculation is the density of states, which for a 3D free gas is $g (ε) \propto ε^{1/2}$ .
The resulting definite integral can be solved using standard mathematical functions, leading to the final expression: $T_{c} = \frac{h ^{2}}{2 π m k _{B}} (\frac{n}{ζ ( 3/2 )})^{2/3}$
The existence of a finite $T_{c}$ is determined by the low-energy behavior of the density of states, providing a general criterion for condensation.

Now that we have a precise value for $T_{c}$ , we are equipped to analyze the behavior of the system below this temperature. In our next lesson, we will calculate the condensate fraction—the fraction of particles that occupy the ground state—as a function of temperature for $T < T_{c}$ , and explore the thermodynamic consequences of this macroscopic quantum phenomenon.

Recommended resources

1. The Defining Equation for the Critical Temperature

2. Calculating the Density of States

3. Evaluating the Integral for Tc​

4. The Final Expression for Tc​

Conclusion

3. Evaluating the Integral for $T_{c}$

4. The Final Expression for $T_{c}$