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As shown in former papers, the nonadiabatic Heisenberg model presents a novel mechanism of Cooper pair formation which is not the result of an attractive electron-electron interaction but can be described in terms of quantum mechanical constraining forces. This mechanism operates in narrow, roughly half-filled superconducting bands of special symmetry and is evidently responsible for the formation of Cooper pairs in all superconductors. Here we consider this new mechanism within an outer magnetic field. We show that in the magnetic field the constraining forces produce Cooper pairs of non-vanishing total momentum with the consequence that an electric current flows within the superconductor. This current satisfies the London equations and, consequently, leads to the Meissner effect. This theoretical result is confirmed by the experimental observation that all superconductors, whether conventional or unconventional, exhibit the Meissner effect.

The nonadiabatic Heisenberg model [

Also within the NHM, the formation of Cooper pairs is mediated by bosons, which, however, bear the crystal spin angular momentum

The aim of this paper is to provide evidence that the constraining forces causing the formation of Cooper pairs in superconducting bands are also responsible for the Meissner effect. When superconductors are cooled below their transition temperature

However, we do not explain the Meissner effect but we restrict ourselves to derive the London equations [

The mechanism of the Cooper pair formation in a narrow, roughly half-filled superconducting band has been described in a former paper [

First we assume no outer magnetic field to be present. The Bloch functions of a superconducting band can be unitarily transformed into optimally localized spin-dependent Wannier functions which are adapted to the symmetry of the electron system [

In a system with k dependent spin directions the electrons couple to crystal- spin-1 boson excitations in order that the total crystal-spin angular-momentum is conserved during the ever-present scattering processes in the electron system, see Section 3.2 of Ref. [

since both states have exactly opposite spin directions. K denotes the operator of time inversion.

At low temperatures, the electrons form Cooper pairs consisting in each case of a Bloch state and its time inverted state. When all the electrons of the superconducting band form Cooper pairs with zero total spin-angular momen- tum, the conservation of spin angular-momentum is satisfied in the electron system alone, see the group-theoretical substantiation in Section 3.2 of Ref. [

The mechanism of Cooper pair formation can be described in terms of constraining forces produced by the crystal-spin-1 boson excitations, see Section 3.3 of Ref. [

Now assume an outer magnetic field to be switched on. An absolutely consistent mathematical description of superconductivity in an outer magnetic field would require to show that the spin-dependent Wannier functions in a superconduct- ing band may be chosen symmetry-adapted even in the presence of an outer magnetic field, as it has been carefully established [

The Hamiltonian of an electron in a solid state and in a uniform external magnetic field has the form

where

is the operator of the generalized momentum, m is the electron mass, e the proton charge,

The translation operators in the magnetic field may be written as

where

we may label the eigenfunctions of

as it was already performed by Onsager to interpret the de Haas-van Alphen Effect [

Consider a superconducting sample within an external magnetic field generated by Helmholtz coils fare away from the sample. As is well-known, the electron system within the sample is invariant under time inversion only if additionally the magnetic field

where

This problem has been overcome for special sheared solids [

without changing the outer magnetic field,

Thus, this operator

In contrast to the standard time inversion

With Equation (3) the Hamiltonian may be written as

showing immediately that

if we continue to neglect the energy of the electron spins in the magnetic field. From this result follows the significant insight that the inner time-inversion

As argued in Section 2.2, the magnetic Wannier functions are adapted to the inner time-inversion just as they are adapted to the standard time inversion in the field-free case. As a consequence, the operator

as it has been shown for the zero-field case in Section 7.3.1 of Ref. [

Since the operator

associated with the same energy, where

With Equations (3) , (9) and (10) we obtain

Remember that the direction of the electron spins depends on p in a narrow, roughly half-filled superconducting band. Just as in the field-free case, the constraining forces produced by the crystal-spin-1 excitations generate Cooper pairs with exactly vanishing total spin-angular momentum. Equation (11) ensures that the spins of the two electrons occupying the states

Hence, in a magnetic field, the total momentum

This equation gives the exact total momentum of a Cooper pair within an outer magnetic field. It shall be interpreted in the following Section 5.

Equation (19) shows that the kinetic momenta of the two Bloch states forming a Cooper pair cancel each other. However, the term

Lorentz force still is active and forces the two electrons to perform a circular motion with the same sense of rotation each. Because the two electrons move on different orbitals, the probability to meet an electron at a certain position r is different for the two electrons and, hence, their average total kinetic momentum

To determine

showing that

of an one-electron state.

Due to this interpretation (21) , the operator

becomes a translation operator commuting with

Thus, the contribution of one Cooper pair to the electric current amounts to

Equation (23) is the result of this paper. It contains both London equations [

This paper provides evidence that the constraining forces causing the formation of Cooper pairs in narrow, roughly half-filled superconducting bands are also responsible for the Meissner effect. In the framework of the nonadiabatic Heisenberg model, the Meissner effect is an intrinsic part of superconductivity.

Hirsch [

I am very indebted to Guido Schmitz for his support of my work.

Krüger, E. (2017) Constraining Forces Causing the Meissner Effect. Journal of Modern Physics, 8, 1134- 1142. https://doi.org/10.4236/jmp.2017.88074