The dependence of the electrical resistance of the conductor on temperature. How does resistance depend on temperature?

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 relation:

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) An ideal gas is 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 of an ideal gas, the particles of which it is composed 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 a car tire chamber is almost evenly 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 almost equal to the speed of the molecule before impact.

Ideal gas. The ideal gas model is used to explain the properties of matter in the gaseous state. The model of an ideal gas 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 strength is the value of the direct current, the action of which will produce the same work (thermal or electrodynamic effect) as the considered alternating current during one period. In modern 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 current oscillations The effective values ​​of EMF and voltage are determined in a similar way.

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 work function φ, leaves the metal: the maximum kinetic energy that an electron has when it leaves the metal.

Laws of the 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 called the "Einstein-Hilbert equations") are the equations of the gravitational field in the general theory of relativity, connecting the metrics of curved space-time with the properties of the matter filling it. The term is also used in the singular: "Einstein's equation", since in tensor notation this 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 a 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) is the minimum temperature limit that a physical body in the Universe can have. 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 the electric field over time.

Electric current has the following manifestations:

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

change in the chemical composition of 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 ideas about acids and bases are important 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, an electric field appears in the surface layer of metals, and the potential increases by a certain value Dj when passing from the outer space into the metal. Accordingly, the potential energy of the 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:

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

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 a theoretical derivation of 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 an 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 wave-particle duality, 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 = 2cm

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 the strong dependence of conductivity on impurity concentration, temperature, and exposure to various types of radiation. The main property of a semiconductor is an increase in electrical conductivity with increasing 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), a huge number of 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.

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 metal 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. Such 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 visible light region), 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 that 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 (see Ref. 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 conventional 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 a new class of metal oxide high-temperature superconductors (HTSCs) (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 of a new area of ​​application for superconductors (see Ref. 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 a Cooper pair turns out to be significantly less than the kinetic energy of 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 spherical mechanisms, 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 an electron 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 away 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 electric 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 generally not a necessary condition for C. 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 finite size superconducting and normal phases are 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.

The electrical resistance of almost all materials depends on temperature. The nature of this dependence is different for different materials.

In metals having a crystalline structure, the free path of electrons as charge carriers is limited by their collisions with ions located at the nodes of the crystal lattice. In collisions, the kinetic energy of the electrons is transferred to the lattice. After each collision, the electrons, under the influence of the electric field forces, pick up speed again and, during the next collisions, give the acquired energy to the ions of the crystal lattice, increasing their oscillations, which leads to an increase in the temperature of the substance. Thus, electrons can be considered intermediaries in the conversion of electrical energy into thermal energy. An increase in temperature is accompanied by an increase in the chaotic thermal motion of particles of matter, which leads to an increase in the number of collisions of electrons with them and makes it difficult for the orderly movement of electrons.

For most metals, within operating temperatures, the resistivity increases linearly

where and - resistivity at initial and final temperatures;

- a coefficient constant for a given metal, called the temperature coefficient of resistance (TCS);

T1 and T2 - initial and final temperatures.

For conductors of the second kind, an increase in temperature leads to an increase in their ionization, so the TCR of this type of conductor is negative.

The values ​​of the resistivity of substances and their TCS are given in reference books. It is customary to give resistivity values ​​at a temperature of +20 °C.

The resistance of the conductor is determined by the expression

R2 = R1
(2.1.2)

Task 3 Example

Determine the resistance of the copper wire of a two-wire transmission line at + 20 ° C and + 40 ° C, if the wire cross section S =

120 mm , and the length of the line is l = 10 km.

Decision

According to the reference tables, we find the resistivity copper at + 20 °C and temperature coefficient of resistance :

= 0.0175 ohm mm /m; = 0.004 deg .

Let's determine the resistance of the wire at T1 = +20 ° С according to the formula R = , considering the length of the forward and reverse wires of the line:

R1=0.0175
2 = 2.917 ohms.

The resistance of the wires at a temperature of + 40 ° C is found by the formula (2.1.2)

R2 \u003d 2.917 \u003d 3.15 ohms.

Exercise

An overhead three-wire line with a length L is made with a wire, the brand of which is given in table 2.1. It is necessary to find the value indicated by the sign "?", using the example given and choosing the option with the data indicated in it in Table 2.1.

It should be noted that the task, unlike the example, provides for calculations related to one wire of the line. In the brands of bare wires, the letter indicates the material of the wire (A - aluminum; M - copper), and the number - the cross section of the wire in mm .

Table 2.1

	Line length L, km	Wire brand	Wire temperature Т, °С	Wire resistance RT at temperature T, Ohm

The study of the material of the topic ends with work with tests No. 2 (TOE-

ETM/PM” and No. 3 (TOE – ETM/IM)

> Dependence of resistance on temperature

Find out how resistance depends on temperature: comparison of the dependence of the resistance of materials and resistivity on temperature, semiconductor.

Resistance and resistivity are based on temperature and are linear.

Learning task

• Compare the temperature dependence of specific and ordinary resistance for large and small fluctuations.

Key Points

• When the temperature changes by 100°C, the resistivity (ρ) changes with ΔT as: p = p 0 (1 + αΔT), where ρ 0 is the initial resistivity and α is the temperature coefficient of resistivity.
• With serious changes in temperature, a non-linear change in resistivity is noticeable.
• The resistance of an object is directly proportional to the specific resistance, therefore it exhibits the same temperature dependence.

Terms

• A semiconductor is a substance with electrical properties that characterize it as a good conductor or insulator.
• The temperature coefficient of resistivity is an empirical value (α) that describes the change in resistance or resistivity with a temperature index.
• Resistivity is the degree to which a material resists electrical flow.

The resistance of materials is based on temperature, so it is possible to trace the dependence of resistivity on temperature. Some are capable of becoming superconductors (zero resistance) at very low temperatures, while others at high temperatures. The rate of vibration of atoms increases at greater distances, so electrons moving through the metal collide more often and increase resistance. Resistivity changes with temperature ΔT:

The resistance of a particular sample of mercury reaches zero at an extremely low temperature index (4.2 K). If the indicator is above this mark, then there is a sudden jump in resistance, and then an almost linear increase with temperature

p = p 0 (1 + αΔT), where ρ 0 is the initial resistivity and α is the temperature coefficient of resistivity. With significant changes in temperature, α can change, and finding p may require a non-linear equation. That is why the suffix of the temperature at which the substance changed is sometimes left (for example, α15).

It is worth noting that α is positive for metals, and resistivity increases with temperature. Typically, the temperature coefficient is +3 × 10 -3 K -1 to +6 × 10 -3 K -1 for metals at about room temperature. There are alloys that are designed specifically to reduce temperature dependence. For example, in manganin, α is close to zero.

Do not forget also that α is negative for semiconductors, that is, their resistivity decreases with increasing temperature. They are excellent conductors at high temperatures because increased temperature mixing increases the amount of free charges available to transport current.

The resistance of an object is also based on temperature, since R 0 is in direct proportion to p. We know that for a cylinder R = ρL/A. If L and A do not change much with temperature, then R has the same temperature dependence as ρ. It turns out:

R = R 0 (1 + αΔT), where R 0 is the initial resistance, and R is the resistance after temperature change T.

Let's look at the resistance of a temperature sensor. A lot of thermometers operate according to this scheme. The most common example is the thermistor. It is a semiconductor crystal with a strong temperature dependence. The device is small, so it quickly goes into thermal balance with the human part that it touches.

Thermometers are based on automatic measurement of thermistor temperature resistance

Based on the classical electronic theory of the conductivity of metals, the Joule-Lenz law can be explained.

The ordered movement of electrons occurs under the action of field forces. As above, we will assume that at the moment of collision with the positive ions of the crystal lattice, the electrons completely transfer their kinetic energy to it. By the end of the free path, the electron's speed is , and the kinetic energy

(14.9)

The power released by a unit volume of metal (power density) is equal to the product of the energy of one electron and the number of collisions per second and on the concentration n of electrons:

(14.10)

Taking into account (14.7), we have

- Joule-Lenz law in differential form.

If we are interested in the energy released by a conductor of length ℓ, cross-sectional area S over a period of time dt, then expression (14.10) must be multiplied by the volume of the conductor V=St and time dt:

Given that
(where R is the resistance of the conductor), we obtain the Joule-Lenz law in the form

§ 14.3 Dependence of the resistance of metals on temperature. Superconductivity. Wiedemann-Franz law

Resistivity depends not only on the type of substance, but also on its state, in particular, on temperature. The dependence of resistivity on temperature can be characterized by setting the temperature coefficient of resistance of a given substance:

(14.11)

It gives a relative increase in resistance with an increase in temperature by one degree.

Figure 14.3

The temperature coefficient of resistance for a given substance is different at different temperatures. This shows that resistivity does not change linearly with temperature, but depends on it in a more complex way.

ρ=ρ 0 (1+αt) (14.12)

where ρ 0 is the resistivity at 0ºС, ρ is its value at a temperature of tºС.

The temperature coefficient of resistance can be either positive or negative. For all metals, the resistance increases with increasing temperature, and therefore for metals

α>0. For all electrolytes, unlike metals, the resistance always decreases when heated. The resistance of graphite also decreases with increasing temperature. For such substances α<0.

Based on the electronic theory of the electrical conductivity of metals, it is possible to explain the dependence of the conductor resistance on temperature. As the temperature rises, its resistivity increases and its electrical conductivity decreases. Analyzing expression (14.7), we see that the electrical conductivity is proportional to the concentration of conduction electrons and the mean free path <ℓ> , i.e. the more <ℓ> , the less interference for the ordered motion of electrons are collisions. Electrical conductivity is inversely proportional to the average thermal velocity < υ τ > . The thermal velocity increases proportionally with increasing temperature
, which leads to a decrease in electrical conductivity and an increase in the resistivity of conductors. Analyzing formula (14.7), one can, in addition, explain the dependence of γ and ρ on the kind of conductor.

At very low temperatures of the order of 1-8ºK, the resistance of some substances drops sharply by billions of times and practically becomes equal to zero.

This phenomenon, first discovered by the Dutch physicist G. Kamerling-Onnes in 1911, is called superconductivity . At present, superconductivity has been established for a number of pure elements (lead, tin, zinc, mercury, aluminum, etc.), as well as for a large number of alloys of these elements with each other and with other elements. On fig. 14.3 schematically shows the dependence of the resistance of superconductors on temperature.

The theory of superconductivity was created in 1958 by N.N. Bogolyubov. According to this theory, superconductivity is the movement of electrons in a crystal lattice without collisions with each other and with lattice atoms. All conduction electrons move as one flow of an inviscid ideal fluid, without interacting with each other and with the lattice, i.e. without experiencing friction. Therefore, the resistance of superconductors is zero. A strong magnetic field, penetrating into the superconductor, deflects the electrons, and, breaking the "laminar flow" of the electron flow, causes the electrons to collide with the lattice, i.e. resistance arises.

In the superconducting state, energy quanta are exchanged between electrons, which leads to the creation of attractive forces between electrons that are greater than the Coulomb repulsive forces. In this case, pairs of electrons (Cooper pairs) are formed with mutually compensated magnetic and mechanical moments. Such pairs of electrons move in the crystal lattice without resistance.

One of the most important practical applications of superconductivity is its use in electromagnets with a superconducting winding. If there were no critical magnetic field that destroys superconductivity, then with the help of such electromagnets it would be possible to obtain magnetic fields of tens and hundreds of millions of amperes per centimeter. It is impossible to obtain such large constant fields with ordinary electromagnets, since this would require enormous power, and it would be practically impossible to remove the heat generated when the winding absorbs such large powers. In a superconducting electromagnet, the power consumption of the current source is negligible, and the power consumption for cooling the winding to helium temperature (4.2ºK) is four orders of magnitude lower than in a conventional electromagnet that creates the same fields. Superconductivity is also used to create memory systems for electronic mathematical machines (cryotron memory elements).

In 1853, Wiedemann and Franz experimentally established that that the ratio of thermal conductivity λ to electrical conductivity γ is the same for all metals at the same temperature and is proportional to their thermodynamic temperature.

This suggests that thermal conductivity in metals, as well as electrical conductivity, is due to the movement of free electrons. We will assume that electrons are similar to a monatomic gas, the thermal conductivity of which, according to the kinetic theory of gases, is equal to

(14.13)

(n is the concentration of atoms, m is the mass of the atom,<ℓ>is the mean free path of an electron, c V is the specific heat capacity).

For a monatomic gas

(k - Boltzmann's constant, M - molar mass).

(14.14)

From equations (14.7) and (14.14) we find the ratio of thermal conductivity and electrical conductivity of the metal:

(14.15)

It is known from the kinetic theory of gases that
, then

(14.16)

(k and e are constant values).

Therefore, the ratio of thermal conductivity and electrical conductivity of the metal is proportional to the thermodynamic temperature, which was established by the Wiedemann-Franz law. Since k \u003d 1.38 ∙ 10 -23 J / K; e \u003d 1.6 ∙ 10 -19 C, then

(14.17)

The Wiedemann-Franz law for most metals is fulfilled at a temperature of 100-400 K, but at a low temperature the law is significantly violated. There are metals (beryllium, manganese) that do not obey the Wiedemann-Franz law at all. A way out of insurmountable contradictions was found in the quantum electronic theory of metals.