Yes. And? It was complicated 50 years ago, and it's still complicated.
To you it is complicated since the model you are believing in, is complicated. As Bernd Matthias said about the BCS model: When you have equations stretching over half a page, the model is wrong: Nature is not so complicated. My model is simple and can be visualised: It explains everything that the BCS model supposedly explains. In addition it explains aspects which BCS cannot and never will be able to explain. For example, the dependence of the critical temperature on the doping density in over-doped p-type diamond; the dependence of the critical temperature on the doping level in the ceramics; the pressure dependence of the critical temperature in the ceramics; the presence of “stripes” in the ceramics; the presence of a pseudogap in the ceramics; etc. etc. According to Occam’s razor, it is clear which model should be preferred.
That's not the complicated part. That's the easy part. And it's quite simple. There are no magic voltage sources, only real voltage sources.
Up to here I agree with you: Especially that there can be no “magic” voltage sources and thus no “magic” electric-fields (take note Ben and Sol).
An applied electric field changes the current, and continues to change the current,
Are you now talking about the electric-field within the superconductor? I will assume that you do. If this is what you are arguing, you are TOTALLY wrong! The resistivity of a superconductor is zero, and therefore a current and an electric-field cannot be simultaneously present within such a material. Superconduction has NOTHING to do with the classical acceleration of charge-carriers within a material: The motion of charge-carriers is a quantum-mechanical effect.
Both my model and the BCS model agree on the latter aspect: Both models state that in order to obtain superconduction you must have a phase transformation so that the charge-carriers, which conduct above the critical temperature Tc, by acceleration and scattering, form a quantum-mechanical lowest-energy condensate within which the charge carriers cannot be accelerated and scattered. The charge-carriers within the condensate can thus not move in a classical manner as modelled by Newton’s laws. According to the BCS model, a charge that enters the superconductor must cause all the charge-carriers (Cooper pairs) within the superconductor to move coherently in a correlated manner (which must be nonsense for a single charge): If they do not move in this manner then according to BCS you will get scattering and thus a resistance. Therefore it is postulated that the momentum of a charge-carrier is supposedly given by the gradient of a phase-angle, and not by an electric-field accelerating charge-carriers in a classical manner. Unfortunately the latter is also nonsense.
How does BCS dovetail with Ben’s model that a single charge-carrier can be “bumped” to move with a constant speed from one contact to the other. To “bump” it the charge-carrier must scatter from an injected charge-carrier: But the BCS model specifically states that a Cooper pair cannot be “bumped” to gain or lose energy. Thus can you not see that what you and Ben are claiming is also violating the BCS model: Or are you agreeing with me that the BCS model is a model without the C?
until the voltage drop outside the superconductor increases to match the applied voltage.
How is the voltage-drop measured “outside”, and the applied voltage “inside”. This sentence is senseless. It is true that when ramping up a superconducting magnet loop, the voltage across the external resistive circuit increases with time until it reaches the full voltage that is being applied across the whole circuit. This is of course caused by a dynamic response of the whole circuit which includes the generation of a back-voltage owing to induction. Such a back-voltage need not be generated by any acceleration of charge-carriers within the superconductor; or by accumulation of MOVING charge-carriers.
You then go ahead and immediately violate you own statement by, in addition, arguing wrongly that charge-carriers can flow with zero resistance through a superconductor-element while an electric-field is present within this element. As you should know, resistivity can only be measured indirectly: i.e. experimentally it has been deduced that a superconductor has zero resistivity BECAUSE a current flows through a superconductor element WITHOUT an electric-field driving the current. Ohm’s law has been invoked to define this zero resistivity, as you all insist must be done; without giving an explanation why Ohm’s law does fit this scenario; as it does: Thus according to your own definition, zero resistivity is ONLY valid when the electric-field is ZERO and NOT when the electric-field is non-zero so that it can accelerate the charge-carriers.
When the electric-field is non-zero, AND a current is flowing, even without scattering of the charge-carriers, you can calculate a resistance R from Ohm’s law. This must be so since, even without scattering within a material-element, you still have essentially the same energy-dissipation when the charge-carriers, which are being accelerated through such a material-element, scatter within the target contact. For example, in the case of a vacuum diode it is said that the diode has a vacuum-resistance. Obviously a vacuum cannot have resistance. But a resistance is measured: The reason for this is that the accelerating charge-carriers can never cancel the applied electric field, and they therefore dissipate all their kinetic energy which they gained by acceleration within the anode: But dissipation of kinetic energy always equals resistance. Thus, even though you have no scattering while the charge-carriers move from the cathode (injecting contact) to the anode (target contact) you still dissipate essentially the same energy as you would have had if they were scattered within the material. Therefore you still measure a resistance. Obviously in this case, this resistance cannot be used to derive a “real” resistivity for the charge-carriers at positions between the injection contact and target contact: But this does not mean that there is no resistance and thus no energy dissipation as one must have in the case of a superconductor.
It is obvious that the voltage can be made smaller and smaller for the same current if one can find a way of increasing the density of the charge-carriers between the cathode and anode within a vacuum diode. This can be done by increasing the temperature of the cathode, and it is then indeed found that the voltage for the same current drops with increasing temperature: Is it possible to increase the density of the charge-carriers to become large enough so that the voltage, and thus the electric field between the cathode and anode, becomes zero? Not in a vacuum diode: But this is what I have achieved with my diamond experiment when sufficiently increasing the donor-dopant density just below the surface of the diamond. A point can then be reached when the density of electrons becomes high enough so that the voltage becomes zero. This, in turn demands that all the electrons between the cathode and anode must become stationary. So how can they now transfer a current; as is experimentally observed that they do? On this point my model and the BCS model again touch sides.
As you increase the density of charge-carriers the electric-field becomes less and the speed with which the electrons reach the anode decreases. Thus on average the kinetic energy of the electrons is decreasing as their density increases: This is in essence the same as cooling a gas by lowering the temperature. When the field becomes zero the electrons have no kinetic energy: They must thus form a condensate as predicted by Bose and Einstein: Note that at this point all the electrons must have the same energy, even though they are fermions. Can fermions form a Bose-Einstein condensate? Well, the electrons can, as long as there is an even number of electrons so that they form an anti-ferromagnetic condensate: Alternatively one can argue that they first pair up with opposite spins, so that these resultant bosons form the condensate; which is what they are forced to do at this point; since they have no other option. According to BCS such a condensate is required for superconduction to occur. According to my experiment, the proof is incontestable that there cannot be an electric–field within this condensate and that the electrons then have no kinetic-energy: Yet charge is transported through the resultant condensate.
Once such a condensate has formed, it is impossible to have separate charge-carriers anymore. One has a single, holistic macro-wave. That this is so is also proved by the properties of Bose-Einstein condensates which have been generated at very low temperatures by cooling suitable gasses (a Nobel Prize has been awarded for this while I do the same with electrons at room temperature: Do you not think that I should get some credit?). So if there are no separate charge-carriers, how does charge transfer from the cathode to the anode through this condensate? It can only happen quantum-mechanically; as both the BCS model and my model claims: What does quantum mechanics tell us? It tells us that there can be quantum fluctuations: i.e. a single wave-entity can borrow energy (delta)E “from the vacuum” as long as it returns this energy within a time-interval (delta)t.
When you do not inject charges into this purely-electron Bose-Einstein Condensate, it is in the lowest energy state that it can be. When you inject an electron, the electron “merges” with the condensate and increases its energy by (delta)E. In order to remain in its lowest energy state, the condensate must then within a time-interval (delta)t spit out an electron at the ejection contact. Note that in this case there is not a charge-carrier moving from the injection contact to the ejection contact; as Ben claims that there must be when superconduction occurs. The transfer of the injected charge is not classical but non-local.
What is important to note is that the charge is transferred NOT because it moves classically with a constant speed through the BEC, but in order for the BEC to maintain its lowest energy state. Since BCS claims that the Cooper pairs form a similar condensate, the transfer of charge through this condensate must also occur for this condensate to maintain its lowest energy state and can therefore not be modelled classically in terms of any of Newton’s laws as you, Ben and Sol have incorrectly claimed all along on this thread.
Within a material, however, the charge-carrier density can never become high enough to form a real BEC similar to the one I have discovered between the diamond and the anode. But just as within the latter phase, the charge-carriers must still form a condensate: However, since their density is too low to merge into becoming a single holistic wave (a real BEC), the charge-carriers are still separately identifiable. Even though this condensate is thus not a “true” BEC, it still requires that all the charge-carriers must each have the same minimum energy.
Since the energy must be a minimum which cannot be lowered any further, or else the charge-carriers will be able to dissipate energy, the charge carriers cannot have kinetic energy. This is only possible when the charge-carriers form an array of localised “anchored” orbitals; just like Mott and Anderson have modelled for many years when you have a phase transition from a metallic state to an insulating state. Thus for a metal to super-conduct it must go through a metal-insulator transition. Note that such a condensate does NOT require that the charge-carriers must be bosons. Since they are separate, like donors within a doped semiconductor, they can all have the same energy while being fermions.
Now how is an injected charge transferred through such an array from the injection contact to the target contact? The clue comes from the real BEC which I formed from electrons between the cathode and the anode when I extracted electrons from a highly-doped n-type diamond. The charge can only be transferred, also in this case, by means of quantum-mechanical fluctuations, each with an energy (delta)E which can appear for a time-interval (delta)t and then totally disappear again. Thus, if this energy is part kinetic energy, this kinetic energy disappears before it has to dissipate by scattering anywhere within the material or the target contact.
If the distance between the anchored orbitals within the insulator-array of orbitals is too large, superconduction cannot occur. When the distance between the orbitals becomes small enough, an orbital can borrow enough energy (delta)E to break free and also have the kinetic energy required to reach the position of an adjacent orbital within the allowed time interval (delta)t. Thus, when injecting a single electron, it reaches a stationary orbital nearest to the injection contact and replaces it: The reason why the resident orbital can be replaced, is that it can move on by a quantum fluctuation [(delta)E for time (delta)t], to replace the next orbital, which then moves on to the next orbital by doing the same, etc. In this manner, a charge is eventually ejected at the target contact without having any kinetic energy that needs to be dissipated. Note that it is thus NOT the same injected charge that moves through the superconductor, nor the charge of the orbital which is “bumped on” by the arrival of the injected charge, but the charge of the last orbital at the target contact.
How quickly is the latter charge ejected at the target orbital after a charge is injected at the injection contact? VERY FAST!!! At a speed between about 10
6 m/s and light speed. Thus for all intent and purposes, the injection of a charge at the injection contact causes the ejection of a charge at the target contact nearly with the speed of light, and this happens without requiring any simultaneous acceleration of all the charge-carriers. But does this mean that a current is flowing through the superconductor at this speed? NO! Note that when you have injected a single charge, this charge now stays trapped at the position of the first orbital. It only moves on when a second orbital is injected to “bump” it to the next orbital position etc. Thus the rate at which charge carriers flow past a position along the superconductor is totally determined by the rate at which charges are being injected at the injection contact.
Thus not all localised charge-carriers immediately move when you inject charges at the injection contact. For a low injection rate only a fraction of the orbitals at the injection contact are replaced and moves on to generate a current. When increasing the injection rate a larger fraction participates: Thus the current is not determined by the acceleration if charge-carriers within the superconductor but solely by the rate at which charges are being injected. This is also why a superconductor has a maximum current: When the injection rate becomes so high that the density of orbitals at the injection contact is too low to handle this injection rate, you get a traffic jam, and current-flow stops. I do not know why I should be explaining this again on this thread, because it is available in detail on my website: See:
www.cathodixx.com/pdfs/ChargeInjection.pdf Oh yes I forgot, YOU, Sol and Ben are all knowing gods who just know when another model is wrong without first studying the model. God help physics!
The fact is, and will remain, that the only experimental result we have is that a current flows through a superconductor without an electric-field being present, and therefore there can be no acceleration of the charge-carriers EVER. Thus, the main issue is to explain how an applied electric field, which is present within the material when a current flows above the critical temperature Tc, is suddenly switched off when the material is cooled through the critical temperature. The final equilibrium current has NOTHING to do with acceleration of charge-carriers through the superconductor; since it is completely determined by the voltage drop V across the external circuit with resistance R(ext). Thus if you have no superconductor element, or superconductor elements of different lengths you will measure the SAME equilibrium current I(eq) for the same external voltage drop V: i.e. I(eq)=V/R(ext).
When you switch on a voltage across a normal conductor, the full voltage will appear immediately when you neglect the inductive response of the conductor. The inductance is usually neglected for a straight piece of wire with resistance. The increase in current through the wire is then mainly determined by the resistance of the wire; and can be modelled by using Drude’s equation: However, when you use a coil of wire, the inductance becomes large, and now you have, in addition, a back-voltage (a voltage supply that is not magic) that has to be overcome before an equilibrium-current is established. Thus, to settle down to an equilibrium current now takes longer. But when equilibrium is reached, the current through the coil is the same as the current will be when the wire is not coiled. Thus the equilibrium current is determined solely by the resistance of the coil. If the circuit containing the coil as an element and the coil have no resistance, the current will increase until the coil burns out.
In your superconducting magnet, you have a large inductive response which gives a back-voltage when you switch on the voltage, and the equilibrium current is obviously achieved when the back-voltage becomes zero: But in this case the coil does NOT burn out since the current through it is totally determined by the resistance of the external circuit which determines the rate at which charge is injected into the superconducting loop. This is also the case during ramping: It is the resistance of the external circuit which determines the current through the superconductor; not the acceleration of free charge-carriers through the superconductor itself. The ramping time is mainly determined by the inductive response of the superconducting coil and has NOTHING to do with acceleration of charge-carriers within the superconductor.
The complicated part is explaining how you GET zero resistance in the first place, along with the other properties of superconductors (like the Meissner effect) which are NOT the direct result of zero resistance.
My model explains it and the explanation is simple as you can see above: And as you would have seen if you took the time to study the excerpts on my website. It is not in the least bit complicated unless you want to use Cooper pairs. That is complicated since it is not a physically possible model.
JFP: I will do this for you: The divergence of an applied conservative electric field while a current is flowing is ALWAYS ZERO.
You: Obviously not. Trivial counterexample: a charging capacitor plate.
There are no charge-carriers between capacitor plates and therefore there can be no charge-carriers flowing everywhere around the circuit. Why do insist on arguing nonsense?
Go to the continuity equation: When a current is flowing through the element (in at one contact and out at the other), then within the element just as many charge-carriers ALWAYS flow into a point as flowing out of a point. Thus the divergence of the current is zero (div)j=0:
That's often (but not always) the case.
It is always the case when you have a current flowing around a circuit without the accumulation of charge at a barrier through which the charge-carriers cannot move; like a capacitor plate. If there are no barriers to the movement of the charge-carriers; then within the circuit this is always so.
The irony here, though, is that one of the cases where this IS the case is during a ramp of a superconducting magnet. Because the current is non-divergent, the continual increase in the current must be happening everywhere simultaneously, which requires that the charge carriers within the superconductor accelerate. Something you claimed couldn't happen. So you've unwittingly contradicted yourself.
As you can see from my discussion above I did not contradict myself. You are contradicting yourself by claiming that both a current and an electric-field can be present within a superconductor. Thus you claim that during ramping there is a voltage V and a current I: According to Ohm’s law this means that there is now a resistance R=V/I. And even if you do not have any scattering within your superconductor, this is a real resistance since your accelerated charges will scatter within the target contact.
As explained above, the increase in current through a superconductor is totally determined by the rate at which charge carriers are being injected into the superconductor, and not with any acceleration of the charge-carriers within the superconductor. As the injection rate increases the current through the superconductor increases and this causes an inductive response which does not require the acceleration of the charge-carriers within the superconductor.
Let us consider the alternative method to generate a current around a superconducting loop. Assume it is a ring and you apply a time-varying magnetic field. According to your reasoning, the charge-carriers will start to move around the ring since they are accelerated by the induced circular electric-field as mandated by Faradays law. This is where you are wrong. They are not moving because they are accelerated.
To prove this, let us first assume that you are correct. One switches on the magnetic field which increases and induces a circular electric-field that accelerates the superconducting charge-carriers. The circular current in turn generates an opposing magnetic field that cancels the applied magnetic field. Once the applied magnetic field becomes constant it is cancelled by the reactive magnetic field generated by the current around the ring. But if we increase the applied magnetic field further, this current must, according to your reasoning, increase owing to the re-appearance of the induced electric-field. This also means that when you now decrease the applied magnetic field an induced circular current will also appear, but now flowing in the opposite direction. Thus this will de-accelerate the charge-carriers and thus decrease the magnetic field that the moving charges are generating in concert with the decrease of the applied magnetic field. This, in turn, mandates that the current becomes zero when the applied magnetic field has been reduced to become zero. Thus a persistent current of charge-carriers cannot remain as is experimentally observed.
This conclusion mandates that the current must be generated by another mechanism than by acceleration. This mechanism is described very clearly in my book, and I am not going to repeat it here since it will make this posting, which is already very long, too long. It is obviously based on the fact that a current will only flow when it is needed to keep the superconducting phase in a lowest energy-state. Suffice to say that the Meissner effect is NOT caused by surface currents but by the absorption of the applied magnetic-field by the stationary charge-carriers. Only under very special conditions will the charge-carriers move to maintain the superconducting state. This happens when the vortex state of type II superconductors is manifesting. But at a lower magnetic field there are no such currents even though the Meissner effect is still manifesting.
Furthermore, if the charge-carriers can be accelerated as you claim, then one must immediately have a current flowing when switching on the external magnetic field. This is not so: A current only starts to flow when a magnetic field is reached at which the charge-carriers have to move in order to maintain a lowest-energy superconducting state. This is why at first there is no current flowing until a critical magnetic field is reached: For a small ring, the magnetic field reaches the value of a flux quantum before a current starts to flow.
JFP: Thus from E=(rho)*j, you must have that (div)E must also be zero everywhere. Thus, it is NOT possible to find a charge density within the material by taking the divergence.
You: That's obviously false. If rho has a gradient, then div(E) can be nonzero even if div(j) is zero. That's simple
vector calculus. You know, that thing that you said you understood and we didn't. Oops. Yet another elementary mistake in a long string of elementary mistakes.
I see that both Sol and Ben made the same comment in the next two posts. It is a good point, and it seems that I made a small mistake: Probably because I did not write for (div) that it si del-dot. I will get back to it. The question to address is, however, whether such an accumulation of charge as you claim can cancel an applied electric field within a material. Even if it can, which I doubt, it does not take away the fact that this argument is modelling superconduction by using classical equations which do not apply at all. See above about condensates which must apply for superconduction to be possible. Thus basically, even if you are correct in this instance, this has nothing to do with superconduction. Furthermore, I am glad that my mistake is an oversight and not so stupid as to claim that the gradient of a differentable scalar field can consitiute a circular vector field. I am not yet a total imbecile.
