Laboratory work No.
STUDYING REGULARITIES IN SPECTRA AND DETERMINING PLANCK'S CONSTANT
Goal of the work: experimental determination of Planck's constant using emission and absorption spectra.
Devices and accessories: spectroscope, incandescent lamp, mercury lamp, cuvette with chromium peak.
THEORETICAL INTRODUCTION
An atom is the smallest particle of a chemical element that determines its basic properties. The planetary model of the atom was substantiated by the experiments of E. Rutherford. At the center of the atom there is a positively charged nucleus with a charge Z∙e (Z– the number of protons in the nucleus, i.e. serial number of a chemical element in the periodic system of Mendeleev; e– the charge of a proton is equal to the charge of an electron). Electrons move around the nucleus in the electric field of the nucleus.
The stability of such an atomic system is justified by Bohr's postulates.
Bohr's first postulate(stationary state postulate): in a stable state of an atom, electrons move in certain stationary orbits without emitting electromagnetic energy; stationary electron orbits are determined by the quantization rule:
. (2)
An electron moving in an orbit around a nucleus is acted upon by the Coulomb force:
.
(3)
For a hydrogen atom Z=1. Then
. (4)
By solving equations (2) and (4) together, we can determine:
a) orbital radius
; (5)
b) electron speed
; (6)
c) electron energy
. (7)
Energy level– the energy possessed by an electron of an atom in a certain stationary state.
A hydrogen atom has one electron. State of the atom with n=1 is called the ground state. Ground state energy 
In its ground state, an atom can only absorb energy.
During quantum transitions, atoms (molecules) jump from one stationary state to another, that is, from one energy level to another. The change in the state of atoms (molecules) is associated with energy transitions of electrons from one stationary orbit to another. In this case, electromagnetic waves of various frequencies are emitted or absorbed.
Bohr's second postulate(frequency rule): when an electron moves from one stationary orbit to another, one photon with energy is emitted or absorbed
, (8)
equal to the energy difference of the corresponding stationary states ( 
And 
- respectively, the energy of the stationary states of the atom before and after radiation or absorption).
Energy is emitted or absorbed in separate portions - quanta (photons), and the energy of each quantum (photon) is associated with frequency ν emitted waves ratio
,
(9)
Where h– Planck’s constant. Planck's constant– one of the most important constants of atomic physics, numerically equal to the energy of one radiation quantum at a radiation frequency of 1 Hz.
Taking this into account, equation (8) can be written as
.
(10)
The totality of electromagnetic waves of all frequencies that a given atom (molecule) emits and absorbs is emission or absorption spectrum of a given substance. Since the atom of each substance has its own internal structure, therefore each atom has an individual, unique spectrum. This is the basis of spectral analysis, discovered in 1859 by Kirchhoff and Bunsen.
Characteristics of emission spectra
The spectral composition of radiation from substances is very diverse. But despite this, all spectra can be divided into three types.
Continuous spectra. The continuous spectrum represents the lengths of all waves. There are no breaks in such a spectrum; it consists of sections of different colors that transform into one another.
Continuous (or continuous) spectra are produced by bodies in a solid or liquid state (incandescent lamp, molten steel, etc.), as well as highly compressed gases. To obtain a continuous spectrum, the body must be heated to a high temperature.
A continuous spectrum is also produced by high-temperature plasma. Electromagnetic waves are emitted by plasma mainly when electrons collide with ions.
Line spectra. Line emission spectra consist of individual spectral lines separated by dark spaces.
Line spectra give all substances in the gaseous atomic state. In this case, light is emitted by atoms that practically do not interact with each other. The presence of a line spectrum means that a substance emits light only at certain wavelengths (more precisely, in certain very narrow spectral intervals).
Striped spectra. Banded emission spectra consist of separate groups of lines so closely spaced that they merge into bands. Thus, the striped spectrum consists of individual bands separated by dark spaces.
Unlike line spectra, striped spectra are created not by atoms, but by molecules that are not bound or weakly bound to each other.
To observe atomic and molecular spectra, the glow of vapor of a substance in a flame or the glow of a gas discharge in a tube filled with the gas under study is used.
Characteristics of absorption spectra.
The absorption spectrum can be observed if, in the path of radiation coming from a source that gives a continuous emission spectrum, a substance is placed that absorbs certain rays of different wavelengths.
In this case, dark lines or stripes will be visible in the field of view of the spectroscope in those places of the continuous spectrum that correspond to absorption. The nature of absorption is determined by the nature and structure of the absorbing substance. The gas absorbs light at precisely the wavelengths it emits when highly heated. Figure 1 shows the emission and absorption spectra of hydrogen.
Absorption spectra, like emission spectra, are divided into continuous, line and striped.
Continuous spectra absorptions are observed when absorbed by a substance in a condensed state.
Line spectra absorptions are observed when an absorbing substance in a gaseous state (atomic gas) is placed between the source of a continuous spectrum of radiation and the spectroscope.
Striped– when absorbed by substances consisting of molecules (solutions).
JUSTIFICATION OF THE RESEARCH METHODOLOGY
To obtain a striped absorption spectrum, an aqueous solution of chromium, that is, potassium dichromium, is used ( 
).
According to quantum theory, atoms, ions and molecules not only emit energy in quanta, but also absorb energy in quanta. Energy of a quantum of emission and absorption for a certain substance (at a certain frequency 
) is the same. Under the influence of light, chemical decomposition of molecules occurs, which can only be caused by a light quantum with the energy 
, sufficient (or greater) for decomposition.
Consider an aqueous solution of potassium dihydroxide 
. In water, its molecules dissociate into ions as follows:
During the reaction, ions appear in the solution 
. If this solution is illuminated with white (achromatic) light, then under the influence of light quanta absorbed by the chromium peak, the ions will disintegrate 
. In this case, each ion will “capture” (“absorb”) one quantum of irradiating radiation with the energy 
. As a result, the spectrum will have an absorption band, the beginning of which corresponds to the frequency 
. The decomposition reaction is written as follows:
.
The energy of this reaction for one kilomole of chromium is known from experiments ( E=2.228·10 8 J/kmol).
According to Avogadro's law, every kilomole of a substance contains the same number of atoms, equal to Avogadro's number N A=6.02 10 26 kmol -1, therefore energy is required for the decay of one ion
. (11)
Consequently, the energy of the absorbed light quantum must be greater than or equal to the energy required to split one ion 
, that is 
. Using equality
(12)
determine the lowest frequency of the quantum that splits the ion:
,
(13)
Where 
- the lowest frequency in the spectral absorption band (the edge of the band from the red light side).
Using the relationship between frequency 
and wavelength 
, expression (13) is written as follows:
,
(14)
where c is the speed of light in vacuum (c=3·10 8 m/s).
From equality (14) we determine Planck’s constant
. (15)
EXPERIMENTAL STUDIES
Wavelength Determination 
the extreme line (right) in the absorption band when observing the spectrum of the chromium peak is carried out in the following sequence:
Calibrate the spectroscope using the radiation spectrum, and then compile and fill out Table 1 to construct a calibration curve.
Table 1
|
Spectrum or line color |
Wavelength, nm |
Position of the boundaries of spectrum sections or lines according to the spectroscope n, division |
|
|
For a continuous spectrum | |||
|
Orange | |||
|
Light green | |||
|
Violet | |||
|
For the line spectrum of mercury vapor |
|||
|
Dark red (medium brightness) | |||
|
Red (medium brightness) | |||
|
Yellow 1 (bright) | |||
|
Yellow 2 (bright) | |||
|
Green (very bright) | |||
|
Violet 1 (very bright) | |||
|
Purple 2 (weak) | |||
|
Violet 3 (medium brightness) | |||
Spectroscope calibration
The spectroscope is calibrated in the following sequence:
A light source is installed in front of the spectroscope slit, the spectrum of which is line (mercury lamp, helium tube, etc.) or continuous (incandescent lamp). Using table 1, note what number n divisions of the spectroscope correspond to a certain line (this is done for all visible lines), that is, values are obtained for each line n and plot them along the x-axis. At the same time, the wavelength values for each line are taken from the table and marked along the ordinate axis 
. The resulting points at the intersection of the corresponding abscissas and ordinates are connected by a smooth curve;
On a large sheet of graph paper, wavelength values are plotted along the ordinate axis. 
in the range of the visible part of the continuous or line spectra (400-750 nm), while respecting the scale, and along the abscissa axis - the values n the total number of divisions of the spectrometer drum, covering the entire range of continuous or line spectra (400-750 nm), taking into account that one revolution of the drum (micrometric screw) corresponds to n=50, that is, fifty divisions.
3. Place a cuvette with a chromium peak in front of the slit of the spectroscope (spectrometer) and point the vertical filament of this spectrometer at the edge of the absorption band (dark band). In this position, the division number is recorded on the spectrometer and, using a calibration curve, the wavelength corresponding to the edge of the absorption band is determined. The experiment is performed four to five times to obtain the average value of Planck's constant 
, as well as for calculating measurement errors.
4. Calculate Planck’s constant for each measurement using formula (15).
5. Determine the absolute error of each measurement, the average value of the absolute error and the relative error:
; (16)
; (17)
. (18)
6. Record the results of measurements and calculations in table 2.
7. Record the measurement result in the form:
8. Check whether the table value of Planck’s constant belongs to the obtained interval (19).
table 2
|
n, division |
|
|
|
|
|
|
|
, nm
, J s
, J s
, J s
, J s
,
%Control questions
Describe the planetary model of the atom.
State Bohr's first postulate. What is the rule for quantizing the electron orbit?
What values can the orbital radius, speed and energy of an electron in an atom take?
What is an energy level?
Formulate Bohr's second postulate.
What is the energy of a photon?
What is the physical meaning of Planck's constant? What is it equal to?
Describe the emission spectra. What types are they divided into? What is needed to observe emission spectra?
Describe the absorption spectra. What types are they divided into? What is needed to observe absorption spectra?
Describe the principle of operation and structure of the spectroscope.
What is the calibration of a spectroscope? What spectra were used for calibration? Using the calibration curve of a spectroscope, how can you determine the wavelength corresponding to the edge of the absorption band?
Describe the procedure for performing the work.
BIBLIOGRAPHICAL LIST
Agapov B.T., Maksyutin G.V., Ostroverkhov P.I. Laboratory workshop in physics. – M.: Higher School, 1982.
Korsunsky M.I. Optics, atomic structure, atomic nucleus. – M.: Fizmatgiz, 1962.
Physical workshop/Ed. I.V. Iveronova. – M.: Fizmatgiz, 1962.
This article, based on the photon concept, reveals the physical essence of the “fundamental constant” of Planck’s constant. Arguments are given to show that Planck's constant is a typical photon parameter that is a function of its wavelength.
Introduction. The end of the 19th and beginning of the 20th centuries was marked by a crisis in theoretical physics, caused by the inability to use the methods of classical physics to substantiate a number of problems, one of which was the “ultraviolet catastrophe.” The essence of this problem was that when establishing the law of energy distribution in the radiation spectrum of an absolutely black body using the methods of classical physics, the spectral energy density of the radiation should increase indefinitely as the wavelength of the radiation shortens. In fact, this problem showed, if not the internal inconsistency of classical physics, then, in any case, an extremely sharp discrepancy with elementary observations and experiment.
Studies of the properties of black body radiation, which took place over almost forty years (1860-1900), culminated in Max Planck’s hypothesis that the energy of any system E when emitting or absorbing electromagnetic radiation frequency ν (\displaystyle ~\nu ) can only change by an amount that is a multiple of the quantum energy:
E γ = hν (\displaystyle ~E=h\nu ) . (1)(\displaystyle ~h)
Proportionality factor h in expression (1) entered into science under the name “Planck’s constant”, becoming main constant quantum theory .
The black body problem was revised in 1905, when Rayleigh and Jeans on the one hand, and Einstein on the other, independently proved that classical electrodynamics could not justify the observed radiation spectrum. This led to the so-called "ultraviolet catastrophe", so designated by Ehrenfest in 1911. The efforts of theorists (together with Einstein's work on the photoelectric effect) led to the recognition that Planck's postulate about the quantization of energy levels was not a simple mathematical formalism, but an important element of understanding about physical reality.
Further development of Planck's quantum ideas - the substantiation of the photoelectric effect using the hypothesis of light quanta (A. Einstein, 1905), the postulate in Bohr's atomic theory of the quantization of the angular momentum of an electron in an atom (N. Bohr, 1913), the discovery of the de Broglie relation between the mass of a particle and its length waves (L. De Broglie, 1921), and then the creation of quantum mechanics (1925 - 26) and the establishment of fundamental uncertainty relations between momentum and coordinate and between energy and time (W. Heisenberg, 1927) led to the establishment of the fundamental status of Planck’s constant in physics .
Modern quantum physics also adheres to this point of view: “In the future it will become clear to us that the formula E / ν = h expresses the fundamental principle of quantum physics, namely the universal relationship between energy and frequency: E = hν. This connection is completely alien to classical physics, and the mystical constant h is a manifestation of the secrets of nature that were not comprehended at that time.”
At the same time, there was an alternative view of Planck’s constant: “Textbooks on quantum mechanics say that classical physics is physics in which h equals zero. But in fact Planck's constant h - this is nothing more than a quantity that actually defines a concept well known in classical physics of the gyroscope. Interpretation to adepts studying physics that h ≠ 0 is a purely quantum phenomenon, which has no analogue in classical physics, and was one of the main elements aimed at strengthening the belief in the necessity of quantum mechanics.”
Thus, the views of theoretical physicists on Planck's constant were divided. On the one hand, there is its exclusivity and mystification, and on the other, an attempt to give a physical interpretation that does not go beyond the framework of classical physics. This situation persists in physics at the present time, and will persist until the physical essence of this constant is established.
The physical essence of Planck's constant. Planck was able to calculate the value h from experimental data on black body radiation: its result was 6.55 10 −34 J s, with an accuracy of 1.2% of the currently accepted value, however, to justify the physical essence of the constant h he could not. Disclosure of the physical essences of any phenomena is not characteristic of quantum mechanics: “The reason for the crisis situation in specific areas of science is the general inability of modern theoretical physics to understand the physical essence of phenomena, to reveal the internal mechanism of phenomena, the structure of material formations and interaction fields, to understand the cause-and-effect relationships between elements, phenomena.” Therefore, apart from mythology, she could not imagine anything else in this matter. In general, these views are reflected in the work: “Planck's Constant h as a physical fact means the existence of the smallest, non-reducible and non-contractible finite amount of action in nature. As a non-zero commutator for any pair of dynamic and kinematic quantities that form the dimension of action through their product, Planck’s constant gives rise to the property of non-commutativity for these quantities, which in turn is the primary and irreducible source of the inevitably probabilistic description of physical reality in any spaces of dynamics and kinematics. Hence the universality and universality of quantum physics.”
In contrast to the views of quantum physics adherents on the nature of Planck's constant, their opponents were more pragmatic. The physical meaning of their ideas was reduced to “calculating by methods of classical mechanics the magnitude of the main angular momentum of the electron P e (angular momentum associated with the rotation of the electron around its own axis) and obtaining a mathematical expression for Planck’s constant “ h "through known fundamental constants." What was the physical essence based on: “ Planck's constant « h » equal to size classical main angular momentum of the electron (associated with the rotation of the electron around its own axis), multiplied by 4 p. ”
The fallacy of these views lies in the misunderstanding of the nature of elementary particles and the origins of the appearance of Planck’s constant. An electron is a structural element of an atom of a substance, which has its own functional purpose - the formation of the physical and chemical properties of the atoms of the substance. Therefore, it cannot act as a carrier of electromagnetic radiation, i.e. Planck’s hypothesis about the transfer of energy by a quantum is not applicable to the electron.
To substantiate the physical essence of Planck's constant, let's consider this problem from a historical aspect. From the above it follows that the solution to the problem of the “ultraviolet catastrophe” was Planck’s hypothesis that the radiation of a completely black body occurs in portions, i.e., in energy quanta. Many physicists of that time initially assumed that energy quantization was the result of some unknown property of matter absorbing and emitting electromagnetic waves. However, already in 1905, Einstein developed Planck's idea, suggesting that energy quantization is a property of electromagnetic radiation itself. Based on the hypothesis of light quanta, he explained a number of patterns of the photoelectric effect, luminescence, and photochemical reactions.
The validity of Einstein's hypothesis was experimentally confirmed by the study of the photoelectric effect by R. Millikan (1914 -1916) and studies of the scattering of X-rays by electrons by A. Compton (1922 - 1923). Thus, it became possible to consider a light quantum as an elementary particle, subject to the same kinematic laws as particles of matter.
In 1926, Lewis proposed the term “photon” for this particle, which was adopted by the scientific community. According to modern concepts, a photon is an elementary particle, a quantum of electromagnetic radiation. Photon rest mass m g is zero (experimental limit m g<5 . 10 -60 г), и поэтому его скорость равна скорости света . Электрический заряд фотона также равен нулю .
If a photon is a quantum (carrier) of electromagnetic radiation, then its electric charge cannot be equal to zero. The inconsistency of this representation of the photon became one of the reasons for the misunderstanding of the physical essence of Planck’s constant.
The insoluble justification for the physical essence of Planck's constant within the framework of existing physical theories can be overcome by the etherodynamic concept developed by V.A Atsyukovsky.
In ether-dynamic models, elementary particles are treated as closed vortex formations(rings), in the walls of which the ether is significantly compacted, and elementary particles, atoms and molecules are structures that unite such vortices. The existence of ring and screw motions corresponds to the presence of a mechanical moment (spin) in the particles, directed along the axis of its free movement.
According to this concept, a photon is structurally a closed toroidal vortex with a circular motion of the torus (like a wheel) and a screw motion inside it. The source of photon generation is a proton-electron pair of atoms of a substance. As a result of excitation, due to the symmetry of its structure, each proton-electron pair generates two photons. Experimental confirmation of this is the process of annihilation of an electron and a positron.
A photon is the only elementary particle that is characterized by three types of motion: rotational motion around its own axis of rotation, rectilinear motion in a given direction and rotational motion with a certain radius R relative to the axis of linear motion. The last movement is interpreted as movement along a cycloid. A cycloid is a periodic function along the x-axis, with a period 2π R (\displaystyle 2\pi r)/…. For a photon, the period of the cycloid is interpreted as the wavelength λ , which is the argument of all other parameters of the photon.
On the other hand, wavelength is also one of the parameters of electromagnetic radiation: a disturbance (change in state) of the electromagnetic field propagating in space. For which the wavelength is the distance between the two points closest to each other in space, in which the oscillations occur in the same phase.
This implies a significant difference in the concepts of wavelength for a photon and electromagnetic radiation in general.
For a photon, wavelength and frequency are related by the relation
ν = u γ / λ, (2)
Where u γ – speed of rectilinear photon motion.
Photon is a concept related to a family (set) of elementary particles, united by common signs of existence. Each photon is characterized by its own specific set of characteristics, one of which is wavelength. At the same time, taking into account the interdependence of these characteristics on each other, in practice it has become convenient to represent the characteristics (parameters) of a photon as a function of one variable. The photon wavelength was defined as the independent variable.
Known value u λ = 299,792,458 ± 1.2/, defined as the speed of light. This value was obtained by K. Evenson and his co-workers in 1972 using the cesium frequency standard of the CH 4 laser, and its wavelength using the krypton frequency standard (approx. 3.39 μm). Thus, the speed of light is formally defined as the linear speed of photons of wavelength λ = 3,39 10 -6 m. Theoretically (\displaystyle 2\pi r)/… it has been established that the speed of motion of (rectilinear) photons is variable and nonlinear, i.e. u λ = f( λ). Experimental confirmation of this is the work related to the research and development of laser frequency standards (\displaystyle 2\pi r)/…. From the results of these studies it follows that all photons for which λ < 3,39 10 -6 m moving faster than the speed of light. The limiting speed of photons (gamma range) is the second sound speed of the ether 3 10 8 m/s (\displaystyle 2\pi r)/….
These studies allow us to draw another significant conclusion that the change in the speed of photons in the region of their existence does not exceed ≈ 0.1%. Such a relatively small change in the speed of photons in the region of their existence allows us to speak of the speed of photons as a quasi-constant value.
A photon is an elementary particle whose integral properties are mass and electric charge. Ehrenhaft's experiments proved that the electric charge of a photon (subelectron) has a continuous spectrum, and from Millikan's experiments it follows that for a photon in the X-ray range, with a wavelength of approximately 10 -9 m, the value of the electric charge is 0.80108831 C (\displaystyle 2\pi r )/….
According to the first materialized definition of the physical essence of electric charge: “ the elementary electric charge is proportional to the mass distributed over the cross section of the elementary vortex“The opposite statement follows that the mass distributed over the cross section of the vortex is proportional to the electric charge. Based on the physical essence of the electric charge, it follows that the photon mass also has a continuous spectrum. Based on the structural similarity of the elementary particles of the proton, electron and photon, the value of the mass and radius of the proton (respectively, m p = 1.672621637(83) 10 -27 kg, rp = 0.8751 10 -15 m (\displaystyle 2\pi r)/…), and also assuming equality of ether density in these particles, the photon mass is estimated at 10 -40 kg, and its circular orbit radius is 0.179◦10 −16 m, The radius of the photon body (the outer radius of the torus) is supposed to be in the range of 0.01 – 0.001 of the radius of the circular orbit, i.e. on the order of 10 -19 – 10 -20 m.
Based on the concepts of photon multiplicity and the dependence of photon parameters on wavelength, as well as from experimentally confirmed facts of the continuity of the spectrum of electric charge and mass, we can assume that e λ , m λ = f ( λ ) , which are quasi-constant.
Based on the above, we can say that expression (1) establishing the relationship between the energy of any system when emitting or absorbing electromagnetic radiation with a frequency ν (\displaystyle ~\nu ) is nothing more than the relationship between the energy of photons emitted or absorbed by a body and the frequency (wavelength) of these photons. And Planck's constant is the coupling coefficient. This representation of the relationship between the photon energy and its frequency removes from the Planck constant the importance of its universality and fundamental nature. In this context, Planck's constant becomes one of the photon parameters, depending on the photon wavelength.
To fully and sufficiently prove this statement, let us consider the energy aspect of the photon. From experimental data it is known that a photon is characterized by an energy spectrum that has a nonlinear dependence: for photons in the infrared range E λ = 0.62 eV for λ = 2 10 -6 m, x-ray E λ = 124 eV for λ = 10 -8 m, gamma range E λ = 124000 eV for λ = 10 -11 m. From the nature of the photon’s motion it follows that the total energy of the photon consists of the kinetic energy of rotation around its own axis, the kinetic energy of rotation along a circular path (cycloid) and the energy of rectilinear motion:
E λ = E 0 λ + E 1 λ+E 2 λ, (3)
where E 0 λ = m λ r 2 γ λ ω 2 γ λ is the kinetic energy of rotation around its own axis,
E 1 λ = m λ u λ 2 is the energy of rectilinear motion, E 2 λ = m λ R 2 λ ω 2 λ is the kinetic energy of rotation along a circular path, where r γ λ is the radius of the photon body, R γ λ is the radius of the circular path , ω γ λ – natural frequency of photon rotation around the axis, ω λ = ν is the circular frequency of rotation of the photon, m λ is the mass of the photon.
Kinetic energy of photon motion in a circular orbit
E 2 λ = m λ r 2 λ ω 2 λ = m λ r 2 λ (2π u λ / λ) 2 = m λ u λ 2 ◦ (2π r λ / λ) 2 = E 1 λ ◦ (2π r λ / λ) 2 .
E 2 λ = E 1 λ ◦ (2π r λ / λ) 2 . (4)
Expression (4) shows that the kinetic energy of rotation along a circular path is part of the energy of rectilinear motion, depending on the radius of the circular path and the wavelength of the photon
(2π r λ / λ) 2 . (5)
Let's estimate this value. For infrared photons
(2π r λ / λ) 2 = (2π 10 -19 m /2 10 -6 m) 2 = π 10 -13.
For gamma-ray photons
(2π r λ / λ) 2 = (2π 10 -19 m /2 10 -11 m) 2 = π 10 -8.
Thus, in the entire region of existence of a photon, its kinetic energy of rotation along a circular path is significantly less than the energy of rectilinear motion and can be neglected.
Let us estimate the energy of rectilinear motion.
E 1 λ = m λ u λ 2 = 10 -40 kg (3 10 8 m/s) 2 =0.9 10 -23 kg m 2 /s 2 = 5.61 10 -5 eV.
The energy of rectilinear motion of a photon in the energy balance (3) is significantly less than the total photon energy, for example, in the infrared region (5.61 10 -5 eV< 0,62 эВ), что указывает на то, что полная энергия фотона фактически определяется собственной кинетической энергией вращения вокруг оси фотона.
Thus, due to the smallness of the energies of rectilinear motion and motion along a circular path, we can say that The energy spectrum of a photon consists of the spectrum of its own kinetic energies of rotation around the photon axis.
Therefore, expression (1) can be represented as
E 0 λ = hν ,
i.e.(\displaystyle ~E=h\nu )
m λ r 2 γ λ ω 2 γ λ = h ν . (6)
h = m λ r 2 γ λ ω 2 γ λ / ν = m λ r 2 γ λ ω 2 γ λ / ω λ . (7)
Expression (7) can be represented as follows
h = m λ r 2 γ λ ω 2 γ λ / ω λ = (m λ r 2 γ λ) ω 2 γ λ / ω λ = k λ (λ) ω 2 γ λ / ω λ .
h = k λ (λ) ω 2 γ λ / ω λ . (8)
Where k λ (λ) = m λ r 2 γ λ is some quasi-constant.
Let us estimate the values of the natural frequencies of photon rotation around the axis: for example,
For λ = 2 10 -6 m (infrared range)
ω 2 γ i = E 0i / m i r 2 γ i = 0.62 · 1.602 · 10 −19 J / (10 -40 kg 10 -38 m 2) = 0.99 1059 s -2,
ω γ i = 3.14 10 29 r/s.
For λ = 10 -11 m (gamma band)
ω γ i = 1.4 10 32 r/s.
Let us estimate the ratio ω 2 γ λ / ω λ for photons in the infrared and gamma ranges. After substituting the above data we get:
For λ = 2 10 -6 m (infrared range) - ω 2 γ λ / ω λ = 6.607 10 44,
For λ = 10 -11 m (gamma range) - ω 2 γ λ / ω λ = 6.653 10 44.
That is, expression (8) shows that the ratio of the square of the frequency of the photon’s own rotation to rotation along a circular path is a quasi-constant value for the entire region of existence of photons. In this case, the value of the frequency of the photon’s own rotation in the region of the photon’s existence changes by three orders of magnitude. From which it follows that Planck’s constant is quasi-constant.
Let us transform expression (6) as follows
m λ r 2 γ λ ω γ λ ω γ λ = h ω λ .
M =h ω λ / ω γ λ , (9)
where M = m λ r 2 γ λ ω γ λ is the photon’s own gyroscopic moment.
From expression (9) follows the physical essence of Planck’s constant: Planck’s constant is a proportionality coefficient that establishes the relationship between the photon’s own gyroscopic moment and the ratio of rotational frequencies (along a circular path and its own), which has the character of a quasi-constant throughout the entire region of the photon’s existence.
Let us transform expression (7) as follows
h = m λ r 2 γ λ ω 2 γ λ / ω λ = m λ r 2 γ λ m λ r 2 γ λ R 2 λ ω 2 γ λ / (m λ r 2 γ λ R 2 λ ω λ) =
= (m λ r 2 γ λ ω γ λ) 2 R 2 λ / (m λ R 2 λ ω λ r 2 γ λ) =M 2 γ λ R 2 λ / M λ r 2 γ λ ,
h = (M 2 γ λ / M λ) (R 2 λ / r 2 γ λ),
h ( r 2 γ λ /R 2 λ), = (M 2 γ λ / M λ) (10)
Expression (10) also shows that the ratio of the square of the photon’s own gyroscopic moment to the gyroscopic moment of motion along a circular path (cycloid) is a quasi-constant value throughout the entire region of existence of the photon and is determined by the expression h ( r 2 γ λ /R 2 λ).
Light is a form of radiant energy that travels through space as electromagnetic waves. In 1900, scientist Max Planck, one of the founders of quantum mechanics, proposed a theory according to which radiant energy is emitted and absorbed not in a continuous wave flow, but in separate portions, which are called quanta (photons).
The energy transferred by one quantum is equal to: E = hv, Where v is the radiation frequency, and h – elementary quantum of action, representing a new universal constant, which soon received the name Planck's constant(according to modern data h = 6.626 × 10 –34 J s).
In 1913, Niels Bohr created a coherent, albeit simplified model of the atom, consistent with the Planck distribution. Bohr proposed a theory of radiation based on the following postulates:
1. There are stationary states in an atom, in which the atom does not emit energy. Stationary states of an atom correspond to stationary orbits along which electrons move;
2. When an electron moves from one stationary orbit to another (from one stationary state to another), a quantum of energy is emitted or absorbed hν = |E i – E n| , Where ν – frequency of the emitted quantum, E i – the energy of the state from which it passes, and E n– the energy of the state into which the electron goes.
If an electron, under any influence, moves from an orbit close to the nucleus to some other more distant one, then the energy of the atom increases, but that requires the expenditure of external energy. But such an excited state of the atom is unstable and the electron falls back towards the nucleus into a closer possible orbit.
And when an electron jumps (falls) into an orbit that lies closer to the nucleus of an atom, the energy lost by the atom turns into one quantum of radiant energy emitted by the atom.
Accordingly, any atom can emit a wide spectrum of interconnected discrete frequencies, which depends on the orbits of the electrons in the atom.
A hydrogen atom consists of a proton and an electron moving around it. If an electron absorbs a portion of energy, the atom goes into an excited state. If an electron gives up energy, then the atom moves from a higher to a lower energy state. Typically, transitions from a higher energy state to a lower energy state are accompanied by the emission of energy in the form of light. However, non-radiative transitions are also possible. In this case, the atom goes into a lower energy state without emitting light, and gives up excess energy, for example, to another atom when they collide.
If an atom, moving from one energy state to another, emits a spectral line with wavelength λ, then, in accordance with Bohr’s second postulate, energy is emitted E equal to: , where h- Planck's constant; c- speed of light.
The set of all spectral lines that an atom can emit is called its emission spectrum.
As quantum mechanics shows, the spectrum of the hydrogen atom is expressed by the formula:
, Where R– constant, called the Rydberg constant; n 1 and n 2 numbers, and n 1
< n 2 .
Each spectral line is characterized by a pair of quantum numbers n 2 and n 1 . They indicate the energy levels of the atom before and after radiation, respectively.
When electrons move from excited energy levels to the first ( n 1 = 1; respectively n 2 = 2, 3, 4, 5...) is formed Lyman series.All lines of the Lyman series are in ultraviolet range.
Transitions of electrons from excited energy levels to the second level ( n 1 = 2; respectively n 2 = 3,4,5,6,7...) form Balmer series. The first four lines (that is, for n 2 = 3, 4, 5, 6) are in the visible spectrum, the rest (that is, for n 2 = 7, 8, 9) in ultraviolet.
That is, visible spectral lines of this series are obtained if the electron jumps to the second level (second orbit): red - from the 3rd orbit, green - from the 4th orbit, blue - from the 5th orbit, violet - from the 6th orbit oh orbits.
Transitions of electrons from excited energy levels to the third ( n 1 = 3; respectively n 2 = 4, 5, 6, 7...) form Paschen series. All lines of the Paschen series are located in infrared range.
Transitions of electrons from excited energy levels to the fourth ( n 1 = 4; respectively n 2 = 6, 7, 8...) form Brackett series. All lines in the series are in the far infrared range.
Also in the spectral series of hydrogen, the Pfund and Humphrey series are distinguished.
By observing the line spectrum of a hydrogen atom in the visible region (Balmer series) and measuring the wavelength λ of the spectral lines of this series, one can determine Planck's constant.
In the SI system, the calculation formula for finding Planck’s constant when performing laboratory work will take the form:
,
Where n 1 = 2 (Balmer series); n 2 = 3, 4, 5, 6.
=
3.2 × 10 -93
λ – wavelength ( nm)
Planck's constant appears in all equations and formulas of quantum mechanics. In particular, it determines the scale from which it comes into force Heisenberg uncertainty principle. Roughly speaking, Planck's constant shows us the lower limit of spatial quantities beyond which quantum effects cannot be ignored. For grains of sand, say, the uncertainty in the product of their linear size and speed is so insignificant that it can be neglected. In other words, Planck’s constant draws the boundary between the macrocosm, where Newton’s laws of mechanics apply, and the microcosm, where the laws of quantum mechanics come into force. Having been obtained only for a theoretical description of a single physical phenomenon, Planck’s constant soon became one of the fundamental constants of theoretical physics, determined by the very nature of the universe.
The work can be performed either on a laboratory installation or on a computer.
Material from the free Russian encyclopedia “Tradition”
|
Values h |
Units |
|
6,626 070 040(81) 10 −34 |
J∙c |
|
4,135 667 662(25) 10 −15 |
eV∙c |
|
6,626 070 040(81) 10 −27 |
erg∙c |
Planck's constant , denoted as h, is a physical constant used to describe the magnitude of the quantum of action in quantum mechanics. This constant first appeared in M. Planck’s works on thermal radiation, and is therefore named after him. It is present as a coefficient between energy E and frequency ν photon in Planck's formula:
Speed of light c related to frequency ν and wavelength λ ratio:
Taking this into account, Planck’s relation is written as follows:
The value is often used
J c,
Erg c,
EV c,
called the reduced (or rationalized) Planck constant or.
The Dirac constant is convenient to use when angular frequency is used ω , measured in radians per second, instead of the usual frequency ν , measured by the number of cycles per second. Because ω = 2π ν , then the formula is valid:
According to Planck's hypothesis, which was later confirmed, the energy of atomic states is quantized. This leads to the fact that the heated substance emits electromagnetic quanta or photons of certain frequencies, the spectrum of which depends on the chemical composition of the substance.
In Unicode, Planck's constant is U+210E (h), and Dirac's constant is U+210F (ħ).
Content
|
Magnitude
Planck's constant has the dimension of energy times time, just like the dimension of action. In the international SI system of units, Planck's constant is expressed in units of J s. The product of impulse and distance in the form N m s, as well as angular momentum, has the same dimension.
The value of Planck's constant is:
J s eV s.
The two digits between the brackets indicate the uncertainty in the last two digits of the value of Planck's constant (data are updated approximately every 4 years).
Origin of Planck's constant
Black body radiation
Main article: Planck's formula
At the end of the 19th century, Planck investigated the problem of black body radiation, which Kirchhoff had formulated 40 years earlier. Heated bodies glow the more strongly, the higher their temperature and the greater the internal thermal energy. Heat is distributed among all the atoms of the body, causing them to move relative to each other and to excite the electrons in the atoms. As electrons transition to stable states, photons are emitted, which can be reabsorbed by atoms. At each temperature, a state of equilibrium between radiation and matter is possible, and the share of radiation energy in the total energy of the system depends on temperature. In a state of equilibrium with radiation, an absolutely black body not only absorbs all the radiation incident on it, but also emits the same amount of energy, according to a certain law of energy distribution over frequencies. The law relating body temperature to the power of total radiated energy per unit surface area of the body is called the Stefan-Boltzmann law and was established in 1879–1884.
When heated, not only does the total amount of emitted energy increase, but the composition of the radiation also changes. This can be seen by the fact that the color of heated bodies changes. According to Wien's displacement law of 1893, based on the principle of adiabatic invariant, for each temperature it is possible to calculate the wavelength of radiation at which the body glows most strongly. Wien made a fairly accurate estimate of the shape of the black body energy spectrum at high frequencies, but was unable to explain either the shape of the spectrum or its behavior at low frequencies.
Planck proposed that the behavior of light is similar to the motion of a set of many identical harmonic oscillators. He studied the change in entropy of these oscillators depending on temperature, trying to substantiate Wien's law, and found a suitable mathematical function for the black body spectrum.
However, Planck soon realized that in addition to his solution, others were possible, leading to other values of the entropy of the oscillators. As a result, he was forced to use statistical physics, which he had previously rejected, instead of a phenomenological approach, which he described as “an act of desperation ... I was ready to sacrifice any previous beliefs in physics.” One of Planck's new conditions was:
interpret U N ( vibration energy of N oscillators ) not as a continuous infinitely divisible quantity, but as a discrete quantity consisting of a sum of limited equal parts. Let us denote each such part in the form of an energy element by ε;
With this new condition, Planck actually introduced the quantization of oscillator energy, saying that it was “a purely formal assumption... I haven’t really thought about it deeply...”, but it led to a real revolution in physics. Application of a new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator. This was the first version of what is now called "Planck's formula":
Planck was able to calculate the value h from experimental data on black body radiation: its result was 6.55 10 −34 J s, with an accuracy of 1.2% of the currently accepted value. He was also able to determine for the first time k B from the same data and his theory.
Before Planck's theory, it was assumed that the energy of a body could be anything, being a continuous function. This is equivalent to the fact that the energy element ε (the difference between allowed energy levels) is zero, therefore must be zero and h. Based on this, one should understand the statements that “Planck’s constant is equal to zero in classical physics” or that “classical physics is the limit of quantum mechanics when Planck’s constant tends to zero.” Due to the smallness of Planck's constant, it almost does not appear in ordinary human experience and was invisible before Planck's work.
The black body problem was revised in 1905, when Rayleigh and Jeans on the one hand, and Einstein on the other, independently proved that classical electrodynamics could not justify the observed radiation spectrum. This led to the so-called "ultraviolet catastrophe", so designated by Ehrenfest in 1911. The efforts of theorists (together with Einstein's work on the photoelectric effect) led to the recognition that Planck's postulate about the quantization of energy levels was not a simple mathematical formalism, but an important element of understanding about physical reality. The first Solvay Congress in 1911 was dedicated to the “theory of radiation and quanta.” Max Planck received the Nobel Prize in Physics in 1918 “for recognition of his services to the development of physics and the discovery of the energy quantum.”
Photo effect
Main article: Photo effect
The photoelectric effect involves the emission of electrons (called photoelectrons) from a surface when light is illuminated. It was first observed by Becquerel in 1839, although it is usually mentioned by Heinrich Hertz, who published an extensive study on the subject in 1887. Stoletov in 1888–1890 made several discoveries in the field of the photoelectric effect, including the first law of the external photoelectric effect. Another important study of the photoelectric effect was published by Lenard in 1902. Although Einstein did not conduct experiments on the photoelectric effect himself, his 1905 work examined the effect based on light quanta. This earned Einstein a Nobel Prize in 1921 when his predictions were confirmed by Millikan's experimental work. At this time, Einstein's theory of the photoelectric effect was considered more significant than his theory of relativity.
Before Einstein's work, each electromagnetic radiation was considered as a set of waves with their own "frequency" and "wavelength". The energy transferred by a wave per unit time is called intensity. Other types of waves, such as a sound wave or a water wave, have similar parameters. However, the transfer of energy associated with the photoelectric effect is not consistent with the wave pattern of light.
The kinetic energy of photoelectrons appearing in the photoelectric effect can be measured. It turns out that it does not depend on the light intensity, but depends linearly on frequency. In this case, an increase in light intensity does not lead to an increase in the kinetic energy of photoelectrons, but to an increase in their number. If the frequency is too low and the kinetic energy of photoelectrons is about zero, then the photoelectric effect disappears, despite the significant intensity of light.
According to Einstein's explanation, these observations reveal the quantum nature of light; Light energy is transferred in small "packets" or quanta, rather than as a continuous wave. The magnitude of these "packets" of energy, which were later called photons, was the same as those of Planck's "elements of energy". This led to the modern form of Planck's formula for photon energy:
Einstein's postulate was proven experimentally: the constant of proportionality between the frequency of light ν and photon energy E turned out to be equal to Planck's constant h.
Atomic structure
Main article: Bohr's postulates
Niels Bohr presented the first quantum model of the atom in 1913, trying to get rid of the difficulties of Rutherford's classical model of the atom. According to classical electrodynamics, a point charge, when rotating around a stationary center, should radiate electromagnetic energy. If such a picture is true for an electron in an atom as it rotates around the nucleus, then over time the electron will lose energy and fall onto the nucleus. To overcome this paradox, Bohr proposed to consider, similarly to what is the case with photons, that the electron in a hydrogen-like atom should have quantized energies E n:
Where R∞ is an experimentally determined constant (Rydberg constant in units of reciprocal length), With– speed of light, n– integer ( n = 1, 2, 3, …), Z– the serial number of a chemical element in the periodic table, equal to one for the hydrogen atom. An electron that reaches the lower energy level ( n= 1), is in the ground state of the atom and can no longer, due to reasons not yet defined in quantum mechanics, reduce its energy. This approach allowed Bohr to arrive at the Rydberg formula, which empirically describes the emission spectrum of the hydrogen atom, and to calculate the value of the Rydberg constant R∞ through other fundamental constants.
Bohr also introduced the quantity h/2π
, known as the reduced Planck constant or ħ, as the quantum of angular momentum. Bohr assumed that ħ determines the angular momentum of each electron in an atom. But this turned out to be inaccurate, despite improvements to Bohr's theory by Sommerfeld and others. The quantum theory turned out to be more correct, in the form of Heisenberg’s matrix mechanics in 1925 and in the form of the Schrödinger equation in 1926. At the same time, the Dirac constant remained the fundamental quantum of angular momentum. If J is the total angular momentum of the system with rotational invariance, and Jz is the angular momentum measured along the selected direction, then these quantities can only have the following values: 
The Uncertainty Principle
Planck's constant is also contained in the expression for Werner Heisenberg's uncertainty principle. If we take a large number of particles in the same state, then the uncertainty in their position is Δ x, and the uncertainty in their momentum (in the same direction), Δ p, obey the relation:
where uncertainty is specified as the standard deviation of the measured value from its mathematical expectation. There are other similar pairs of physical quantities for which the uncertainty relation is valid.
In quantum mechanics, Planck's constant appears in the expression for the commutator between the position operator and the momentum operator:
where δ ij is the Kronecker symbol.
Bremsstrahlung X-ray spectrum
When electrons interact with the electrostatic field of atomic nuclei, bremsstrahlung radiation appears in the form of X-ray quanta. It is known that the frequency spectrum of bremsstrahlung X-rays has a precise upper limit, called the violet limit. Its existence follows from the quantum properties of electromagnetic radiation and the law of conservation of energy. Really,
where is the speed of light,
– wavelength of X-ray radiation,
– electron charge,
– accelerating voltage between the electrodes of the X-ray tube.
Then Planck's constant will be equal to:
Physical constants related to Planck's constant
The list of constants below is based on 2014 data CODATA. . Approximately 90% of the uncertainty in these constants is due to uncertainty in the determination of Planck's constant, as can be seen from the square of the Pearson correlation coefficient ( r 2 >
0,99, r> 0.995). Compared with other constants, Planck's constant is known to an accuracy of the order of 
with measurement uncertainty 1 σ
.This accuracy is significantly better than that of the universal gas constant.
Electron rest mass
Typically, the Rydberg constant R∞ (in reciprocal length units) is determined in terms of mass m e and other physical constants:
The Rydberg constant can be determined very precisely ( 
) from the spectrum of a hydrogen atom, while there is no direct way of measuring the electron mass. Therefore, to determine the mass of an electron, the formula is used:
Where c is the speed of light and α
There is . The speed of light is determined quite accurately in SI units, as is the fine structure constant ( 
). Therefore, the inaccuracy in determining the electron mass depends only on the inaccuracy of Planck’s constant ( r 2 >
0,999).
Avogadro's constant
Main article: Avogadro's number
Avogadro's number N A is defined as the ratio of the mass of one mole of electrons to the mass of one electron. To find it, you need to take the mass of one mole of electrons in the form of the “relative atomic mass” of the electron A r(e), measured in Penning trap
(
), multiplied by unit molar mass M u, which in turn is defined as 0.001 kg/mol. The result is:
Dependence of Avogadro's number on Planck's constant ( r 2 > 0.999) is repeated for other constants related to the amount of matter, for example, for the atomic mass unit. Uncertainty in the value of Planck's constant limits the values of atomic masses and particles in SI units, that is, in kilograms. At the same time, the particle mass ratios are known with better accuracy.
Elementary charge
Sommerfeld originally determined the fine structure constant α So:
Where e there is an elementary electric charge, ε 0 – (also called dielectric constant of vacuum), μ 0 – magnetic constant or magnetic permeability of vacuum. The last two constants have fixed values in the SI system of units. Meaning α can be determined experimentally by measuring the g-factor of the electron g e and subsequent comparison with the value resulting from quantum electrodynamics.
Currently, the most accurate value of the elementary electric charge is obtained from the above formula:
Bohr magneton and nuclear magneton
Main articles: Bohr magneton , Nuclear magneton
The Bohr magneton and nuclear magneton are units used to describe the magnetic properties of the electron and atomic nuclei, respectively. The Bohr magneton is the magnetic moment that would be expected for an electron if it behaved like a rotating charged particle according to classical electrodynamics. Its value is derived through the Dirac constant, the elementary electric charge and the mass of the electron. All these quantities are derived through Planck’s constant, the resulting dependence on h ½ ( r 2 > 0.995) can be found using the formula:
A nuclear magneton has a similar definition, with the difference that the proton is much more massive than the electron. The ratio of electron relative atomic mass to proton relative atomic mass can be determined with great accuracy ( 
). For the connection between both magnetons, we can write:
Determination from experiments
|
Method |
Meaning h, |
Accuracy |
|
|
Power balance |
6,626 068 89(23) |
3,4∙10 –8 |
|
|
X-ray crystal density |
6,626 074 5(19) |
2,9∙10 –7 |
|
|
Josephson constant |
6,626 067 8(27) |
4,1∙10 –7 |
|
|
Magnetic resonance |
6,626 072 4(57) |
8,6∙10 –7 |
[ 20 ] |
|
Faraday's constant |
6,626 065 7(88) |
1,3∙10 –6 |
|
|
CODATA 20
10
|
6,626 06 9 57 (29 ) |
4 , 4 ∙10 –8 |
[ 22 ] |
|
Nine recent measurements of Planck's constant are listed for five different methods. If there is more than one measurement, the weighted average is indicated h according to the CODATA method. |
|||
Planck's constant can be determined from the spectrum of a radiating black body or the kinetic energy of photoelectrons, as was done in the early twentieth century. However, these methods are not the most accurate. Meaning h according to CODATA based on the basis of three measurements by the power balance method of the product of quantities K J2 R K and one interlaboratory measurement of the molar volume of silicon, mainly by the power balance method until 2007 in the USA at the National Institute of Standards and Technology (NIST). Other measurements listed in the table did not affect the result due to lack of accuracy.
There are both practical and theoretical difficulties in determining h. Thus, the most accurate methods for balancing the power and X-ray density of a crystal do not fully agree with each other in their results. This may be a consequence of the overestimation of accuracy in these methods. Theoretical difficulties arise from the fact that all methods, except for X-ray crystal density, are based on the theoretical basis of the Josephson effect and the quantum Hall effect. With some possible inaccuracy of these theories, there will also be an inaccuracy in determining Planck's constant. In this case, the obtained value of Planck’s constant can no longer be used as a test to test these theories in order to avoid a vicious logical circle. The good news is that there are independent statistical ways to test these theories.
Josephson constant
Main article: Josephson effect
Josephson constant K J relates the potential difference U, arising in the Josephson effect in "Josephson contacts", with a frequency ν microwave radiation. The theory quite strictly follows the expression:
The Josephson constant can be measured by comparison with the potential difference across a bank of Josephson contacts. To measure the potential difference, compensation of the electrostatic force by the force of gravity is used. From the theory it follows that after replacing the electric charge e to its value through fundamental constants (see above Elementary charge ), expression for Planck's constant through K J:
Power balance
This method compares two types of power, one of which is measured in SI units in watts, and the other is measured in conventional electrical units. From the definition conditional watt W 90, it gives the measure for the product K J2 R K in SI units, where R K is the Klitzing constant, which appears in the quantum Hall effect. If the theoretical interpretation of the Josephson effect and the quantum Hall effect is correct, then R K= h/e 2, and measurement K J2 R K leads to the definition of Planck's constant:
Magnetic resonance
Main article: Gyromagnetic ratio
Gyromagnetic ratio γ is the proportionality coefficient between frequency ν nuclear magnetic resonance (or electron paramagnetic resonance for electrons), and an applied magnetic field B: ν = γB. Although there is difficulty in determining the gyromagnetic ratio due to measurement inaccuracy B, for protons in water at 25 °C it is known with better accuracy than 10 –6. Protons are partially “screened” from the applied magnetic field by the electrons of water molecules. The same effect leads to chemical shift in nuclear magnetic spectroscopy, and is indicated by a prime next to the gyromagnetic ratio symbol, γ′ p. The gyromagnetic ratio is related to the magnetic moment of the shielded proton μ′ p, spin quantum number S (S=1/2 for protons) and the Dirac constant:
Screened proton magnetic moment ratio μ′ p to the magnetic moment of the electron μ e can be measured independently with high accuracy, since the inaccuracy of the magnetic field has little effect on the result. Meaning μ e, expressed in Bohr magnetons, is equal to half the electron g-factor g e. Hence,
Further complication arises from the fact that to measure γ′ p measurement of electric current is required. This current is independently measured in conditional amperes, so a conversion factor is required to convert to SI amperes. Symbol Γ′ p-90 denotes the measured gyromagnetic ratio in conventional electrical units (the permitted use of these units began in early 1990). This quantity can be measured in two ways, the “weak field” method and the “strong field” method, and the conversion factor in these cases is different. Typically, the high field method is used to measure Planck's constant and the value Γ′ p-90(hi):
After the replacement, we obtain an expression for Planck’s constant through Γ′ p-90(hi):
Faraday's constant
Main article: Faraday's constant
Faraday's constant F is the charge of one mole of electrons equal to Avogadro's number N A multiplied by the elementary electric charge e. It can be determined by careful electrolysis experiments, by measuring the amount of silver transferred from one electrode to another in a given time at a given electric current. In practice, it is measured in conventional electrical units, and is designated F 90. Substituting values N A and e, and moving from conventional electrical units to SI units, we obtain the relation for Planck’s constant:
X-ray crystal density
The X-ray crystal density method is the main method for measuring Avogadro's constant N A, and through it Planck’s constant h. To find N A is the ratio between the volume of the unit cell of a crystal, measured by X-ray diffraction analysis, and the molar volume of the substance. Silicon crystals are used because they are available in high quality and purity thanks to technology developed in semiconductor manufacturing. The unit cell volume is calculated from the space between two crystal planes, denoted d 220. Molar volume V m(Si) is calculated through the density of the crystal and the atomic weight of the silicon used. Planck's constant is given by:
Planck's constant in SI units
Main article: Kilogram
As stated above, the numerical value of Planck's constant depends on the system of units used. Its value in the SI system of units is known with an accuracy of 1.2∙10 –8, although it is determined in atomic (quantum) units exactly(in atomic units, by choosing the units of energy and time, it is possible to ensure that the Dirac constant as a reduced Planck constant is equal to 1). The same situation occurs in conventional electrical units, where Planck’s constant (written h 90 in contrast to the designation in SI) is given by the expression:
Where K J–90 and R K–90 are precisely defined constants. Atomic units and conventional electrical units are convenient to use in the relevant fields, since the uncertainties in the final result depend only on the uncertainties of measurements, without requiring an additional and inaccurate conversion factor into the SI system.
There are a number of proposals to modernize the values of the existing system of basic SI units using fundamental physical constants. This has already been done for the meter, which is determined through a given value of the speed of light. A possible next unit for revision is the kilogram, whose value has been fixed since 1889 by the mass of a small cylinder of platinum-iridium alloy stored under three glass bells. There are about 80 copies of these mass standards, which are periodically compared with the international unit of mass. The accuracy of secondary standards varies over time through their use, down to values in the tens of micrograms. This roughly corresponds to the uncertainty in the determination of Planck's constant.
At the 24th General Conference on Weights and Measures on October 17-21, 2011, a resolution was unanimously adopted, in which, in particular, it was proposed that in a future revision of the International System of Units (SI) the SI units of measurement should be redefined so that Planck's constant would be equal to exactly 6.62606X 10 −34 J s, where X stands for one or more significant figures to be determined based on the best CODATA recommendations. . The same resolution proposed to determine in the same way the exact values of Avogadro's constant, and .
Planck's constant in the theory of infinite nesting of matter
Unlike atomism, the theory does not contain material objects—particles with minimal mass or size. Instead, it is assumed that matter is endlessly divisible into ever smaller structures, and at the same time the existence of many objects significantly larger in size than our Metagalaxy. In this case, matter is organized into separate levels according to mass and size, for which it arises, manifests itself and is realized.
Just like Boltzmann's constant and a number of other constants, Planck's constant reflects the properties inherent in the level of elementary particles (primarily nucleons and components that make up matter). On the one hand, Planck's constant relates the energy of photons and their frequency; on the other hand, it, up to a small numerical coefficient 2π, in the form ħ, specifies the unit of orbital momentum of an electron in an atom. This connection is not accidental, since when emitted from an atom, an electron reduces its orbital angular momentum, transferring it to the photon during the period of existence of the excited state. During one period of revolution of the electron cloud around the nucleus, the photon receives such a fraction of energy that corresponds to the fraction of angular momentum transferred by the electron. The average frequency of a photon is close to the frequency of rotation of the electron near the energy level where the electron goes during radiation, since the radiation power of the electron increases rapidly as it approaches the nucleus.
Mathematically it can be described as follows. The equation of rotational motion has the form:
Where K - moment of power, L – angular momentum. If we multiply this ratio by the increment in the rotation angle and take into account that there is a change in the electron rotation energy, and there is the angular frequency of the orbital rotation, then it will be:
In this ratio the energy dE can be interpreted as an increase in the energy of an emitted photon when its angular momentum increases by the amount dL . For the total photon energy E and the total angular momentum of the photon, the value ω should be understood as the average angular frequency of the photon.
In addition to correlating the properties of emitted photons and atomic electrons through angular momentum, atomic nuclei also have angular momentum expressed in units of ħ. It can therefore be assumed that Planck's constant describes the rotational motion of elementary particles (nucleons, nuclei and electrons, orbital motion of electrons in an atom), and the conversion of the energy of rotation and vibrations of charged particles into radiation energy. In addition, based on the idea of particle-wave dualism, in quantum mechanics all particles are assigned an accompanying material de Broglie wave. This wave is considered in the form of a wave of the amplitude of the probability of finding a particle at a particular point in space. As for photons, the Planck and Dirac constants in this case become proportionality coefficients for a quantum particle, entering the expressions for the particle momentum, for energy E and for action S :
constant bar, what is the constant bar equal to
Planck's constant(quantum of action) is the main constant of quantum theory, a coefficient that connects the energy value of a quantum of electromagnetic radiation with its frequency, as well as in general the value of the energy quantum of any linear oscillatory physical system with its frequency. Links energy and impulse to frequency and spatial frequency, actions to phase. Is a quantum of angular momentum. It was first mentioned by Planck in his work on thermal radiation, and therefore named after him. The usual designation is Latin. J s erg s. eV c.
The value often used is:
J s, erg s, eV s,
called the reduced (sometimes rationalized or reduced) Planck constant or Dirac constant. The use of this notation simplifies many formulas of quantum mechanics, since these formulas include the traditional Planck constant in the form divided by a constant.
At the 24th General Conference on Weights and Measures on October 17-21, 2011, a resolution was unanimously adopted, in which, in particular, it was proposed that in a future revision of the International System of Units (SI) the SI units of measurement should be redefined so that Planck's constant would be equal to exactly 6.62606X 10−34 J s, where X stands for one or more significant figures to be determined based on the best CODATA recommendations. The same resolution proposed to determine in the same way the Avogadro constant, the elementary charge and the Boltzmann constant as exact values.
- 1 Physical meaning
- 2 History of discovery
- 2.1 Planck's formula for thermal radiation
- 2.2 Photoelectric effect
- 2.3 Compton effect
- 3 Measurement methods
- 3.1 Use of the laws of the photoelectric effect
- 3.2 Analysis of the X-ray bremsstrahlung spectrum
- 4 Notes
- 5 Literature
- 6 Links
Physical meaning
In quantum mechanics, impulse has the physical meaning of a wave vector, energy - frequency, and action - wave phase, but traditionally (historically) mechanical quantities are measured in other units (kg m/s, J, J s) than the corresponding wave ones (m −1, s−1, dimensionless phase units). Planck's constant plays the role of a conversion factor (always the same) connecting these two systems of units - quantum and traditional:
(impulse) (energy) (action)
If the system of physical units had been formed after the advent of quantum mechanics and had been adapted to simplify the basic theoretical formulas, Planck's constant would probably simply have been made equal to one, or, in any case, to a more round number. In theoretical physics, the system of units c is very often used to simplify formulas, in it
.Planck's constant also has a simple evaluative role in delimiting the areas of applicability of classical and quantum physics: in comparison with the magnitude of the action or angular momentum characteristic of the system under consideration, or the product of a characteristic impulse by a characteristic size, or a characteristic energy by a characteristic time, it shows how applicable classical mechanics to this physical system. Namely, if is the action of the system, and is its angular momentum, then at or the behavior of the system is described with good accuracy by classical mechanics. These estimates are quite directly related to the Heisenberg uncertainty relations.
History of discovery
Planck's formula for thermal radiation
Main article: Planck's formulaPlanck's formula is an expression for the spectral power density of black body radiation, which was obtained by Max Planck for the equilibrium radiation density. Planck's formula was obtained after it became clear that the Rayleigh-Jeans formula satisfactorily describes radiation only in the long-wave region. In 1900, Planck proposed a formula with a constant (later called Planck's constant), which was in good agreement with experimental data. At the same time, Planck believed that this formula was just a successful mathematical trick, but had no physical meaning. That is, Planck did not assume that electromagnetic radiation is emitted in the form of individual portions of energy (quanta), the magnitude of which is related to the frequency of the radiation by the expression:
The proportionality coefficient was later called Planck's constant, = 1.054·10−34 J·s.
Photo effect
Main article: Photo effectThe photoelectric effect is the emission of electrons by a substance under the influence of light (and, generally speaking, any electromagnetic radiation). condensed substances (solid and liquid) produce external and internal photoelectric effects.
The photoelectric effect was explained in 1905 by Albert Einstein (for which he received the Nobel Prize in 1921, thanks to the nomination of the Swedish physicist Oseen) on the basis of Planck's hypothesis about the quantum nature of light. Einstein's work contained an important new hypothesis - if Planck suggested that light is emitted only in quantized portions, then Einstein already believed that light exists only in the form of quantized portions. From the law of conservation of energy, when representing light in the form of particles (photons), Einstein’s formula for the photoelectric effect follows:
where - so-called work function (the minimum energy required to remove an electron from a substance), - the kinetic energy of the emitted electron, - the frequency of the incident photon with energy, - Planck's constant. This formula implies the existence of the red limit of the photoelectric effect, that is, the existence of the lowest frequency below which the photon energy is no longer sufficient to “knock out” an electron from the body. The essence of the formula is that the energy of a photon is spent on ionizing an atom of a substance, that is, on the work necessary to “tear out” an electron, and the remainder is converted into the kinetic energy of the electron.
Compton effect
Main article: Compton effectMeasurement methods
Using the laws of the photoelectric effect
This method of measuring Planck's constant uses Einstein's law for the photoelectric effect:
where is the maximum kinetic energy of photoelectrons emitted from the cathode,
The frequency of the incident light, - the so-called. electron work function.
The measurement is carried out like this. First, the cathode of the photocell is irradiated with monochromatic light at a frequency, while a blocking voltage is applied to the photocell so that the current through the photocell stops. In this case, the following relationship takes place, which directly follows from Einstein’s law:
where is the electron charge.
Then the same photocell is irradiated with monochromatic light with a frequency and is similarly locked using voltage
Subtracting the second expression term by term from the first, we get
whence follows
Analysis of the X-ray bremsstrahlung spectrum
This method is considered the most accurate of the existing ones. It takes advantage of the fact that the frequency spectrum of bremsstrahlung X-rays has a precise upper limit, called the violet limit. Its existence follows from the quantum properties of electromagnetic radiation and the law of conservation of energy. Really,
where is the speed of light,
The wavelength of the X-ray radiation, - the charge of the electron, - the accelerating voltage between the electrodes of the X-ray tube.
Then Planck's constant is
Notes
- 1 2 3 4 Fundamental Physical Constants - Complete Listing
- On the possible future revision of the International System of Units, the SI. Resolution 1 of the 24th meeting of the CGPM (2011).
- Agreement to tie kilogram and friends to fundamentals - physics-math - 25 October 2011 - New Scientist
Literature
- John D. Barrow. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. - Pantheon Books, 2002. - ISBN 0-37-542221-8.
- Steiner R. History and progress on accurate measurements of the Planck constant // Reports on Progress in Physics. - 2013. - Vol. 76. - P. 016101.
Links
- Yu. K. Zemtsov, Course of lectures on atomic physics, dimensional analysis
- History of refinement of Planck's constant
- The NIST Reference on Constants, Units and Uncertainty
constant bar, what is the constant bar equal to