A new variational method has been proposed for studying the equilibrium states of the interacting particle system to have been statistically described by using the density matrix. This method is used for describing conductivity electrons and their behavior in metals. The electron energy has been expressed by means of the density matrix. The interaction energy of two εkk ′ electrons dependent on their wave vectors k and k’ has been found. Energy εkk ′ has two summands. The first energy I summand depends on the wave vectors to be equal in magnitude and opposite in direction. This summand describes the repulsion between electrons. Another energy J summand describes the attraction between the electrons of equal wave vectors. Thus, the equation of wavevector electron distribution function has been obtained by using the variational method. Particular solutions of the equations have been found. It has been demonstrated that the electron distribution function exhibits some previously unknown features at low temperatures. Repulsion of the wave vectors k and – k electrons results in anisotropy of the distribution function. This matter points to the electron superconductivity. Those electrons to have equal wave vectors are attracted thus producing pairs and creating an energy gap. It is considered the influence of magnetic field on the superconductor. This explains the phenomenon of Meissner and Ochsenfeld. We consider a new possibility of penetration of the external magnetic field into the superconductor.
Kamerlingh and Onnes were those who could discover the phenomenon of superconductivity in 1911 [
where
The plane (H, T) is represented by a superconductive state phase diagram. The substance in superconductive state S is found under the curve (1.1) and that to be in normal state N―above the curve. Any superconductor introduced by such state diagram is known as the type-I superconductor.
Brothers Fritz and Heinz London could develop the first macroscopic theory of superconductivity in 1935 [
These facts were acknowledged a priori. For the superconductor magnetic field, the brothers obtained the following equation:
where
Value l is called a London length of the magnetic field penetration in a superconductor.
Let us consider a superconductor occupying the half-space y > 0, then the region y < 0 is filled with vacuum where a magnetic field B0 can be found as directed along the boundary surface (see
The Equation (1.4) may be expressed as follows:
It should be noted that such function does not agree with the Meissner-Ochsenfeld effect. When magnetic- field strength produced on conductor surface exceeds its critical value Hc(T), the superconductivity vanishes. But, on the whole, the function (1.5) does not depend on any critical field.
The Meissner-Ochsenfeld effect is an experimental fact. If the function (1.5) does not depend on the critical field, then the Londons theory is not true. Moreover, the superconductivity shall be referred to equilibrium state of any matter, i.e. all values are not to be dependent on time t. But the very dependences can be used for deriving the London’s equation.
Any quantum mechanics system can be much closely described by using the density matrix [
where
The two-particle density matrix
Taking into account the above equality let us use the following approximate expression named as the mean field approximation for the two-particle density matrix:
Internal energy of the identical particle system may be exactly expressed through the use of the density matrices
where
The coordinate representation to which the Hamiltonians commonly assigned passes to a certain α-represen- tation through the use of system of the orthonormal wave functions
where the above sign of integration is a symbol of both the coordinate integration and spin variable summation; Φ12 is a Slater two-particle wave function:
On substituting such function in the formula (3.4), we can obtain an antisymmetric matrix as follows:
where
With the expression (2.3) substituted in the formula (3.1), the following formula may be obtained:
that agrees with the mean field approximation.
Any single-particle density matrix may be represented by a diagonal element and, particularly, it can be formulated as follows:
where n is a set of quantum numbers that defines state of newly represented single particle;
The n-representation passes to the α-representation to which specific matrix elements of the Hamiltonians
where
Now, we transform the energy (3.8) by the formula (4.3) to obtain the expression below:
where
―n-state kinetic particle energy,
―double-particle interaction energy containing one n-state particle and another n′-state particle;
If to use the distribution function
For finding the fermion state distribution function
where μ is a chemical potential. The variational principle may be expressed as follows for the distribution function
where
The arrangement of atoms within a given type of crystal can be described in terms of the Bravais lattice with location of the atoms in an isolated unit cell specified. We will determine position of one of the atoms in the unit cell using vector R. Such description agrees with simple crystal lattices including those where type-I superconductivity can be observed. Let us consider s as a set of quantum numbers defining the wave function of one of the states of a valence electron located within the neighborhood of the atom, the position of which is determined by vector R. Now, we can write the orthonormal system of wave functions that describe localized electron states by using the form shown below:
where
The normalizing condition (7.2) may be expressed by:
Should a system of electrons in metal be brought to a state of thermodynamic equilibrium, any of the States
The probability
where
Let is assume that number of electron states localized in site R, as described by the Wannier functions
is formulated as follows:
where NL is number of crystal lattice sites,
―degree of state filling,
―number of states within one site.
Let us assume the density matrix
where the summation is performed by using wave vectors k of the first Brillouin zone; wk is a wave-vector electron distribution function that satisfies the following normalizing condition:
It follows from the equalities (7.5) and (7.10) that the density matrix
Thus, the density matrix
where
that the state κ, as described by the wave function
Let us write the Equation (6.2) for the electron wave-vector distribution function
where
The expression for two-electron and wave vector k and k′ interaction energy
herein, we use I and J that have particular dimension energy. Let us substitute this expression in the formula (8.2). Now, we will obtain as follows:
Using the above formula (8.4) it is possible to transform the Equation (8.1) as follows:
This equation describes all superconductor properties [
The Equation (8.5) contains two unknown quantities―i.e.
Taking into account that any kinetic energy is an even function:
As applies to the equations (8.5) and (9.1), the unknown functions wk and w−k are composite ones wherein the kinetic electron energy εk is used as an intervening variable:
The functions
are solutions of the equations
where
is expressed by the I and J energy relation as defined by the following parameter:
In this Section, we will consider the case when parameter J = 3I. Wherein f = 1/2.
Any solutions of the Equations (9.3) may be rapidly found by computation. A plot of the w(ε, τ) distribution function is demonstrated in
As shown in
The electron system energy in approximation of a medium field takes the form as follows:
If the distribution function ω = ω(ε) is known, it is possible to find a chemical potential and electron energy at T = 0. Such computation results have been demonstrated in the papers [
This function is featured with anisotropy at
Taking into account normalizing conditions, we can express the average electron travel velocity as follows:
If the distribution function is isotropic, the average velocity of electron v will be zero. As applies to some anisotropic functions, the distributed formula (11.1) may produce nonzero electron directed motion velocity
values ? i.e. such distribution functions are those that describe electric current. If current states are rather stable and can exist even in the absence of any external fields such traveling electron system states are to be considered as superconductive.
Let us view the state of electron gas to be described by the anisotropic distribution function (10.3). Any value taken on by the average electron velocity function may be determined against the nature of initial electron gas state. The average velocity will be zero when pairs of free and occupied states (with vectors k and ?k) are distributed over the layer S randomly, i.e. no supercurrent flow will be induced in a conductor at
When all the states to be considered in one half of the layer S (e.g. at
The distribution function should be single-valued. This means that a single distribution function value corre- sponds to a single energy value. As concerns the anisotropic distribution function, it shall be a double-valued one. It simultaneously defines two k and ?k wave vector values that agree with two values of the distribution function
For detecting any superconductivity, low current flow is supplied along a conductor. Then, temperature is decreased. A circuit should be closed when temperature drops below a specified value. Steady current is conserved and keeps flowing as long as it can. Let us consider that current flows along axis x. The pattern of the k wave-vector space electron distribution function at T = 0 is shown in
Dependence
Such dependence, as rated at various temperature τ values, is graphically represented in
The above curves are featured with the following specific characteristics. A “well” is demonstrated on each curve of the dependence
Value
according to which borders of the “well”, as demonstrated on the curve of dependence
is also extended from zero to value I at T = 0.
Let us find the wave function
where
Since an electron has a spin, the Hamiltonian
where
where
Let us write the Schrödinger equation for the wave function
and on substituting the product (13.3), we will obtain:
If the coordinate function satisfies the equation
than a spin function shall be used for solving the following equation:
where
As follows from the Equation (13.6), the proper value
As a result, we will obtain:
Now, we will accept the functions
where
where
Let us consider the energies possessed by an electron when its spin is valued differently. As follows from the formula (13.9), when an electron spin is equal to -1/2, its energy is taken as
and when an electron spin is equal to 1/2, its energy is taken as
Thus,
Therefore, the probability
Let us consider that two values of electron energy, as determined by the formula (13.9), correspond to each of the states specified in site R. So, matrix elements of the single-particle Hamiltonian may be expressed as:
As a result, we will obtain:
Since
here
In this case
where
Thus, the following formula is obtained:
As applies to the coordinate function (13.11) the normalizing condition will take up the following form:
when a system of electrons in metal is at the state of thermodynamic equilibrium, any of the states
As such, the density matrix
where
The probability
Let us substitute the formula (14.4) in the following expression for noninteracting electron energy:
Upon simple transformation we can obtain the formula shown below:
Now, we write:
where
Let us write the energy E1 as follows:
The interacting electron energy will take up the below form:
where an appropriate electron interaction energy will be formulated as:
On summing up the energies (16.4) and (16.5), we will obtain the following electron energy expression:
On minimizing the thermodynamic potential Ω, as referred to the energy (16.6), we obtain the following nonlinear equation to be applied for finding the distribution function
Let us consider
than the Equation (17.1) will take up its previous solution:
Let us consider, for example,
This means that a curve of the electron wave-vector distribution function is displaced on its right by value Λ as compared with the graphical representation without a magnetic field. As follows from the Equation (17.3), the critical temperature is equal to:
As provided by computational results, the equilibrium distribution function applicable in a wave-vector region is multiple-valued. For selecting a real distribution function, it is necessary to find the least electron energy. Now, we shall consider the case when a magnetic field destroys the superconductivity at T = 0. Let us
The graphical representation of the distribution function is displaced on its write by the above value (see
We can bring the upper integration level to ∞ since the probability of occupancy of states with the energy ε to be at a ceiling of the conductivity zone is virtually equal to zero. Now, the normalizing condition (10.1) may be expressed as:
In such a case any energy of electrons (10.2) distributed isotropically may be expressed as:
The superconductivity disappears when the distribution function is taken as:
As a result, we can obtain the following normalizing condition:
Thus
where
On calculating the above integral we will obtain:
As a result, the following formula is derived:
But such superconducting state may in any way survive and the distribution function takes up the form as:
As applies to the normalizing condition, the equation takes up the form as:
The equation gives the following chemical potential:
An electron energy may be calculated by the formula:
Let us suppose that J = 3I and Λ = I. And the following form will be obtained:
As provided by the above equation, we can obtain:
Now, we can find the difference in energy:
As we can see, the energy of state
This means that the superconductivity can vanish under magnetic field effect when its strength, as defined on a surface of a superconductor, is equal to a critical field. This condition meets Meissner-Ochsenfeld effect.
Let us consider that parameter Λ is equal to I. In this case, the distribution pattern is displaced on its right. Now, we can find the critical magnetic-field strength, as rated at T = 0, through the use of the formulas (16.2) and (18.1):
If temperature τ exceeds zero, the critical magnetic-field strength subject to the formula (16.2) is formulated as:
where
With the graphical representation of the mean electron energy dependence applied against its kinetic energy shown in
By what way does the width of the energy well behave at constant temperature (τ = const), provided that the external magnetic-field strength jumps from zero to value H? In case of absence of the magnetic field, the width of a well will be rated at
Let us substitute
When the magnetic-field strength H is equal to the critical value
Any superconductor current flow is featured with its density matrix shown in
Now, we substitute the sum (20.1) for the following integral:
Using the above formula, we calculate mean velocity for T = 0 with the formula (18.2) considered as true.
This formula makes it possible to find radii k1 and k2 of the spheres shown in
As a result, we will obtain as follows:
The layer thickness, as specified in
In case of absence of external magnetic fields, when an electron distribution pattern is described by the formula (18.9), the largest value of mean electron velocity produced in the flow of supercurrent will be formulated as:
As it is follows the respective electric current density vector can be expressed as:
With the external magnetic field applied while its strength is kept on the level of less than the critical value, the following mean electron velocity can be obtained:
If to substitute the formula (19.4), we will obtain the expression shown below:
Now, the current flowing along the superconductor will have the density determined by the current velocity (20.6):
As a result, we obtain the formula expressed as:
where
It is need to remind that we investigate equilibrium state of the superconductor. Time independent functions are to be considered only in this state. These functions satisfy the equations expressed as:
Let us consider the magnetic field effect produced on a planar border of the superconductor, provided that the magnetic field vector H runs parallel to the border. We arrange axes x lengthwise the conductor border and parallel to vector H and axis y―perpendicularly to the border (see
In this case, the Equation (21.1) takes up the form as:
Since any superconducting current is produced by a negative component
substituting the value (20.7) in this equation we will obtain the formula expressed as:
where
In view of the superposition principle, the magnetic field vector H can be considered as equal to the sum of the external and magnetic fields to be created by supercurrents:
Let us consider that any supercurrent flowing along axis z, as demonstrated in
Such current may flow along the superconductor as long as it can. As concerns to electrons, they achieve their mean velocity. The supercurrent produces am self-magnetic field featured with the strength of
Any strength produced by the supercurrent will be equal to zero at the superconductor border:
Since the strength satisfies the very condition it can be expressed as:
For the graphical representation of this function (see
Now, we create the external magnetic field running lengthwise axis x with its strength represented by
Let us substitute the complete field (21.10) in the Equation (21.4). We will obtain the formula shown below:
We subtract the Equation (21.7) from the Equation (21.11). And the following external field equation is obtained:
The external field strength, as determined at the border of the superconductor, will be equal to the given value ±H0 to be subject to change throughout an experiment:
Since the function
This function is graphically represented in
Thus, we could obtain the strength of the external magnetic field going down inside the superconductor. The
complete field (21.10) is equal to the magnetic field produced by supercurrent and to the external field, provided that
When the external field strength takes up the following form
the function (21.15) will be equal to the critical field:
With
So, it is proved that when value H0 belongs to the interval
the superconductivity will be applicable in the conductor under the Meissner-Achsenfeld effect. But if value
the superconductivity vanishes.
We have viewed in details the case when the strengths
At
As follows from the above, the Meissner-Ochsenfeld effect cannot be applicable. To remedy such condition, it is necessary to specify the external field as that changing direction of supercurrent flow. Thus, the direction of the superconductor self-magnetic field vector is considered as that matching the direction of the external magnetic field strength. As provided by the given calculations, the external magnetic field goes down inside the superconductor thanks to presence of the magnetic field produced by supercurrent.
Using the formula (20.7), let us find the supercurrent density as dependent on coordinate y. We substitute the formula (21.15) in the formula (20.7). As a result, we will obtain the following formula:
where
As follows form the above formula, the density of current goes down from the superconductor border to an exponent. In case when the external magnetic field is absent (H0 = 0), the density will be equal to
At
Let us now consider behavior of the magnetic field both inside and outside a spherically shaped superconductor. The superconductor at T > Tc, in case of absence of the superconductivity, is demonstrated in
In case when the temperature is brought to the level of below the critical value ? i.e. T < Tc, superconductivity is created in the superconductor. This very case is shown in
Now, we will deactivate the external magnetic field in the state when temperature is less than the critical one: T < Tc. In this case, the superconductivity it is not vanished. The supercurrent is not changed but remains the same as before. And the structure of self-magnetic field lines is to be changed just a little (see
Let is consider the quantum capture effect. Such capture effect is produced by any permanent magnet fixed at the constant height level. A disk-shaped superconductor is positioned below it less any supports. For a graphical layout (see
Let us assume that the external field vector H(external), superconductor self-magnetic field vector H(c) and superconductor magnetic moment vector Pm have the same orientation. According to the physical theory it is
known that in this case the potential superconductor energy produced in an external field will be equal to the following:
The superconductor obtains the capacity to move in the space. The strength that makes effect on the superconductor is coupled with its potential energy formulated as follows:
The external magnetic field is not homogeneous by nature and it has excess magnetic induction near a permanent magnet. Any force acting on the superconductor will make it move to those space regions wherein the energy (23.1) is reduced. If we direct the external magnetic field oppositely, it makes the supercurrent flow being reoriented to another direction along with the self-magnetic field. Thus, the external magnetic field, self- magnetic field and vector Pm will have the same orientation with the strength remained unchanged.
As provided by (23.1), the superconductor has the energy reduced at those areas where magnetic induction is exceeded. Therefore, the superconductor will be pulled into the region where an external field is stronger than that in other regions at x > x0, where x is a distance from a permanent magnet to the superconductor, x0 is an equilibrium position of the superconductor.
According to the formula (21.22) the supercurrent density will be approximately equal to as follows:
where
The graph in
Now consider thequantumlevitation, when the superconductoris located above thefixedpermanent magnet (see.
The potential energy when
net, the value of
function (see
Differentiating the potential energy in xand from (23.2), we construct a graph of the power of location. The graphpower is shown in
The electron model in metal, as described in this paper, may be used as a basis of an alternative superconductivity theory. Such model significantly varies from those ones applied in the modern superconductivity theory.
Here, the superconductivity is caused by repulsion of k and −k wave vector electrons but electron pairs and energy gap―by attracting of equal wave vector electrons. Such electrons are statically described in scope of a particular density matrix formalism featured with its simplicity typical for the appearance of mathematical formulas and their physical content. The problem under consideration demonstrates advantages of the density matrix method.
Influence of magnetic field effects on the superconductor has been studied in the context of this theory. The following aspects have been proved:
1. Any magnetic field is able to displace the electron wave vector distribution function pattern towards increased kinetic energy values.
2. The critical magnetic field
3. It has been found that there are dependences on penetration of the supercurrent external field strength and self-magnetic field strength into the superconductor.
4. The dependence of the supercurrent density on the depth of penetration, temperature, and strength of external and critical fields has been found.
Boris V. Bondarev, (2016) New Theory of Superconductivity. Magnetic Field in Superconductor. Effect of Meissner and Ochsenfeld. Open Access Library Journal,03,1-27. doi: 10.4236/oalib.1102418