Superconductivity and Superfluidity

Universal Journal of Physics and Application 2(1): 22-35, 2014 DOI: 10.13189/ujpa.2014.020106 http://www.hrpub.org

Superconductivity and Superﬂuidity-Part II: Superconductivity as a Consequence of Ordering of Zero-point Oscillations in Electron Gas

Boris V.Vasiliev

∗Corresponding Author: [email protected]

Abstract Currently there is a common belief that and the superfluidity about thirty years later. the explanation of superconductivity phenomenon lies However, despite the attention of many scientists to in understanding the mechanism of the formation the study of these phenomena, they have been the great of electron pairs. Paired electrons, however, cannot mysteries in condensed matter physics for a long time. form a superconducting condensate spontaneously. This mystery attracted the best minds of the twentieth These paired electrons perform disorderly zero-point century. oscillations and there are no force of attraction in their The mystery of the superconductivity phenomenon ensemble. In order to create a unified ensemble of par- has begun to drop in the middle of the last century when ticles, the pairs must order their zero-point fluctuations the effect of magnetic flux quantization in superconduct- so that an attraction between the particles appears. ing cylinders was discovered and investigated. This phe- As a result of this ordering of zero-point oscillations in nomenon was predicted even before the WWII by broth- the electron gas, superconductivity arises. This model ers F. London and H. London, but its quantitative study of condensation of zero-point oscillations creates the were performed only two decades later. possibility of being able to obtain estimates for the crit- By these measurements it became clear that at the ical parameters of elementary superconductors, which formation of the superconducting state, two free elec- are in satisfactory agreement with the measured data. trons are combined into a single boson with zero spin On the another hand, the phenomenon of superfluidity and zero momentum. in He-4 and He-3 can be similarly explained, due to Around the same time, it was observed that the sub- the ordering of zero-point fluctuations. It is therefore stitution of one isotope of the superconducting element established that both related phenomena are based on to another leads to a changing of the critical temperature the same physical mechanism. of superconductors: the phenomenon called an isotope- effect [1], [2]. This effect was interpreted as the direct Keywords Superconductivity Superfluidity Zero- proof of the key role of phonons in the formation of the point Oscillations superconducting state. Following these understandings, L. Cooper proposed the phonon mechanism of electron pairing on which base the microscopic theory of super- 1 Superconductivity as a conse- conductivity (so called BCS-theory) was built by N. Bo- golyubov and J. Bardeen, L. Cooper and J. Shrieffer quence of ordering of zero- (probably it should be named better the Bogolyubov- point oscillations BCS-theory). However the B-BCS theory based on the phonon Superfluidity and superconductivity, which can be re- mechanism brokes a hypothetic link between supercon- garded as the superfluidity of the electron gas, are re- ductivity and superfluidity as in liquid helium there are lated phenomena. The main feature of these phenomena no phonons for combining atoms. can be seen in a fact that a special condensate in super- Something similar happened with the description of conductors as well as in superfluid helium is formed from superfluidity. particles interconnected by attraction. Soon after discovery of superfluidity, L.D. Landau This mutual attraction does not allow a scattering of in- in his first papers on the subject immediately demon- dividual particles on defects and walls, if the energy of strated that the superfluidity should be considered as a this scattering is less than the energy of attraction. Due result of condensate formation consisting of macroscopic to the lack of scattering, the condensate acquires ability number of atoms in the same quantum state and obey- to move without friction. ing quantum laws. It gave the possibility to describe the Superconductivity was discovered over a century ago, main features of this phenomenon: the temperature de- Universal Journal of Physics and Application 2(1): 22-35, 2014 23 pendence of the superfluid phase density, the existence was proved to be fruitful. At the consideration of super- of the second sound, etc. But it does not gave an an- phenomena as consequences of the zero-point oscillations swer to the question which physical mechanism leads to ordering, one can construct theoretical mechanisms en- the unification of the atoms in the superfluid condensate abling to give estimations for the critical parameters of and what is the critical temperature of the condensate, these phenomena which are in satisfactory agreement i.e. why the ratio of the temperature of transition to with measurements. the superfluid state to the boiling point of helium-4 is As result, one can see that as the critical temperatures almost exactly equals to 1/2. of (type-I) superconductors are equal to about 10−6 from On the whole, the description of both super- the Fermi temperature for superconducting metal, which phenomena, superconductivity and superfluidity, to the is consistent with data of measurements. At this the de- beginning of the struction of superconductivity by application of critical twenty first century induced some feeling of dissatisfac- magnetic field occurs when the field destroys the coher- tion primarily due to the fact that a common mechanism ence of zero-point oscillations of electron pairs. This is of their occurrence has not been understood. in good agreement with measurements also. More than fifty years of a study of the B-BCS-theory A such-like mechanism works in superfluid liquid he- has shown that this theory successfully describes the lium. The problem of the interaction of zero-point os- general features of the phenomenon, but it can not be cillations of the electronic shells of neutral atoms in the developed in the theory of superconductors. It explains s-state, was considered yet before the World War II by general laws such as the emergence of the energy gap, the F.London. He has shown that this interaction is respon- behavior of specific heat capacity, the flux quantization, sible for the liquefaction of helium. The closer analysis of etc., but it can not predict the main parameters of the interactions of zero-point oscillations for helium atomic individual superconductors: their critical temperatures shells shows that at first at the temperature of about 4K and critical magnetic fields. More precisely, in the B- only, one of the oscillations mode becomes ordered. As BCS-theory, the expression for the critical temperature a result, the forces of attraction appear between atoms of superconductor obtains an exponential form which which are need for helium liquefaction. To create a sin- exponential factor is impossible to measure directly and gle quantum ensemble, it is necessary to reach the com- this formula is of no practical interest. plete ordering of atomic oscillations. At the complete Recent studies of the isotopic substitution showed that ordering of oscillations at about 2K, the additional en- zero-point oscillations of the ions in the metal lattice are ergy of the mutual attraction appears and the system of not harmonic. Consequently the isotopic substitution helium-4 atoms transits in superfluid state. To form the affects the interatomic distances in a lattice, and as the superfluid quantum ensemble in Helium-3, not only the result, they directly change the Fermi energy of a metal zero-point oscillations should be ordered, but the mag- [3]. netic moments of the nuclei should be ordered too. For Therefore, the assumption developed in the middle of this reason, it is necessary to lower the temperature be- the last century, that the electron-phonon interaction is low 0.001K. This is also in agreement with experiment. the only possible mechanism of superconductivity was Thus it is possible to show that both related super- proved to be wrong. The direct effect of isotopic substi- phenomena, superconductivity and superfluidity, are tution on the Fermi energy gives a possibility to consider based on the single physical mechanism: the ordering the superconductivity without the phonon mechanism. of zero-point oscillations. Furthermore, a closer look at the problem reveals that The roles of zero-point oscillations in formation of the the B-BCS-theory describes the mechanism of electron superconducting state have been previously considered pairing, but in this theory there is no mechanism for in papers [4]-[6]. combining pairs in the single super-ensemble. The necessary condition for the existence of superconductivity is formation of a unique ensemble of particles. By this 2 The electron pairing mechanism, a very small amount of electrons are combined in super-ensemble, on the level 10 in minus fifth J.Bardeen was first who turned his attention toward a power from the full number of free electrons. This fact possible link between superconductivity and zero-point also can not be understood in the framework of the B- oscillations [7]. BCS theory. The special role of zero-point vibrations exists due An operation of the mechanism of electron pairing to the fact that at low temperatures all movements of and turning them into boson pairs is a necessary but electrons in metals have been frozen except for these not sufficient condition for the existence of a supercon- oscillations. ducting state. Obtained pairs are not identical at any Superconducting condensate formation requires two such mechanism. They differ because of their uncorre- mechanisms: first, the electrons must be united in bo- lated zero-point oscillations and they can not form the son pairs, and then the zero-point fluctuations must be condensate at that. ordered (see Fig.(1)). At very low temperatures, that allow superfluidity in The energetically favorable pairing of electrons in the helium and superconductivity in metals, all movements electron gas should occur above the critical temperature. of particles are freezed except for their zero-point os- Possibly, the pairing of electrons can occur due to the cillations. Therefore, as an alternative, we should con- magnetic dipole-dipole interaction. sider the interaction of super-particles through electro- For the magnetic dipole-dipole interaction, to merge magnetic fields of zero-point oscillations. This approach two electrons into the singlet pair at the temperature of Superconductivity and Superfluidity-Part II: Superconductivity as a Consequence 24 of Ordering of Zero-point Oscillations in Electron Gas

Figure 2. Two ions placed on the distance L and centers of their electronic clouds. Figure 1. The schematic representation of the energy levels of conducting electrons in a superconducting metal 3 The condensate of ordered about 10K, the distance between these particles must be zero-point oscillations of elec- small enough: tron gas 2 1/3 ≈ r < (µB/kTc) aB, (1) 3.1 The interaction of zero-point oscilla- ~2 tions where aB = 2 is the Bohr radius. mee That is, two collectivized electrons must be localized The principal condition for the superconducting state in one lattice site volume. It is agreed that the supercon- formation is the ordering of zero-point oscillations. It ductivity can occur only in metals with two collectivized is realized because the paired electrons obeying Bose- electrons per atom, and cannot exist in the monovalent Einstein statistics attract each other. alkali and noble metals. The origin of this attraction can be explained as fol- It is easy to see that the presence of magnetic moments lows. on ion sites should interfere with the magnetic combina- tion of electrons. This is confirmed by the experimental Let two ion A and B be located on the z axis at the fact: as there are no strong magnetic substances among distance L from each other. Two collectivized electrons superconductors, so adding of iron, for example, to tra- create clouds with centers at points 1 and 2 in the vicin- ditional superconducting alloys always leads to a lower ity of each ions (Figure2). Let r1 be the radius-vector critical temperature. of the center of the first electronic cloud relative to the On the other hand, this magnetic coupling should not ion A and r2 is the radius-vector of the second electron be destroyed at the critical temperature. The energy relative to the ion B. of interaction between two electrons, located near one Following the Born-Oppenheimer approximation, lattice site, can be much greater. This is confirmed by slowly oscillating ions are assumed fixed. Let the tem- experiments showing that throughout the period of the perature be low enough (T → 0), so only zero-point magnetic flux quantization, there is no change at the fluctuations of electrons would be taken into considera- transition through the critical temperature of supercon- tion. ductor [8], [9]. In this case, the Hamiltonian of the system can be The outcomes of these experiments are evidence that written as: the existence of the mechanism of electron pairing is a necessary but not a sufficient condition for the existence ′ of superconductivity. H = H0 + H The magnetic mechanism of electronic pairing pro- ( ) ~2 2 2 posed above can be seen as an assumption which is con- H0 = − ∇ + ∇ − 4me 1 2 sistent with the measurement data and therefore needs − 4e2 − 4e2 (2) r r a more detailed theoretic consideration and further re- 1 2 finement. ′ 4e2 4e2 − 4e2 − 4e2 H = L + r r r On the other hand, this issue is not very important 12 1B 2A in the grander scheme, because the nature of the mechanism that causes electron pairing is not of a signifi- Eigenfunctions of the unperturbed cant importance. Instead, it is important that there is Hamiltonian describes two ions a mechanism which converts the electronic gas into an surrounded by electronic clouds without interactions ensemble of charged bosons with zero spin in the con- between them. Due to the fact that the distance sidered temperature range (as well as in a some range of between the ions is large compared with the size of ≫ ′ temperatures above Tc). the electron clouds L r , the additional term H If the temperature is not low enough, the electronic characterizing the interaction can be regarded as a pairs still exist but their zero-point oscillations are dis- perturbation. ordered. Upon reaching the Tc, the interaction between zero- If we are interested in the leading term of the interac- point oscillations should cause their ordering and there- tion energy for L, the function H′ can be expanded in a fore a superconducting state is created. series in powers of 1/L and we can write the first term: Universal Journal of Physics and Application 2(1): 22-35, 2014 25

{ [ which obey Bose-Einstein statistics, interact with each 2 − H′ = 4e 1 + 1 + 2(z2 z1) + other. L L At they interact, their amplitudes, frequencies and ]−1/2 2 2 2 (x2−x1) +(y2−y1) +(z2−z1) phases of zero-point oscillations become ordered. + 2 − L (3) Let an electron gas has density ne and its Fermi- ( )−1/2 r2 energy be EF . Each electron of this gas can be con- − − 2z1 1 − 1 + 2 1 L L } sidered as fixed inside a cell with linear dimension λF : ( )−1/2 r2 − 2z2 2 1 + L + L2 . 1 λ3 = (10) F n After combining the terms in this expression, we get: e which corresponds to the de Broglie wavelength: 4e2 H′ ≈ (x x + y y − 2z z ) . (4) L3 1 2 1 2 1 2 2π~ λF = . (11) pF This expression describes the interaction of two dipoles d1 and d2, which are formed by fixed ions and electronic Having taken into account (11), the Fermi energy of the clouds of the corresponding instantaneous configuration. electron gas can be written as Let us determine the displacements of electrons which p2 e2a lead to an attraction in the system . E F 2 B F = = 2π 2 . (12) Let zero-point fluctuations of the dipole moments 2me λF formed by ions with their electronic clouds occur with However, a free electron interacts with the ion at its the frequency Ω0, whereas each dipole moment can zero-point oscillations. If we consider the ions system as be decomposed into three orthogonal projection dx = a positive background uniformly spread over the cells, ex, dy = ey and dz = ez, and fluctuations of the second the electron inside one cell has the potential energy: clouds are shifted in phase on φx, φy and φz relative to fluctuations of the first. e2 As can be seen from Eq.(4), the interaction of z- E ≃ − . (13) p λ components is advantageous at in-phase zero-point os- F cillations of clouds, i.e., when φz = 2π. As zero-point oscillations of the electron pair are quan- Since the interaction of oscillating electric dipoles is tized by definition, their frequency and amplitude are due to the occurrence of oscillating electric field gener- related ~ ated by them, the phase shift on 2π means that attract- 2 mea Ω0 ≃ . (14) ing dipoles are placed along the z-axis on the wavelength 0 2 Λ0: Therefore, the kinetic energy of electron undergoing c Lz = Λ0 = . (5) zero-point oscillations in a limited region of space, can 2πΩ0 be written as: ~2 As follows from (4), the attraction of dipoles at the in- E ≃ k 2 . (15) teraction of the x and y-component will occur if these 2mea0 oscillations are in antiphase, i.e. if the dipoles are sepa- In accordance with the virial theorem [10], if a par- rated along these axes on the distance equals to half of ticle executes a finite motion, its potential energy Ep the wavelength: should be associated with its kinetic energy Ek through the simple relation |Ep| = 2Ek. Λ0 c Lx,y = = . (6) In this regard, we find that the amplitude of the zero- 2 4πΩ 0 point oscillations of an electron in a cell is: In this case √ ( ) a0 ≃ 2λF aB. (16) ′ − 2 x1x2 y1y2 z1z2 H = 4e 3 + 3 + 2 3 . (7) Lx Ly Lz 3.3 The condensation temperature Assuming that the electronic clouds have isotropic oscil- Hence the interaction energy, which unites particles lations with amplitude a0 for each axis into the condensate of ordered zero-point oscillations

2 x1 = x2 = y1 = y2 = z1 = z2 = a0 (8) ≡ ′ 3 3 e aB ∆0 H = 18π α 2 , (17) λF we obtain e2 1Of course, the electrons are quantum particles and their fixa- H′ = 576π3 Ω3a2. (9) c3 0 0 tion cannot be considered too literally. Due to the Coulomb forces of ions, it is more favorable for collectivized electrons to be placed near the ions for the shielding of ions fields. At the same time, 3.2 The zero-point oscillations ampli- collectivized electrons are spread over whole metal. It is wrong tude to think that a particular electron is fixed inside a cell near to a particular ion. But the spread of the electrons does not play a The principal condition for the superconducting state fundamental importance for our further consideration, since there are two electrons near the node of the lattice in the divalent metal formation, that is the ordering of zero-point oscillations, at any given time. They can be considered as located inside the is realized due to the fact that the paired electrons, cell as averaged. Superconductivity and Superfluidity-Part II: Superconductivity as a Consequence 26 of Ordering of Zero-point Oscillations in Electron Gas where α = 1 is the fine structure constant. Table 1. The comparison of the calculated values of supercon- 137 ductors critical temperatures with measured Fermi temperatures Comparing this association energy with the Fermi energy (12), we obtain Tc superconductor Tc,K TF ,K TF ∆0 3 ≃ · −5 Eq(22) = 9πα 1.1 10 . (18) − E Cd 0.51 1.81 · 105 2.86 · 10 6 F − Zn 0.85 3.30 · 105 2.58 · 10 6 − Ga 1.09 1.65 · 105 6.65 · 10 6 − Assuming that the critical temperature below which Tl 2.39 4.67 · 105 5.09 · 10 6 − In 3.41 7.22 · 105 4.72 · 10 6 the possible existence of such condensate is approxi- − Sn 3.72 7.33 · 105 5.08 · 10 6 − Hg 4.15 1.05 · 106 3.96 · 10 6 mately equal − 1 ∆ Pb 7.19 1.85 · 106 3.90 · 10 6 T ≃ 0 (19) c 2 k Table 2. The comparison of the calculated values of supercon- (the coefficient approximately equal to 1/2 corresponds ductors critical temperatures with measurement data to the experimental data, discussed below in the section (4.6)). super- Tc, γ, Tc,K erg T (calc) con- (measur) (calc) c cm3K2 Tc(meas) After substituting obtained parameters, we have ductors K Eq.(23) Cd 0.517 532 0.77 1.49 −6 Zn 0.85 718 1.41 1.65 Tc ≃ 5.5 · 10 TF (20) Ga 1.09 508 0.70 0.65 Tl 2.39 855 1.99 0.84 In 3.41 1062 3.08 0.90 The experimentally measured ratios Tc for I-type su- Sn 3.72 1070 3.12 0.84 TF Hg 4.15 1280 4.48 1.07 perconductors are given in Table (1) and in Fig.(3). Pb 7.19 1699 7.88 1.09 The straight line on this figure is obtained from Eq.(20), which as seen defines an upper limit of critical temperatures of I-type superconductors. 4.2 The relation of critical parameters of type-I superconductors 4 The condensate of zero-point The phenomenon of condensation of zero-point oscillations in the electron gas has its characteristic features. oscillations and type-I super- There are several ways of destroying the zero-point conductors oscillations condensate in electron gas: Firstly, it can be evaporated by heating. In this case, 4.1 The critical temperature of type-I evaporation of the condensate should possess the prop- superconductors erties of an order-disorder transition. Secondly, due to the fact that the oscillating electrons In order to compare the critical temperature of the carry electric charge, the condensate can be destroyed condensate of zero-point oscillations with measured crit- by the application of a sufficiently strong magnetic field. ical temperatures of superconductors, at first we should For this reason, the critical temperature and critical make an estimation on the Fermi energies of supercon- magnetic field of the condensate will be interconnected. ductors. For this we use the experimental data for the This interconnection should manifest itself through Sommerfeld‘s constant through which the Fermi energy the relationship of the critical temperature and critical can be expressed: field of the superconductors, if superconductivity occurs ( ) as result of an ordering of zero-point fluctuations. 2 2 ( )2/3 2 π k ne 1 · π k 1/3 Let us assume that at a given temperature T < T γ = E = ~ mene (21) c 4 F 2 3 the system of vibrational levels of conducting electrons consists of only two levels: So on the basis of Eqs.(12) and (21), we get: firstly, basic level which is characterized by an anti- ( ) ( ) 2 2 ~2 3 phase oscillations of the electron pairs at the distance pF (γ) ≃ 12 2 kTF (γ) = 2 γ . (22) Λ0/2, and 2me k 2me On base of these calculations we obtain possibility to relate directly the critical temperature of a superconductor with the experimentally measurable parameter: with its electronic specific heat. Taking into account Eq.(20), we have:

2 ∆0 ≃ Θγ , (23) where the constant [ ] π2 α~2 3 K4cm6 Θ ≃ 31 ≃ 6.65 · 10−22 . (24) k kme erg

The comparison of the calculated parameters and measured data ([12],[11]) is given in Table (1)-(2) Figure 3. The comparison of critical temperatures Tc of type-I and in Fig.(3) and (8). superconductors with their Fermi temperatures TF . The straight line is obtained from Eq.(20). Universal Journal of Physics and Application 2(1): 22-35, 2014 27

secondly, an excited level characterized by in-phase oscillation of the pairs. Let the population of the basic level be N0 particles and the excited level has N1 particles. Two electron pairs at an in-phase oscillations have a high energy of interaction and therefore cannot form the condensate. The condensate can be formed only by the particles that make up the difference between the populations of levels N0 − N1. In a dimensionless form, this difference defines the order parameter: N N Ψ = 0 − 1 . (25) N0 + N1 N0 + N1 In the theory of superconductivity, by definition, the order parameter is determined by the value of the energy gap Ψ = ∆T /∆0. (26) Figure 4. The temperature dependence of the value of the gap When taking a counting of energy from the level ε0, we in the energetic spectrum of zero-point oscillations calculated on obtain Eq.(28).

∆ N − N e2∆T /kT − 1 T = 0 1 ≃ = th(2∆ /kT ). (27) 2∆ /kT T ∆0 N0 + N1 e T + 1 Passing to dimensionless variables δ ≡ ∆T , t ≡ kT and ∆0 kTc β ≡ 2∆0 we have kTc eβδ/t − 1 δ = = th(βδ/t). (28) eβδ/t + 1 This equation describes the temperature dependence of the energy gap in the spectrum of zero-point oscillations. It is similar to other equations describing other physical phenomena, that are also characterized by the existence of the temperature dependence of order parame- E ters [13],[14]. For example, this dependence is similar to Figure 5. The comparison of the critical energy densities T (Eq.(30)) and E (Eq.(31)) for the type-I superconductors. temperature dependencies of the concentration of the su- H perfluid component in liquid helium or the spontaneous magnetization of ferromagnetic materials. This equation As a result, we acquire the condition: is the same for all order-disorder transitions (the phase 1 H2 transitions of 2nd-type in the Landau classification). n ⟨∆ ⟩ = c . (32) The solution of this equation, obtained by the itera- 2 0 0 8π tion method, is shown in Fig.(4). This creates a relation of the critical temperature to the This decision is in a agreement with the known tran- critical magnetic field of the zero-point oscillations con- scendental equation of the BCS, which was obtained by densate of the charged bosons. the integration of the phonon spectrum, and is in a sat- The comparison of the critical energy densities ET and isfactory agreement with the measurement data. E for type-I superconductors are shown in Fig.(5). As After numerical integrating we can obtain the averag- H shown, the obtained agreement between the energies ET ing value of the gap: (Eq.(30)) and EH (Eq.(31)) is quite satisfactory for type- ∫ 1 I superconductors [11],[12]. A similar comparison for ⟨ ⟩ ∆ = ∆0 δdt = 0.852 ∆0 . (29) type-II superconductors shows results that differ by a 0 factor two approximately. The reason for this will be To convert the condensate into the normal state, we considered below. The correction of this calculation, has must raise half of its particles into the excited state (ac- not apparently made sense here. The purpose of these cording to Eq.(27), the gap collapses under this condi- calculations was to show that the description of super- tion). To do this, taking into account Eq.(29), the unit conductivity as the effect of the condensation of ordered volume of condensate should have the energy: zero-point oscillations is in accordance with the avail- ( )3/2 able experimental data. This goal is considered reached 1 0.85 me 5/2 ET ≃ n0⟨∆0⟩ ≈ ∆ , (30) in the simple case of type-I superconductors. 2 2 2π2α~2 0 On the other hand, we can obtain the normal state of an electrically charged condensate when applying a mag- 4.3 The critical magnetic field of superconductors netic field of critical value Hc with the density of energy: H2 The direct influence of the external magnetic field E = c . (31) H 8π of the critical value applied to the electron system is Superconductivity and Superfluidity-Part II: Superconductivity as a Consequence 28 of Ordering of Zero-point Oscillations in Electron Gas

This band of energy can be ﬁlled by N∆ particles: !''# ()*+ ∫ & EF

$ % E E E N∆ = 2 F ( )D( )d . (35) EF −∆

# " E 1 ! Where F ( ) = E−µ is the Fermi-Dirac function and e τ +1 D(E) is number of states per an unit energy interval, a deuce front of the integral arises from the fact that there

are two electron at each energy level. To find the density of states D(E), one needs to find the difference in energy of the system at T = 0 and finite

temperature: ∫ ∫ ! ∞ EF ∆E = F (E)ED(E)dE − ED(E)dE. (36) 0 0 Figure 6. The comparison of the calculated energy of super- For the calculation of the density of states D(E), we conducting pairs in the critical magnetic field with the value of the superconducting gap. Here, the following key applies: filled must note that two electrons can be placed on each level. triangles - type-II superconductors, empty triangles - type-I su- Thus, from the expression of the Fermi-energy Eq.(12) perconductors. On vertical axis - logarithm of the product of the we obtain calculated value of the oscillating dipole moment of an electron pair on the critical magnetic field is plotted.On horizontal axis - 1 · dne 3ne 3γ the value of the gap is shown. D(EF ) = = = 2 2 , (37) 2 dEF 4EF 2k π where too weak to disrupt the dipole-dipole interaction of two ( ) 2 2 ( )3/2 2 paired electrons: π k ne 1 · π k 1/3 γ = E = ~ mene (38) µBHc ≪ kTc. (33) 4 F 2 3

In order to violate the superconductivity, the ordering is the Sommerfeld constant 2. of the electron zero-point oscillations must be destroyed. Using similar arguments, we can calculate the number For this the presence of relatively weak magnetic field is of electrons, which populate the levels in the range from required. EF − ∆ to EF . For an unit volume of material, Eq.(35) At combing of Eqs.(32),(30) and (16), we can express can be rewritten as: the gap through the critical magnetic field and the mag- ∫ 0 dx nitude of the oscillating dipole moment: · E n∆ = 2kT D( F ) x . (39) − ∆0 (e + 1) kTc 1 ∆ ≈ e a H . (34) 0 0 c By supposing that for superconductors ∆0 = 1.86, as 2 kTc a result of numerical integration we obtain The properties of the zero-point oscillations of the elec- ∫ 0 trons should not be dependent on the characteristics of dx − x 0 ≈ x = [x ln(e + 1)]−1.86 1.22. (40) the mechanism of association and also on the condition − ∆0 (e + 1) of the existence of electron pairs. Therefore, we should kTc expect that this equation would also be valid for type-I Thus, the density of electrons, which throw up above the superconductors, as well as for II-type superconductors Fermi level in a metal at temperature T = Tc is (for II-type superconductor Hc = Hc1 is the first critical ( ) field) 3γ ne(Tc) ≈ 2.44 kTc. (41) An agreement with this condition is illustrated on the k2π2 Fig.(6). Where the Sommerfeld constant γ is related to the volume unit of the metal. 4.4 The density of superconducting car- From Eq.(6) it follows riers λF L0 ≃ (42) Let us consider the process of heating the electron πα gas in metal. When heating, the electrons from levels and this forms the ratio of the condensate particle den- slightly below the Fermi-energy are raised to higher lev- sity to the Fermi gas density: els. As a result, the levels closest to the Fermi level, from which at low temperature electrons were forming n λ3 0 = F ≃ (πα)3 ≃ 10−5. (43) bosons, become vacant. 3 ne L0 At critical temperature T , all electrons from the lev- c 2 els of energy bands from E − ∆ to E move to higher It should be noted that because on each level two electrons can F F be placed, the expression for the Sommerfeld constant Eq.(38) levels (and the gap collapses). At this temperature su- contains the additional factor 1/2 in comparison with the usual perconductivity is therefore destroyed completely. formula in literature [14] Universal Journal of Physics and Application 2(1): 22-35, 2014 29

Table 3. The comparison of the superconducting carriers density λF n0 Table 4. The ratios Λ and n for type-I superconductors at T = 0 with the density of thermally activated electrons at 0 e T = Tc ( ) 3 λF n0 λF super- λ , Λ0, = F Λ0 ne Λ0 − − supercon- 10 8cm 10 6cm -ductor Eq(47) Eq(6) 2n0 − − con- n0 ne(Tc) Cd 3.1 1.18 2.6 · 10 2 1.8 · 10 5 ne(Tc) − − Zn 2.3 0.92 2.5 · 10 2 1.5 · 10 5 − − ductor Ga 3.2 0.81 3.9 · 10 2 6.3 · 10 5 − − 17 18 Tl 1.9 0.55 3.4 · 10 2 4.3 · 10 5 Cd 6.11 · 10 1.48 · 10 0.83 − − In 1.5 0.46 3.2 · 10 2 3.8 · 10 5 − − · 18 · 18 Sn 1.5 0.44 3.4 · 10 2 4.3 · 10 5 Zn 1.29 10 3.28 10 0.78 − − Hg 1.3 0.42 3.1 · 10 2 2.9 · 10 5 18 18 − − Ga 1.85 · 10 2.96 · 10 1.25 Pb 1.0 0.32 3.1 · 10 2 2.9 · 10 5 Al 2.09 · 1018 8.53 · 1018 0.49 Tl 6.03 · 1018 1.09 · 1019 1.10 In 1.03 · 1019 1.94 · 1019 1.06 Sn 1.18 · 1019 2.14 · 1019 1.10 Hg 1.39 · 1019 2.86 · 1019 0.97 Pb 3.17 · 1019 6.58 · 1019 0.96

When using these equations, we can ﬁnd a linear dimension of localization for an electron pair: Λ 1 L = 0 ≃ . (44) 0 1/3 2 πα(ne) or, taking into account Eq.(16), we can obtain the relation between the density of particles in the condensate and the value of the energy gap:

~2 ≃ 2 2/3 ∆0 2π α n0 (45) me Figure 7. The comparison of the number of superconducting carriers at T = 0 with the number of thermally activated electrons or ( ) 1 m 3/2 at T = Tc. n = = e ∆ . (46) 0 3 2 ~2 0 L0 2π α It should be noted that the obtained ratios for the This comparison is shown in Table (3) and Fig.7. zero-point oscillations condensate (of bose-particles) dif- (Data has been taken from the tables [11], [12]). fer from the corresponding expressions for the bose- From the data described above, we can obtain the con- condensate of particles, which can be obtained in many dition of destruction of superconductivity, after heating courses (see eg [13]). The expressions for the ordered for superconductors of type-I, as written in the equation: condensate of zero-point oscillations have an additional n (T ) ≃ 2n (48) coeﬃcient α on the right side of Eq.(45). e c 0

The de Broglie wavelengths of Fermi electrons ex- 4.5 The sound velocity of the zero-point pressed through the Sommerfelds constant oscillations condensate The wavelength of zero-point oscillations Λ in this 2π~ π k2m 0 λ = ≃ · e (47) model is an analogue of the Pippard coherence length in F p (γ) 3 ~2γ F the BCS. As usually accepted [12], the coherence length ~ ξ = vF . The ratio of these lengths, taking into account are shown in Tab.3. 4∆0 In accordance with Eq.(42), which was obtained at Eq.(20), is simply the constant: the zero-point oscillations consideration, the ratio λF ≃ Λ0 Λ0 2 2 −3 · −2 ≈ 8π α ≈ ·10 . (49) 2.3 10 . ξ In connection with this ratio, the calculated ratio of the zero-point oscillations condensate density to the density of fermions in accordance with Eq.(43) should be The attractive forces arising between the dipoles lo- −5 Λ0 near to 10 . cated at a distance 2 from each other and vibrating in It can be therefore be seen, that calculated estima- opposite phase, create pressure in the system: tions of the condensate parameters are in satisfactory 2 agreement with experimental data of superconductors. ≃ d∆0 ≃ dΩ P 6 . (50) dV L0 Based on these calculations, it is interesting to com- In this regard, sound into this condensation should prop- pare the density of superconducting carriers n at T = 0, 0 agate with the velocity: which is described by Eq.(46), with the density of normal √ carriers n (T ), which are evaporated on levels above E e c F ≃ 1 dP at T = T and are described by Eq.(41). cs . (51) c 2me dn0 Superconductivity and Superﬂuidity-Part II: Superconductivity as a Consequence 30 of Ordering of Zero-point Oscillations in Electron Gas

Table 5. The value of ratio ∆0/kTc obtained experimentally for Table 6. The external electron shells of elementary type-II su- type-I superconductors perconductors

superconductors electron shells ∆0 2 2 superconductor Tc,K ∆0,mev T i 3d 4s kTc V 3d3 4s2 Cd 0.51 0.072 1.64 Zr 4d2 5s2 Zn 0.85 0.13 1.77 Nb 4d3 5s2 Ga 1.09 0.169 1.80 Mo 4d4 5s2 Tl 2.39 0.369 1.79 T c 4d5 5s2 In 3.41 0.541 1.84 Ru 4d6 5s2 Sn 3.72 0.593 1.85 La 5d1 6s2 Hg 4.15 0.824 2.29 Hf 5d2 6s2 Pb 7.19 1.38 2.22 T a 5d3 6s2 W 5d4 6s2 Re 5d5 6s2 After the appropriate substitutions, the speed of sound Os 5d6 6s2 in the condensate can be expressed through the Fermi Ir 5d7 6s2 velocity of electron gas √ 2 3 −2 characteristic interaction of free electrons with the elec- cs ≃ 2π α vF ≃ 10 vF . (52) trons of the unﬁlled d-shell.

The condensate particles moving with velocity cS have Since the heat capacity of the ionic lattice of metals is the kinetic energy: negligible at low temperatures, only the electronic subsystem is thermally active . 2 ≃ 2mecs ∆0. (53) At T = 0 the superconducting careers populates the energetic level EF − ∆0. During the destruction of su- Therefore, by either heating the condensate to the crit- perconductivity through heating, an each heated career ical temperature when each of its volume obtains the increases its thermal vibration. If the effective velocity energy E ≈ n0∆0, or initiating the current of its par- of vibration is vt, its kinetic energy: ticles with a velocity exceeding cS, can achieve the de- 2 mvt struction of the condensate. (Because the condensate of Ek = ≃ ∆0 (55) charged particles oscillations is considered, destroying 2 its coherence can be also obtained at the application of Only a fraction of the heat energy transferred to the a sufficiently strong magnetic field. See below.) metal is consumed in order to increase the kinetic energy of the electron gas in the transition metals. 4.6 The relationship ∆ /kT Another part of the energy will be spent on the mag- 0 c netic interaction of a moving electron. From Eq.(48) and taking into account Eqs.(23),(41) At contact with the d-shell electron, a freely moving and (46), which were obtained for condensate, we have: electron induces onto it the magnetic field of the order of value: e v ∆0 H ≈ . (56) ≃ 1.86. (54) 2 rc c kTc The magnetic moment of d-electron is approximately This estimation of the relationship ∆0/kTc obtained equal to the Bohr magneton. Therefore the energy of for condensate has a satisfactory agreement with the the magnetic interaction between a moving electron of measured data [11], for type-I superconductors as listed conductivity and a d-electron is approximately equal to: in Table (5).3 e2 v Eµ ≈ . (57) 2rc c 5 The estimation of properties of This energy is not connected with the process of destruc- type-II superconductors tion of superconductivity. Whereas, in metals with a filled d-shell (type-I su- In the case of type-II superconductors the situation is perconductors), the whole heating energy increases the more complicated. kinetic energy of the conductivity electrons and only a In this case, measurements show that these metals small part of the heating energy is spent on it in transi- have an electronic specific heat that has an order of value tion metals: E mv greater than those calculated on the base of free electron k ≃ t E E aB. (58) gas model. µ + k h The peculiarity of these metals is associated with the So approximately specific structure of their ions. They are transition met- Ek aB als with unfilled inner d-shell (see Table 6). ≃ . (59) E + E L It can be assumed that the increase in the electronic µ k 0 specific heat of these metals should be associated with a 3In the BCS-theory ∆0 ≃ 1.76. kTc Universal Journal of Physics and Application 2(1): 22-35, 2014 31

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Figure 9. The comparison of the calculated parameter Θγ2 with Figure 8. The comparison of the calculated values of critical the measurement of the critical temperatures of elementary super- temperatures of superconductors with measurement data. Circles conductors and some superconducting compounds. relate to type-I superconductors, squares show type-II superconductors. On the abscissa, the measured values of critical temperatures are plotted, on ordinate, the calculated estimations are the specific heat of superconductors I and II-types are plotted. The calculations of critical temperatures for type-I su- differing considerably. perconductors were made by using Eq.(23) and the estimations for type-II superconductors was obtained by using Eq.(60). There are some difficulties by determining the answer this way: as we do not precisely know the density of the electron gas in high-temperature superconductors. Therefore, whereas the dependence of the gap in type- However, the densities of atoms in metals do not dif- I superconductors from the heat capacity is defined by fer too much and we can use Eq.(23) for the solution Eq.(23), it is necessary to take into account the relation of the problem of the I- and II-types superconductors Eq.(59) in type-II superconductors for the determination distinguishing. of this gap dependence. As a result of this estimation, If parameters of type-I superconductors are inserted we can obtain: into this equation, we obtain quite a satisfactory esti- ( ) ( ) mation of the critical temperature (as was done above, E a 1 ∆ ≃ Θγ2 k ≃ Θγ2 B , (60) see Fig.8). For the type-II superconductors‘ values, 0 E + E L 2 µ k 0 this assessment gives an overestimated value due to the where 1/2 is the fitting parameter. fact that type-II superconductors’ specific heat has additional term associated with the magnetization of d- The comparison of the results of these calculations electrons. with the measurement data (Fig.(8)) shows that for the majority of type-II superconductors the estimation This analysis therefore, illustrates a possibility where Eq.(60) can be considered quite satisfactory.4 we can divide all superconductors into two groups, as is evident from the Fig.(9). It is generally assumed that we consider alloys Nb3Sn 6 Alloys and high-temperature and V3Si as the type-II superconductors. This assumption seems quite normal because they are placed in close superconductors surroundings of Nb. Some excess of the calculated critical temperature over the experimentally measured value In order to understand the mechanism of high tem- for ceramics T a Ba Ca Cu O can be attributed to perature superconductivity, it is important to establish 2 2 2 3 10 the measured heat capacity that may have been created whether the high-T ceramics are the I or II-type su- c by not only conductive electrons, but also perconductors, or whether they are a special class of non-superconducting elements (layers) of ceramics. It is superconductors. already known that it, as well as ceramics Y Ba Cu O , In order to determine this, we need to look at the 2 3 7 belongs to the type-II superconductors. However, ce- above established dependence of critical parameters ramics (LaSr) Cu , Bi-2212 and Tl-2201, according to from the electronic specific heat and also consider that 2 4 this figure should be regarded as type-I superconductors, 4The lowest critical temperature was measured for Mg. It is which is unusual. approximately equal to 1mK. Mg-atoms in the metallic state are given two electrons into the electron gas of conductivity. It is confirmed by the fact that the pairing of these electrons, which manifests itself in the measured value of the flux quantum [9], is 7 About the London penetration observed above Tc. It would seem that in view of this metallic Mg-ion must have electron shell like the Ne-atom. Therefore it depth is logical to expect that the critical temperature of Mg can be calculated by the formula for I-type superconductors. But actually 7.1 The magnetic energy of a moving in order to get the value of Tc ≈ 1mK, the critical temperature of Mg should be calculated by the formula (60), which is applicable electron to the description of metals with an unfilled inner shell. This suggests that the ionic core of magnesium metal apparently is not To avoid these incorrect results, let us consider a bal- as simple as the completely filled Ne-shell. ance of magnetic energy in a superconductor within Superconductivity and Superfluidity-Part II: Superconductivity as a Consequence 32 of Ordering of Zero-point Oscillations in Electron Gas magnetic field. This magnetic energy is composed of 7.2 The magnetic energy and the Lon- energy from a penetrating external magnetic field and don penetration depth magnetic energy of moving electrons. By using of the standard formulas of electrodynamics The energy of external magnetic field into volume dv: [15], let us estimate the ratio of the magnetic and kinetic H2 energy of the electron (the charge of e and the mass me) E = dv. (69) while moving rectilinearly with a velocity v ≪ c. 8π The density of the electromagnetic field momentum is At a density of superconducting carriers ns, their mag- expressed by the equation: netic energy per unit volume in accordance with (68):

1 2 2 g = [EH] (61) m2v mejs 4πc EH ≃ αns = α , (70) 2 2nse While moving with a velocity v, the electric charge carrying the electric field with intensity E creates a mag- where js = 2ensvs is the density of a current of super- netic field conducting carriers. 1 Taking into account the Maxwell equation H = [Ev] (62) c 4π rotH = j , (71) with the density of the electromagnetic field momentum c s (at v ≪ c) the magnetic energy of moving carriers can be written 1 1 ( ) g = [E[vE]] = vE2 − E(v · E) (63) as 4πc2 4πc2 e2 Λ 2 EH ≃ (rotH) , (72) As a result, the momentum of the electromagnetic field 8π of a moving electron where we introduce the notation ∫ √ G = gdV = 2 √ V( ∫ ∫ ) e mec 1 2 − (64) Λ = α = αΛ . (73) = 2 v E dV E E v cosϑ dV 2 L 4πc V V 4πnse The integrals are taken over the entire space, which is oc- In this case, part of the free energy of the superconduc- cupied by particle fields, and ϑ is the angle between the tor connected with the application of a magnetic field is particle velocity and the radius vector of the observation equal to: point. By calculating the last integral in the condition of ∫ ( ) the axial symmetry with respect to v, the contributions 1 F = H2 + Λe2(rotH)2 dv. (74) from the components of the vector E, which is perpen- H 8π dicular to the velocity, cancel each other for all pairs of V elements of the space (if they located diametrically op- At the minimization of the free energy, after some simple posite on the magnetic force line). Therefore, according transformations we obtain to Eq.(64), the component of the field which is collinear to v H + Λe2rotrotH = 0, (75) E cosϑ · v (65) v thus Λe is the depth of magnetic field penetration into can be taken instead of the vector E. By taking this the superconductor. information into account, going over to the spherical co- In view of Eq.(46) from Eq.(73) we can estimate the ordinates and integrating over angles, we can obtain values of London penetration depth (see table (7.2)). The consent of the obtained values with the measure- ∫ ∞ v 2 · 2 ment data can be considered quite satisfactory. G = 2 E 4πr dr (66) 4πc r Table 7.2 If we limit the integration of the field by the Compton ~ 5 ≪ −6 e −6 electron radius rC = , then v c, we can obtain: super- λL,10 cm Λ,10 cm mec e conductors measured [17] calculated Λ/λL ∫ Eq.(73) v ∞ v e2 G = E2 · 4πr2 dr = . (67) Tl 9.2 11.0 1.2 4πc2 c2 r In 6.4 8.4 1.3 rC C Sn 5.1 7.9 1.5 Hg 4.2 7.2 1.7 In this case by taking into account Eq.(62), the magnetic Pb 3.9 4.8 1.2 energy of a slowly moving electron pair is equal to: The resulting refinement may be important for es- 2 2 2 E vG v e mev timates within the frame of Ginzburg-Landau theory, = = 2 = α . (68) 2 c 2rC 2 where the London penetration depth is used as a comparison of calculations and specific parameters of super- 5 Such effects as the pair generation force us to consider the ra- conductors. dius of the ”quantum electron” as approximately equal to Comp- ton radius [16]. Universal Journal of Physics and Application 2(1): 22-35, 2014 33

8 Three words to experimenters 8.1 Why a creation of room- temperature superconductors are hardly probably? The understanding of the mechanism of the superconducting state should open a way towards finding a solution to the technological problem. This problem was just a dream in the last century - the dream to fabricate a superconductor that was easily produced (in the sense of ductility) and had high critical temperature. In order to move towards achieving this goal, it im- Figure 10. The schematic representation of the dependence of portant to firstly understand the mechanism that limits critical temperature on the speed of sound in superconductors. On of the critical properties of superconductors. the ordinate, the logarithm of the critical temperature of super- Let us consider a superconductor with a large limiting conductor is shown. On the abscissa, the logarithm of the square current. The length of their localisation determines the of the speed of sound is shown (for Sn and Pb - the transverse velocity of sound is shown, because it is smaller. The speed of limiting momentum of superconducting carriers: sound in a film was used for yttrium-123 ceramics. The dashed ~ line shows the value of the transverse velocity of sound in sapphire, 2π as some estimation of the limit of its value. It can be seen that pc ≃ . (76) L0 this estimation leads to the restriction on the critical temperature in the range of 0oC - the dot-dashed line. Therefore, by using the Eq.(53), we can compare the critical velocity of superconducting carriers with the sound velocity: in order to obtain a thin wire, we require a plastic su- pc vc = ≃ cs (77) perconductor. 2me A solution of this problem would be to find a material and both these velocities are about a hundred times that possesses an acceptably high critical temperature smaller than the Fermi velocity. (above 80K) and also experiences a phase transition at The sound velocity in the crystal lattice of metal vs, an even higher temperature of heat treatment. It would in accordance with the Bohm-Staver relation [19], has be possible to make a thin wire from a superconductor approximately the same value: near the point of phase transition, as the elastic modules are typically not usually very strong at this stage. kTD −2 vs ≃ vF ≃ 10 vF . (78) EF This therefore, makes it possible to consider supercon- 8.2 Magnetic electron pairing ductivity being destroyed as a superconducting carrier This considered formation of mechanism for the su- overcomes the sound barrier. That is, if they moved perconducting state provides a possibility of obtaining without friction at a speed that was less than that of the estimations of the critical parameters of supercon- sound, after it gained speed and the speed of sound was ductors, which in most cases is in satisfactory agreement surpassed, it then acquire a mechanism of friction. with measured data. For some superconductors, this Therefore, it is conceivable that if the speed of sound agreement is stronger, and for other, such as Ir, Al, V in the metal lattice vs < cs, then it would create a re- (see Fig.(8)), it is expedient to carry out further theoret- striction on the limiting current in the superconductor. ical and experimental studies due to causes of deviations. If this is correct, then superconductors with high criti- The mechanism of magnetic electron pairing is also of cal parameters should have not only a high Fermi energy fundamental interest in order to further clarify this. of their electron gas, but also have a high speed of sound As was found earlier, in the cylinders made from cer- in their lattice. tain superconducting metals (Al[8] and Mg[9]), the ob- It is in agreement with the fact that ceramics have served magnetic flux quantization has exactly the same higher elastic moduli compared to metals and alloys and period above Tc and that below Tc. The authors of these also poss much higher critical temperatures (Fig.10). studies attributed this to the influence of a special effect. It seems more natural to think that the stability of the period is a result of the pairing of electrons due The dependence of the critical temperature on the to magnetic dipole-dipole interaction continuing to exist square of the speed of sound [18] is illustrated in at temperatures above Tc, despite the disappearance of Fig.(10). the material’s superconducting properties. At this tem- This figure, which can be viewed only as a rough esti- perature the coherence of the zero-point fluctuations is mation due to the lack of necessary experimental data, destroyed, and with it so is the superconductivity. shows that the elastic modulus of ceramics with a crit- The pairing of electrons due to dipole-dipole inter- ical temperature close to room temperature should be action should be absent in the monovalent metals. In close to the elastic modulus of sapphire, which is very these metals, the conduction electrons are localized in difficult to achieve. the lattice at very large distances from each other. In addition, such ceramics would be deprived of yet It is therefore interesting to compare the period of an other important quality - their adaptability. Indeed, quantization in the two cases. In a thin cylinder made Superconductivity and Superfluidity-Part II: Superconductivity as a Consequence 34 of Ordering of Zero-point Oscillations in Electron Gas

of a superconductor, such as Mg, above Tc the quanti- others, it displays different values, and therefore in a 2π~c zation period is equal to 2e . In the same cylinder of a general case it can be described by introducing of the noble metal (such as gold), the sampling period should parameter a: a be twice as large. Mi Tc = Const. (80) At taking into account Eq.(20), we can write 8.3 The effect of isotopic substitution on ∼ E ∼ 2/3 the condensation of zero-point oscil- Tc F ne . (81) lations The parameter l which characterizes the ion lattice The attention of experimentalists could be attracted obtains an increment ∆l with an isotope substitution: to the isotope effect in superconductors, which served as ∆l a ∆M a starting point of the BCS theory. In the ’50s it had = − · i , (82) been experimentally established that there is a depen- l 2 Mi dence of the critical temperature of superconductors due where Mi and ∆Mi are the mass of isotope and its in- to the mass of the isotope. Because the effect depends crement. on the ion mass, this is considered to be because of the It is generally accepted that in an accordance with the fact that it is based on the vibrational (phonon) process. ≈ 1 terms of the phonon mechanism, the parameter a 2 The isotope effect for a number of I-type supercon- for mercury. However, the analysis of experimental data ductors - Zn, Sn, In, Hg, P b - can be described by the [1]-[2] (see Fig.(Part I-4)) shows that this parameter is relationship: √ actually closer to 1/3. Accordingly, one can expect that MiTc = const, (79) the ratio of the mercury parameters is close to: where M - the mass of the isotope, T is the critical ( ) i c ∆l 1 temperature. The isotope effect in other superconduc- ( l ) ≈ − . (83) tors can either be described by other dependencies, or is ∆Mi 6 Mi absent altogether. In recent decades, however, the effects associated with the replacement of isotopes in the metal lattice have been studied in detail. It was shown that the zero-point REFERENCES oscillations of ions in the lattice of many metals are non- harmonical. Therefore, the isotopic substitution can di- [1] Maxwell E. : Phys.Rev.,78,p 477(1950). rectly affect the lattice parameters, the density of the lattice and the density of the electron gas in the metal, [2] Serin et al : Phys.Rev.B,78,p 813(1950). on its Fermi energy and on other properties of the electron subsystem. [3] Inyushkin A.V. : Chapter 12 in ”Isotops” (Editor The direct study of the effect of isotopic substitution Baranov V.Yu), PhysMathLit, 2005 (In Russian). on the lattice parameters of superconducting metals has [4] Vasiliev B.V. : Superconductivity as a consequence not been carried out. of an ordering of the electron gas zero-point oscil- The results of measurements made on Ge, Si, dia- lations, Physica C, 471,277-284 (2011) mond and light metals, such as Li ([3],[20])(researchers prefer to study crystals, where the isotope effects are [5] Vasiliev B.V. : Superconductivity and condensa- large and it is easier to carry out appropriate measure- tion of ordered zero-point oscillations, Physica C, ments) show that there is square-root dependence of the 471,277-284 (2012) force constants on the isotope mass, which was required by Eq.(79). The same dependence of the force constants [6] Vasiliev B.V. : ”Superconductivity, Superfluidity on the mass of the isotope has been found in tin [21]. and Zero-Point Oscillations” in ”Recent Advances Unfortunately, no direct experiments of the effect of in Superconductivity Research” , pp.249-280, Nova isotopic substitution on the electronic properties (such Publisher,NY(2013) as the electronic specific heat and the Fermi energy), on [7] Bardeen J.: Phys.Rev.,79,p. 167-168(1950). metals of interest to us have so far been conducted. Let us consider what should be expected in such mea- [8] Shablo A.A. et al: Letters JETPh, v.19, 7,p.457- surements. A convenient choice for the superconductor 461 (1974) is mercury, as it has many isotopes and their isotope effect has been carefully measured back in the 50s of the [9] Sharvin D.Iu. and Sharvin Iu.V.: Letters JETPh, last century as aforementioned. v.34, 5, p.285-288 (1981) The linear dependence of the critical temperature of [10] Vasiliev B.V. and Luboshits V.L.: a superconductor on its Fermi energy (Eq.(20)) and also Physics-Uspekhi, 37, 345, (1994) the existence of the isotope effect suggests the dependence of the ion density in the crystal lattice from the [11] Pool Ch.P.Jr : Handbook of Superconductivity, mass of the isotope. Let us consider what should be Academic Press, (2000) expected in such measurements. Even then, it was found that the isotope effect is de- [12] Ketterson J.B. and Song S.N.: Superconductivity, scribed by Eq.(79) in only a few superconductors. In Cambridge (1999) Universal Journal of Physics and Application 2(1): 22-35, 2014 35

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