Abstract: Absolutely black body. Pure black body

33.Thermal radiation. Radiation spectra of a completely black body at different temperatures. Laws of thermal radiation (Kirchhoff, Wien and Boltzmann). Planck's formula.

THERMAL RADIATION OF BODIES

The emission of electromagnetic waves by a substance occurs due to intra-atomic and intramolecular processes. Energy sources and, therefore, the type of glow can be different: a TV screen, a fluorescent lamp, an incandescent lamp, a rotting tree, a firefly, etc. From the whole variety of electromagnetic radiation, visible or invisible to the human eye, one can be distinguished, which is inherent in all bodies: radiation from heated bodies, or thermal radiation. It occurs at any temperature above 0 K, therefore it is emitted by all bodies. Depending on the body temperature, the radiation intensity and spectral composition, therefore, thermal radiation is not always perceived by the eye as a glow.

CHARACTERISTICS OF THERMAL RADIATION. BLACK BODY

The average radiation power over a time significantly longer than the period of light oscillations is taken as the radiation flux F. In the SI system, it is expressed in watts (W).

The radiation flux emitted by 1 m 2 of surface is called energetic luminosity R e . It is expressed in watts per square meter (W/m2).

A heated body emits electromagnetic waves of various wavelengths. Let us highlight a small interval of wavelengths from גּ to גּ + dגּ. The energetic luminosity corresponding to this interval is proportional to the width of the interval:

where r is the spectral density of energy luminosity

body, equal to the ratio of the energy luminosity of a narrow section of the spectrum to the width of this section, W/m 3.

The dependence of the spectral density of energetic luminosity on the wavelength is called the emission spectrum of the body.

Having integrated, we obtain an expression for the energetic luminosity of the body:

The ability of a body to absorb radiation energy is characterized by an absorption coefficient equal to the ratio of the radiation flux absorbed by a given body to the radiation flux incident on it: a = F absorb / F pad

Since the absorption coefficient depends on the wavelength, (27.3) is written for fluxes of monochromatic radiation, and then this ratio determines the monochromatic absorption coefficient: a גּ = F absorb(גּ)/ F down(גּ) .

It follows that absorption coefficients can take values ​​from 0 to 1. Black bodies absorb radiation especially well: black paper, fabrics, velvet, soot, platinum black, etc.; Bodies with a white surface and mirrors do not absorb well.

A body whose absorption coefficient is equal to unity for all frequencies is called black. It absorbs all radiation falling on it. There are no black bodies in nature; this concept is a physical abstraction. The black body model is a small hole in a closed opaque cavity. A beam entering this hole, reflected many times from the walls, will be almost completely absorbed. In what follows, we will take this model as a black body. A body whose absorption coefficient is less than unity and does not depend on the wavelength of light incident on it is called gray.

There are no gray bodies in nature, but some bodies in a certain wavelength range emit and absorb as gray bodies. For example, the human body is sometimes considered gray, having an absorption coefficient of approximately 0.9 for the infrared region of the spectrum.

KIRCHHOFF'S LAW

There is a certain relationship between the spectral density of energetic luminosity and the monochromatic absorption coefficient of bodies, which can be explained using the following example.

In a closed adiabatic shell there are two different bodies in conditions of thermodynamic equilibrium, and their temperatures are the same. Since the state of the bodies does not change, each of them emits and absorbs the same energy. The radiation spectrum of each body must coincide with the spectrum of electromagnetic waves absorbed by it, otherwise the thermodynamic equilibrium would be disrupted. This means that if one of the bodies emits any waves, for example red ones, more than the other, then it must absorb more of them.

The quantitative relationship between radiation and absorption was established by G. Kirchhoff in 1859: at the same temperature, the ratio of the spectral density of energy luminosity to the monochromatic absorption coefficient is the same for any bodies, including black ones (Kirchhoff’s law).

Using Kirchhoff's law and knowing from experiment the spectrum of a black body and the dependence of the monochromatic absorption coefficient of the body on the wavelength, we can find the emission spectrum of the body r גּ = f(גּ).

LAWS OF BLACK BODY RADIATION

Black body radiation has a continuous spectrum. Graphs of emission spectra for different temperatures are shown in Fig. There is a maximum spectral density of energy luminosity, which shifts towards short waves.

In classical physics, the emission and absorption of radiation by a body was considered as a continuous process. Planck came to the conclusion that it is precisely these basic provisions that do not allow one to obtain the correct relationship. He expressed a hypothesis from which it followed that a black body emits and absorbs energy not continuously, but in certain discrete portions - quanta.

Stefan-Boltzmann law: The energetic luminosity of a black body is proportional to the fourth power of its thermodynamic temperature. The quantity a is called the Stefan-Boltzmann constant. The Stefan-Boltzmann law can be qualitatively illustrated on different bodies (oven, electric stove, metal blank, etc.): as they heat up, more and more intense radiation is felt.

From here we find Wien's displacement law: גּ m ах =b/Т, where גּ m ах is the wavelength at which the maximum spectral density of the energy luminosity of the black body falls; b = = 0.28978*10 -2 m-K - Wien's constant. This law is also true for gray bodies.

The manifestation of Wien's law is known from everyday observations. At room temperature, the thermal radiation of bodies is mainly in the infrared region and is not perceived by the human eye. If the temperature rises, the body begins to glow with a dark red light, and at a very high temperature - white with a bluish tint, the feeling of the body being heated increases.

The Stefan-Boltzmann and Wien laws make it possible, by measuring the radiation of bodies, to determine their temperatures (optical pyrometry).

A completely black body that completely absorbs electromagnetic radiation of any frequency, when heated, emits energy in the form of waves evenly distributed over the entire frequency spectrum

TO end of the 19th century centuries scientists studying the interaction electromagnetic radiation(in particular, light) with atoms of matter, faced serious problems that could only be solved within the framework of quantum mechanics, which, in many ways, arose due to the fact that these problems arose. To understand the first and perhaps most serious of these problems, imagine a large black box with a mirrored interior surface, and in one of the walls there is a small hole made. A ray of light penetrating into a box through a microscopic hole remains inside forever, endlessly reflecting off the walls. An object that does not reflect light, but completely absorbs it, appears black, which is why it is commonly called a black body. (Absolutely black body - like many other conceptual physical phenomena- the object is purely hypothetical, although, for example, a hollow, uniformly heated sphere mirrored from the inside, into which light penetrates through a single tiny hole, is a good approximation.)

Absolutely black bodies do not exist in nature, so in physics a model is used for experiments. It is an opaque closed cavity with a small hole, the walls of which have the same temperature. Light entering through this hole will be completely absorbed after repeated reflections, and the hole will appear completely black from the outside. But when this cavity is heated, it will develop its own visible radiation. Since the radiation emitted by the inner walls of the cavity, before it leaves (after all, the hole is very small), in the overwhelming majority of cases will undergo a huge amount of new absorption and radiation, we can say with confidence that the radiation inside the cavity is in thermodynamic equilibrium with the walls. (In fact, the hole is not important for this model at all, it is only needed to emphasize the fundamental observability of the radiation located inside; the hole can, for example, be completely closed, and quickly opened only when equilibrium has already been established and the measurement is being carried out).

You, however, have probably seen quite close analogues of a black body in reality. In a fireplace, for example, it happens that several logs are stacked almost tightly together, and a rather large cavity burns out inside them. The outside of the logs remains dark and does not glow, while inside the burnt cavity heat (infrared radiation) and light accumulate, and these rays are reflected repeatedly from the walls of the cavity before escaping outside. If you look into the gap between such logs, you will see a bright yellow-orange high-temperature glow and from there you will literally be blazing with heat. The rays were simply trapped for some time between the logs, just as light is completely trapped and absorbed by the black box described above.

The model of such a black box helps us understand how the light absorbed by a black body behaves, interacting with the atoms of its substance. Here it is important to understand that light is absorbed by an atom, immediately emitted by it and absorbed by another atom, again emitted and absorbed, and this will happen until the state of equilibrium saturation is reached. When a black body is heated to an equilibrium state, the intensities of emission and absorption of rays inside the black body are equalized: when a certain amount of light of a certain frequency is absorbed by one atom, another atom somewhere inside simultaneously emits the same amount of light of the same frequency. Thus, the amount of absorbed light of each frequency within a black body remains the same, although different atoms of the body absorb and emit it.

Until this moment, the behavior of the black body remains quite understandable. Problems within classical physics (by “classical” here we mean physics before the advent of quantum mechanics) began when trying to calculate the radiation energy stored inside a black body in equilibrium state. And two things soon became clear:

1. the higher the wave frequency of the rays, the more of them accumulate inside the black body (that is, the shorter the wavelengths of the studied part of the spectrum of radiation waves, the more rays of this part of the spectrum inside the black body are predicted by the classical theory);
2. The higher the frequency of the wave, the more energy it carries and, accordingly, the more of it is stored inside the black body.

Taken together, these two conclusions led to an unthinkable result: the radiation energy inside a black body should be infinite! This evil mockery of the laws of classical physics was dubbed the ultraviolet catastrophe, since high-frequency radiation lies in the ultraviolet part of the spectrum.

The German physicist Max Planck managed to restore order (see Planck's constant) - he showed that the problem is removed if we assume that atoms can absorb and emit light only in portions and only at certain frequencies. (Later, Albert Einstein generalized this idea by introducing the concept of photons - strictly defined portions of light radiation.) According to this scheme, many frequencies of radiation predicted by classical physics simply cannot exist inside a black body, since atoms are unable to absorb or emit them; Accordingly, these frequencies are excluded from consideration when calculating the equilibrium radiation inside a black body. By leaving only permissible frequencies, Planck prevented the ultraviolet catastrophe and set science on the path to a correct understanding of the structure of the world at the subatomic level. In addition, he calculated the characteristic frequency distribution of equilibrium black body radiation.

This distribution received worldwide fame many decades after its publication by Planck himself, when cosmologists discovered that the cosmic microwave background radiation they discovered exactly obeys the Planck distribution in its spectral characteristics and corresponds to the radiation of a completely black body at a temperature of about three degrees above absolute zero.

Encyclopedia by James Trefil “The Nature of Science. 200 laws of the universe."
James Trefil is a professor of physics at George Mason University (USA), one of the most famous Western authors of popular science books.

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One of the facts of the subatomic world is that its objects - such as electrons or photons - are not at all similar to the usual objects of the macroworld. They behave neither like particles nor like waves, but like completely special formations that exhibit both wave and corpuscular properties depending on the circumstances. It is one thing to make a statement, but quite another to connect together the wave and particle aspects of the behavior of quantum particles, describing them with an exact equation. This is exactly what was done in the de Broglie relation.

IN Everyday life There are two ways to transfer energy in space - through particles or waves. IN everyday life There are no visible contradictions between the two mechanisms of energy transfer. So, a basketball is a particle, and sound is a wave, and everything is clear. However, in quantum mechanics things are not so simple. Even from the simplest experiments with quantum objects, it very soon becomes clear that in the microworld the principles and laws of the macroworld that we are familiar with do not apply. Light, which we are accustomed to thinking of as a wave, sometimes behaves as if it consists of a stream of particles (photons), and elementary particles, such as an electron or even a massive proton, often exhibit the properties of a wave.

There are a number of types of electromagnetic radiation, ranging from radio waves to gamma rays. Electromagnetic rays of all types propagate in a vacuum at the speed of light and differ from each other only in wavelengths.

The dual particle-wave nature of quantum particles is described by a differential equation.

Max Planck, one of the founders of quantum mechanics, came to the ideas of energy quantization, trying to theoretically explain the process of interaction between recently discovered electromagnetic waves and atoms and, thereby, solve the problem of black body radiation. He realized that to explain the observed emission spectrum of atoms, it is necessary to take for granted that atoms emit and absorb energy in portions (which the scientist called quanta) and only at individual wave frequencies.

The word “quantum” comes from the Latin quantum (“how much, how much”) and the English quantum (“quantity, portion, quantum”). “Mechanics” has long been the name given to the science of the movement of matter. Accordingly, the term “quantum mechanics” means the science of the movement of matter in portions (or, in modern terms scientific language science of the movement of quantized matter). The term “quantum” was coined by the German physicist Max Planck to describe the interaction of light with atoms.

Most of all, Einstein protested against the need to describe the phenomena of the microworld in terms of probabilities and wave functions, and not from the usual position of coordinates and particle velocities. That's what he meant by "rolling the dice." He recognized that describing the movement of electrons in terms of their speeds and coordinates contradicts the uncertainty principle. But, Einstein argued, there must be some other variables or parameters, taking into account which the quantum mechanical picture of the microworld will return to the path of integrity and determinism. That is, he insisted, it only seems to us that God is playing dice with us, because we do not understand everything. Thus, he was the first to formulate the hidden variable hypothesis in the equations of quantum mechanics. It lies in the fact that in fact electrons have fixed coordinates and speed, like Newton’s billiard balls, and the uncertainty principle and the probabilistic approach to their determination within the framework of quantum mechanics are the result of the incompleteness of the theory itself, which is why it does not allow them for certain define.

Light is the basis of life on our planet. Answering the questions “Why is the sky blue?” and “Why is the grass green?” you can give a definite answer - “Thanks to the light.” This an integral part of our lives, but we are still trying to understand the phenomenon of light...

Waves are one of two ways of energy transfer in space (the other way is corpuscular, using particles). Waves usually propagate in some medium (for example, waves on the surface of a lake propagate in water), but the direction of movement of the medium itself does not coincide with the direction of movement of the waves. Imagine a float bobbing on the waves. Rising and falling, the float follows the movements of the water as the waves pass by it. The phenomenon of interference occurs when two or more waves of the same frequency, propagating in different directions, interact.

The basics of the phenomenon of diffraction can be understood by referring to Huygens' principle, according to which each point along the path of propagation of a light beam can be considered as a new independent source of secondary waves, and the further diffraction pattern is determined by the interference of these secondary waves. When a light wave interacts with an obstacle, some of the secondary Huygens waves are blocked.

The concept of an “absolute black body” was introduced by the German physicist Gustav Kirchhoff in mid-19th century. The need to introduce such a concept was associated with the development of the theory of thermal radiation.

An absolutely black body is an idealized body that absorbs all electromagnetic radiation incident on it in all wavelength ranges and does not reflect anything.

Thus, the energy of any incident radiation is completely transferred to the black body and converted into its internal energy. Simultaneously with absorption, the blackbody also emits electromagnetic radiation and loses energy. Moreover, the power of this radiation and its spectral range are determined only by the temperature of the black body. It is the temperature of the black body that determines how much radiation it emits in the infrared, visible, ultraviolet and other ranges. Therefore, the blackbody, despite its name, at a sufficiently high temperature will emit in the visible range and visually have color. Our Sun is an example of an object heated to a temperature of 5800°C, with properties close to the black body.

Absolutely black bodies do not exist in nature, so in physics a model is used for experiments. Most often it is a closed cavity with a small entrance hole. The radiation entering through this hole is completely absorbed by the walls after repeated reflections. No part of the radiation entering the hole is reflected back from it - this corresponds to the definition of a blackbody (total absorption and no reflection). In this case, the cavity has its own radiation corresponding to its temperature. Since the own radiation of the inner walls of the cavity also performs a huge number of new absorptions and emissions, we can say that the radiation inside the cavity is in thermodynamic equilibrium with the walls. The characteristics of this equilibrium radiation are determined only by the cavity temperature (CBT): the total (at all wavelengths) radiation energy according to the Stefan-Boltzmann law, and the distribution of radiation energy over wavelengths is described by Planck’s formula.

There are no absolutely black bodies in nature. There are examples of bodies that are only closest in their characteristics to completely black. For example, soot can absorb up to 99% of the light falling on it. Obviously, the special surface roughness of the material makes it possible to reduce reflections to a minimum. It is thanks to multiple reflections followed by absorption that we see objects such as black velvet black.

I once met an object very close to the blackbody at the production of Gillette razor blades in St. Petersburg, where I had the opportunity to work even before taking up thermal imaging. Classic double-sided razor blades in technological process They are collected into “knives” with up to 3000 blades in a pack. Side surface, consisting of many sharpened blades pressed tightly together, is velvety black in color, although each individual steel blade has a shiny, sharpened steel edge. A block of blades left on a windowsill in sunny weather could reach temperatures of up to 80°C. At the same time, individual blades practically did not heat up, as they reflected most radiation. Threads on bolts and studs have a similar surface shape; their emissivity is higher than on a smooth surface. This property is often used in thermal imaging testing of electrical equipment.

Scientists are working to create materials with properties close to those of absolute black bodies. For example, significant results have been achieved in the optical range. In 2004, an alloy of nickel and phosphorus was developed in England, which was a microporous coating and had a reflectance of 0.16–0.18%. This material was listed in the Guinness Book of Records as the blackest material in the world. In 2008, American scientists set a new record - the thin film they grew, consisting of vertical carbon tubes, almost completely absorbs radiation, reflecting it by 0.045%. The diameter of such a tube is from ten nanometers and a length from ten to several hundred micrometers. The created material has a loose, velvety structure and a rough surface.

Each infrared device is calibrated according to the black body model(s). Temperature measurement accuracy can never be better than calibration accuracy. Therefore, the quality of calibration is very important. During calibration (or verification) using reference emitters, temperatures from the entire measurement range of the thermal imager or pyrometer are reproduced. In practice, reference thermal emitters are used in the form of a black body model of the following types:

Cavity models of the blackbody. They have a cavity with a small inlet hole. The temperature in the cavity is set, maintained and measured with high accuracy. Such emitters can produce high temperatures.

Extended or planar models of the black body. They have a platform painted with a composition with a high emissivity (low reflectance). The site temperature is set, maintained and measured with high accuracy. Low negative temperatures can be reproduced in such emitters.

When searching for information about imported black body models, use the term “black body”. It is also important to understand the difference between testing, calibrating and verifying a thermal imager. These procedures are described in detail on the website in the section on thermal imagers.

Materials used: Wikipedia; TSB; Infrared Training Center (ITC); Fluke Calibration

Absolutely black body

Absolutely black body- a physical idealization used in thermodynamics, a body that absorbs all electromagnetic radiation incident on it in all ranges and does not reflect anything. Despite the name, a completely black body can itself emit electromagnetic radiation of any frequency and visually have color. The radiation spectrum of an absolutely black body is determined only by its temperature.

The importance of an absolutely black body in the question of the spectrum of thermal radiation of any (gray and colored) bodies in general, in addition to the fact that it represents the simplest non-trivial case, also lies in the fact that the question of the spectrum of equilibrium thermal radiation of bodies of any color and reflection coefficient is reduced by the methods of classical thermodynamics to the question of the radiation of an absolutely black body (and historically this was already done by the end of the 19th century, when the problem of radiation of an absolutely black body came to the fore).

The blackest real substances, for example, soot, absorb up to 99% of incident radiation (that is, they have an albedo of 0.01) in the visible wavelength range, but they absorb infrared radiation much worse. Among the bodies of the Solar System, the Sun has the properties of an absolutely black body to the greatest extent.

The term was introduced by Gustav Kirchhoff in 1862.

Practical model

Black body model

Absolutely black bodies do not exist in nature (except for black holes), so in physics a model is used for experiments. It is a closed cavity with a small hole. Light entering through this hole will be completely absorbed after repeated reflections, and the hole will appear completely black from the outside. But when this cavity is heated, it will develop its own visible radiation. Since the radiation emitted by the inner walls of the cavity, before it leaves (after all, the hole is very small), in the overwhelming majority of cases will undergo a huge amount of new absorption and radiation, we can say with confidence that the radiation inside the cavity is in thermodynamic equilibrium with the walls. (In fact, the hole is not important for this model at all, it is only needed to emphasize the fundamental observability of the radiation located inside; the hole can, for example, be completely closed, and quickly opened only when equilibrium has already been established and the measurement is being carried out).

Laws of black body radiation

Classic approach

Initially, purely classical methods were applied to solve the problem, which gave a number of important and correct results, but they did not allow the problem to be completely solved, ultimately leading not only to a sharp discrepancy with experiment, but also to an internal contradiction - the so-called ultraviolet disaster.

The study of the laws of black body radiation was one of the prerequisites for the emergence of quantum mechanics.

Wien's first law of radiation

In 1893, Wilhelm Wien, using, in addition to classical thermodynamics, the electromagnetic theory of light, derived the following formula:

Where uν - radiation energy density,

ν - radiation frequency,

T- temperature of the radiating body,

f- a function that depends only on frequency and temperature. The form of this function cannot be established based only on thermodynamic considerations.

Wien's first formula is valid for all frequencies. Any more specific formula (for example, Planck's law) must satisfy Wien's first formula.

From Wien's first formula one can derive Wien's displacement law (maximum law) and Stefan-Boltzmann law, but one cannot find the values ​​of the constants included in these laws.

Historically, it was Wien’s first law that was called the displacement law, but currently the term “Wien’s displacement law” refers to the maximum law.

Wien's second law of radiation

In 1896, Wien derived the second law based on additional assumptions:

Where C 1 , C 2 - constants. Experience shows that Wien's second formula is valid only in the limit of high frequencies (short wavelengths). It is a special case of Wien's first law.

Later, Max Planck showed that Wien's second law follows from Planck's law for high quantum energies, and also found the constants C 1 and C 2. Taking this into account, Wien's second law can be written as:

Where h- Planck's constant,

k- Boltzmann constant,

c- speed of light in vacuum.

Rayleigh-Jeans law

An attempt to describe the radiation of a completely black body based on the classical principles of thermodynamics and electrodynamics leads to the Rayleigh-Jeans law:

This formula assumes a quadratic increase in the spectral density of radiation depending on its frequency. In practice, such a law would mean the impossibility of thermodynamic equilibrium between matter and radiation, since according to it all thermal energy would have to be converted into radiation energy in the short-wave region of the spectrum. This hypothetical phenomenon was called an ultraviolet catastrophe.

Nevertheless, the Rayleigh-Jeans radiation law is valid for the long-wave region of the spectrum and adequately describes the nature of the radiation. The fact of such correspondence can be explained only by using a quantum mechanical approach, according to which radiation occurs discretely. Based on quantum laws, we can obtain Planck's formula, which will coincide with the Rayleigh-Jeans formula for .

This fact is an excellent illustration of the principle of correspondence, according to which a new physical theory must explain everything that the old one was able to explain.

Planck's law

Dependence of black body radiation power on wavelength.

The radiation intensity of an absolutely black body, depending on temperature and frequency, is determined by Planck's law:

where is the radiation power per unit area of ​​the emitting surface in a unit frequency interval in the perpendicular direction per unit solid angle (dimension in SI: J s −1 m −2 Hz −1 sr −1).

Equivalently,

where is the radiation power per unit area of ​​the emitting surface in a unit wavelength interval in the perpendicular direction per unit solid angle (SI dimension: J s −1 m −2 m −1 sr −1).

The total (i.e. emitted in all directions) spectral radiation power per unit surface of an absolutely black body is described by the same formulas accurate to the coefficient π: ε(ν, T) = π I(ν, T), ε(λ, T) = π u(λ, T).

Stefan-Boltzmann law

The total energy of thermal radiation is determined by the Stefan-Boltzmann law, which states:

The radiation power of an absolutely black body (integrated power over the entire spectrum) per unit surface area is directly proportional to the fourth power of the body temperature:

where is the power per unit area of ​​the radiating surface, and

W/(m²·K 4) ​​- Stefan-Boltzmann constant.

Thus, an absolutely black body at = 100 K emits 5.67 watts from a square meter of its surface. At a temperature of 1000 K, the radiation power increases to 56.7 kilowatts per square meter.

For non-black bodies we can approximately write:

where is the degree of blackness (for all substances, for an absolutely black body).

The Stefan-Boltzmann constant can be theoretically calculated only from quantum considerations, using Planck's formula. At the same time, the general form of the formula can be obtained from classical considerations (which does not eliminate the problem of the ultraviolet catastrophe).

Wien's displacement law

The wavelength at which the radiation energy of a completely black body is maximum is determined by Wien's displacement law:

where is the temperature in Kelvin, and is the wavelength with maximum intensity in meters.

So, if we assume as a first approximation that human skin is close in properties to an absolutely black body, then the maximum of the radiation spectrum at a temperature of 36 °C (309 K) lies at a wavelength of 9400 nm (in the infrared region of the spectrum).

The apparent color of completely black bodies at different temperatures is shown in the diagram.

Blackbody radiation

Electromagnetic radiation that is in thermodynamic equilibrium with a blackbody at a given temperature (for example, radiation inside a cavity in a blackbody) is called blackbody (or thermal equilibrium) radiation. Equilibrium thermal radiation is homogeneous, isotropic and non-polarized, there is no energy transfer in it, all its characteristics depend only on the temperature of the absolutely blackbody emitter (and, since blackbody radiation is in thermal equilibrium with this body, this temperature can be attributed to radiation). The volumetric energy density of blackbody radiation is equal to its pressure is equal to Very close in its properties to blackbody radiation is the so-called relict radiation, or cosmic microwave background - radiation filling the Universe with a temperature of about 3 K.

Blackbody chromaticity

Colors are given in comparison to diffuse daylight. The actual perceived color may be distorted by the eye's adaptation to lighting conditions.

Kirchhoff's radiation law

Kirchhoff's radiation law is a physical law established by the German physicist Kirchhoff in 1859.

In its modern formulation, the law reads as follows:

The ratio of the emissivity of any body to its absorption capacity is the same for all bodies at a given temperature for a given frequency and does not depend on their shape and chemical nature.

It is known that when electromagnetic radiation falls on a certain body, part of it is reflected, part is absorbed, and part can be transmitted. The fraction of radiation absorbed at a given frequency is called absorption capacity body. On the other hand, every heated body emits energy according to some law called emissivity of the body.

The values ​​of and can vary greatly when moving from one body to another, however, according to Kirchhoff’s law of radiation, the ratio of emissivity and absorption abilities does not depend on the nature of the body and is a universal function of frequency (wavelength) and temperature:

By definition, an absolutely black body absorbs all radiation incident on it, that is, for it. Therefore, the function coincides with the emissivity of an absolutely black body, described by the Stefan-Boltzmann law, as a result of which the emissivity of any body can be found based only on its absorption capacity.

Real bodies have an absorption capacity less than unity, and therefore an emissivity less than that of an absolutely black body. Bodies whose absorption capacity does not depend on frequency are called gray. Their spectrum has the same appearance as that of an absolutely black body. In the general case, the absorption capacity of bodies depends on frequency and temperature, and their spectrum can differ significantly from the spectrum of an absolutely black body. The study of the emissivity of different surfaces was first carried out by the Scottish scientist Leslie using his own invention - the Leslie cube.

Pure black body- this is a body for which the absorption capacity is identically equal to unity for all frequencies or wavelengths and for any temperature, i.e.:

From the definition of an absolutely black body it follows that it must absorb all radiation incident on it.

The concept of "absolutely black body" is a model concept. Absolute black bodies do not exist in nature, but it is possible to create a device that is a good approximation to an absolutely black body - black body model .

Black body model- this is a closed cavity with a small hole compared to its size (Fig. 1.2). The cavity is made of a material that absorbs radiation quite well. The radiation entering the hole is reflected many times from the inner surface of the cavity before leaving the hole.

With each reflection, part of the energy is absorbed, as a result, the reflected flux dФ comes out of the hole, which is a very small part of the radiation flux dФ that entered it. As a result, the absorption capacity holes in the cavity will be close to unity.

If the inner walls of the cavity are maintained at temperature T, then radiation will emerge from the hole, the properties of which will be very close to the properties of black body radiation. Inside the cavity, this radiation will be in thermodynamic equilibrium with the cavity matter.

By definition of energy density, the volumetric energy density w(T) of equilibrium radiation in a cavity is:

where dE is the radiation energy in the volume dV. Spectral distribution of volume density is given by the functions u(λ,T) (or u(ω,T)), which are introduced similarly to the spectral density of energetic luminosity ((1.6) and (1.9)), i.e.:

Here dw λ and dw ω are the volumetric energy density in the corresponding interval of wavelengths dλ or frequencies dω.

Kirchhoff's law states that the relationship emissivity body ((1.6) and (1.9)) to its absorption capacity (1.14) is the same for all bodies and is universal function frequency ω (or wavelength λ) and temperature T, i.e.:

It is obvious that the absorption capacity aω (or a λ) is different for different bodies, then from Kirchhoff’s law it follows that the stronger a body absorbs radiation, the stronger it should emit this radiation. Since for an absolute black body aω ≡ 1 (or aλ ≡ 1), then it follows that in the case of a completely black body:

In other words, f(ω,T) or φ(λ,T) , is nothing more than the spectral energy luminosity density (or emissivity) of a completely black body.

The function φ(λ,T) and f(ω,T) are related to the spectral energy density of black body radiation by the following relations:

where c is the speed of light in vacuum.

Installation diagram for experimental determination of the dependence φ(λ,T) is shown in Figure 1.3.

Radiation is emitted from the opening of a closed cavity, heated to a temperature T, then hits a spectral device (prism or grating monochromator), which emits radiation in the frequency range from λ to λ + dλ. This radiation hits a receiver, which allows the radiation power incident on it to be measured. By dividing this power per interval from λ to λ + dλ by the area of ​​the emitter (the area of ​​the hole in the cavity!), we obtain the value of the function φ(λ,T) for a given wavelength λ and temperature T. The experimental results obtained are reproduced in Figure 1.4.

Results of lecture No. 1

1. German physicist Max Planck in 1900 put forward a hypothesis according to which electromagnetic energy is emitted in portions, energy quanta. The magnitude of the energy quantum (see (1.2):

ε = h v,

where h=6.6261·10 -34 J·s is Planck’s constant, v- frequency of oscillations of an electromagnetic wave emitted by a body.

This hypothesis allowed Planck to solve the problem of black body radiation.

2. And Einstein, developing Planck’s concept of energy quanta, introduced in 1905 the concept of “quantum of light” or photon. According to Einstein, quantum of electromagnetic energy ε = h v moves in the form of a photon localized in a small region of space. The idea of ​​photons allowed Einstein to solve the problem of the photoelectric effect.

3. English physicist E. Rutherford, based on experimental studies, carried out in 1909-1910, built a planetary model of the atom. According to this model, at the center of the atom there is a very small nucleus (r I ~ 10 -15 m), in which almost the entire mass of the atom is concentrated. The nuclear charge is positive. Negatively charged electrons move around the nucleus like planets solar system in orbits whose size is ~ 10 -10 m.

4. The atom in Rutherford’s model turned out to be unstable: according to Maxwell’s electrodynamics, electrons, moving in circular orbits, should continuously emit energy, as a result of which they should fall onto the nucleus in ~ 10 -8 s. But all our experience testifies to the stability of the atom. This is how the problem of atomic stability arose.

5. The problem of atomic stability was solved in 1913 by the Danish physicist Niels Bohr on the basis of two postulates he put forward. In the theory of the hydrogen atom, developed by N. Bohr, Planck's constant plays a significant role.

6. Thermal radiation is electromagnetic radiation emitted by a substance due to its internal energy. Thermal radiation can be in thermodynamic equilibrium with surrounding bodies.

7. The energetic luminosity of a body R is the ratio of the energy dE emitted during a time dt by the surface dS in all directions to dt and dS (see (1.5)):

8. Spectral density of energy luminosity r λ (or emissivity of a body) is the ratio of energy luminosity dR, taken in an infinitesimal wavelength interval dλ, to the value dλ (see (1.6)):

9. Radiation flux Ф is the ratio of the energy dE transferred by electromagnetic radiation through any surface to the transfer time dt, which significantly exceeds the period of electromagnetic oscillations (see (1.13)):

10. Body absorption capacity a λ is the ratio of the radiation flux dФ λ "absorbed by a body in the wavelength interval dλ to the flux dФ λ incident on it in the same interval dλ, (see (1.14):

11. An absolutely black body is a body for which the absorption capacity is identically equal to unity for all wavelengths and for any temperature, i.e.

A completely black body is a model concept.

12. Kirchhoff’s law states that the ratio of the emissivity of a body r λ to its absorption capacity a λ is the same for all bodies and is a universal function of wavelength λ (or frequency ω) and temperature T (see (1.17)):

LECTURE N 2

The problem of blackbody radiation. Planck's formula. Stefan-Boltzmann law, Wien's law

§ 1. The problem of black body radiation. Planck's formula

The problem with black body radiation was to theoretically get addictedφ(λ,T)- the spectral density of the energy luminosity of an absolutely black body.

It seemed that the situation was clear: at a given temperature T, the molecules of the substance of the radiating cavity have a Maxwellian velocity distribution and emit electromagnetic waves in accordance with the laws of classical electrodynamics. Radiation is in thermodynamic equilibrium with matter, which means that the laws of thermodynamics and classical statistics can be used to find the spectral radiation energy density u(λ,T) and the associated function φ(λ,T).

However, all attempts by theorists to obtain the law of black body radiation based on classical physics have failed.

Partial contributions to the solution of this problem were made by Gustav Kirchhoff, Wilhelm Wien, Joseph Stefan, Ludwig Boltzmann, John William Rayleigh, James Honwood Jeans.

The problem of blackbody radiation was solved by Max Planck. To do this, he had to abandon classical concepts and make the assumption that a charge oscillating with a frequency v, can receive or give energy in portions, or quanta.

The magnitude of the energy quantum in accordance with (1.2) and (1.4):

where h is Planck's constant; v- frequency of oscillations of an electromagnetic wave emitted by an oscillating charge; ω = 2π v- circular frequency.

Based on the concept of energy quanta, M. Planck, using the methods of statistical thermodynamics, obtained an expression for the function u(ω,T), giving distribution of energy density in the radiation spectrum of an absolute black body:

The derivation of this formula will be given in Lecture No. 12, § 3 after we become acquainted with the basics of quantum statistics.

To go to the spectral density of energy luminosity f(ω,T), we write the second formula (1.19):

Using this relation and Planck’s formula (2.1) for u(ω,T), we obtain that:

This is Planck's formula for spectral density of energetic luminosity f(ω ,T).

Now we get Planck's formula for φ(λ,T). As we know from (1.18), in the case of a completely black body f(ω,T) = r ω, and φ(λ,T) = r λ.

The relationship between r λ and r ω is given by formula (1.12), applying it we get:

Here we expressed the argument ω of the function f(ω,T) in terms of the wavelength λ. Substituting here Planck’s formula for f(ω,T) from (2.2), we obtain Planck’s formula for φ(λ,T) - the spectral density of energy luminosity depending on the wavelength λ:

The graph of this function coincides well with the experimental graphs of φ(λ,T) for all wavelengths and temperatures.

This means that the problem of black body radiation has been solved.

§ 2. Stefan-Boltzmann law and Wien's law

From (1.11) for an absolutely black body, when r ω = f(λ,T), we obtain the energy luminosity R(T) , integrating the function f(ω,Т) (2.2) over the entire frequency range.

Integration gives:

Let us introduce the notation:

then the expression for the energetic luminosity R will take the following form:

That's what it is Stefan-Boltzmann law .

M. Stefan, based on an analysis of experimental data, came to the conclusion in 1879 that the energetic luminosity of any body is proportional to the fourth power of temperature.

L. Boltzmann in 1884 found from thermodynamic considerations that such a dependence of energetic luminosity on temperature is valid only for an absolutely black body.

The constant σ is called Stefan-Boltzmann constant . Its experimental significance:

Calculations using the theoretical formula give a result for σ that is in very good agreement with the experimental one.

Note that graphically the energetic luminosity is equal to the area limited by the graph of the function f(ω,T), this is illustrated in Figure 2.1.

The maximum of the graph of the spectral density of energy luminosity φ(λ,T) shifts to the region of shorter waves with increasing temperature (Fig. 2.2). To find the law according to which the maximum φ(λ,T) shifts depending on temperature, it is necessary to study the function φ(λ,T) to the maximum. Having determined the position of this maximum, we obtain the law of its movement with temperature change.

As is known from mathematics, to study a function to its maximum, you need to find its derivative and equate it to zero:

Substituting here φ(λ,Т) from (1.23) and taking the derivative, we obtain three roots algebraic equation relative to the variable λ. Two of them (λ = 0 and λ = ∞) correspond to zero minima of the function φ(λ,T). For the third root, an approximate expression is obtained:

Let us introduce the notation:

then the position of the maximum of the function φ(λ,T) will be determined by a simple formula:

That's what it is Wien's displacement law .

It is named after V. Wien, who theoretically obtained this ratio in 1894. The constant in Wien's displacement law has the following numerical value:

Results of lecture No. 2

1. The problem of black body radiation was that all attempts to obtain, on the basis of classical physics, the dependence φ(λ,T) - the spectral density of the energy luminosity of a black body failed.

2. This problem was solved in 1900 by M. Planck on the basis of his quantum hypothesis: a charge oscillating with a frequency v, can receive or give out energy in portions or quanta. Energy quantum value:

here h = 6.626 10 -34 is Planck’s constant, the value J s is also called Planck's constant ["ash" with a bar], ω is the circular (cyclic) frequency.

3. Planck’s formula for the spectral density of the energy luminosity of an absolutely black body has the following form (see (2.4):

here λ is the wavelength of electromagnetic radiation, T is the absolute temperature, h is Planck’s constant, c is the speed of light in vacuum, k is Boltzmann’s constant.

4. From Planck’s formula follows the expression for the energy luminosity R of an absolutely black body:

which allows us to theoretically calculate the Stefan-Boltzmann constant (see (2.5)):

the theoretical value of which coincides well with its experimental value:

in the Stefan-Boltzmann law (see (2.6)):

5. From Planck’s formula follows Wien’s displacement law, which determines λ max - the position of the maximum of the function φ(λ,T) depending on the absolute temperature (see (2.9):

For b - the Wien constant - the following expression is obtained from Planck’s formula (see (2.8)):

Wien's constant has the following value b = 2.90 ·10 -3 m·K.

LECTURE N 3

Photoelectric effect problem . Einstein's equation for the photoelectric effect

§ 1. The photoelectric effect problem A

The photoelectric effect is the emission of electrons by a substance under the influence of electromagnetic radiation.

This photoelectric effect is called external. This is what we will talk about in this chapter. There is also internal photoelectric effect . (see lecture 13, § 2).

In 1887, German physicist Heinrich Hertz discovered that ultraviolet light shining on the negative electrode in a spark gap facilitated the passage of the discharge. In 1888-89 Russian physicist A. G. Stoletov is engaged in a systematic study of the photoelectric effect (a diagram of its installation is shown in the figure). The research was carried out in a gas atmosphere, which greatly complicated the processes taking place.

Stoletov discovered that:

1) ultraviolet rays have the greatest impact;

2) the current increases with increasing intensity of light illuminating the photocathode;

3) charges emitted under the influence of light have a negative sign.

Further studies of the photoelectric effect were carried out in 1900-1904. German physicist F. Lenard in the highest vacuum achieved at that time.

Lenard was able to establish that the speed of electrons escaping from the photocathode does not depend on light intensity and directly proportional to its frequency . This is how I was born photoelectric effect problem . It was impossible to explain the results of Lenard's experiments on the basis of Maxwell's electrodynamics!

Figure 3.2 shows a setup that allows you to study the photoelectric effect in detail.

Electrodes, photocathode And anode , placed in balloon, from which the air has been pumped out. Light is supplied to the photocathode through quartz window . Quartz, unlike glass, transmits ultraviolet rays well. The potential difference (voltage) between the photocathode and anode measures voltmeter . The current in the anode circuit is measured by a sensitive microammeter . To regulate voltage power battery connected to rheostat with a midpoint. If the rheostat motor is opposite the midpoint connected through a microammeter to the anode, then the potential difference between the photocathode and the anode is zero. When the slider is shifted to the left, the anode potential becomes negative relative to the cathode. If the rheostat slider is moved to the right from the midpoint, then the anode potential becomes positive.

The current-voltage characteristic of the installation for studying the photoelectric effect allows one to obtain information about the energy of electrons emitted by the photocathode.

The current-voltage characteristic is the dependence of the photocurrent i on the voltage between the cathode and anode U. When illuminated with light, the frequency v which is sufficient for the photoelectric effect to occur, the current-voltage characteristic has the form of the graph shown in Fig. 3.3:

From this characteristic it follows that at a certain positive voltage at the anode, the photocurrent i reaches saturation. In this case, all electrons emitted by the photocathode per unit time fall on the anode during the same time.

At U = 0, some electrons reach the anode and create a photocurrent i 0 . At some negative voltage at the anode - U back - the photocurrent stops. At this voltage value, the maximum kinetic energy of the photoelectron at the photocathode (mv 2 max)/2 is completely spent on doing work against the forces electric field:

In this formula, m e is the mass of the electron; v max - its maximum speed at the photocathode; e is the absolute value of the electron charge.

Thus, by measuring the retarding voltage U back, you can find the kinetic energy (and speed of the electron) immediately after its departure from the photocathode.

Experience has shown that

1)the energy of the electrons emitted from the photocathode (and their speed) did not depend on the light intensity! When the frequency of light changes v U back also changes, i.e. maximum kinetic energy of electrons leaving the photocathode;

2)maximum kinetic energy of electrons, at the photocathode,(mv 2 max)/2 , is directly proportional to the frequency v of the light illuminating the photocathode.

Problem, as in the case of black body radiation, was that theoretical predictions made for the photoelectric effect based on classical physics (Maxwellian electrodynamics) contradicted the experimental results. Light intensity I in classical electrodynamics is the energy flux density of a light wave. Firstly, from this point of view, the energy transferred by a light wave to an electron must be proportional to the intensity of the light. Experience does not confirm this prediction. Secondly, in classical electrodynamics there are no explanations for direct proportionality kinetic energy electrons,(mv 2 max)/2 , light frequency v.