RU2327972C2 - Process of boltzmann constant calculation - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: invention involves simultaneous irradiation of at least two objects of identical chemical composition and physical properties, placed into thermostatic medium. Phonon mode frequency is varied by radiation angle change, energy densities of thermal phonons in high- and low-temperature objects are equated, and the sought Boltzmann constant is calculated.

EFFECT: lower fractional error of Boltzmann constant calculation and higher manufacturability of the process.

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Description

The invention relates to the field of measurement technology and can be used for precision measurements of absolute temperature in studies of the process of formation of intra-sea ice in sea water, for calculating the thermodynamic potentials characterizing the process of electrical conductivity of sea water, for clarifying the values of the reference points of the International practical temperature scale.

A known method for determining the Boltzmann constant (see A. Cook. Metrology and fundamental constants. Collection of articles "Quantum metrology and fundamental constants" / Edited by R.N. Faustov and V.P. Shelest. Translated from English by M. : World, 1981. P.106), based on the application of the law of the gas state, expressed by the equation:

where P, N / m ² is the gas pressure inside the vessel;

V, m ³ is the volume of gas;

T, K - absolute gas temperature;

R, J · mol ^-1 · K ^-1 is the universal gas constant.

The universal gas constant R is calculated from the measured values of P, V, and T. The sought value of the Boltzmann constant k is obtained by dividing the value of R by the Avogadro number N ₀ , i.e. k = R / N ₀ .

To implement the method, a gas thermometer is used (see, for example, E. Mason, T. Sperling. The virial equation of state. M .: Mir, 1972. P.83), which includes a glass vessel with a constant known volume V filled with gas , and a mercury pressure gauge connected to the vessel using a tube. The vessel is placed in a thermostat at a temperature of 273.16 K and the pressure P in it is measured with a manometer.

The disadvantages of the described method include the occurrence of measurement errors due to deviations in the behavior of the gas in which the measurements are made from ideal, as well as difficulties in measuring temperature in large volumes of gas.

There is also a method of determining the Boltzmann constant (see A.N. Matveev. Molecular Physics. M: Higher School, 1987. P.89), based on the calculation of the named constant from the Boltzmann equation, which describes the energy distribution of particles (atoms, molecules) .

To implement the method, a laboratory setup is used, which includes a glass cuvette with the test mixture, a microscope with a depth of field of 10 ^-6 m, an analytical balance for weighing particles, and a pycnometer for determining the density of a liquid. Measurements are carried out in a layer with a thickness of 10 ^-4 m.

To experimentally determine the Boltzmann constant by this method, balls (particles) with a diameter of approximately 10 ^-6 m made of gummigut tree resin are placed in a vessel with a liquid, measured by independent methods: total number of particles A in the system, particle density ρ, kg / m ³ ; the density of the liquid ρ _W , and the liquid is chosen such that its density is slightly lower than the density of the particles; volume ν, m ³ occupied by all particles located at a height of H in the liquid; the number of particles n _i located at a height of H and having energy m _i · g · H,

where m _i is the mass of particles located at a height of H;

g is the acceleration of gravity,

absolute temperature T, K of the test mixture, and according to the formula representing the Boltzmann distribution

where ν · (ρ-ρ _W ) = m _i ,

k = 1,380662 (44) · 10 ^-23 J · K - Boltzmann constant;

k T, J · K ^-1 - energy of thermal motion of particles,

the desired value of k is calculated.

The disadvantages of the described method include the complexity of the particle counting process, carried out visually using a microscope, as well as the low accuracy of determining the height of the layer in which the particles are counted (relative error of the order of 10 ^-2 ), introducing the predominant component into the measurement error.

There is a method of determining the Boltzmann constant (see T. Quin. Temperature. M .: Mir, 1985. P.27), based on the application of the Stefan-Boltzmann law, which describes the dependence of the total energy of thermal radiation E (T) emitted by a black body on its temperature T, mathematically expressed as:

where E (T), J is the total energy of thermal radiation emitted by the black body;

s, m / s - speed of light;

h = 6.626176 (38) · 10 ^-34 - Planck's constant;

T, K is the absolute temperature of the radiating body and connecting the named physical quantities with the Boltzmann constant.

Analysis of the information selected in the search process showed that the above analogue is the closest in technical essence and the achieved result to the claimed method and can be taken for its prototype.

To implement the method, a device is used that includes a source and a receiver of thermal radiation, a device for supplying a radiation source, an ammeter and a voltmeter for measuring the current and voltage of the source of radiation, a focusing lens, a photocurrent meter of the radiation receiver, calibrated in degrees Kelvin.

The method includes the following operations: supply power to a radiation source, which can be used, for example, a special incandescent lamp with a filament of tungsten; measure the voltage drop U on it and the current I through the mentioned thread, i.e. consumed electric power proportional to the total energy of thermal radiation emitted by the black body in accordance with the formula (4)

where S, mm ² - the known area of the radiating surface of the thread;

σ = 5.67032 (71) · 10 ^-8 W · m ^-2 · K ^-4 , J - Stefan-Boltzmann constant.

An optical pyrometer is used as a radiation receiver. On the pyrometer scale, graduated in degrees Kelvin, determine the value of the absolute temperature T of the radiating body. Using the formula (4), calculate the value of the total energy of thermal radiation and using the obtained values of T and E (T), use the formula (3) to calculate the Boltzmann constant k.

The disadvantage of the considered method is the need to take into account the geometric factor in formula (3) due to the inability to experimentally measure the radiation of a non-black body emitted by the hemisphere, which introduces a significant error in determining the Boltzmann constant.

The inventive method is based on the use of a phenomenon known in physics - scattering of photons (quanta of electromagnetic energy) by phonons (quanta of acoustic energy) or thermal fluctuations in the density of matter on the so-called Mandelstam-Brillouin scattering, and also on the use of the fact known in physics that the dependence of the energy density of both electromagnetic radiation and acoustic radiation on frequency obeys the Planck distribution law of the energy density of electromagnetic radiation over frequencies, having the form:

where E · dν, J / m ³ is the energy density of electromagnetic radiation;

ν, Hz - frequency of a quantum of electromagnetic radiation

and linking the energy density with the radiation frequency and the Boltzmann constant.

The inventive method allows to obtain a new technical result compared to the prototype, which consists in reducing the relative error in determining the Boltzmann constant and achieving high adaptability of the measurement process. The achieved result is due to the fact that the process of determining the named constant is associated with measuring the frequency that is most accurately measured at the present time and the technological physical quantity, which corresponds to one of the tasks of fundamental metrology - to connect measurements of fundamental constants with frequency measurement. As calculations showed, the relative error in determining the Boltzmann constant by the proposed method does not exceed 6 · 10 ^-7 , i.e. about two orders of magnitude lower than its most accurately measured value - 2.5 · 10 ^-5 .

The essence of the proposed method lies in the fact that in it, as well as in the prototype, physical quantities are measured that characterize the energy state of the substance of the object under study, connected with each other and with the Boltzmann constant by a mathematical equation, from which the sought value of the Boltzmann constant is determined.

But, unlike the prototype, in the proposed method:

simultaneously irradiate at least two studied objects, identical in chemical composition and physical properties, and place them in a thermostatic medium with a given constant temperature, and one object is placed in an environment with low cryogenic temperatures, and the other higher.

_{satisfy the} equality condition hν _sн = k _n T _n in the low-temperature object,

where hν _sn - phonon energy (quantum of acoustic energy);

k _n T _n - average kinetic energy of the thermal motion of phonons;

ν _s is the phonon frequency;

the index "n" means that the physical quantity belongs to a low-temperature object,

why in it the frequency ν _{SH of the} excited phonon mode is varied by changing the irradiation angle θ _{n of the} said object,

when the named equality is reached, the named frequency is measured with a spectrometer,

operate energy inequality condition for high temperature object to hν _sre << _{n T,} where the subscript "in" refers to the physical quantity belonging to high-temperature object, by varying the frequency ν _sre change _{in the} irradiation angle θ of said object,

when the above inequality is fulfilled, the frequency ν _{sv is} measured using a spectrometer,

establish equal energy densities of thermal phonons E _n · dν _SH = E · dν _{in SB} in high- and low-temperature objects,

under all the above conditions, by the formula below calculate the desired value of the Boltzmann constant

where Λ and Λ _{n in} - phonon oscillation wavelengths in the respective objects.

The invention is illustrated in the drawing,

where figure 1 shows a functional diagram of a device that implements the method,

figure 2 presents graphs of the dependence of the energy density of phonon noise on frequency.

The device (Fig. 1) contains: a laser 1, a translucent mirror 2, a mirror 3, a polarizer 4, test objects 5 and 7 made of acousto-optical material, for example, fused silica, a cryostat 6, thermostat 8, focusing lenses 9 and 10, a photomultiplier 11 and an electronic spectrometer 12.

The proposed method is carried out in the following sequence: measuring elements 5 and 7 are placed in a thermostatic medium at a constant but different temperature, for example, at the temperature of phase equilibria in the so-called reference points, which serve as reference temperature points of the International Practical Temperature Scale MGGTS - 68. The measuring object 5 is placed in a cryostat 6 at low - cryogenic temperature, for example, at a temperature of λ - ³ He helium point, i.e. at T ₅ = 2.17 K, and measuring element 7 to the thermostat 8 at a temperature of a triple point of water T ₇ = 273.16 K.

The monochromatic radiation of laser 1, for example, in the ultraviolet (UV) part of the optical spectrum, transmitted through objects 5 and 7 and scattered by the thermal phonons of their substance, is focused by lenses 9 and 10 onto a photomultiplier 11. The electric signal from the photomultiplier 11 is fed to an electronic spectrometer 12.

The fluctuation spectrum of the photocurrent of photomultiplier 11 contains difference frequencies ν _s5 and ν _s7 , which are the frequencies of thermal phonons in the material of the studied objects at temperatures T ₅ and T ₇ , respectively.

In object 5, a phonon mode is excited with a frequency ν _s5 satisfying the equality hν _s5 = k ₅ T ₅ (point 1 of FIG. 2), where hν _s5 is the phonon energy (quantum of acoustic energy); k ₅ T ₅ - the average kinetic energy of the thermal motion of phonons. The specified frequency is set by changing the angle 5 of the irradiation of element 5 when the laser 1 is rotated relative to the irradiation point O ₁ .

The value of k _{5 is} taken equal to the known, most accurately measured, value - 1380662 · 10 ^-23 J · K ^-1 (see Quantum metrology and fundamental constants / Ed. By the doctor of physical and mathematical sciences R.N. Faustov - and Corresponding Member of the Academy of Sciences of the USSR V.P. Shelest.Lane from English.M .: Mir, 1981. P.401).

In element 7, one of the phonon modes is excited with a frequency ν _s7 satisfying the inequality hν _s7 << k ₅ T ₇ . In this example, this condition is satisfied by all frequencies ν _s - the values of the argument of the function E _s · dν _s = f (ν _s ) represented by curve I in FIG. 2

Under the two above-mentioned conditions, the energy densities of thermal phonons are equal in objects 5 and 7 E _s5 · dν _s5 = E _s7 · dν _s7 .

where E _s5 and Es ₇ , J - kinetic energy of phonons in objects 5 and 7;

dν _s5 and dν _s7 , Hz is the phonon frequency band in objects 5 and 7, for which the frequency ν _{s7 is} varied by changing the irradiation angle θ _{7 of} object 7, and to change the named angle, mirror 3 is moved along an arc and / or rotated around the normal axis to the plane of the figure (figure 1).

The achievement of the above equality is controlled by the fulfillment of the relation

ν _s7 / ν _s5 = I _s7 · Λ ₅ / ΛI _s5 · Λ ₇ ,

where I _s5 , I _s7 and Λ ₅ , Λ ₇ are the intensities and wavelengths of phonon vibrations in objects 5 and 7, respectively,

by changing the intensity I _s5 and / or I _s7 when rotating polarizer 4.

The intensities I _s5 and I _s7 , the frequencies ν _s7 and ν _s5 and the wavelengths Λ ₅ and Λ ₇ of the phonon vibrations are measured by an electronic spectrometer 12. Under the three conditions: hν _s5 = k ₅ T ₅ , hν _s7 << k ₅ T ₇ and E _s5 · dν _s5 = E _s7 · dν _s7 solve the system of two equations with respect to k ₇ , which are the Planck equation, written as the frequency distribution of the phonon energy density in the studied objects 5 and 7

where E _s5 · dν _s5 and E _s7 · dν _s7 , J / m ³ is the energy density of thermal phonon noise;

C _s , m / s — phonon propagation velocity — sound velocity in measuring elements;

the factor "12" is due to the fact that three directions of polarization of the phonon are possible, then the necessary substitutions and transformations are made taking into account the relations:

_{C s = Λ Ω / 2π,} Ω2π · ν s and

where Ω, Hz is the angular frequency of the phonon;

Λ, m - wavelength of phonon vibrations;

λ _l, m - length of laser radiation;

Θ ° - the angle of exposure of the object,

and by the formula (6) calculate the desired value of the Boltzmann constant k ₇

Next, to obtain a more accurate value of the desired constant, the procedure is repeated, while the calculated value of the Boltzmann constant k _{7 is} used as a known one, for which:

establish for the object 5 the equality hν _s5 ⁽¹⁾ = k ₅ ⁽¹⁾ T ₅ ,

where k ₅ ⁽¹⁾ is equal to the calculated value of k ₇ ,

a ν _s5 ⁽¹⁾ is the new value of the phonon mode in the substance of object 5, satisfying this equality,

Then, a phonon mode with a frequency V _s7 ⁽¹⁾ satisfying the inequality hν _s7 ⁽¹⁾ << k ₇ T _{7 is} excited in object ₇ ,

establish the equality of the energy densities of thermal phonons in objects 5 and 7 under the two above conditions E _s5 ⁽¹⁾ · dν _s5 ⁽¹⁾ = E _s7 ⁽¹⁾ · dν _s7 ⁽¹⁾ ,

where E _s5 ⁽¹⁾ and E _s7 ⁽¹⁾ are the new values of the kinetic energy of thermal phonons corresponding to the value of the Boltzmann constant k ₇ ⁽¹⁾

and by formula (6), substituting the corresponding values

calculating a current value of the constant k Boltzmann ₇ ^(1).

To test the possibility of industrial applicability of the invention, the Boltzmann constant was determined by calculation. The calculation results make it possible to attribute the method to solutions that meet the specified criterion.

Example.

For the given example, the following initial data were taken: fused silica was chosen as the substance from which the studied objects 5 and 7 were made, a laser with a wavelength in the ultraviolet (UV) part of the optical spectrum λ _l = 2 · 10 was chosen as the source of photon radiation ^-7 m. In this case, the temperature of the thermostatic liquid T ₇ in thermostat 8 is taken equal to 273.16 K, and T ₅ in the cryostat 6 - 2.17 K.

For objects 5 and 7, according to formulas (7a) and (7b), the values of E _s · dν _s = f (ν _s , θ) are calculated for the energy density of the phonon frequency and the irradiation angle at E = Const. The calculation results are shown in table 1.

According to the table, the dependences Es · dνs = f (ν _s , θ) are constructed (Fig. 2), with the help of which the frequency ν _{s of the} phonon modes is determined depending on the irradiation angle at a constant temperature.

Next, the condition for the equality of energies hν _sн = k _n T _n in the object at low temperature is fulfilled. In the example, the position of point 1 is determined on curve II, for which, from the condition hν _s5 = k ₅ T ₅ , the frequency ν _{s5 of the} phonon mode is determined

which corresponds to the angle of exposure of the object, located between 50 and 60 degrees (table 1 lines 13 and 14). The value of the Boltzmann constant in this case is taken to be known, equal to the most accurately measured value of k ₅ = 1380662 · 10 ^-23 J · K ^-1 .

The inequality condition hν _sв << k _н T _{в is} fulfilled for an object located at a high temperature. In the example, at the maximum phonon frequency ν _s7 ≈20 · 10 ⁹ Hz, and therefore for all frequencies in the range from 0 to 20 · 10 ⁹ Hz (see Fig. 2, curve 1), the above inequality is fulfilled.

Valid: 6.626176 · 10 ^-34 << 1.38662 · 10 ^-23 · 273.16.

The equality of the energy densities of the thermal phonons E _s7 · dν _s7 = E _s5 · dν _{s5 is established} , for which the example determines the position of point 2 on curve I at which the above equality is fulfilled, and its projection onto the abscissa gives the value of the frequency of the phonon mode ν _s7 = 9.412 Hz. The mentioned point is at the intersection of curve 1 with a straight line parallel to the abscissa axis passing through point 1 on curve II.

The intersection point of the named line with the ordinate axis gives the value of the energy density of vibration of the phonon modes of objects 7 and 5, which satisfies the established equality.

From the found values of Λ ₇ , Λ ₅ , ν _s7 , ν _s5 , dν _s5 , dν _s7 and the values of the constant values T ₅ , T ₇ and k ₅ , using the formula (6), calculate the desired constant k ₇ :

The relative error in determining the Boltzmann constant by the claimed method was determined by the analysis of equation (6), from which it follows:

where is the wavelength (measured by a typical precision spectrometer with a relative error of the order of 10 ^-7 .

Since the relative measurement error of the frequencies ν _s7 and ν _s5 is of the order of 10 ^-10 , i.e. three orders of magnitude lower than the relative error in determining the phonon wavelength and is negligible, the relative error in determining the Boltzmann constant is

If necessary, to obtain a more accurate value of the Boltzmann constant, the calculation is repeated, while the calculated value of the constant k _{7 is} used as a known one.

Claims (1)

The method of determining the Boltzmann constant by measuring physical quantities characterizing the energy state of the substance of the studied objects, functionally related to each other and with the named constant mathematical equation, from which the specified constant is calculated, characterized in that at least two studied objects are placed in a thermostatic medium, identical in chemical composition and physical properties made of acousto-optical material, for example, fused quartz, and one object is placed in medium with low cryogenic temperatures, and another with higher temperatures, then these objects are simultaneously irradiated with monochromatic laser radiation with a wavelength equal, for example, to the wavelength located in the ultraviolet part of the optical spectrum, to excite phonons with a frequency ν _s in them, then they achieve equality of energies hν _SH and k _H Т _H in the object located at low temperature, where ν _SH is the frequency of the phonon mode in the element at low temperature; k _H is the known value of the Boltzmann constant; T _H , k is the known value of low temperature; h, J · s - Planck constant, why the frequency ν _{SH is} varied by changing the irradiation angle θ _{H of an} object located at a low temperature, while the named frequency is measured with a spectrometer, the energy inequality condition for an object located at a high temperature is fulfilled hν _SB << k _H T _B , where ν _SB is the frequency of the phonon mode in an object at a high temperature; T _in , k is the known value of heat, by varying the frequency ν _{SB by} changing the irradiation angle θ _{B of the} object at high temperature, then establish the equality of the energy densities of the thermal phonons E _H · dν _SH and E _B · dν _SB , where Е _H and Е _B - kinetic energy of phonons in objects located at low and high temperatures, respectively; dν _SH and dν _SB are the frequency band of phonon vibrations in objects at low and high temperatures, respectively, why they vary the frequency ν _SB by changing the angle θ _{В of the} object at high temperature, and at the same time control the achievement of the named equality by fulfilling the relation ν _SB / ν _SH = I _SB · Λ _H / I _SH · Λ _B , where I _SH and I _SB are the intensities of phonon vibrations in objects at low and high temperatures, respectively; Λ _H and Λ _B are the wavelengths of oscillations of phonons in objects, by changing the intensity of I _SH and / or I _SB of phonon vibrations in objects using a polarizer, and then using the spectrometer to measure the frequency ν _SB and ν _SH , the wavelengths Λ _B and Λ _H , and by the formula

the sought value of the Boltzmann constant k _{B is} calculated.

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