How does the resistance of a conductor change with increasing temperature. Joule-Lenz law in classical electron theory

Temperature dependence of resistance

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The resistance R of a homogeneous conductor of constant cross section depends on the properties of the conductor's substance, its length and cross section as follows:

Where ρ is the resistivity of the material of the conductor, L is the length of the conductor, and S is the cross-sectional area. The reciprocal of resistivity is called conductivity. This value is related to temperature by the Nernst-Einstein formula:

T is the temperature of the conductor;

D is the diffusion coefficient of charge carriers;

Z is the number of electric charges of the carrier;

e - elementary electric charge;

C - concentration of charge carriers;

Boltzmann's constant.

Therefore, the resistance of a conductor is related to temperature by the following relationship:

The resistance can also depend on the parameters S and I, since the cross section and length of the conductor also depend on temperature.

2) Ideal gas - a mathematical model of a gas, in which it is assumed that: 1) the potential energy of the interaction of molecules can be neglected in comparison with their kinetic energy; 2) the total volume of gas molecules is negligible; 3) forces of attraction or repulsion do not act between molecules, collisions of particles between themselves and with the walls of the vessel are absolutely elastic; 4) the interaction time between molecules is negligible compared to the average time between collisions. In the extended model ideal gas the particles of which it consists are in the form of elastic spheres or ellipsoids, which makes it possible to take into account the energy of not only translational, but also rotational-oscillatory motion, as well as not only central, but also non-central collisions of particles.

Gas pressure:

A gas always fills a volume bounded by impenetrable walls. So, for example, a gas cylinder or chamber car tire almost uniformly filled with gas.

In an effort to expand, the gas exerts pressure on the walls of the cylinder, tire chamber or any other body, solid or liquid, with which it comes into contact. If we do not take into account the action of the Earth's gravitational field, which, with the usual dimensions of vessels, only negligibly changes the pressure, then at equilibrium, the pressure of the gas in the vessel seems to us to be completely uniform. This remark refers to the macrocosm. If we imagine what happens in the microcosm of the molecules that make up the gas in the vessel, then there can be no question of any uniform distribution of pressure. In some places on the surface of the wall, gas molecules hit the walls, while in other places there are no impacts. This picture changes all the time in a chaotic way. Gas molecules hit the walls of the vessels, and then fly off at a speed of almost equal speed molecules before impact.

Ideal gas. The ideal gas model is used to explain the properties of matter in the gaseous state. The ideal gas model assumes the following: the molecules have a negligible volume compared to the volume of the vessel, there are no attractive forces between the molecules, and when molecules collide with each other and with the walls of the vessel, repulsive forces act.

Task for Ticket No. 16

1) Work equals power * time = (voltage squared) / resistance * time

Resistance = 220 volts * 220 volts * 600 seconds / 66000 joules = 440 ohms

1. Alternating current. The effective value of the current and voltage.

2. Photoelectric effect. Laws of the photoelectric effect. Einstein's equation.

3. Determine the speed of red light = 671 nm in glass with a refractive index of 1.64.

Answers to Ticket No. 17

Alternating current is an electric current that changes in magnitude and direction over time, or, in a particular case, changes in magnitude, keeping its direction in the electrical circuit unchanged.

The effective (effective) value of the alternating current is called the value direct current, the action of which will produce the same work (thermal or electrodynamic effect) as the considered alternating current during one period. AT contemporary literature the mathematical definition of this quantity is more often used - the root mean square value of the alternating current strength.

In other words, the effective value of the current can be determined by the formula:

For harmonic vibrations current In a similar way, the effective values ​​​​of EMF and voltage are determined.

Photoelectric effect, Photoelectric effect - the emission of electrons by a substance under the action of light (or any other electromagnetic radiation). In condensed (solid and liquid) substances, external and internal photoelectric effects are distinguished.

Stoletov's laws for the photoelectric effect:

Formulation of the 1st law of the photoelectric effect: The strength of the photocurrent is directly proportional to the density of the light flux.

According to the 2nd law of the photoelectric effect, the maximum kinetic energy of electrons ejected by light increases linearly with the frequency of light and does not depend on its intensity.

3rd law of the photoelectric effect: for each substance there is a red border of the photoelectric effect, that is, the minimum frequency of light (or the maximum wavelength λ0) at which the photoelectric effect is still possible, and if then the photoelectric effect no longer occurs. The theoretical explanation of these laws was given in 1905 by Einstein. According to him, electromagnetic radiation is a stream of individual quanta (photons) with energy hν each, where h is Planck's constant. With the photoelectric effect, part of the incident electromagnetic radiation is reflected from the metal surface, and part penetrates into the surface layer of the metal and is absorbed there. Having absorbed a photon, the electron receives energy from it and, doing the work function φ, leaves the metal: the maximum kinetic energy that an electron has when it leaves the metal.

Laws external photoelectric effect

Stoletov's law: with a constant spectral composition of electromagnetic radiation incident on the photocathode, the saturation photocurrent is proportional to the energy illumination of the cathode (otherwise: the number of photoelectrons knocked out of the cathode in 1 s is directly proportional to the radiation intensity):

And the maximum initial speed of photoelectrons does not depend on the intensity of the incident light, but is determined only by its frequency.

For each substance there is a red limit of the photoelectric effect, that is, the minimum frequency of light (depending on the chemical nature of the substance and the state of the surface), below which the photoelectric effect is impossible.

Einstein's equations (sometimes the name "Einstein-Hilbert equations" is found) - the equations of the gravitational field in general theory relativity, connecting the metrics of curved space-time with the properties of the matter filling it. The term is also used in singular: "Einstein's equation", since in tensor notation it is one equation, although in components it is a system of partial differential equations.

The equations look like this:

Where is the Ricci tensor, which is obtained from the space-time curvature tensor by convolving it over a pair of indices, R is the scalar curvature, that is, the convoluted Ricci tensor, the metric tensor, o

cosmological constant, a is the energy-momentum tensor of matter, (π is the number pi, c is the speed of light in vacuum, G is Newton's gravitational constant).

Task for Ticket No. 17

k \u003d 10 * 10 in 4 \u003d 10 in 5 n / m \u003d 100000 n / m

F=k*delta L

delta L = mg/k

answer 2 cm

1. The Mendeleev-Clapeyron equation. Thermodynamic temperature scale. Absolute zero.

2. Electric current in metals. Fundamentals of the electronic theory of metals.

3. What speed does the rocket acquire in 1 minute, moving from a state of rest with an acceleration of 60 m / s2?

Answers to Ticket No. 18

1) The equation of state of an ideal gas (sometimes the Clapeyron equation or the Mendeleev-Clapeyron equation) is a formula that establishes the relationship between pressure, molar volume and absolute temperature of an ideal gas. The equation looks like:

P-pressure

Vm - molar volume

R is the universal gas constant

T is the absolute temperature, K.

This form of writing is named after the equation (law) of Mendeleev - Clapeyron.

The equation derived by Clapeyron contained a certain non-universal gas constant r, the value of which had to be measured for each gas:

Mendeleev also found that r is directly proportional to u proportionality coefficient R he called the universal gas constant.

THERMODYNAMIC TEMPERATURE SCALE (Kelvin scale) - an absolute temperature scale that does not depend on the properties of a thermometric substance (the reference point is the absolute zero temperature). The construction of the thermodynamic temperature scale is based on the second law of thermodynamics and, in particular, on the independence of the efficiency of the Carnot cycle from the nature of the working fluid. The unit of thermodynamic temperature, the kelvin (K), is defined as 1/273.16 of the thermodynamic temperature of the triple point of water.

Absolute zero temperature (more rarely - absolute zero temperature) - the minimum temperature limit that a physical body in the Universe. Absolute zero serves as the reference point for an absolute temperature scale, such as the Kelvin scale. In 1954, the X General Conference on Weights and Measures established a thermodynamic temperature scale with one reference point - the triple point of water, the temperature of which is taken to be 273.16 K (exactly), which corresponds to 0.01 ° C, so that on the Celsius scale absolute zero corresponds to temperature -273.15°C.

Electric current - directed (ordered) movement of charged particles. Such particles can be: in metals - electrons, in electrolytes - ions (cations and anions), in gases - ions and electrons, in vacuum under certain conditions - electrons, in semiconductors - electrons and holes (electron-hole conductivity). Sometimes electric current is also called the displacement current resulting from a change in time electric field.

Electric current has the following manifestations:

heating of conductors (there is no heat release in superconductors);

change chemical composition conductors (observed mainly in electrolytes);

creation of a magnetic field (manifested in all conductors without exception)

Theories of acids and bases are a set of fundamental physical and chemical concepts that describe the nature and properties of acids and bases. All of them introduce definitions of acids and bases - two classes of substances that react with each other. The task of the theory is to predict the products of the reaction between the acid and the base and the possibility of its occurrence, for which the quantitative characteristics of the strength of the acid and base are used. The differences between theories lie in the definitions of acids and bases, the characteristics of their strength and, as a result, in the rules for predicting the reaction products between them. All of them have their own area of ​​applicability, which areas partially intersect.

The main provisions of the electronic theory of interaction metals are extremely common in nature and are widely used in scientific and industrial practice. Theoretical concepts of acids and bases have importance in the formation of all conceptual systems of chemistry and have a versatile influence on the development of many theoretical concepts in all major chemical disciplines. Based on the modern theory of acids and bases, such sections of chemical sciences as the chemistry of aqueous and non-aqueous electrolyte solutions, pH-metry in non-aqueous media, homo- and heterogeneous acid-base catalysis, the theory of acidity functions, and many others have been developed.

Task for Ticket No. 18

v=at=60m/s2*60s=3600m/s

Answer: 3600m/s

1. Current in a vacuum. Cathode-ray tube.

2. Planck's quantum hypothesis. The quantum nature of light.

3. The hardness of the steel wire is 10000 N/m. how long will the cable lengthen if a weight of 20 kg is hung from it.

Answers to Ticket No. 19

1) To obtain an electric current in a vacuum, the presence of free carriers is necessary. They can be obtained by emitting electrons from metals - electron emission (from the Latin emissio - release).

As you know, at ordinary temperatures, electrons are held inside the metal, despite the fact that they perform thermal motion. Consequently, near the surface there are forces acting on electrons and directed inside the metal. These are the forces that arise due to the attraction between electrons and positive ions of the crystal lattice. As a result, in the surface layer of metals appears electric field, and the potential increases by a certain value Dj when passing from the outer space into the metal. Respectively potential energy electron decreases by eDj.

A kinescope is a cathode-ray device that converts electrical signals into light. It is widely used in the device of televisions, until the 1990s televisions were used exclusively on the basis of a kinescope. The name of the device reflected the word "kinetics", which is associated with moving figures on the screen.

Main parts:

electron gun, designed to form an electron beam, in color kinescopes and multibeam oscilloscope tubes are combined into an electron-optical spotlight;

a screen coated with a phosphor - a substance that glows when an electron beam hits it;

deflecting system controls the beam in such a way that it forms the desired image.

2) Planck's hypothesis - a hypothesis put forward on December 14, 1900 by Max Planck and consisting in the fact that during thermal radiation, energy is emitted and absorbed not continuously, but in separate quanta (portions). Each such portion-quantum has an energy E proportional to the frequency ν of the radiation:

where h or the coefficient of proportionality, later called Planck's constant. Based on this hypothesis, he proposed theoretical conclusion the relationship between the temperature of a body and the radiation emitted by this body - Planck's formula.

Planck's hypothesis was later confirmed experimentally.

The advancement of this hypothesis is considered the moment of the birth of quantum mechanics.

The quantum nature of light is an elementary particle, a quantum of electromagnetic radiation (in the narrow sense - light). It is a massless particle that can exist in vacuum only by moving at the speed of light. The electric charge of a photon is also equal to zero. A photon can only be in two spin states with a spin projection on the direction of motion (helicity) ±1. In physics, photons are denoted by the letter γ.

Classical electrodynamics describes a photon as electromagnetic wave with circular right or left polarization. From the point of view of classical quantum mechanics, a photon as a quantum particle is characterized by corpuscular-wave dualism, it simultaneously exhibits the properties of a particle and a wave.

Task for Ticket No. 19

F=k*delta L

delta L = mg/k

delta L = 20kg*10000n/kg / 100000n/m = 2 cm

answer 2 cm

1. Electric current in semiconductors. Intrinsic conductivity of semiconductors on the example of silicon.

2. Laws of reflection and refraction of light.

3. What work does the electric field do to move 5x10 18 electrons in a circuit section with a potential difference of 20 V.

Answers to Ticket No. 20

Electric current in semiconductors is a material that, in terms of its conductivity, occupies an intermediate position between conductors and dielectrics and differs from conductors in its strong dependence of conductivity on impurity concentration, temperature, and exposure. various kinds radiation. The main property of a semiconductor is the increase electrical conductivity with an increase in temperature.

Semiconductors are substances whose band gap is on the order of a few electron volts (eV). For example, diamond can be attributed to wide-gap semiconductors, and indium arsenide - to narrow-gap ones. Semiconductors include many chemical elements (germanium, silicon, selenium, tellurium, arsenic and others), great amount alloys and chemical compounds(gallium arsenide, etc.). Almost all inorganic substances of the world around us are semiconductors. The most common semiconductor in nature is silicon, which makes up almost 30% of the earth's crust.

In metals that do not possess superconductivity, at low temperatures, due to the presence of impurities, a region is observed 1 - the area of ​​\u200b\u200bresidual resistance, almost independent of temperature (Fig. 10.5). Residual resistance- r rest the less, the purer the metal.

Rice. 10.5. Dependence of metal resistivity on temperature

Rapid increase in resistivity at low temperatures down to the Debye temperature Q dcan be explained by the excitation of new frequencies of thermal vibrations of the lattice, at which the scattering of charge carriers occurs - the region 2 .

At T>Q d, when the spectrum of oscillations is fully excited, an increase in the amplitude of oscillations with increasing temperature leads to a linear increase in resistance to approximately T pl - region 3 . If the periodicity of the structure is violated, the electron experiences scattering, which leads to a change in the direction of motion, finite mean free paths and conductivity of the metal. The energy of conduction electrons in metals is 3–15 eV, which corresponds to wavelengths of 3–7 Å. Therefore, any violations of the periodicity due to impurities, defects, the surface of the crystal or thermal vibrations of atoms (phonons) cause an increase in the resistivity of the metal.

Let's spend qualitative analysis of the temperature dependence of the resistivity of metals. The electron gas in metals is degenerate, and the main mechanism of electron scattering at high temperatures is scattering by phonons.

Attemperature drops to absolute zero, the resistance of normal metals tends to a constant value- residual resistance. An exception to this rule are superconducting metals and alloys, in which the resistance disappears below a certain critical temperature. T sv (temperature of transition to the superconducting state).

With increasing temperature, the deviation of resistivity from a linear dependence for most metals occurs near the melting point T pl. Some deviation from the linear dependence can be observed in ferromagnetic metals, in which there is an additional scattering of electrons on violations of the spin order.

When the melting temperature is reached and the transition to a liquid state, most metals experience a sharp increase in resistivity and some decrease it. If the melting of a metal or alloy is accompanied by an increase in volume, then the resistivity increases by a factor of two to four (for example, for mercury, by a factor of 4).

In metals, the volume of which decreases during melting, on the contrary, there is a decrease in resistivity (for gallium by 53%, antimony -29% and bismuth -54%). Such an anomaly can be explained by an increase in density and compressibility modulus during the transition of these metals from the solid to the liquid state. For some molten (liquid) metals, the resistivity ceases to increase with increasing temperature at a constant volume, in others it grows more slowly than in the solid state. Such anomalies, apparently, can be associated with the phenomena of lattice disorder, which occur differently in different metals during their transition from one state of aggregation to another.

An important characteristic of metals is temperature coefficient electrical resistivity, showing the relative change in resistivity for a change in temperature of one Kelvin (degree)

a-r- positive when the resistivity increases with increasing temperature. It is obvious that the value a r is also a function of temperature. In area 3 linear dependence r( T) (see figure 10.3) the relation is fulfilled:

where r 0 and a r - resistivity and temperature coefficient of resistivity at temperatureT 0 , and r - resistivity at temperatureT. Experimental data show that for most metals a r at room temperature about 0.004 To-1. For ferromagnetic metals, the value a r is somewhat higher.

Residual resistivity of metals . As mentioned above, the resistance of normal metals tends to a constant value - the residual resistance, as the temperature decreases to absolute zero. In normal metals (not superconductors), the residual resistance arises from the scattering of conduction electrons by static defects

The overall purity and perfection of a metallic conductor can be determined by the ratio of resistances r= R 273 /R 4,2 K. For standard 99.999 purity copper, this ratio is 1000. More values r can be achieved by additional zone remelting and preparation of samples in the form of single crystals.

Extensive experimental material contains numerous data on the measurement of resistance in metals, caused by the presence of impurities in them. The following most characteristic changes in metals caused by alloying can be noted. First, apart from phonon perturbations, the impurity is a local violation of the ideality of the lattice, perfect in all other respects. Second, doping affects the band structure by shifting the Fermi energy and changing the density of state and effective mass, i.e. parameters that partially determine the ideal resistance of the metal. Third, doping can change the elastic constants and, accordingly, the vibrational spectrum of the lattice, affecting the ideal resistance.

Total Conductor Resistivity at temperatures above 0K, the sum of the residual resistance r rest and resistivity due to scattering on thermal vibrations of the lattice - r T

This relation is known as Mathyssen's rule of additivity of resistivity. Often, however, significant deviations from the Mathyssen rule are observed, and some of these deviations may not speak in favor of the applicability of the main factors affecting the resistance of metals when impurities are introduced into them. However, the second and third factors noted at the beginning of this section also make a significant contribution. But, nevertheless, the first factor has a stronger effect on the resistance of dilute solid solutions.

Change in residual resistance by 1 at. % impurity for monovalent metals can be found by Linde's rule, according to which

where a and b- constants depending on the nature of the metal and the period that it occupies in Periodic system impurity atom elements;Δ Ζ - the difference between the valences of the solvent metal and the impurity atom. Of considerable practical interest are calculations of the resistance due to vacancies and interstitial atoms. Such defects easily arise when a sample is irradiated with high-energy particles, for example, neutrons from a reactor or ions from an accelerator.

One of the characteristics of any conductive material is the dependence of resistance on temperature. If it is depicted as a graph on where time intervals (t) are marked along the horizontal axis, and the value of ohmic resistance (R) along the vertical axis, then a broken line will be obtained. The dependence of resistance on temperature schematically consists of three sections. The first corresponds to a slight heating - at this time, the resistance changes very slightly. This happens until a certain moment, after which the line on the chart goes up sharply - this is the second section. The third and last component is a straight line going up from the point at which the growth of R stopped, at a relatively small angle to the horizontal axis.

physical meaning This graph is as follows: the dependence of the resistance on the temperature of the conductor is described as simple until the amount of heating exceeds some value that is characteristic of this particular material. Let's give an abstract example: if at a temperature of +10°C the resistance of a substance is 10 Ohm, then up to 40°C the value of R will practically not change, remaining within the measurement error. But already at 41 ° C there will be a jump in resistance up to 70 ohms. If the further increase in temperature does not stop, then for each subsequent degree there will be an additional 5 ohms.

This property is widely used in various electrical devices, therefore it is natural to give data on copper as one of the most common materials in So, for a copper conductor, heating by each additional degree leads to an increase in resistance by half a percent of the specific value (can be found in the reference tables, given for 20°C, 1 m length with a section of 1 sq. mm).

When an electric current appears in a metal conductor - directional movement elementary particles that have a charge. The ions located in the nodes of the metal are not able to keep the electrons in their outer orbits for a long time, so they freely move throughout the entire volume of the material from one node to another. This chaotic movement is due to external energy - heat.

Although the fact of movement is obvious, it is not directed, therefore it is not considered as a current. When an electric field appears, the electrons orient themselves in accordance with its configuration, forming a directed movement. But since the thermal effect has not disappeared anywhere, the randomly moving particles collide with the directed field. The dependence of the resistance of metals on temperature shows the amount of interference with the passage of current. The higher the temperature, the higher the R of the conductor.

The obvious conclusion: by reducing the degree of heating, you can reduce the resistance. (about 20°K) is precisely characterized by a significant decrease in the thermal chaotic motion of particles in the structure of matter.

The considered property of conductive materials has found wide application in electrical engineering. For example, the dependence of conductor resistance on temperature is used in electronic sensors. Knowing its value for any material, it is possible to manufacture a thermistor, connect it to a digital or analog reading device, perform the appropriate scale graduation and use it as an alternative. Most modern temperature sensors are based on this principle, because the reliability is higher, and the design is simpler.

In addition, the dependence of resistance on temperature makes it possible to calculate the heating of electric motor windings.

Resistivity, and hence the resistance of metals, depends on temperature, increasing with its growth. The temperature dependence of the conductor resistance is explained by the fact that

1. the intensity of scattering (number of collisions) of charge carriers increases with increasing temperature;

2. their concentration changes when the conductor is heated.

Experience shows that at not too high and not too low temperatures, the dependences of resistivity and conductor resistance on temperature are expressed by the formulas:

where ρ 0 , ρ t - specific resistances of the conductor substance, respectively, at 0 ° C and t°C; R 0 , R t - conductor resistance at 0 °С and t°С, α - temperature coefficient of resistance: measured in SI in Kelvin to the minus first power (K ​​-1). For metallic conductors, these formulas are applicable from a temperature of 140 K and above.

Temperature coefficient The resistance of a substance characterizes the dependence of the change in resistance during heating on the type of substance. It is numerically equal to the relative change in resistance (resistivity) of the conductor when heated by 1 K.

hαi=1⋅ΔρρΔT,

where hαi is the average value of the temperature coefficient of resistance in the interval Δ Τ .

For all metallic conductors α > 0 and slightly changes with temperature. For pure metals α \u003d 1/273 K -1. In metals, the concentration of free charge carriers (electrons) n= const and increase ρ occurs due to an increase in the intensity of scattering of free electrons on the ions of the crystal lattice.

For electrolyte solutions α < 0, например, для 10%-ного раствора поваренной соли α \u003d -0.02 K -1. The resistance of electrolytes decreases with increasing temperature, since the increase in the number of free ions due to the dissociation of molecules exceeds the increase in the scattering of ions during collisions with solvent molecules.

Dependency formulas ρ and R on temperature for electrolytes are similar to the above formulas for metal conductors. It should be noted that this linear dependence is preserved only in a small temperature range, in which α = const. At large intervals of temperature change, the dependence of the resistance of electrolytes on temperature becomes non-linear.

Graphically, the dependences of the resistance of metal conductors and electrolytes on temperature are shown in Figures 1, a, b.

At very low temperatures, close to absolute zero (-273 °C), the resistance of many metals abruptly drops to zero. This phenomenon has been named superconductivity. The metal goes into a superconducting state.

The dependence of the resistance of metals on temperature is used in resistance thermometers. Usually, a platinum wire is taken as the thermometric body of such a thermometer, the dependence of the resistance of which on temperature has been sufficiently studied.

Changes in temperature are judged by the change in wire resistance, which can be measured. These thermometers can measure very low and very high temperatures when conventional liquid thermometers are unsuitable.

The phenomenon of superconductivity

SUPERCONDUCTIVITY- the phenomenon that many chem. elements, compounds, alloys (called superconductors) when cooled below a certain value. (characteristic for this material) temperature T s there is a transition from normal to so-called. superconducting state, in which their electric. DC resistance current is completely absent. In this transition, the structural and optical (in the area of visible light) the properties of superconductors remain virtually unchanged. Electric and magn. the properties of a substance in the superconducting state (phase) differ sharply from the same properties in the normal state (where they are, as a rule, metals) or from the properties of other materials, which do not pass into the superconducting state at the same temperature.

The phenomenon of S. was discovered by G. Kamerlingh-Onnes (N. Kamerlingh-Onnes, 1911) in the study of the low-temperature course of the resistance of mercury. He found that when the mercury wire is cooled below 4 K, its resistance jumps to zero. The normal state can be restored by passing a sufficiently strong current through the sample [exceeding critical current I C (T)] or placing it in a sufficiently strong ext. magn. field [exceeding critical magnetic field H C (T)].

In 1933, F. W. Meissner and R. Ochsenfeld discovered another important property characteristic of superconductors (cf. Meissner effect:) ext. magn. field less than some critical. value (depending on the type of substance) does not penetrate deep into the superconductor, which has the form of an infinite solid cylinder, the axis of which is directed along the field, and differs from zero only in a thin surface layer. This discovery allowed F. and G. London (F. London, H. London, 1935) to formulate phenomenological. theory describing the magnetostatics of superconductors (see Londons equation), but the nature of S. remained unclear.

The discovery of superfluidity in 1938 and the explanation of this phenomenon by L. D. Landau on the basis of the criterion he formulated (see Landau’s theory of superfluidity) for systems of Bose particles gave reason to assume that superfluidity can be interpreted as the superfluidity of an electron liquid, but the Fermi nature of electrons and the Coulomb the repulsion between them did not allow simply transferring the theory of superfluidity to S. In 1950, V. L. Ginzburg and Landau, on the basis of the theory of phase transitions of the 2nd kind (see Landau theory), formulated a phenomenological. ur-tion, describing thermodynamics and e-magn. properties of superconductors near critical. temp. T s. Building a microscopic theory (see below) substantiated the Ginzburg-Landau theory and clarified the phenomenological elements included in it. ur-tion constant. Opening dependency critical. temp. T s transition to the superconducting state of the metal from its isotopic composition (isotope effect, 1950) testified to the influence of crystalline. lattices on C. This allowed X. Frohlich (H. Frohlich) and J. Bardeen (J. Bardeen) to demonstrate the possibility of occurrence between electrons in the presence of crystalline. lattices of specific attraction, which can prevail over their Coulomb repulsion, and subsequently to L. Cooper (L. Cooper, 1956) - the possibility of the formation of bound states by electrons - Cooper pairs (Cooper effect).

In 1957, J. Bardin, L. Cooper and J. Shrpffer (J. Schrieffer) formulated microscopic. S.'s theory, which explained this phenomenon on the basis of Bose condensation of Cooper pairs of electrons, and also made it possible to describe many others within the framework of a simple model (see Bardeen - Cooper - Schrieffer model, BCS model). properties of superconductors.

Practical the use of superconductors was limited by low critical values. fields (~1 kOe) and temperature (~20 K). In 1952, A. A. Abrikosov and N. N. Zavaritskii, on the basis of an analysis of experiments. critical data. magn. fields of thin superconducting films indicated the possibility of the existence of a new class of superconductors (L. V. Shubnikov encountered their unusual magnetic properties back in 1937, one of the most important differences from ordinary superconductors is the possibility of the flow of a superconducting current with incomplete displacement of the magnetic field from the volume of the superconductor into wide range of magnetic fields). This discovery further determined the division of superconductors into superconductors of the first kind and superconductors of the second kind. The use of superconductors of the second kind subsequently made it possible to create superconducting systems with high criticality. fields (of the order of hundreds kOe).

Search for superconductors with high criticality. pace-rami stimulated the study of new types of materials. Many have been researched. classes of superconducting systems, organic superconductors and magnetic superconductors were synthesized, but up to 1986 max. critical temp-pa was observed for the Nb 3 Ge alloy ( T s 23 K). In 1986 J. G. Bednorz and K. A. Muller discovered new class metal oxide high-temperature superconductors (HTSC) (see Oxide high-temperature superconductors), critical. the temp-pa to-rykh over the next two years was "raised" from 30-35 K to 120-125 K. These superconductors are being intensively studied, new ones are being searched, and technologies are being improved. properties of existing ones, on the basis of which certain devices are already being created.

An important achievement in the field of S. was the discovery in 1962 josephson effect tunneling Cooper pairs between two superconductors through a thin dielectric. layer. This phenomenon formed the basis new area applications of superconductors (see Weak superconductivity, Cryoelectronic devices).

Nature superconductivity. The phenomenon of S. is due to the appearance of a correlation between electrons, as a result of which they form Cooper pairs that obey Bose statistics, and the electron liquid acquires the property of superfluidity. In the phonon model of S. pairing of electrons occurs as a result of a specific, associated with the presence of crystalline. phonon attraction gratings. Even with abs. zero temperature, the grating oscillates (see Fig. Zero vibrations, Crystal lattice dynamics). El - static. the interaction of an electron with lattice ions changes the nature of these oscillations, which leads to the appearance of an addition. attractive force acting on other electrons. This attraction can be considered as an exchange of virtual phonons between electrons. This attraction binds electrons in a narrow layer near the boundary Fermi surfaces. The thickness of this layer in energetic. scale is determined by max. phonon energy , where wD is the Debye frequency, v s- speed of sound, o - lattice constant(see Debye temperature ; ) in momentum space, this corresponds to a layer of thickness , where v F is the electron velocity near the Fermi surface. The uncertainty relation gives the characteristic scale of the phonon interaction region in the coordinate space:
where M is the mass of the core ion, t is the mass of the electron. The quantity cm, i.e., the phonon attraction turns out to be long-range (compared to interatomic distances). The Coulomb repulsion of electrons usually somewhat exceeds the phonon attraction in magnitude, but due to screening at interatomic distances, it is effectively weakened and the phonon attraction can prevail, combining electrons into pairs. The relatively small binding energy of the Cooper pair turns out to be significantly less kinetic energy electrons, therefore, according to quantum mechanics, bound states should not have arisen. However, in this case we are talking about the formation of pairs not from free isolates. electrons in three-dimensional space, but from quasiparticles of a Fermi liquid with a large Fermi surface filled. This leads to actual replacement of a three-dimensional problem by a one-dimensional one, where bound states arise at an arbitrarily weak attraction.

In the BCS model, electrons with opposite momenta are paired R and - R(the total momentum of the Cooper pair is 0). The orbital momentum and the total spin of the pair are also equal to 0. Theoretically, for certain nonphonon mechanisms of spin, pairing of electrons with a nonzero orbital momentum is also possible. Apparently, pairing into such a state occurs in superconductors with heavy fermions (eg, CeCu 2 Si 2 , CeCu 6 , UB 13 , CeA1 3 ).

In a superconductor at a temperature T < T s some of the electrons combined into Cooper pairs form a Bose condensate (see Fig. Bose-Einstein condensation). All electrons in the Bose condensate are described by a single coherent wave function. The remaining electrons are in excited over-condensate states (Fermi quasi-particles), and their energy. the spectrum is rearranged in comparison with the spectrum of electrons in a normal metal. In the isotropic BCS model, the dependence of the electron energy e on the momentum R in a superconductor has the form ( p F - Fermi momentum):

Rice. Fig. 1. Rearrangement of the energy spectrum of electrons in a superconductor (solid line) in comparison with a normal metal (dashed line).

Rice. 2. Temperature dependence of the energy gap in the BCS model.

Thus, near the Fermi level (Fig. 1), an energy gap appears in the spectrum (1). In order to excite electronic system with such a spectrum, it is necessary to break at least one Cooper pair. Since two electrons are formed in this case, each of them has an energy no less than , so the binding energy of the Cooper pair makes sense. The size of the gap significantly depends on the temperature (Fig. 2), with she behaves like T = 0 reaches max. values, and

where is the density of one-electron states near the Fermi surface, g- eff. interelectronic attraction constant.

In the BCS model, the coupling between electrons is assumed to be weak and critical. temp-pa turns out to be small compared to the characteristic phonon frequencies . However, for a number of substances (eg, Pb) this condition is not met and the parameter (strong bond). Even the approximation is discussed in the literature. Superconductors with a strong bond between electrons are described by the so-called. Eliashberg’s equations (G. M. Eliashberg, 1968), from which it is clear that the value T s there are no fundamental restrictions.

The presence of a gap in the electron spectrum leads to exponential. dependences in the region of low temperatures of all quantities determined by the number of these electrons (for example, electronic heat capacity and thermal conductivity, sound absorption coefficients and low-frequency el-magn. radiation).

Far from Fermi level expression (1) describes the energetic. the electron spectrum of a normal metal, i.e., the pairing effect affects electrons with momenta in a region of width . The spatial scale of the Cooper correlation (the "size" of the pair) . The correlation length is cm (the lower limit is realized by HTSC), but usually much exceeds the period of the crystal. gratings.

Al-dynamic the properties of superconductors depend on the relationship between the standard correlation. length and characteristic thickness of the surface layer, in which the magnitude of the e-magn. changes significantly. fields where n s is the concentration of superconducting (paired) electrons, e is the charge of an electron. If (such a region always exists near T s, because at ), then the Cooper pairs can be considered as point pairs, so the el-dynamics of the superconductor is local and the superconducting current is determined by the value of the vector potential BUT at the considered point of the superconductor (London equation). At , the coherent properties of the condensate of Cooper pairs appear, the el-dynamics becomes nonlocal - the current at a given point is determined by the values BUT in an entire region of size ( Pippard equation). This is usually the situation in massive pure superconductors (at a sufficient distance from their surface).

The transition of a metal from a normal to a superconducting state in the absence of a magnetic field. field is a second-order phase transition. This transition is characterized by a complex scalar order parameter - the wave function of the Bose condensate of Cooper pairs , where r- spatial coordinate. In the BCS model [for T = T s , and when T = O ]. The phase of the wave function is also essential: the superconducting current density j s is determined through the gradient of this phase:

where the * sign denotes complex conjugation. The value of the current density j s also vanishes when T = T s. The phase transition normal metal - superconductor can be considered as a result of spontaneous symmetry breaking with respect to the group symmetryU(l) gauge transformations of the wave function . Physically, this corresponds to the violation below T s conservation of the number of electrons in connection with their pairing, and is mathematically expressed by the appearance of non-zero cf. order parameter values

The gap in the energy. spectrum of electrons does not always coincide with the modulus of the order parameter (as is the case in the BCS model) and is not at all necessary condition C. So, for example, when a paramagnet is introduced into a superconductor. impurities in a certain range of their concentrations, gapless S. can be realized (see below). A peculiar picture of S. in two-dimensional systems, where thermodynamic. fluctuations in the phase of the order parameter destroy the long-range order (see Fig. Mermin-Wagner theorem), and yet S. takes place. It turns out that a necessary condition for the existence of a superconducting current j s is not even the presence of a long-range order (a finite average value of the order parameter ), but a weaker condition for the power-law decrease of the correlation function

thermal properties. The heat capacity of a superconductor (as well as a normal metal) consists of the electron Ces and lattice Cps component. Index s refers to the superconducting phase, P- to normal e- to the electronic component, R- to the lattice.

During the transition to the superconducting state, the lattice part of the heat capacity almost does not change, while the electronic part increases abruptly. Within the framework of the BCS theory for an isotropic spectrum

When value Ces decreases exponentially (Fig. 3) and the heat capacity of the superconductor is determined by its lattice part Cps ~ T 3. Characteristic exponential dependence Ces allows for direct measurement. The absence of this dependence indicates that at certain points on the Fermi surface, the energy gap goes to zero. In all likelihood, the latter is due to the non-phonon mechanism of electron attraction (for example, in systems with heavy fermions, where at low temperatures for UB 13 and for CeCuSi 2).

Rice. 3. Heat capacity jump during the transition to the superconducting state.

The thermal conductivity of the metal during the transition to the superconducting state does not experience a jump, i.e. . Dependence is caused by a number of factors. On the one hand, the electrons themselves contribute to the thermal conductivity, which decreases as the temperature decreases and Cooper pairs are formed. On the other hand, the phonon contribution m ps begins to increase somewhat, since the mean free path of phonons increases with a decrease in the number of electrons (electrons combined into Cooper pairs do not scatter phonons and do not themselves transfer heat). Thus, , while . In pure metals, where higher T s the electronic part of thermal conductivity prevails, it remains decisive even during the transition to the superconducting state; as a result, at all temperatures below T s. In alloys, on the contrary, the thermal conductivity is determined mainly by its phonon part and, upon passing through, begins to increase due to a decrease in the number of unpaired electrons.

Magnetic properties. Due to the possibility of non-dissipative superconducting currents flowing in the superconductor, it, when determined. experimental conditions exhibits the Meissner effect, i.e., behaves in the presence of a not too strong external. magn. fields as an ideal diamagnet (magnetic susceptibility). So, for a sample having the shape of a long solid cylinder in a homogeneous ext. magn. field H applied along its axis, the magnetization of the sample . Extrusion ext. magn. field from the bulk of the superconductor leads to a decrease in its free energy. In this case, screening superconducting currents flow in a thin surface layer cm. This value also characterizes the penetration depth of the external. magn. fields in the sample.

According to their behavior in sufficiently strong fields, superconducting materials are divided into two groups: superconductors of the 1st and 2nd kind (Fig. 4). Beginning the portion of the magnetization curves (where ) corresponds to the full Meissner effect. The further course of the curves for superconductors of the 1st and 2nd kind differs significantly.

Rice. 4. Dependence of the magnetization on the external magnetic field for superconductors of the 1st and 2nd kind.

Superconductors of the 1st kind lose their S. abruptly (phase transition of the 1st kind): either upon reaching the critical value corresponding to the given field. temp. T C (N), or with an increase in ext. fields to critical values H C (T)(thermodynamic critical field). At the point of the phase transition occurring in the magnetic. field, in energetic. In the spectrum of a type 1 superconductor, a gap of finite size immediately appears. Critical field H C (T) determines the difference between beats. free energy superconductor F s and normal F p phases:

Hidden ud. heat of phase transition

where S n and S s- ud. entropies of the corresponding phases. Beat jump heat capacity at T = T with

In the absence of external magn. fields at T = T s magnitude Q= Oh, that is, a transition of the 2nd kind occurs.

According to the BCS model, thermodynamic critical the field is associated with critical. temp-swarm ratio

and its temperature dependence in the limiting cases of high and low temperatures has the form:

Rice. 5. Temperature dependence of the thermodynamic critical magnetic field Hc.

Both limit f-ly are close to empirical. relation , which describes well typical experiments. data (Fig. 5). In the case of non-cylindrical geometry of experience when exceeding ext. magn. field defined quantities H 0 = (1 - N)H C (N - demagnetizing factor) a type 1 superconductor passes into an intermediate state : the sample is divided into layers of normal and superconducting phases, the ratio between the volumes of which depends on the value H. The transition of the sample to the normal state occurs gradually, by increasing the proportion of the corresponding phase.

An intermediate state can also arise when a current flows through a superconductor that exceeds a certain critical value. meaning I s, corresponding to the creation on the surface of the sample critical. magn. fields N s.

The formation of an intermediate state in a type 1 superconductor and the alternation of layers of the superconducting and normal phases of finite size turn out to be possible only on the assumption that the interface between these phases has a positive surface energy . The magnitude and sign depend on the relationship between

The relation called parameter Ginzburg - Landau and plays an important role in the phenomenological. theory C. The sign (or value of x) makes it possible to strictly determine the type of superconductor: for a superconductor of the 1st kind and; for a type 2 superconductor and Type 2 superconductors include pure Nb, most superconducting alloys, organic and high-temperature superconductors.

For type 2 superconductors, therefore, a type 1 phase transition to the normal state is impossible. The intermediate state is not realized, since the surface at the phase boundaries would have a negative value. energy and would no longer play the role of a factor restraining infinite fragmentation. For sufficiently weak fields and in type 2 superconductors, the Mensner effect takes place. Upon reaching the lower critical fields H C1(in the case ), which turns out to be less than formally calculated in this case H S becomes energetically favorable penetration of the magnetic. fields into a superconductor in the form of single vortices (see Quantized vortices) containing one magnetic flux quantum each. A superconductor of the 2nd kind passes into a mixed state.

« Physics - Grade 10 "

What physical quantity is called resistance
On what and how does the resistance of a metal conductor depend?

Different substances have different resistivities. Does the resistance depend on the state of the conductor? from his temperature? The answer must come from experience.

If you pass current from the battery through a steel spiral, and then start heating it in a burner flame, then the ammeter will show a decrease in current strength. This means that as the temperature changes, the resistance of the conductor changes.

If at a temperature equal to 0 ° C, the resistance of the conductor is R 0, and at a temperature t it is equal to R, then the relative change in resistance, as experience shows, is directly proportional to the change in temperature t:

The coefficient of proportionality α is called the temperature coefficient of resistance.

Temperature coefficient of resistance- a value equal to the ratio of the relative change in the resistance of the conductor to the change in its temperature.

It characterizes the dependence of the resistance of a substance on temperature.

The temperature coefficient of resistance is numerically equal to the relative change in the resistance of the conductor when heated by 1 K (by 1 °C).

For all metallic conductors, the coefficient α > 0 and changes slightly with temperature. If the interval of temperature change is small, then the temperature coefficient can be considered constant and equal to its average value over this temperature range. For pure metals

In electrolyte solutions, the resistance does not increase with increasing temperature, but decreases. For them α< 0. Например, для 10%-ного раствора поваренной соли α = -0,02 К -1 .

When the conductor is heated, its geometric dimensions change slightly. The resistance of a conductor changes mainly due to changes in its resistivity. You can find the dependence of this resistivity on temperature, if you substitute the values ​​​​in formula (16.1) The calculations lead to the following result:

ρ = ρ 0 (1 + αt), or ρ = ρ 0 (1 + αΔT), (16.2)

where ΔT is the change in absolute temperature.

Since a changes little with a change in the temperature of the conductor, we can assume that the resistivity of the conductor depends linearly on temperature (Fig. 16.2).

The increase in resistance can be explained by the fact that with increasing temperature, the amplitude of ion oscillations at the nodes of the crystal lattice increases, so free electrons collide with them more often, losing their direction of motion. Although the coefficient a is quite small, taking into account the dependence of resistance on temperature when calculating the parameters of heating devices is absolutely necessary. So, the resistance of the tungsten filament of an incandescent lamp increases by more than 10 times when a current passes through it due to heating.

For some alloys, for example, for an alloy of copper and nickel (Konstantin), the temperature coefficient of resistance is very small: α ≈ 10 -5 K -1; the specific resistance of Konstantin is large: ρ ≈ 10 -6 Ohm m. Such alloys are used for the manufacture of reference resistors and additional resistors for measuring instruments, i.e., in cases where it is required that the resistance does not noticeably change with temperature fluctuations.

There are also such metals, for example, nickel, tin, platinum, etc., whose temperature coefficient is much higher: α ≈ 10 -3 K -1 . The dependence of their resistance on temperature can be used to measure the temperature itself, which is carried out in resistance thermometers.

Devices made of semiconductor materials are also based on the dependence of resistance on temperature - thermistors. They are characterized by a large temperature coefficient of resistance (tens of times higher than this coefficient for metals), stability of characteristics over time. The nominal resistance of thermistors is much higher than that of metal resistance thermometers, it is usually 1, 2, 5, 10, 15 and 30 kΩ.

Usually, platinum wire is taken as the main working element of a resistance thermometer, the temperature dependence of which is well known. Changes in temperature are judged by the change in resistance of the wire, which can be measured. Such thermometers can measure very low and very high temperatures, when conventional liquid thermometers are unsuitable.

Superconductivity.

The resistance of metals decreases with decreasing temperature. What happens when the temperature approaches absolute zero?

In 1911, the Dutch physicist X. Kamerling-Onnes discovered a remarkable phenomenon - superconductivity. He found that when mercury is cooled in liquid helium, its resistance first changes gradually, and then at a temperature of 4.1 K drops very sharply to zero (Fig. 16.3).

The phenomenon of a conductor resistance dropping to zero at a critical temperature is called superconductivity.

Discovery of Kamerling-Onnes, for which he was awarded in 1913 Nobel Prize, led to the study of the properties of substances at low temperatures. Later, many other superconductors were discovered.

The superconductivity of many metals and alloys is observed at very low temperatures - starting from about 25 K. Reference tables give the transition temperatures to the superconducting state of some substances.

The temperature at which a substance becomes superconducting is called critical temperature.

The critical temperature depends not only on the chemical composition of the substance, but also on the structure of the crystal itself. For example, gray tin has a diamond structure with a cubic crystal lattice and is a semiconductor, and white tin has a tetragonal unit cell and is a silver-white, soft, ductile metal capable of passing into a superconducting state at a temperature of 3.72 K.

Substances in the superconducting state had sharp anomalies in their magnetic, thermal, and a number of other properties, so it would be more correct to speak not of a superconducting state, but of a special state of matter observed at low temperatures.

If a current is created in a ring conductor, which is in a superconducting state, and then the current source is removed, then the strength of this current does not change for an arbitrarily long time. In an ordinary (non-superconducting) conductor, the electric current in this case stops.

Superconductors are widely used. Thus, powerful electromagnets with a superconducting winding are being built, which create a magnetic field for long periods of time without consuming energy. After all no heat is generated in the superconducting winding.

However, it is impossible to obtain an arbitrarily strong magnetic field using a superconducting magnet. A very strong magnetic field destroys the superconducting state. Such a field can also be created by the current in the superconductor itself. Therefore, for each conductor in the superconducting state, there is a critical value of the current strength, which cannot be exceeded without violating the superconducting state.

Superconducting magnets are used in elementary particle accelerators, magnetohydrodynamic generators that convert the mechanical energy of a jet of hot ionized gas moving in a magnetic field into electrical energy.

An explanation of superconductivity is possible only on the basis of quantum theory. It was given only in 1957 by the American scientists J. Bardeen, L. Cooper, J. Schrieffer and the Soviet scientist, Academician N. N. Bogolyubov.

In 1986, high-temperature superconductivity was discovered. Complex oxide compounds of lanthanum, barium and other elements (ceramics) have been obtained with a transition temperature to the superconducting state of about 100 K. This is higher than the boiling point of liquid nitrogen at atmospheric pressure(77 K).

High-temperature superconductivity in the near future will certainly lead to a new technical revolution in all electrical engineering, radio engineering, and computer design. Now progress in this area is hampered by the need to cool the conductors to the boiling temperatures of the expensive gas - helium.

The physical mechanism of superconductivity is rather complicated. In a very simplified way, it can be explained as follows: electrons are combined into a regular line and move without colliding with a crystal lattice consisting of ions. This movement is very different from the usual thermal motion, at which the free electron moves randomly.

It is to be hoped that it will be possible to create superconductors at room temperature as well. Generators and electric motors will become extremely compact (several times smaller) and economical. Electricity can be transmitted over any distance without loss and accumulated in simple devices.