Superfluids are a unique state of matter that exhibit frictionless flow and zero viscosity, making the measurement of their velocity a fascinating and challenging task. In this comprehensive guide, we will delve into the theoretical background, formulas, and practical techniques used to determine the velocity in these remarkable fluids.
Understanding Superfluidity
Superfluidity is a phenomenon that occurs in certain quantum fluids, such as liquid helium-4 (4He) and ultracold atomic gases, at extremely low temperatures. This state of matter is characterized by the presence of a critical velocity, below which the flow is laminar and above which turbulence occurs. The superfluidity is quantified by imposing twisted boundary conditions on the system, which results in a phase gradient that diverges when the velocity decreases.
Landau’s Criteria of Superfluidity
The superfluid critical velocity, which is the threshold velocity below which the flow is laminar, can be calculated using the Landau’s criteria of superfluidity. According to this criteria, the relative velocity between the fluid and the capillary is smaller than the critical value, v < vc. The critical velocity is given by the sound velocity for weakly interacting Bose gases, while for strongly interacting fluids such as liquid 4He, the critical velocity is smaller than the sound velocity due to the complicated excitation spectrum.
The Landau’s criteria of superfluidity is expressed mathematically as:
vc = min(E(p) / p)
where vc is the critical velocity, E(p) is the energy of the excitation with momentum p, and the minimum is taken over all possible excitations.
Measuring Velocity in Superfluids
To measure the velocity in superfluids, researchers have developed several methods and techniques, each with its own advantages and limitations. Let’s explore these approaches in detail:
Mechanical Oscillatory Resonators
Mechanical oscillatory resonators create an oscillatory superflow that can be used to create and detect turbulence via the interactions between the flow and the oscillator. These devices allow for the measurement of the driving force and peak velocity in helium at all currently achievable temperatures. The ease of use of these resonators makes them a popular choice for velocity measurements in superfluids.
The velocity in a mechanical oscillatory resonator can be calculated using the following formula:
v = 2πf A
where v is the velocity, f is the frequency of the oscillation, and A is the amplitude of the oscillation.
Laser Tracking of Tracer Molecules
Another method used to measure the velocity in superfluids is the laser tracking of tracer molecules. In this technique, quantum vortices trap the tracer molecules, and the laser tracks the movement of the tracers and hence the vortices themselves. This method has been used to picture tracer molecules in 4He using a laser, which has allowed for the visualization of quantum turbulence at extremely turbulent states.
The velocity of the tracer molecules can be calculated using the following formula:
v = Δx / Δt
where v is the velocity, Δx is the displacement of the tracer molecule, and Δt is the time interval.
Visualization of Vortex Filaments
Superfluids experience turbulence, which is made up of a tangle of string-like vortex filaments. These vortex filaments offer a much more conceptually intuitive system to the chaotic dance of classical turbulence. By visualizing the vortex filaments, researchers can infer the velocity of the superfluid flow.
The velocity of the vortex filaments can be calculated using the following formula:
v = Γ / (2πr)
where v is the velocity, Γ is the circulation of the vortex, and r is the radius of the vortex core.
Examples and Numerical Problems
To further illustrate the concepts and techniques discussed, let’s consider some examples and numerical problems:
Example 1: Calculating the critical velocity in liquid 4He
Given:
– The sound velocity in liquid 4He is 238 m/s
– The excitation spectrum of liquid 4He is complicated
Using the Landau’s criteria of superfluidity, the critical velocity in liquid 4He is smaller than the sound velocity, due to the complicated excitation spectrum.
Example 2: Measuring the velocity in a mechanical oscillatory resonator
Given:
– The frequency of the oscillation is 100 Hz
– The amplitude of the oscillation is 0.1 mm
Using the formula v = 2πf A, the velocity in the mechanical oscillatory resonator is:
v = 2π × 100 Hz × 0.1 mm = 62.8 mm/s
Example 3: Calculating the velocity of tracer molecules in 4He
Given:
– The displacement of the tracer molecule is 0.5 mm
– The time interval is 0.01 s
Using the formula v = Δx / Δt, the velocity of the tracer molecule is:
v = 0.5 mm / 0.01 s = 50 mm/s
Example 4: Determining the velocity of vortex filaments in superfluid 4He
Given:
– The circulation of the vortex is 9.97 × 10^-8 m^2/s
– The radius of the vortex core is 1 μm
Using the formula v = Γ / (2πr), the velocity of the vortex filament is:
v = (9.97 × 10^-8 m^2/s) / (2π × 1 × 10^-6 m) = 1.59 m/s
These examples demonstrate the application of the theoretical formulas and techniques discussed earlier, providing a practical understanding of how to measure the velocity in superfluids.
Conclusion
In this comprehensive guide, we have explored the various methods and techniques used to determine the velocity in superfluids, including the theoretical background, formulas, and practical examples. From mechanical oscillatory resonators to laser tracking of tracer molecules and visualization of vortex filaments, each approach offers unique insights and challenges in the quest to understand the remarkable properties of these quantum fluids. By mastering these techniques, physicists and researchers can unlock the secrets of superfluidity and push the boundaries of our understanding of the quantum world.
References
- Superfluidity – an overview | ScienceDirect Topics
- Velocity Statistics in Holographic Fluids: Magnetized Quark-Gluon Plasma
- Superfluidity – an overview | ScienceDirect Topics
- New Methods of Measurements in Superfluid Helium
- Superfluidity
- Superfluidity
- Superfluidity
- Superfluidity
- Superfluidity
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