- Email: [email protected]
Internal spinor fields and London's equation
Physica C 369 (2002) 370–373 www.elsevier.com/locate/physc
Internal spinor fields and London’s equation Valerio Dallacasa a,b,* a
Laboratory for Materials Analysis, Scientific and Technological Department, Verona University, Verona, Italy b INFM, Physics Department, Parma University, Parma, Italy
Abstract An equivalence between Dirac’s equation of an electron subject to an applied magnetic and Maxwell’s equations by introducing electric and magnetic fields linearly related to the spinor wavefunctions. The internal magnetic field adds to the applied field and screens it due to its diamagnetic character. For a superconductor this screening is the origin of the expulsion of magnetic vortices and turns into a non-linear Ampere’s law, of which London’s equation is an approximate linearized form obtained on fixing the value of the internal field to its average value. This equation can be used to predict the critical field, intermediate normal/superconductor phases as in type I materials and non-linear effects beyond London’s state. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Internal spinor fields; London equation; Dirac spinors
1. Introduction In this paper we examine a possible origin and significance of the London’s equation, which is the basis for the comprehension of the expulsion of magnetic fields from a superconductor. The London’s phenomenological equation states that the current density is proportional to the vector potential and such a result is normally justified on the basis of a certain rigidity of the wavefunction, i.e. that the magnetic field has no effect on the wavefunction which therefore remains equal to the one in the absence of the field [1]. The expulsion and creation of vortices within a superconductor and
* Address: Laboratory for Materials Analysis, Scientific and Technological Department, Verona University, Verona, Italy. Tel.: +39-045-802-7936; fax: +39-045-802-7928. E-mail address: [email protected] (V. Dallacasa).
their dynamics are of general interest for the understanding of type I and II materials in a magnetic field, but novel techniques of investigation including single vortex visualization techniques by electron microscopy and analyses at extreme length scales, i.e. in submicron and nano-structured materials, or conditions, i.e. extremely high magnetic fields, suggest a particular importance of the understanding of the single vortex dynamics and structure, also in connection with organized states like liquid and solid vortex lattice and their confinement for high Tc . A vortex should coincide with the quantum motion, therefore we introduce the concept of internal electric and magnetic fields as consequence of the quantum motion and find that the origin of the cancellation of an applied magnetic field in the interior of the superconductor is the internal magnetic field which turns out to opposed to the applied field and then able to screen it to a certain
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 2 7 9 - 5
V. Dallacasa / Physica C 369 (2002) 370–373
extent. There is also an internal electric field which however does not couple to the motion since it turns out to be perpendicular to the current. The current itself turns out to be perpendicular to the internal magnetic field and parallel to the vector potential. The internal magnetic field gives also an explanation of the existence of a critical field and the transition to the normal state when the applied field is big enough to overcome the diamagnetic internal field. London’s state is found as the solution of a linearized form of the non-linear equation describing the magnetic state. 2. The equivalence between internal fields and Dirac’s spinors The method we use to study this matter is to find an equivalence between Dirac’s and Maxwell’s equations, which renders the quantum motion equivalent to an electromagnetic field. by mapping the wavefunction into the field. The mapping exists only under certain conditions in which the electric field, the magnetic field and the vector potential are orthogonal each other and under these circumstances a current density and charge density, i.e. the constitutive equations, remain defined. Within this method each vortex is put in one to one correspondence with the wavefunction. Under the conditions e ~ ~ 2m0 c ð A ^ BÞ z þ Ez ¼ 0; hc h ð1Þ ~ A~ E ¼ 0; A^~ EÞ ¼ 0 ð~ A^~ EÞ ¼ ð~ x
x
where ~ A is any real vector, m0 a scalar and ~ E, ~ B are fields, on introducing the mapping W ¼ aðBz ; Bx iBy ; iEz ; Ey þ iEx Þeimc
2 t= h
ð2Þ
where a is an arbitrary constant and with sources of the form 4p 2m0 c e Jx ¼ Ey þ ð~ A^~ BÞ y ; c h hc 4p 2m0 c e Jy ¼ Ex ð~ A^~ BÞ x ; c h hc 4p e Jz ¼ ð~ A ~ BÞ; c hc 4p e ~ ~ q ¼ ðA ^ E Þz c hc
ð3Þ
371
the eight Maxwell’s equations can be written as four equations in the compact Dirac’s form e oW A þ m0 c2 W ¼ ih ð4Þ c~ a ~ p ~ c ot for a particle with rest mass m0 in a magnetic field generated by the vector potential ~ A. Conversely it can be shown that from the Dirac’s equation in a magnetic field described by the vector potential ~ A, fields ~ E and ~ B can be defined obeying Maxwell’s equations if the constitutive equations, the mapping and conditions above are valid. The reality of the fields restricts the equivalence only for certain bi-spinor solutions W of the Dirac’s equation. The correspondence discussed here makes use of a linear relationship between the fields and the wavefunction; therefore the fields have spinor character and are a direct realization of the wavefunction (comprising its phase). Attempts to introduce real fields for the structure of the electron are well known [2]. In the more common scheme a relation quadratic in the wavefunction [3] is found which generates a tensorial electromagnetic field. We can view the spinor fields as fields associated at any point of space where the motion (including internal motions) of the electron occurs, while the tensorial fields are those produced by the particle in the space around it. This fact implies that a possible measure of the wavefunction in visualization techniques can be achieved. As seen from the constitutive equations the current density has two contributions, a magnetic term proportional to the vector potential and an electric term, this latter arising from the zitterbewegung, through the frequency term 2m0 c2 =h of this motion [3]. As a whole, the associated spinor electromagnetic field can be interpreted as an internal electromagnetic field generated by the quantum motion of the electron in a magnetic field BA deriving from the vector potential ~ A. In general, for consistency, this field should include the internal field ~ B and the external field ~ B0 , i.e. the relation will hold ~ BA ¼ ~ Bþ~ B0 . The field ~ B0 will then be screened in the interior of the system, due to the internal field. In the particular case considered here the Dirac’s hamiltonian includes only a coupling with a magnetic field and no electric coupling.
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V. Dallacasa / Physica C 369 (2002) 370–373
Therefore the internal magnetic field should contribute to the magnetic coupling whereas the electric field should not give rise to any electric-type coupling.
4. The London ansatz, the conditions for its existence and generalizations Using the results above, we can calculate the total magnetic field through Ampere’s law and we find
3. An expression for the internal fields rot ~ BA ¼ In the case we are studying in which there is no electric field term in the Dirac’s equation, and on simplifying to the case in which the motion of the Dirac’s particle occurs only along planes perpendicular to the z-direction, the Dirac’s field can be calculated as W ¼ ðE þ H ÞU and for up and down spin can be reduced to the forms cðPx þ iPy Þ/ w" ¼ 0; /; ;0 ; E þ mc2 ð5Þ cðPx iPy Þ/ ;0 w# ¼ 0; /; E þ mc2 where Px ¼ px ðe=cÞAx , Py ¼ py ðe=cÞAy and U" ¼ ð/; 0; 0; 0Þ, U# ¼ ð0; /; 0; 0Þ are solutions of the second-order equation [3]. For a magnetic field pointing towards the positive(negative) z-axis the states with spin up(down) will correspond to the ground state; therefore we shall consider the case of W" as appropriate for BA > 0 and W# for BA < 0. For simplicity we shall examine the case of spin up, for spin down similar results are obtained. We can use a gauge in which the vector potential has only the component Ax along z. The correspondence (2) then gives for the fields c e oBz A Ey ¼ B þ h ; x z E þ m 0 c2 c oy ð6Þ Bz ¼ U with Ex ¼ 0 and for the current density 4p 2m0 c2 e oBz e Jx ¼ A B þ þ Ax Bz x z c hc E þ m0 c2 hc oy ð7Þ with Jy ¼ 0. Since the electric field is perpendicular to the current flow, there will be no electric coupling, and the internal magnetic field is along x. The electric field and magnetic fields are the small and large components respectively.
e A: Bz ðBA Þ~ hc
ð8Þ
This result could be obtained in the non-relativistic limit ignoring the small component, i.e. the electric field, of the wavefunction, but in fact is exact. We have explicitly indicated that the internal field is a function of BA as obtained from the solution of the Dirac’s equation in the field BA . The resulting Ampere’s equation is therefore in general a highly non-linear equation to be solved coupled with the Dirac’s equation. We can use this equation to discuss the Meissner effect in a superconductor in a time-independent magnetic field. In such a case, the Dirac’s hamiltonian with no electric coupling is appropriate since there is no dissipation along the direction of the current (infinite conductivity). Eq. ( ) would have the form of London’s equation, if it were linear, i.e. if Bz were independent on BA . Therefore it appears as a generalization: first the penetration factor k2 ¼ ðe=hcÞBz in general depends on the coordinates and exists provided that Bz < 0, i.e. the internal magnetic field has diamagnetic character, secondly it predicts the solution for the normal state, since by inspection one notes that Bz ¼ 0, B ¼ B0 is a possible solution, therefore it can also describe intermediate phases as combinations of normal and London’s solutions. The usual London’s expression k2 ¼ 4pe2 n=mc2 is found as a convenient approximation which linearizes the equation on reducing the internal field to its average value, i.e. the field generated by a vortex motion occurring in a volume r3 : hBz i ¼ nz ð4p=cÞej~ v ^~ rj=r3 , where nz is the fraction of the particles within the volume, i.e. nz ¼ nr3 , with n the density taken as a constant and ~ v their velocity. Since the quantization condition m0 vr ¼ h holds irrespective of the type of motion, this can be written as hBz i ¼ 4penh=mc. From Ampere’s law we can then pass to the second-order equation, i.e.
V. Dallacasa / Physica C 369 (2002) 370–373
D2 BA ¼ k2 BA to eliminate the vector potential, with k now a constant. This linearization loses the prediction of the normal state since such an equation does not admit a constant solution BA ¼ B0FS . The London’s ansatz also fails to predict a critical field and intermediate phases as in type II superconductors. We suggest that this further information is available from the more general Eq. (8). An improvement is obtained on relaxing the approximation of constant internal magnetic field in passing from this equation to the second-order equation. In fact, from Ampere’s law in the general case one finds the second-order equation for our case D2 BA ¼
e 1 2 Bz BA þ ðdBA =dyÞ : hc Bz
ð9Þ
By inspection it is seen that the London’s result is obtained if the first derivative term of the field can be ignored with respect to the other terms, being in fact exact for a rigorously constant internal field. Using the London’s solution BA ¼ B0 eky the condition B0 < jBz jeky results for its validity. The more restrictive criterion B0 < B1cr ¼ jBz j can be used, obtained by putting y ¼ 0, for then the condition is always satisfied. In a natural way the existence of an upper field coincident with the internal field is established indicating the compatibility of London’s state with externally applied fields lower than the internal
373
field. In the opposite case, B0 > jBz j we can choose the solution Bz ¼ 0 an B ¼ B0 which is admitted by Eq. (9). Hence Bz can be identified with the critical field and as a whole the solution describes a type I superconductor. The order of magnitude for a typical situation n 1022 cm3 with rc ¼ 1011 cm is Bz 100 G, which is right for a type I material. Eq. (9) can be used to discuss type II superconductors, by matching piecewise solutions of the form BA ¼ B0 (normal) and BA ¼ BA Lond (superconductor) between adjacent regions, where the London’s solutions are taken as combinations of e ky . For a superconductor segment ð0 . . . LÞ matched to a normal region otherwise the resulting critical field is of the order Bcrit hBz iðL=kÞ2 for L=k 1 and Bcrit hBz i for L=k 1 and hence always higher than the London’s value. Generalized states involving non-linearities can also be studied through the solutionof the nonlinear equation.
References [1] J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, 1965 (Chapter 11). [2] A.O. Barut, A.J. Bracken, Foundations of Physics 22 (1992) 1267. [3] M.E. Rose, Relativistic Electron Theory, John Wiley & Sons, New York, 1961.
Internal spinor fields and London’s equation Valerio Dallacasa a,b,* a
Laboratory for Materials Analysis, Scientific and Technological Department, Verona University, Verona, Italy b INFM, Physics Department, Parma University, Parma, Italy
Abstract An equivalence between Dirac’s equation of an electron subject to an applied magnetic and Maxwell’s equations by introducing electric and magnetic fields linearly related to the spinor wavefunctions. The internal magnetic field adds to the applied field and screens it due to its diamagnetic character. For a superconductor this screening is the origin of the expulsion of magnetic vortices and turns into a non-linear Ampere’s law, of which London’s equation is an approximate linearized form obtained on fixing the value of the internal field to its average value. This equation can be used to predict the critical field, intermediate normal/superconductor phases as in type I materials and non-linear effects beyond London’s state. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Internal spinor fields; London equation; Dirac spinors
1. Introduction In this paper we examine a possible origin and significance of the London’s equation, which is the basis for the comprehension of the expulsion of magnetic fields from a superconductor. The London’s phenomenological equation states that the current density is proportional to the vector potential and such a result is normally justified on the basis of a certain rigidity of the wavefunction, i.e. that the magnetic field has no effect on the wavefunction which therefore remains equal to the one in the absence of the field [1]. The expulsion and creation of vortices within a superconductor and
* Address: Laboratory for Materials Analysis, Scientific and Technological Department, Verona University, Verona, Italy. Tel.: +39-045-802-7936; fax: +39-045-802-7928. E-mail address: [email protected] (V. Dallacasa).
their dynamics are of general interest for the understanding of type I and II materials in a magnetic field, but novel techniques of investigation including single vortex visualization techniques by electron microscopy and analyses at extreme length scales, i.e. in submicron and nano-structured materials, or conditions, i.e. extremely high magnetic fields, suggest a particular importance of the understanding of the single vortex dynamics and structure, also in connection with organized states like liquid and solid vortex lattice and their confinement for high Tc . A vortex should coincide with the quantum motion, therefore we introduce the concept of internal electric and magnetic fields as consequence of the quantum motion and find that the origin of the cancellation of an applied magnetic field in the interior of the superconductor is the internal magnetic field which turns out to opposed to the applied field and then able to screen it to a certain
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 2 7 9 - 5
V. Dallacasa / Physica C 369 (2002) 370–373
extent. There is also an internal electric field which however does not couple to the motion since it turns out to be perpendicular to the current. The current itself turns out to be perpendicular to the internal magnetic field and parallel to the vector potential. The internal magnetic field gives also an explanation of the existence of a critical field and the transition to the normal state when the applied field is big enough to overcome the diamagnetic internal field. London’s state is found as the solution of a linearized form of the non-linear equation describing the magnetic state. 2. The equivalence between internal fields and Dirac’s spinors The method we use to study this matter is to find an equivalence between Dirac’s and Maxwell’s equations, which renders the quantum motion equivalent to an electromagnetic field. by mapping the wavefunction into the field. The mapping exists only under certain conditions in which the electric field, the magnetic field and the vector potential are orthogonal each other and under these circumstances a current density and charge density, i.e. the constitutive equations, remain defined. Within this method each vortex is put in one to one correspondence with the wavefunction. Under the conditions e ~ ~ 2m0 c ð A ^ BÞ z þ Ez ¼ 0; hc h ð1Þ ~ A~ E ¼ 0; A^~ EÞ ¼ 0 ð~ A^~ EÞ ¼ ð~ x
x
where ~ A is any real vector, m0 a scalar and ~ E, ~ B are fields, on introducing the mapping W ¼ aðBz ; Bx iBy ; iEz ; Ey þ iEx Þeimc
2 t= h
ð2Þ
where a is an arbitrary constant and with sources of the form 4p 2m0 c e Jx ¼ Ey þ ð~ A^~ BÞ y ; c h hc 4p 2m0 c e Jy ¼ Ex ð~ A^~ BÞ x ; c h hc 4p e Jz ¼ ð~ A ~ BÞ; c hc 4p e ~ ~ q ¼ ðA ^ E Þz c hc
ð3Þ
371
the eight Maxwell’s equations can be written as four equations in the compact Dirac’s form e oW A þ m0 c2 W ¼ ih ð4Þ c~ a ~ p ~ c ot for a particle with rest mass m0 in a magnetic field generated by the vector potential ~ A. Conversely it can be shown that from the Dirac’s equation in a magnetic field described by the vector potential ~ A, fields ~ E and ~ B can be defined obeying Maxwell’s equations if the constitutive equations, the mapping and conditions above are valid. The reality of the fields restricts the equivalence only for certain bi-spinor solutions W of the Dirac’s equation. The correspondence discussed here makes use of a linear relationship between the fields and the wavefunction; therefore the fields have spinor character and are a direct realization of the wavefunction (comprising its phase). Attempts to introduce real fields for the structure of the electron are well known [2]. In the more common scheme a relation quadratic in the wavefunction [3] is found which generates a tensorial electromagnetic field. We can view the spinor fields as fields associated at any point of space where the motion (including internal motions) of the electron occurs, while the tensorial fields are those produced by the particle in the space around it. This fact implies that a possible measure of the wavefunction in visualization techniques can be achieved. As seen from the constitutive equations the current density has two contributions, a magnetic term proportional to the vector potential and an electric term, this latter arising from the zitterbewegung, through the frequency term 2m0 c2 =h of this motion [3]. As a whole, the associated spinor electromagnetic field can be interpreted as an internal electromagnetic field generated by the quantum motion of the electron in a magnetic field BA deriving from the vector potential ~ A. In general, for consistency, this field should include the internal field ~ B and the external field ~ B0 , i.e. the relation will hold ~ BA ¼ ~ Bþ~ B0 . The field ~ B0 will then be screened in the interior of the system, due to the internal field. In the particular case considered here the Dirac’s hamiltonian includes only a coupling with a magnetic field and no electric coupling.
372
V. Dallacasa / Physica C 369 (2002) 370–373
Therefore the internal magnetic field should contribute to the magnetic coupling whereas the electric field should not give rise to any electric-type coupling.
4. The London ansatz, the conditions for its existence and generalizations Using the results above, we can calculate the total magnetic field through Ampere’s law and we find
3. An expression for the internal fields rot ~ BA ¼ In the case we are studying in which there is no electric field term in the Dirac’s equation, and on simplifying to the case in which the motion of the Dirac’s particle occurs only along planes perpendicular to the z-direction, the Dirac’s field can be calculated as W ¼ ðE þ H ÞU and for up and down spin can be reduced to the forms cðPx þ iPy Þ/ w" ¼ 0; /; ;0 ; E þ mc2 ð5Þ cðPx iPy Þ/ ;0 w# ¼ 0; /; E þ mc2 where Px ¼ px ðe=cÞAx , Py ¼ py ðe=cÞAy and U" ¼ ð/; 0; 0; 0Þ, U# ¼ ð0; /; 0; 0Þ are solutions of the second-order equation [3]. For a magnetic field pointing towards the positive(negative) z-axis the states with spin up(down) will correspond to the ground state; therefore we shall consider the case of W" as appropriate for BA > 0 and W# for BA < 0. For simplicity we shall examine the case of spin up, for spin down similar results are obtained. We can use a gauge in which the vector potential has only the component Ax along z. The correspondence (2) then gives for the fields c e oBz A Ey ¼ B þ h ; x z E þ m 0 c2 c oy ð6Þ Bz ¼ U with Ex ¼ 0 and for the current density 4p 2m0 c2 e oBz e Jx ¼ A B þ þ Ax Bz x z c hc E þ m0 c2 hc oy ð7Þ with Jy ¼ 0. Since the electric field is perpendicular to the current flow, there will be no electric coupling, and the internal magnetic field is along x. The electric field and magnetic fields are the small and large components respectively.
e A: Bz ðBA Þ~ hc
ð8Þ
This result could be obtained in the non-relativistic limit ignoring the small component, i.e. the electric field, of the wavefunction, but in fact is exact. We have explicitly indicated that the internal field is a function of BA as obtained from the solution of the Dirac’s equation in the field BA . The resulting Ampere’s equation is therefore in general a highly non-linear equation to be solved coupled with the Dirac’s equation. We can use this equation to discuss the Meissner effect in a superconductor in a time-independent magnetic field. In such a case, the Dirac’s hamiltonian with no electric coupling is appropriate since there is no dissipation along the direction of the current (infinite conductivity). Eq. ( ) would have the form of London’s equation, if it were linear, i.e. if Bz were independent on BA . Therefore it appears as a generalization: first the penetration factor k2 ¼ ðe=hcÞBz in general depends on the coordinates and exists provided that Bz < 0, i.e. the internal magnetic field has diamagnetic character, secondly it predicts the solution for the normal state, since by inspection one notes that Bz ¼ 0, B ¼ B0 is a possible solution, therefore it can also describe intermediate phases as combinations of normal and London’s solutions. The usual London’s expression k2 ¼ 4pe2 n=mc2 is found as a convenient approximation which linearizes the equation on reducing the internal field to its average value, i.e. the field generated by a vortex motion occurring in a volume r3 : hBz i ¼ nz ð4p=cÞej~ v ^~ rj=r3 , where nz is the fraction of the particles within the volume, i.e. nz ¼ nr3 , with n the density taken as a constant and ~ v their velocity. Since the quantization condition m0 vr ¼ h holds irrespective of the type of motion, this can be written as hBz i ¼ 4penh=mc. From Ampere’s law we can then pass to the second-order equation, i.e.
V. Dallacasa / Physica C 369 (2002) 370–373
D2 BA ¼ k2 BA to eliminate the vector potential, with k now a constant. This linearization loses the prediction of the normal state since such an equation does not admit a constant solution BA ¼ B0FS . The London’s ansatz also fails to predict a critical field and intermediate phases as in type II superconductors. We suggest that this further information is available from the more general Eq. (8). An improvement is obtained on relaxing the approximation of constant internal magnetic field in passing from this equation to the second-order equation. In fact, from Ampere’s law in the general case one finds the second-order equation for our case D2 BA ¼
e 1 2 Bz BA þ ðdBA =dyÞ : hc Bz
ð9Þ
By inspection it is seen that the London’s result is obtained if the first derivative term of the field can be ignored with respect to the other terms, being in fact exact for a rigorously constant internal field. Using the London’s solution BA ¼ B0 eky the condition B0 < jBz jeky results for its validity. The more restrictive criterion B0 < B1cr ¼ jBz j can be used, obtained by putting y ¼ 0, for then the condition is always satisfied. In a natural way the existence of an upper field coincident with the internal field is established indicating the compatibility of London’s state with externally applied fields lower than the internal
373
field. In the opposite case, B0 > jBz j we can choose the solution Bz ¼ 0 an B ¼ B0 which is admitted by Eq. (9). Hence Bz can be identified with the critical field and as a whole the solution describes a type I superconductor. The order of magnitude for a typical situation n 1022 cm3 with rc ¼ 1011 cm is Bz 100 G, which is right for a type I material. Eq. (9) can be used to discuss type II superconductors, by matching piecewise solutions of the form BA ¼ B0 (normal) and BA ¼ BA Lond (superconductor) between adjacent regions, where the London’s solutions are taken as combinations of e ky . For a superconductor segment ð0 . . . LÞ matched to a normal region otherwise the resulting critical field is of the order Bcrit hBz iðL=kÞ2 for L=k 1 and Bcrit hBz i for L=k 1 and hence always higher than the London’s value. Generalized states involving non-linearities can also be studied through the solutionof the nonlinear equation.
References [1] J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, 1965 (Chapter 11). [2] A.O. Barut, A.J. Bracken, Foundations of Physics 22 (1992) 1267. [3] M.E. Rose, Relativistic Electron Theory, John Wiley & Sons, New York, 1961.