CC BY
48
Н. П. Саргаева, П. М. Саргаев
РАВНОВЕСИЕ БЭК-КВАНТОВЫЙ ГАЗ И СТРУКТУРА H2O ЖИДКОСТИ
Предложена концепция проявления бозе-эйнштейновского конденсата (БЭК) в спектре, полученном как геометрическое среднее двух масс-спектров, один из которых соответствует частицам при температуре появления БЭК, а другой — равновесию квантовый газ-конденсат на уровне энергии идеального квантового газа по Эйнштейну (1) или тепловой длины волны (2). Масс-спектры равновесия БЭК-квантовый газ H2O на линии насыщения жидкости (m) моделируются системами из двух и более кластеров, содержащих резонансы с N = 2-12. Используя т, удается успешно интерпретировать экстремумы температурной функции конфигурационной теплоемкости тождественных частиц и рассчитать межмолекулярные расстояния и другие характеристики структуры H2O по дифракции и интерференции волн де Бройля: в свойствах и структуре метастабильной H2O жидкости обнаруживается проявление БЭК вплоть до критической температуры.
Ключевые слова: Н2О, жидкая фаза, равновесие БЭК-квантовый газ, квантовые волновые масс-спектры, резонансы Ефимова с N=2-12, квантовый газ тепловой длины волны, квантовый газ по Эйнштейну, масс-спектры и экстремумы теплоемкости; бозонные пики тождественных частиц; дифракция и интерференция волн де Бройля; межмолекулярные расстояния, тетраэдраическая, гексагональная и пентагональная координация.
N. Sargaeva, P Sargaev
THE BEC-QUANTUM GAS EQUILIBRIUM AND THE STRUCTURE OF H2O LIQUID
The concept of a bose-einstein condensate (BEC) displaying in a spectrum received as of geometrical mean of two mass spectra, one of which corresponds to particles at temperature of BEC occurrence, and another corresponds to equilibrium ideal quantum gas by Einstein (1) or with thermal wavelength (2) is offered. Spectra of H2O liquid (m) are modulated by systems of two or more clusters (atoms, “resonances” with N = 2-12). Using m, the temperature function extremes of configuration heat capacity of identical particles are successfully interpreted and the intermolecular distances and other characteristics of structure H2O in the phenomenon of diffraction and interference of de Broglie waves are designed. It is revealed, that BEC is shown in properties and structure of a metastable and stable H2O liquid down to critical temperature.
Keywords: H2O, liquid phase, BEC-ideal quantum gas equilibrium, quantum wave mass spectra, Efimov resonance at N = 2-12; thermal wavelength quantum gas, Einstein quantum gas, mass spectra and extremums of heat capacity; identical particles heat capacity boson peaks; de Broglie waves diffraction and interference; tetrahedral, hexagonal and pentagonal coordination.
The Einstein predicted transition of a substance from a gas condition into a Bose-Einstein condensate (BEC) [10; 13] takes place in ultracold gases at temperatures 10-6 — 10-9 K [1; 8; 16]. The influence of Efimov resonances [12; 17] in a cluster condition on recombination speed of the cold gases Bose-Einstein condensate is being investigated [24]. The BEC becomes apparent within the superfluidity phenomena of liquid helium at 2.17 K [7; 15; 18]. It also appears in the superconductivity of solid phase [18], which takes place at temperatures up to and above 135 K [19].
Particle mass, which corresponds to the critical temperature of the BEC formation (Tc) [1], has been estimated for liquid ethane [23]. In the case where (Tc) temperatures coincide with the temperature (T), the results do not contradict with the masses of particles involved in the quantum
gas — condensate reversible transitions. Moreover, at certain conditions the quantitative ratio can be found for compared masses [23].
Within the framework of reversible transitions Einstein quantum gas — condensate model [13] the appearance of proton pairs is revealed in liquid H2O [22]. The temperature function extremes of heat capacity of liquid ethane are successfully interpreted [23]. These can be a basis for application of the BEC concept towards properties and structure of liquids. H2O as most investigated [4-5; 9; 11; 14; 20-22; 25; 26] and full of puzzles [9; 22; 25; 26] liquid is suitable for approbation of techniques and equations determined within the framework of such concept.
The BEC — quantum gas equilibrium concept
The nature of identical particles was found to appear in configuration fluctuations accompanied by reversible transitions quantum gas — condensate [22-23]. The mass of a cluster, containing n resonantly but not chemically interacting particles, is used as a criterion to identify particles and to evaluate possible formation of Efimov resonances. The coherent movement of identical particles results in an increase of a cluster mass. Two or more interacting clusters constitute a system. The total mass of a system is smaller than the mass of a light cluster, and it decreases with the increasing number of clusters, which could have the same mass. The mass of particles participating in considered processes depends on the energy of quantum gas equilibrium. The temperature function of the mass of particles is called mass — spectrum [23]. There are four levels of energy used for gradation of quantum gas and mass — spectra (m1, m2, m3, m4 [22-23]). The mass of particles at the temperature of BEC occurrence (Tc, [1]) are selected in a separate spectrum (m11) [23]. Furthermore, mass-spectra were found to transition from one to another. Such transitions could be caused by interactions between clusters as well as with particles of an environment [23]. These interactions were classified as an “interaction of clusters” [24] within the mass-spectrum; however, the interaction between mass-spectra was not considered and, accordingly, the results of such interaction were not discussed.
In this work the possibility of mass — spectra interaction is proposed. The mass of liquid particles, which is given as a result of such interaction, coincides with the geometric mean of interacting masses. The use of geometric mean is necessary to account for the sum of masses as well as for their inverse values. There is a special case with two interacting mass-spectra: when one (m11) is the mass-spectrum of particles at the temperature of BEC occurrence, and the other one corresponds to the ideal quantum gas equilibrium (m1, m2, m3, m4). In this work, such case will be called the BEC-quantum gas equilibrium. In the indication of mass the indexes of initial mass-spectra will be taken into account, for example, m411 = (m4 • m11)1/2, m211 = (m2 • m11)1/2.
Formalism of the mass of particles computation in mass-spectra
The mass of particles at the temperature of BEC occurrence is described by equation
mn = (h2 / (2-n-k))3/5 • (p / 2.612)2/5 • T3/5, (1)
where p is density; h, k are Planck and Boltzmann constants, respectively. Equation (1) is derived [23] by equating of the temperature (T) and the critical temperature of BEC occurrence Tc = (h2 / (2-n-m-k)) • (n / 2.612)2/3, where m and n are molecular weight and concentration, respectively, of particles of the system (gas) [1, p. 30].
The mass of particles in a quantum gas — condensate equilibrium can be determined according to the following equation
m = (Cp/Cy)-k-T / C2s, (2)
where Cs is a speed of sound in liquid; Cp/Cv — adiabatic component; Cp and Cv — isobaric and isochoric heat capacities of quantum gas. Equation (2) is derived based on the basis of the equilibrium condition Ec = Eig., where Ec = m-C2s is the energy of configuration fluctuations of particles of a liquid participating in quantum gas — condensate reversible transitions; Eig. = (Cp/Cv) k-T — fluctuation energy of equilibrium quantum gas particles [23]. There are four levels of energy in equation (2) used for differentiation of the mass of particles (m) of a liquid (and for notation index) [23]. Certain adiabatic component of equilibrium quantum gas corresponds to each level of energy:
(Cp/Cv)i = 1 is the level (1) of quantum gas with critical adiabat; (Cp/Cv)2 = 3/2 and (Cp/Cv)3 = (5/3) — levels of saturated (2) and fully unsaturated (3) ideal monatomic quantum gases by Einstein [13]; (Cp/Cv)4 = 2-n — the level (4) of quantum gas with thermal wavelength of particles.
The use of speed of sound in equation (2) does not contradict with the BEC — quantum gas equilibrium concept: the speed of sound is used for comparison of Bose-gas and Bose-liquid properties. It is also used in the theory of helium superfluidity [18].
The particle mass values for the BEC — quantum gas equilibrium will be defined as geometric mean of masses determined from equations (1) and (2). Only Einstein saturated ideal
quantum gas at BEC equilibrium (2) and quantum gas formed by particles with thermal wave-
length (4) will be considered. The case (4) is used to study ultracold gases [8; 16-17; 24], as well as to derive equation for the critical temperature of BEC occurrence [3, p. 614], and for criterion of the ideal gas degeneration [3, p. 608]. The condition of the Einstein saturated ideal quantum gas (2) can not be reached in usual gases [13]. However, in the studied case the condensed condition of substance on a liquid saturation line is considered. Such a condition corresponds to saturated rather than unsaturated quantum gas.
The mass of particles corresponding to the reversible transitions of the BEC — equilibrium of saturated monatomic ideal quantum gas by Einstein is
m211 = (m2-m11)12. (3)
The mass of particles participating in the BEC — quantum gas equilibrium at a level of energy of quantum gas with a thermal wavelength of particles, is calculated by the equation
m411 = (m4-mn)1/2. (4)
The mass-spectra comparison
The significance of m11, m2, m4, m211 and m411 for H2O liquid saturation (obtained by equations (1)-(4) using data from NIST Standard Reference Database [21]) and [9; 25-26], is represented in Fig. 1. Figure 1 shows dependency of m (atomic mass units) on the temperature (T, K). The temperature interval is chosen from the supercooled (metastable) liquid state up to the critical temperature. The calculated mass-spectra results are given in comparison with the mass of proton (1mp), the proton pair (2mp) and the 3-proton Efimov resonance (3mp). The proton mass value (1.00739) corresponds to the natural isotopic composition of hydrogen. The values of configuration heat capacity of identical particles (Cc2, J/ (mol-K)) are also provided in fig. 1 from the previous work [22]. The values of Cc2 were calculated by the equation Cc2 = Cv - Cvib - Cc1 - Cinf, where Cv, Cvib, Cc1, Cinf — isochoric, vibrational, configuration distinct particles and infinite heat capacity of H2O liquid saturation respectively [21; 22].
О
ОС
VO
в
тг
(N
О
(N
■
T, K
Fig. 1. Dependence on temperature (T, K) found by the equations (1) — (4) modeling masses (m) of particles of H2O liquid saturation for the energy levels of Einstein saturated quantum gas (m2) and thermal wavelength of particles (m4), at temperature of BEC occurrence (m11) and BEC — quantum gas equilibrium (m211, m411): (Cc2 — configuration heat capacity of identical particles (J/ (mol-K)) by data [22]; Ttrp — triple point temperature)
The values of m11 decrease smoothly while the temperature increases, which can be noticed within m411(T) and m211(T) functions. These functions depend on the temperature less than m4 and m2 do. In a temperature course of functions m411(T) and m211(T) the constancy of the mass ratio m411 / m211 = (m4 / m2)05 = (4n / 3)05 « 2.04665 is also shown. In a wide temperature range the values of m211 are nearly equal to the proton mass (1mp), and values of m411 are nearly equal to the mass of proton pair (2mp) and m2. The temperature area, in which the values of m411 and m2 are nearly equal to the proton pair mass, is more extensive in the m411(T) case than in the m2(T) case. At temperatures 239-240K of m211(T) and m411(T) functions have maximum. In this temperature range the configuration heat capacity of identical particles (Cc2(T), see fig. 1) has a sharp maximum. In the field of temperatures 350-630K the Cc2(T) function becomes negative. In this temperature range m411 and m211 increase with the temperature. The greatest masses of m411 and m211 correspond to the critical temperature.
Mass-spectra and configuration capacity of identical particles
Features of functions m211 (T) and m411 (T) are used for interpretation of extreme in temperature dependence of a configuration heat capacity of identical particles of H2O liquid saturation Cc2(T).
200
400
600
The temperature range of 273-300 K
There is a minimum of Cc2(T) function at the temperature of triple point of H2O water (see fig. 1). There is a maximum of Cc2(T) at 285K. Both extremes are poorly expressed and their presence is challenged because Cc2 values are derived based on a difference of large numbers. However, at an example when the pressure is 100 MPa and the temperature is 285 K there is a maximum in a temperature dependence of isochoric heat capacity Cv(T) [21]. The Cc2(T) function has an extreme on a line of a liquid saturation provides that the function Cc2(T) is more sensitive to changes of liquid properties, rather than an isochoric heat capacity temperature function Cv(T). This, therefore, can be the basis to consider H2O extremes for a discussion.
The temperature of Cc2(T) maximum (285 K) practically coincides with the temperature (286 K), at which the value of m411 equals to the mass of proton pair. The mass of proton pair can be derived using a modeling system containing one cluster (2H). It can also be derived based on the system containing two clusters (4H; 4H). It means, that the Cc2(T) maximum corresponds to modeling systems containing bosons. Transitions from even-even system (4H; 4H) to the nearest even-odd systems ((3H; 4H) or (4H; 5H)) result in an increase of the fermion content and in a decrease of configuration heat capacity of Cc2 identical particles. Thus, the Cc2(T) maximum at the temperature of 285 K can be classified as a boson peak. The obtained result within the framework of the BEC-quantum gas equilibrium concept coincides with those, obtained on the basis of the analysis of frequencies of configuration vibrations in equilibrium of a quantum gas — condensate [22].
At the temperature of 273.16 K, corresponding to the Cc2(T) minimum, the m211 value is greater than the mass of proton. According to the BEC-quantum gas equilibrium concept it is possible to explain the deviation of m211 from the proton mass at the temperature of the Cc2 minimum. In considered temperature range m211(T) rises with the temperature decrease. The proton mass can be obtained based on (1H) and (2H; 2H) modeling systems. The last system (2H; 2H) is classified as even-even modeling system, which contains bosons. The presence of bosons causes the values of configuration heat capacity Cc2 to be positive (increased comparatively to the values at 273.16 K). The decrease of the heat capacity Cc2, observed at temperatures lower then 275 K is related to an increase of the fermion content in the modeling system. Indeed, it is possible to obtain the mass exceeding the proton mass in an even-odd modeling system (2H; 3H). In such system the fermion content is higher, than in (2H; 2H) system. The mass close to m211 = 1,01425 corresponding to the Cc2(T) minimum can be obtained from various modeling systems. This characterizes such modeling systems at 273.16K as variable. For example, modeling systems (2H; 3H; 1HO; 1H4O; 1H4O), (2H; 3H; 1M; 1M; 2M; 3OH), and (2H; 3H; 1M; 1M; 2OH; 3M), where M = H2O, correspond to masses 1.0143, 1.0146, and 1.0141. The variability testifies high probability of modeling system realization.
The temperature range of 300-400 K
The mass of m411 = 1,9459, corresponding to the m411(T) minimum at 329K, can be obtained as given in the model system, in which the least cluster contains two or greater number of protons, which is in agreement with proton pairs in a H2O liquid found earlier [22]. In the range of 329K the Cc2 heat capacity is positive, therefore proton pairs could be used in the modeling of the minimal value (0,9508) of the m211(T) function. For example, the even-even (2H; 2H; 1OH) model system used in calculation of the mass of the proton H+ and OH- ion gives a value of 0,951, which is close to the desired value.
When the temperature rises above 329K the mass of particles m2n and m4n increases and even-even system are giving way to even-odd model systems. This leads to an increase of fermion clusters contribution to configurational heat capacity of identical particles Cc2 as well as its transition to negative values. Fermions, by definition, [6], interfere with a negative sign, which manifests itself in the configuration of the heat capacity of identical particles Cc2(T). The role of all clusters of the model system appears in the process of the Cc2 decrease. For example, at 350K the m4n = 1,957 value is modeled by the system (3H; 1O; 1HO; 1HO), which contains one bosonic cluster (1O) formed by O2- particles among three fermion clusters. At higher temperatures (361-362K) the m411 = 1,971 value is modeled by the fermionic system (3H; 1HO; 1HO; 1HO).
The temperature range of 200-273 K
High temperature (right) wing of the Cc2(T) maximum is observed at temperatures from 239-240 K to 273,16 K. The mass of 1,2088698 corresponds to the model system (2H; 3H). This mass exceeds the maximum value of m211 (1,19272), which corresponds to 239 K. In this regard, the m211 values of all of the temperatures of the right wing of the Cc2(T) maximum can be calculated as at the temperature of 273,16 K based on the even-odd system (2H; 3H). For example the mass of 1,1929, close to the value of m211 at the maximum Cc2(T), corresponds to the system (2H; 3H; 5M). Even-odd (2H; 3H) systems are also suitable for describing the temperature dependence of the m211 in the left (low temperature) wing of the Cc2(T) maximum. However, (2H; 3H) systems can not explain the reasons for sharp Cc2(T) rise as it approaches the maximum in the configurational heat capacity of identical particles. The problem can be solved by increasing hydrogen parity in the process of changing the composition of model systems. The even-odd system (2H; 3H) has to be consistently replaced with (2H; 4H) and (2H; 6H) even-even model systems as the temperature approaches the Cc2(T)maximum. At the Cc2(T) maximum the following systems can be realized (2H; 4H; 1O; 2O) and (2H; 6H; 1OH; 1OH; 1OH), which correspond to 1,1930 and 1,1931 values. In the case of ethane the (2H; 4H) model was attributed to the number of critical systems [23]. In this case the (2H; 4H) system is also critical, since the next even-even (2H; 6H) system corresponds to an equivalent (3H; 3H; 1OH; 1OH; 1OH) odd-odd model in which all clusters are fermionic.
At large values of the inductive capacity of water, and at differences in the electronegativity of hydrogen and oxygen, all the clusters (3H; 3H; 1OH; 1OH; 1OH) in odd-odd model system are electrically charged particles ((3H)3+, OH-) and, accordingly, are fermions. The possibility of transition of model system containing predominantly boson clusters to the system of a fermionic structure is characteristic to liquid H2O. Further, it is one of the reasons for the formation of spiked maximums of configurational heat capacity of identical particles Cc2 (T) at 239-240 K. Boson-fermion contrasts reflect, for example, (2H; 4H; 1O; 2O) and (3H; 3H; 1OH; 1OH; 1OH) systems, which simulate the m211-mass values within Cc2(T) peak at 239-240 K temperatures.
The temperature range of 400-620 K
The shape of an extensive Cc2(T) minimum can be explained by the equilibrium BEC-quantum gas concept in the temperature range of 400-620 K. From Fig. 1 it follows that the trend line of Cc2(T) function has a wave-like shape in these three segments of temperatures. A weak maximum, corresponding to 510-520 K temperatures, is surrounded by two nearly symmetrically located minima. The first of these is located at 460-470 K, and the second is in the temperature range of 560-570 K.
The m211(T) function at temperatures from 402 K to 484 K has the same values as at 275 K to 239-240K temperatures. Therefore, the m211(T) are modeled by (2H; 3H) even-odd systems. (2H; 3H; 1OH) and (2H; 3H; 1M) model systems, for example, correspond to m211(T) values in the temperature range of 460-470 K. In the temperature range of 484 K the Cc2(T) function increases with higher temperatures. In this case, the mass m211 = 1,1927 is simulated by model system with two equivalent forms (3H; 3H; 1OH; 1OH; 1OH) and (2H; 6H; 1OH; 1OH; 1OH). All the components of one system are classified as fermions. They are classified as bosons (2H; 6H) in another system. In the temperature range of 470-520 K the m211 mass can also be obtained by other containing bosons model systems. The implementation of (2H; 4H) even-even systems such as (2H; 4H; 1HO); (2H; 4H; 1M), (2H; 4H; 2O) and alike, contributes to the maximum of Cc2(T) in the temperature range of 510-520 K. A significant contribution to the formation of the Cc2(T) maximum is also made by (4H) systems. With further increase in temperature the contribution of even-odd model systems to m211 and m411 masses increases, which leads to some decrease in Cc2 and the observation of a minimum of Cc2(T) at 560-570 K. Such factors as high variability of model systems, the possibility of smooth and continuous temperature change in the mass of particles in mass spectra of m211(T) and m411 (T), and dissociation of molecules into ions allow smoothing the boson-fermion variation of Cc2(T) on the H2O liquid saturation line in the temperature range of400-620 K.
The near-critical temperature region
At the critical temperature (~647,27K) the highest value m411 ~ 9,8 is achieved. In order to model this number by using hydrogen clusters, they must contain at least 10 protons. In such model system (10H; 20M), along with 10-atomic resonances (10H), it is necessary to use the 20-molecule (20M) Efimov resonance. To reduce the size of the second cluster it is necessary to increase the mass of the first cluster. The m411 = 9,8 value can be modeled by, for example, (12H; 3HO) system, in which the 12H cluster mass commensurate with 2/3 of the mass of water molecules. If using only H2O molecules in the cluster, the model system will be (1M; 2M; 3M). It is known that the mass of the liquid particles increases at the critical temperature. However, in this case, the (3M) cluster is a classic three-body Efimov resonance, when the (2M) cluster has the resonance nature, formed by the coherent motion of particles. Bosonic nature of the molecules characterizes high meanings of configurational heat capacity as the boson peak. However, the configurational heat capacity of H2O is less than this of D2O [22]. This fact was attributed to the manifestation of fermions in the configurational heat capacity of identical particles of H2O. Indeed, for example, at the critical temperature, the m211 (4,78) value can be simulated by systems containing, in addition to bosons, fermion clusters: (5H; 6O) = (10H; 10H; 6O), (1M; 1M; 1M; 3O; 3O), (1HO; 1M; 1M; 3HO; 3HO). The possibility of representing m411 and m211 masses in the form of models of different composition characterizes model systems at critical temperature as a variable. The transformation of proton resonances to molecular resonances is one of the factors which reduces the scale of BEC manifestations in liquid H2O, as opposed to, for example, cold gases [24].
Mass spectra and the structure
Features of m211(T) and m411(T) mass spectra are used for quantitative estimation of H2O liquid structure on the saturation line. Mass spectra contain resonances that are formed due to the coherent motion of identical particles of liquid. The de Broglie waves, related to the BEC-
quantum gas equilibrium, propagate with the speed of sound (Cs). In this case, the de Broglie wavelength (X) is comparable to the intermolecular and interatomic distances (d) of the liquid. The phenomenon of wave diffraction and interference can be used to study the liquid structure. To realize this possibility the particle mass (m) of de Broglie wave sources and receivers will be compared with m2n and m4n masses.
To determine the mass of particles (m) the de Broglie wavelength X = h / (m-Cs) is used in Bragg's law Equation 2-d-sin(a)= n-X, where ais a slip angle, and n is the order of diffraction. In this case, the mass is dependent on the angle a:
m = n - h / (2 d Cs Sin(a)). (5)
Diffracted waves interfere. In the calculations of fluctuation intensity (J, Ji, J2) in accor-
1/2
dance with the interference: J = J1 + J2 + 2-(J1-J2) -cos(O), where O — the difference of phase fluctuations, an additional angle (O) has to be considered. To account for possible angle variations (a and O) the structural features of liquids will be taken into account. In these liquids, unlike solids, the short-range order takes place, which undergoes temperature changes. In this regard, «the composition» and structure of the first coordination sphere of the molecules we mainly taken into account.
The molecules of the first coordination sphere of water H2O are divided [14, 20] into fractions: f1 and f2 — for intermolecular distances 0,28 and 0,33 nm. Multiplication of fractions f1 and f2 on the coordination number z gives the number of molecules of the first (n1) and the second (n2) type in the first coordination sphere of water z = n1 + n2.
It was found [5] that in a wide temperature range n1 remains constant and equal to the value (2,36), the reverse value is equal to the percolation threshold for the tetrahedral lattice sites (0,43 [2, p. 436]). Since, by definition, the percolation threshold is n1 = z1 - 1, where z1 — a coordination number of molecules in lattice sites, then z1 = n1 + 1. By analogy, z2 = n2 + 1 is the coordination number of molecules in position 2 (in the «interstices»). The number z is the vector sum of z1 and z2 [5], in contrast to the algebraic sum of z = n1 + n2. Thus, z, z1, and z2 form a vector triangle. Parameters of the triangle associated with the composition and structure of the first coordination sphere.
The vector triangle parameters will be used to describe the phenomena of diffraction and interference of de Broglie waves. Let the straight line to coincide with the interatomic or intermolecular distance d at the right angle of one side of the triangle. The intensity will be represented in mass units as the product of mass m in (5) and fractions. The indexes (0, 1, 2) will be used to describe fractions (f0 = f1 + f2; f1; f2) and angles (F0; F1; F2) at the opposite sides of z, z1 and z2. Indexes will be assigned to letters (i = 0; j = 1, k = 2). Angles (a and O) will be equated to corners of the vector triangle. The chosen succession let us to account for all the angles of the vector triangle in each computation of the mass of particles involved in the phenomena of diffraction and interference of de Broglie waves.
The result of interference described by equation (taking into account the summation of mass (m) to (5) and their reciprocals (m)-1):
mijk = ((mi)b-fj + (mj)b-fi + (-1)c-2-((mi)b-fj-(mj)b-fi)1/ 2 -Cos(Fk))b, (6)
where b can be +1 and -1; c — may be 1 and 2, the first and second indexes (mijk) — characterize the slip angle and the fraction in the first term, as well as the fraction and the slip angle in the second term. The third index characterizes phase (in the third term) of the right-hand side of the equation (6). After index permutation the first index transfers to the place of the third one. As a
result, equation (6) gives 12 meanings of the mass of particles, which are associated with de Broglie waves involved in the phenomena of diffraction and interference.
Comparing computational results of the equation (6) with the results of equations (1) - (4) the distance d can be estimated.
Mass spectra and intermolecular distances
In this paper, based on the results obtained from (6) a geometric mean (mg.m.) of mass was found when b = +1, b = -1 and b = (+1 and -1) with a parameter c = 1; c = 2; c = (1 and 2). Such an averaging gives 15 masses (mg.m.) of particles. Configuration fluctuations of the fluid particles reduce the difference between fractions f1 and f2. In this paper f1 = f2 = 0,5 was accepted. Computational results of mg.m. involved in the diffraction and interference, was compared with the m211 mass. Using this method, the distance d (nm) for H2O on the liquid saturation line was determined, which is shown in Fig. 2 as d(T) functions (lines 1-15). The effective radius of the molecules (r) and various data in the literature [4, 14, 20, 25, 26] are also presented.
From Fig. 2 it follows that most of the calculated d(T) functions (lines 1-15) are convex to the temperature axis (T, K).
Fig. 2. Intermolecular distances (d, nm) of H2O on the saturation line of the liquid at temperatures (T, K) according to equation (6) calculations, when fi = f2 = 0.5 and mg.m. = m211, (lines 1-15) and literature data: ([G; N; P; Tu; X] — in [14, 20, 4, 25, 26]; r1, r2, rc, 2r1, 2r2, 2rc — radius and intermolecular distances of the first and second types of molecules and critical fluid; rOH, rOHg — the distance between the atoms of OH-liquid and H2O-gas; r — effective radius of the liquid molecules; Ttrp — triple-point temperature)
The lowest meanings of d(T) (lines 14-15) are in the region of interatomic distances of liquid (гон) and gas (rOHg). The greatest meanings of d(T) (lines 1-3) are located in the intermolecular distances of the first (2r1) and second (2r2) types of molecules. Lines 10-11 practically possess the same features of the temperature dependence of the molecular radius (r2) and (r1). An agreement of obtained d(T) distances was found for stable, supercooled and critical liquids.
The angular characteristics (F (T)) of molecules and their relative position in the liquid can be estimated based on d(T) of, for example, adjacent lines. From these estimations it follows, that the angle of H2O gaseous molecules (104,52o) is the limit of the F(T) function at the critical temperature. While, characteristic angles of tetrahedral (109,47o), hexagonal (120o), as well as the adjacent angle of pentagonal (108o) coordination appear as the F(T) function limit at low temperatures only.
From the analysis of the temperature dependence of the configurational heat capacity of identical particles Cc2(T) and intermolecular distances d(T), calculated on the basis of our proposed concept of equilibrium BEC-quantum gas, it follows that within the properties and structure of H2O on the saturation line of liquid the Bose-Einstein condensate is manifested up to the critical temperature. In addition, such a manifestation is confirmed by the expression of classical (3-body) Efimov resonances and by resonances that contain a large number of particles (protons, H2O molecules).
СПИСОК ЛИТЕРАТУРЫ
1. Гринштейн Дж., Зайонц А. Квантовый вызов. Современные исследования оснований квантовой механики. Долгопрудный: Интеллект. 2008. 400 с.
2. Займан Дж. М. Модели беспорядка. М.: Мир. 1982. 591 с.
3. Левич В. Г., ВдовинЮ. А., Мямлин В. А. Курс теоретической физики. Т. 2. М.: Наука 1971. 936 с.
4. Путинцев Н. М. Физические свойства вещества (лед, вода, пар). Мурманск: Изд-во МГАРФ. 1995. 255 с.
5. Саргаева Н. П., Саргаев П. М. Координационно-угловое распределение молекул воды в растворах // Всероссийский симпозиум «Эффекты среды и процессы комплексообразования в растворах»: Тезисы докладов. Красноярск. 2006. С. 193-194.
6. Фейнман Р., Лейтон Р., Сэндс Р. Фейнмановские лекции (вып. 8, 9). М.: Мир. 1978. 526 c.
7. Allen J. F., Misener A. D. Flow of liquid helium ii // Nature. 1938. V. 141. P. 75.
8. Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E., Cornell E. A. Observation of BoseEinstein Condensation in a Dilute Atomic Vapor // Science. 1995. Vol. 269. 14 July. P. 198-201.
9. Angell C. A. Insights into phases of liquid water from study of its unusual glass-forming properties // Science. 2008. Vol. 319. P 582-587.
10. Bose S. N. Abhandlung Plancks Gesetz und Lichtquantenhypothese // Zeitschrift fur Physik. 1924. V. 26. P. 178-181.
11. Debye P and Scherrer P. Interferenzen an regellos orientierten Teilchen im Rontgenlicht. I // Physikalische Zeitschrift. 1916. №. 17 P. 277-283.
12. Efimov V Energy Levels Arising from Resonant Two-Body Forces in a Three-Body System // Phys. Lett. 1970. Vol. B33. No. 8. P. 563-564.
13. Einstein A. Quantentheorie des einatomigen idealen Gases // Sitzungsber. preuss. Akad. Wiss., Phys.-math. K1., 1925. P. 3-14.
14. Gorbaty Y. E., Bondarenko G V. Water in the supercritical state // Supercritical fluids. Theory and practice. 2007. Vol. 2. №. 2. P. 5-19.
15. Kapitza P. Viscosity of liquid helium below the X point // Nature. 1938. Vol. 141. P. 74.
16. Ketterle W. Nobel lecture: When atoms behave as waves: bose-einstein condensation and the atom laser // Rev. modern phys. 2002. Vol. 74. P. 1131-1151.
17. Knoop S., Ferlaino F., Mark M., Berninger M., Schöbel H., Nägerl H.-C. and Grimm R. Observation of an Efimov-like trimer resonance in ultracold atom-dimer scattering // Nature Physics. 2009. Vol. 5. P. 227-230.
18. Landau L. D. On the Theory of Superfluidity of Helium II // J. Phys. USSR. 1947. Vol. 11. P. 91-94.
19. Lokshin K. A., Pavlov D. A., KovbaM. L., Putilin S. N., Antipov E. V., Bryntse I. Synthesis and investigation of (Hgi-xCux)Ba2Ca2Cu3O8+d // Phys. C. 2002. V. 366. P. 263-269.
20. Narten A. H., Danford M. D., Levy H. A. X-Ray Diffraction Study of Liquid Water in the Temperature Range 4-200 oC // Disc. Farad. Soc. 1967. № 43. P. 97-107.
21. NIST Standard Reference Database Number 69, June 2005 Release.
22. Sargaeva N. P., Baryshev A. N., Puchkov L. V., Sargaev P. M. Heat capacity bose-fermi contrasts of D2O and H2O liquids // XVII International Conference on Chemical Thermodynamics in Russia. Abstracts. Vol.
1. Kazan, Russian Federation. June 29 - July 3, 2009. P. 210.
23. Sargaeva N., Sargaev P. The synergy of structural units and ideal quantum gas — condensate reversible transitions of liquid ethane // Izvestia: Herzen University Journal of Humanities & Sciences. 2011. №138. P. 65-76.
24. Thogersen M., Fedorov D. V., and Jensen A. S. N-body Efimov states of trapped Bosons // Europhys. Lett. 2008. Vol. 83. P. 30012.
25. Tulk C. A., Benmore C. J., Urquidi J., Klug D. D., Neuefeind J., Tomberli B. and Egelstaff P. A. Structural studies of several distinct meta-stable forms of amorphous ice // Science. 2002. Vol. 297. P. 1320-1323.
26. Xu L., Buldyrev S. V., Angell C. A., Stanley H. E. Thermodynamics and dynamics of the two-scale spherically symmetric Jagla ramp model of anomalous liquids // Phys. Rev. E 2006. Vol. 74. P. 031108.
REFERENCES
1. Grinstein G., Zaionz A. Kvantovyi vyzov. Sovremennye issledovaniya osnovanyi kvantovoi mek-haniki. Dolgoprudnyi: Intellekt. 2008. 400 s.
2. Zayman G M. Modeli Besporyadka. M.: Mir. 1982. 591 s.
3. Levich V G, Vdovin U. A., Myamlin V A. Kurs teoreticheskoi fiziki. T. 2, M.: Nauka 1971. 936 s.
4. Putintsev N. M. Fizicheskye svoistva veshestva (led, voda, par). Murmansk: Izd-vo MGARF. 1995.
255 s.
5. Sargaeva N. P Sargaev P. M. Koordinatsionno-uglovoe raspredelenie molekul vody v rasstvorah // Vserossiiskyi simposium «Effekty sredy i processy kompleksoobrazovaniya v rasstvorah»: Tezisy dokladov. Krasnoyarsk, 2006. S. 193-194.
6. Feinman R., Leiton R., Sends R. Feinmanovskie lekcii (vyp. 8, 9). M.: Mir. 1978. 526 s.
7. Allen J. F., Misener A. D. Flow of liquid helium ii // Nature. 1938. V. 141. P. 75.
8. Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E., Cornell E. A. Observation of BoseEinstein Condensation in a Dilute Atomic Vapor // Science. 1995. Vol. 269. 14 July. P. 198-201.
9. Angell C. A. Insights into phases of liquid water from study of its unusual glass-forming properties // Science. 2008. Vol. 319. P 582-587.
10. Bose S. N. Abhandlung Plancks Gesetz und Lichtquantenhypothese // Zeitschrift fur Physik. 1924. V. 26. P. 178-181.
11. Debye P. and Scherrer P. Interferenzen an regellos orientierten Teilchen im Röntgenlicht. I // Physikalische Zeitschrift. 1916. No. 17. P. 277-283.
12. Efimov V. Energy Levels Arising from Resonant Two-Body Forces in a Three-Body System // Phys. Lett. 1970. Vol. B33. № 8. P. 563-564.
13. Einstein A. Quantentheorie des einatomigen idealen Gases // Sitzungsber. preuss. Akad. Wiss., Phys.-math. K1., 1925. P. 3-14.
14. Gorbaty Y. E., Bondarenko G V. Water in the supercritical state // Supercritical fluids. Theory and practice. 2007. Vol. 2. № 2. P. 5-19.
15. Kapitza P. Viscosity of liquid helium below the X point // Nature. 1938. Vol. 141. P. 74.
16. Ketterle W. Nobel lecture: When atoms behave as waves: bose-einstein condensation and the atom laser // Rev. modern phys. 2002. Vol. 74. p. 1131-1151.
17. Knoop S., Ferlaino F., Mark M., Berninger M., Schöbel H., Nägerl H.-C. and Grimm R. Observation of an Efimov-like trimer resonance in ultracold atom-dimer scattering // Nature Physics. 2009. Vol. 5. P. 227-230.
18. Landau L. D. On the Theory of Superfluidity of Helium II // J. Phys. USSR. 1947. Vol. 11. P. 91-94.
19. Lokshin K A., Pavlov D. A., KovbaM. L., Putilin S. N., Antipov E. V., Bryntse I. Synthesis and investigation of (Hg!-xCux)Ba2Ca2Cu3O8+d // Phys. C. 2002. V 366. P. 263-269.
20. Narten A. H., Danford M. D., Levy H. A. X-Ray Diffraction Study of Liquid Water in the Temperature Range 4-200 oC // Disc. Farad. Soc. 1967. № 43. P. 97-107.
21. NIST Standard Reference Database Number 69, June 2005 Release.
22. Sargaeva N. P., Baryshev A. N., Puchkov L. V., Sargaev P. M. Heat capacity bose-fermi contrasts of D2O and H2O liquids // XVII International Conference on Chemical Thermodynamics in Russia. Abstracts. Vol. 1. Kazan, Russian Federation. June 29 — July 3, 2009. P. 210.
23. Sargaeva N., Sargaev P. The synergy of structural units and ideal quantum gas — condensate reversible transitions of liquid ethane // Izvestia: Herzen University Journal of Humanities & Sciences. 2011. №138. P. 65-76.
24. Thogersen M., Fedorov D. V., and Jensen A. S. N-body Efimov states of trapped Bosons // Europhys. Lett. 2008. Vol. 83. P. 30012.
25. Tulk C. A., Benmore C. J., Urquidi J., Klug D. D., Neuefeind J., Tomberli B. and Egelstaff P. A. Structural studies of several distinct meta-stable forms of amorphous ice // Science. 2002. Vol. 297. P. 1320-1323.
26. Xu L., Buldyrev S.K, Angell C.A., Stanley H. E. Thermodynamics and dynamics of the two-scale spherically symmetric Jagla ramp model of anomalous liquids // Phys. Rev. E 2006. Vol. 74. P. 031108.
