Indeed in a magnetic field a force should act upon the vortices and leads them

Indeed in a magnetic field a force should act upon

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Indeed, in a magnetic field a force should act upon the vortices and leads them to move. But moving vortices should produce non-steady magnetic fields and, consequently, energy loss. To make the estimates assume that there is one vortex with the current j and the exter- nal current density is equal to j ex , the total current being j + j ex . Assuming that the corre- sponding contribution to the free energy is n s mv 2 s / 2 = ( n s m/ 2)( j/n s e ) 2 = (4 π/c 2 δ 2 L ) j 2 / 2 we get for the interaction energy U = 4 π c 2 δ 2 L Z j ex · j d V . Then we remember that j depends only on the difference r - r L (where r is the 2D co- ordinate in the plane perpendicular to the line). The force is F k = - U ∂r Lk ∝ - X i Z d V j ex i ∂j i ∂r Lk = X i Z d V j ex i ∂j i ∂r k =
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348 CHAPTER 17. MAGNETIC PROPERTIES -TYPE II = X i Z d V j ex i ∂j i ∂r k - ∂j k ∂r i + X i Z d V j ex i ∂j k ∂r i . The last item vanishes because by integration by parts we get div j ex = 0 . Thus F = 4 π c 2 δ 2 L Z [ j ex × curl j ] d V . Substituting the expression for curl j we get f L = Φ 0 c [ j ex × z ] for a separate vortex. The total force acting upon the vortex structure is just the Lorentz force F L = 1 c [ j ex × B ] . Now let us assume that a vortex moves with a given velocity v L and there is a viscous braking force f v = - η v L . As a result of force balance, v L = Φ 0 ηc [ j ex × z ] . We see that v L j ex B . According to the laws of electrodynamics, such a motion produces the electric field E = 1 c [ B × v L ] = Φ 0 B ηc 2 j ex . We observe the Ohm’s law with the resistivity ρ = Φ 0 B ηc 2 . If we assume that at B = H c 2 ρ = ρ n (i.e. to the resistivity of the normal phase) we get η ( H c 2 ) = Φ 0 H c 2 ρ n c 2 ρ = ρ n B H c 2 . This expression is only a very rough order-of-magnitude estimate. In fact the viscosity η is a very complicated and interesting function of both the temperature and magnetic field. From this point of view we get the conclusion that the superconductivity is destroyed at H = H c 1 . Fortunately, this statement is wrong. In real materials there is pinning , i.e. the vortices become pinned by the defects. One kind of pinning is the surface barrier we have discussed earlier. It is clear that large-scale defects with size greater than ζ should be very effective. To get a simple estimate let us consider a cavity in the SC with d. ζ. Suppose that the core is in the normal state that leads to the extra energy H 2 c ζ 2 per unit length. If the
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17.4. NON-EQUILIBRIUM PROPERTIES. PINNING. 349 vortex passes through the cavity, this energy is absent. Consequently, there is attraction between the line and the cavity the force being of the order f p H 2 c ζ (we have taken into account that at the distance ζ the vortex collides with its image). Combining this expression with the expression for the Lorentz force we find the critical current density able to start the motion j c H c c δ L (we have used the relations f L = j ex Φ 0 /c and H c Φ 0 L ζ ).
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  • Spring '10
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  • Physics, Cubic crystal system, periodic structures, Reciprocal lattice, Lattice Vibrations

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