In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation: for an arbitrary real number α (the order). The most common and important special case is where α is an integer, n. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel func ...

Including:

• Bessel function - Applications
• Bessel function - Definitions
• Bessel function - Bessel functions of the first kind
• Bessel function - Bessel functions of the second kind
• Bessel function - Hankel functions
• Bessel function - Modified Bessel functions
• Bessel function - Spherical Bessel functions
• Bessel function - Riccati-Bessel functions
• Bessel function - Asymptotic forms
• Bessel function - Properties

Read more here: » Bessel function: Encyclopedia - Bessel function

The Seebeck effect is the conversion of heat differences directly into electricity. This effect was first discovered, accidentally, by the Estonian physicist Thomas Johann Seebeck in 1821, who found that a voltage existed between two ends of a metal bar when a temperature gradient existed in the bar. He also discovered that a compass needle would be deflected when a closed loop was formed of two metals with a temperature difference between the junctions. This is because the metals respond differently to the heat difference, whi ...

See also:

Peltier–Seebeck effect, Peltier–Seebeck effect - Seebeck effect, Peltier–Seebeck effect - Thermopower, Peltier–Seebeck effect - Charge carrier diffusion, Peltier–Seebeck effect - Phonon drag, Peltier–Seebeck effect - Peltier effect, Peltier–Seebeck effect - Thomson effect, Peltier–Seebeck effect - Patents

Read more here: » Peltier–Seebeck effect: Encyclopedia II - Peltier–Seebeck effect - Seebeck effect

Peter "Pie" Debye was born in Maastricht and after attending local schools in Maastricht went to the University of Aachen, Germany, only 30 km from Maastricht, in 1901. He studied mathematics and classical physics, and in 1905 received a degree in electrical engineering. In 1907 he published his first paper, a mathematically elegant solution of a problem involving eddy currents. At Aachen he studied under the theoretical physicist Arnold Sommerfeld, who later claimed that his ...

See also:

Peter Debye, Peter Debye - Early life, Peter Debye - Scientific contributions prior to the Nobel Prize, Peter Debye - His Nobel Prize, Peter Debye - Later life, Peter Debye - A list of accomplishments named for Peter Debye

Read more here: » Peter Debye: Encyclopedia II - Peter Debye - Early life

The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical. Consider a cube of side L. From the particle in a box article, the resonating modes of the sonic disturbances inside the box (considering for now only those aligned with one axis) have wavelengths given by where n is an integer. The ...

See also:

Debye model, Debye model - Derivation, Debye model - Debye's derivation, Debye model - Low temperature limit, Debye model - High temperature limit, Debye model - Debye versus Einstein, Debye model - Debye temperature table

Read more here: » Debye model: Encyclopedia II - Debye model - Derivation

The basic operating principle of an ADR is the use of a strong magnetic field to control the entropy of a sample of material, often called the "refrigerant." Magnetic field constrains the orientation of magnetic dipoles in the refrigerant. The stronger the magnetic field, the more aligned the dipoles are, and this corresponds to lower entropy and heat capacity because the material has (effectively) lost some of its internal degrees of freedom. If the refrigerant is kept at a constant temperature through thermal contact with a heat sink (usua ...

See also:

Adiabatic demagnetization, Adiabatic demagnetization - Basic technique, Adiabatic demagnetization - Implementations, Adiabatic demagnetization - Paramagnetic salts, Adiabatic demagnetization - Nuclear demagnetization

Read more here: » Adiabatic demagnetization: Encyclopedia II - Adiabatic demagnetization - Basic technique

According to kinetic theory, there should be no movement of individual molecules at absolute zero, so any material at this temperature would be solid. In a monatomic gas, most of the energy is in the form of translational motion, and the temperature can be measured in terms of the distribution of this motion, with slower speeds corresponding to lower temperatures, perhaps even down to absolute zero. But this is contrary to experimental evidence, and it is predicted that helium will nev ...

See also:

Absolute zero, Absolute zero - Kinetic theory and motion, Absolute zero - Cryogenics, Absolute zero - Thermodynamics near absolute zero, Absolute zero - Absolute temperature scales, Absolute zero - Negative temperatures, Absolute zero - Notes

Read more here: » Absolute zero: Encyclopedia II - Absolute zero - Kinetic theory and motion

Scientific units named after people - No longer in use. Franklin, electric charge – Benjamin Franklin ...

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Scientific units named after people, Scientific units named after people - SI base units, Scientific units named after people - SI derived unit, Scientific units named after people - Centimeter-gram-second system of units, Scientific units named after people - No longer in use, Scientific units named after people - Others, Scientific units named after people - No longer in use

Read more here: » Scientific units named after people: Encyclopedia II - Scientific units named after people - Centimeter-gram-second system of units

Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below. Bessel function - Bessel functions of the first kind. Bessel functions of the first kind, denoted with Jα(x), are solutions of Bessel's differential equation which are finite at x = 0 for α an integer or α non-negative. The s ...

See also:

Bessel function, Bessel function - Applications, Bessel function - Definitions, Bessel function - Bessel functions of the first kind, Bessel function - Bessel functions of the second kind, Bessel function - Hankel functions, Bessel function - Modified Bessel functions, Bessel function - Spherical Bessel functions, Bessel function - Riccati-Bessel functions, Bessel function - Asymptotic forms, Bessel function - Properties

Read more here: » Bessel function: Encyclopedia II - Bessel function - Definitions

Lars Onsager was born in Christiania (now Oslo), Norway. His father was a lawyer. After completing secondary school in Oslo, he attended the Norwegian Institute of Technology (NTH) in Trondheim, graduating as a chemical engineer in 1925. In 1925 he arrived at a correction to the Debye-Hückel theory of electrolytic solutions, to take care of Brownian movement of ions in solution, and in 1926 published it. He made a trip to Zürich, where Peter Debye was teaching, and confronted Debye, telling him his theory was wrong. He so thoroughly ...

See also:

Lars Onsager, Lars Onsager - His life before coming to the United States, Lars Onsager - At Johns Hopkins, Lars Onsager - At Brown, Lars Onsager - Yale and later

Read more here: » Lars Onsager: Encyclopedia II - Lars Onsager - His life before coming to the United States

The first theoretical treatment of screening, due to Debye and Hückel (1923), dealt with a stationary point charge embedded in a fluid. This is known as electrostatic screening. Consider a fluid of electrons in a background of heavy, positively-charged ions. For simplicity, we ignore the motion and spatial distribution of the ions, approximating them as a uniform background charge. This is permissible since the electrons are lighter and more mobile than the ions, and provided we consider distances much larger than the ionic separation ...

See also:

Electric field screening, Electric field screening - Electrostatic screening

Read more here: » Electric field screening: Encyclopedia II - Electric field screening - Electrostatic screening

The Seebeck effect is the conversion of heat differences directly into electricity. This effect was first discovered, accidentally, by the Estonian physicist Thomas Johann Seebeck in 1821, who found that a voltage existed between two ends of a metal bar when a temperature gradient existed in the bar. He also discovered that a compass needle would be deflected when a closed loop was formed of two metals with a temperature difference between the junctions. This is because the metals respond differently to the heat difference, which ...

See also:

Peltier-Seebeck effect, Peltier-Seebeck effect - Seebeck effect, Peltier-Seebeck effect - Thermopower, Peltier-Seebeck effect - Charge carrier diffusion, Peltier-Seebeck effect - Phonon drag, Peltier-Seebeck effect - Peltier effect, Peltier-Seebeck effect - Thomson effect, Peltier-Seebeck effect - Patents

Read more here: » Peltier-Seebeck effect: Encyclopedia II - Peltier-Seebeck effect - Seebeck effect

According to kinetic theory, there should be no movement of individual molecules at absolute zero, so any material at this temperature would be solid. In a monatomic gas, most of the energy is in the form of translational motion, and the temperature can be measured in terms of the distribution of this motion, with slower speeds corresponding to lower temperatures, perhaps even down to absolute zero. But this is contrary to experimental evidence, and it is predicted that helium will nev ...

See also:

Absolute zero, Absolute zero - Kinetic theory and motion, Absolute zero - Cryogenics, Absolute zero - Thermodynamics near absolute zero, Absolute zero - Absolute temperature scales, Absolute zero - Negative temperatures

Read more here: » Absolute zero: Encyclopedia II - Absolute zero - Kinetic theory and motion

On leaving JHU, he took a position (involving the teaching of statistical mechanics to graduate students in chemistry) at Brown University in Providence, Rhode Island, where it became clear that he was no better at teaching advanced students than freshmen, but he made significant contributions to statistical mechanics and thermodynamics. The only graduate student who could really understand his lectures on electrolyte systems, Raymond Fuoss, worked under him and eventually joined him on the Yale chemistry faculty. In 1933, when the Great Dep ...

See also:

Lars Onsager, Lars Onsager - His life before coming to the United States, Lars Onsager - At Johns Hopkins, Lars Onsager - At Brown, Lars Onsager - Yale and later

Read more here: » Lars Onsager: Encyclopedia II - Lars Onsager - At Brown

It can be shown from the laws of thermodynamics that absolute zero can never be achieved, though it is possible to reach temperatures arbitrarily close to it through the use of cryocoolers. This is the same principle that ensures no machine can be 100% efficient. At very low temperatures in the vicinity of absolute zero, matter exhibits many unusual properties including superconductivity, superfluidity, and Bose-Einstein condensation. In order to study such phenomena, scientists ha ...

See also:

Absolute zero, Absolute zero - Kinetic theory and motion, Absolute zero - Cryogenics, Absolute zero - Thermodynamics near absolute zero, Absolute zero - Absolute temperature scales, Absolute zero - Negative temperatures

Read more here: » Absolute zero: Encyclopedia II - Absolute zero - Cryogenics

Thomson effect, named for William Thomson (Lord Kelvin), describes the heating or cooling of a current-carrying conductor with a temperature gradient. Any current-carrying conductor, with a temperature difference between two points, will either absorb or emit heat, depending on the material. If a current density J is passed through a homogeneous conductor, heat production per unit volume is where ρ is the resistivity of the material dT/dx is the temperature gradient a ...

See also:

Peltier–Seebeck effect, Peltier–Seebeck effect - Seebeck effect, Peltier–Seebeck effect - Thermopower, Peltier–Seebeck effect - Charge carrier diffusion, Peltier–Seebeck effect - Phonon drag, Peltier–Seebeck effect - Peltier effect, Peltier–Seebeck effect - Thomson effect, Peltier–Seebeck effect - Patents

Read more here: » Peltier–Seebeck effect: Encyclopedia II - Peltier–Seebeck effect - Thomson effect

Scientific units named after people - No longer in use. Mercalli Intensity Scale, earthquake effects – Giuseppe Mercalli Degree Rankine, temperature – William John Macquorn Rankine Réaumur, temperature – René Antoine Ferchault de Réaumur Torr, pressure – Evangelista Torricelli ...

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Scientific units named after people, Scientific units named after people - SI base units, Scientific units named after people - SI derived unit, Scientific units named after people - Centimeter-gram-second system of units, Scientific units named after people - No longer in use, Scientific units named after people - Others, Scientific units named after people - No longer in use

Read more here: » Scientific units named after people: Encyclopedia II - Scientific units named after people - Others

From 1934 to 1939 Debye was director of the prestigious Max Planck Institute in Berlin. In January 2006, documents surfaced that seem to establish that in this period, Debye was actively involved in cleansing German science institutions from jewish and other "non-arian elements". For this reason, Albert Einstein actively tried to prevent Debye from being appointed in the United States. Nevertheless, Debye was offered a chance to give a series of lectures at Cornell University in Ithaca, New York, traveled to the United States of America. He ...

See also:

Peter Debye, Peter Debye - Early life, Peter Debye - Scientific contributions prior to the Nobel Prize, Peter Debye - His Nobel Prize, Peter Debye - Later life, Peter Debye - A list of accomplishments named for Peter Debye

Read more here: » Peter Debye: Encyclopedia II - Peter Debye - Later life

So how closely do the Debye and Einstein models correspond to experiment? -- Surprisingly close, but Debye is correct at low temperatures whereas Einstein is not. How different are the models? To answer that question one would naturally plot the two on the same set of axes... except one can't. Both the Einstein model and the Debye model provide a functional form for the heat capacity. They are models, and no model is without a scale. A scale relates the model to its real-world counterpart. One can see that the scale of the E ...

See also:

Debye model, Debye model - Derivation, Debye model - Debye's derivation, Debye model - Low temperature limit, Debye model - High temperature limit, Debye model - Debye versus Einstein, Debye model - Debye temperature table

Read more here: » Debye model: Encyclopedia II - Debye model - Debye versus Einstein

As mentioned, absolute or thermodynamic temperature is conventionally measured in Kelvins (Celsius-size degrees), and increasingly rarely in the Rankine scale (Fahrenheit-size degrees). Absolute temperature is uniquely determined up to a multiplicative constant which specifies the size of the "degree", so the ratios of two absolute temperatures, T2/T1, are the same in all scales. The most transparent definition comes from the classical Maxwell-Boltzmann distribution over energies, or from the quantu ...

See also:

Absolute zero, Absolute zero - Kinetic theory and motion, Absolute zero - Cryogenics, Absolute zero - Thermodynamics near absolute zero, Absolute zero - Absolute temperature scales, Absolute zero - Negative temperatures

Read more here: » Absolute zero: Encyclopedia II - Absolute zero - Absolute temperature scales

At 0 K, (nearly) all molecular motion ceases and ΔS = 0 for any adiabatic process. Pure substances can (ideally) form perfect crystals as T → 0. Planck's strong form of the third law of thermodynamics states that the entropy of a perfect crystal vanishes at absolute zero. However, if the lowest energy state is degenerate (more than one microstate), this cannot be true. The original Nernst heat theorem makes the weaker and less controversial claim that the entropy change for any isothermal proce ...

See also:

Absolute zero, Absolute zero - Kinetic theory and motion, Absolute zero - Cryogenics, Absolute zero - Thermodynamics near absolute zero, Absolute zero - Absolute temperature scales, Absolute zero - Negative temperatures

Read more here: » Absolute zero: Encyclopedia II - Absolute zero - Thermodynamics near absolute zero