Related Experiment Video
Updated: Feb 1, 2026
09:23
Harmonic Nanoparticles for Regenerative Research
Published on: May 1, 2014
12.1K
Thermal echo in a finite one-dimensional harmonic crystal.
A S Murachev1, A M Krivtsov1,2, D V Tsvetkov1
1Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, Russia.
Summary
A thermal echo phenomenon was discovered in finite crystals, where temperature oscillations sharply increase periodically after initial damping. This echo
Area of Science:
- Condensed matter physics
- Statistical mechanics
- Solid-state physics
Background:
- Previous studies analyzed thermal perturbations in infinite harmonic crystals.
- Kinetic temperature oscillations in infinite crystals are damped and described by Bessel functions.
Purpose of the Study:
- Investigate thermal perturbation dynamics in finite harmonic crystals.
- Characterize the novel 'thermal echo' phenomenon observed in finite systems.
Main Methods:
- Analytical derivation of kinetic temperature time-dependence.
- Mathematical modeling using Bessel and Airy functions.
- Analysis of phenomena in finite and thermodynamic limit crystal systems.
Main Results:
- Finite crystals exhibit a 'thermal echo'—a periodic sharp increase in temperature oscillation amplitude.
- The thermal echo occurs periodically, with period proportional to crystal length.
- Time-dependence is described by sums of Bessel functions; in the thermodynamic limit, by the Airy function.
Conclusions:
- The thermal echo is a distinct behavior of finite harmonic crystals under thermal perturbation.
- This phenomenon introduces periodic temperature amplifications not seen in infinite systems.
- The mathematical descriptions (Bessel and Airy functions) provide a quantitative framework for understanding thermal echoes.
Related Concept Videos
01:06Echo
957
The human ear cannot distinguish between two sources of sound if they happen to reach within a specific time interval, typically 0.1 seconds apart. More than this, and they are perceived as separate sources.
Imagine the sound is reflected back to the ears. Assuming that the source is very close to the human, the difference between hearing the two sounds—the emitted sound and the reflected sound—may be more than the minimum time for perceiving distinct sounds. If this is the case,...
Imagine the sound is reflected back to the ears. Assuming that the source is very close to the human, the difference between hearing the two sounds—the emitted sound and the reflected sound—may be more than the minimum time for perceiving distinct sounds. If this is the case,...
957
01:09Harmonic Mean
3.7K
The arithmetic mean is usually skewed towards the larger values in the data set. Therefore, to avoid this inherent bias towards smaller values, the harmonic mean is used.
Take the example of the speed of a car, which is the measure of the rate of distance traveled. If the vehicle traverses the same distance back-and-forth, its average speed equals the total distance traveled divided by the total time taken. However, if the car moves with varying speeds, then the arithmetic mean is more skewed...
Take the example of the speed of a car, which is the measure of the rate of distance traveled. If the vehicle traverses the same distance back-and-forth, its average speed equals the total distance traveled divided by the total time taken. However, if the car moves with varying speeds, then the arithmetic mean is more skewed...
3.7K
02:42Ionic Crystal Structures
17.0K
Ionic crystals consist of two or more different kinds of ions that usually have different sizes. The packing of these ions into a crystal structure is more complex than the packing of metal atoms that are the same size.
Most monatomic ions behave as charged spheres, and their attraction for ions of opposite charge is the same in every direction. Consequently, stable structures for ionic compounds result (1) when ions of one charge are surrounded by as many ions as possible of the opposite...
Most monatomic ions behave as charged spheres, and their attraction for ions of opposite charge is the same in every direction. Consequently, stable structures for ionic compounds result (1) when ions of one charge are surrounded by as many ions as possible of the opposite...
17.0K
01:21Simple Harmonic Motion
15.1K
Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke's law. If the net force can be described by Hooke's law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position. To derive an equation for period and frequency, the equation of motion is used. The period of a simple harmonic oscillator is given...
15.1K
01:25Crystal Growth: Principles of Crystallization
5.0K
Crystallization is a phase transformation process in which crystals are precipitated from a supersaturated solution or formed from other sources. During crystallization, atoms or molecules arrange themselves into a well-defined, rigid crystal lattice to minimize energy.
Initiating crystallization involves manipulating the concentration of the solute and the temperature of the solution. Since crystal growth occurs when the ratio of concentration and solubility of the solute in the solvent...
Initiating crystallization involves manipulating the concentration of the solute and the temperature of the solution. Since crystal growth occurs when the ratio of concentration and solubility of the solute in the solvent...
5.0K
01:23Energy in Simple Harmonic Motion
12.9K
To determine the energy of a simple harmonic oscillator, consider all the forms of energy it can have during its simple harmonic motion. According to Hooke's Law, the energy stored during the compression/stretching of a string in a simple harmonic oscillator is potential energy. As the simple harmonic oscillator has no dissipative forces, it also possesses kinetic energy. In the presence of conservative forces, both energies can interconvert during oscillation, but the total energy remains...
12.9K