An Attempt to Improve Understanding of the Physics behind Superconductor Phase Transitions and Stability
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An Attempt to Improve Understanding of the Physics behind Superconductor Phase Transitions and Stability
Harald Reiss
Department of Physics University of Wuerzburg, Am Hubland, D-97074 Wuerzburg, FRG [email protected]
Abstract
Focus of this paper is on superconductors under disturbances, when the materials are already close to a phase transition. Numerical simulations are applied to multi-filament, BSCCO 2223 and to thin film, coated YBaCuO 123 superconductors. The results shall contribute to improve understanding of the physics behind superconductor stability and quench, and to which extent stability is coupled to time dependence of phase transitions. Temporal localisability of events (like simple, local temperature variations under disturbances) and images thereof created by transport processes most probably cannot be realised in case the superconductor is non-transparent to radiation. The paper questions, in this situation, the existence of a sharply defined critical temperature. Superconductor stability and occurrence of a quench then might become unpredictable. Keywords Superconductor; transient temperature field; critical temperature; phase transition; relaxation; stability; temporal localisability; thermal diffusivity
Außerplanmäßiger (Associate) Professor
1 Survey: Stability of Superconductors against Quench
Under disturbances, a superconductor may experience a sudden, most undesirable phase transition (a quench) from superconducting to normal conducting state.
Quench may lead to damage or even to catastrophic conductor failure. A superconductor is stable if it does not quench. Disturbances are responsible for a variety of losses, like transformation of released mechanical to thermal energy under conductor movement by Lorentz forces, or absorption of high energy, particle radiation, or fault currents and cooling failure. Disturbances frequently are transient, but there are also permanent disturbances like flux flow losses if transport current density exceeds critical current density, or, under AC currents, hysteretic and, in multi-filamentary conductors, coupling losses. See Wilson [1], Chap. 5 for a comprehensive catalogue of superconductor disturbances. In a rough picture, the physics behind quench is sudden conversion of stored mechanical or electromagnetic energy, the latter originating from screening and transport currents, to thermal energy in a magnetic field. Quench can be avoided by application of stability models for design of superconductor geometry (cross section of wires, filaments, thin films) and for safety during conductor manufacture, handling, winding and operation of a superconducting device. The literature on superconductor stability, published since about the 1980s, is very large, and a variety of standard, analytic stability models has been suggested. The models usually assume worst case conditions, or apply safety margins to make sure, from practical reasoning and from experience, the conductor will not quench. The models apply algebraic expressions, see Wilson [1], Dresner [2] or Seeger (Handbook of Superconductivity [3], and essentially are energy balances. From all-day practice, application of standard stability models is successful if the superconductor is far from its critical states. But the probability that a superconductor experiences a quench increases the more, the closer the superconductor approaches or exceeds its critical values of temperature, magnetic field or current density. It is this situation that shall be analysed in this paper. A discussion of properties, benefits, problems and of also risks of standard stability models, in relation to numerical investigation of superconductor stability, has been presented recently, see [4-7]. Quench is a short-time physics problem; it proceeds on time scales of milliseconds and less. Numerical calculation of heat transfer and of transient temperature distribution in the conductor cross section are more suitable than standard, analytical stability models and yield a first key for successful analysis of the stability problem. Superconductor temperature (in rather exceptional cases its local temperature distribution) can be measured as average values without spatial resolution with sensors that are mechanically or radiatively connected to filaments or to thin films. Instead, by analytic or numerical simulations of superconductor temperature, local temperature T(x,y,t) is obtained, with high resolution from solutions of Fourier's differential equation using the thermal diffusivity of the superconductor material. Both procedures reflect the “ phonon aspect” of the transient stability problem. The phonon contribution in the total thermal diffusivity constitutes its overwhelming part. Standard stability analysis considers the decrease, dJ Crit [T(x,y,t > t )]/dt, of critical current density during a warm-up period in parallel to the increase dT(x,y,t)/dt of local "phonon" temperature, t, of the superconductor. But at very low temperature, the superconducting electron system (that is responsibly for a non-zero value of J Crit ) is largely decoupled from the lattice. The electron system reflects its own dynamic response to disturbances. The question thus is whether decay or generation of electron pairs, the “ electron aspect” manifested by dJ
Crit [T(x,y,t > t )]/dt ≠ 0, proceeds on another timescale t’, under disturbances, during decay and subsequent recombination of excited electron states to a new dynamic equilibrium, carrier concentration. It is not clear that this timescale might really be identical with the traditional (phononic) timescale, t. A shift, Δt Ph/El , between both time scales and its impact on critical current density and stability functions has been investigated recently [4]. The shift is not constant in the conductor cross section or over its length. It can be rather large, between 1 ms and 5s in NbTi, but is very small in YBaCuO. Non-zero values of the shift lead to the more general problem of unique identification of time scales, see Sects. 7 to 8 of the present paper.
2 Modeling Heat Transfer within Superconductors
Radiative contributions to heat transfer in bulk solid material are vanishingly small. In thin films, however, the situation may be different. This aspect will be investigated in the present paper. In a superconductor, near its phase transition, even tiny temperature fluctuations arising under any heat transfer mechanism, including radiative, can locally drive the superconductor into normal conducting state, which means a ll existing channels have to be considered in temperature field and, accordingly, stability calculations. But non-transparency enormously simplifies numerical analysis of the radiative transfer problem. Non-transparency is expected for optical thickness of at least τ = E D = 15, with E the extinction coefficient and D the sample thickness. The τ ≥ 15 criterion results from experience. It concerns direct transmission of radiation (under the well-known Beer's transmission law). The critical value τ ≥ 15 does not mean there is no radiation in the material. Solid conduction heat transfer, besides other transport mechanisms, is responsible for local sample temperature, which means there is local absorption/remission. Radiation, if any, in non-transparent materials accordingly is a local, non-zero property, too (non-zero because of emission of Black-Body radiation under multiple heat transfer). Non-transparency allows application of the so called Additive Approximation of conductivities. Conditions for applicability of this approximation have been investigated recently [8, 9]. Non-transparency of the superconductor is the second key to successfully simulate stability of superconductors. We accordingly have to check whether the criterion τ ≥ 15 is really fulfilled in filaments and thin film superconductors to apply the Additive Approximation. Strictly speaking, the check has to be performed at all relevant mid-IR wavelengths (mid-IR in case of High Temperature Superconductors). It has been shown [4] that extinction coefficients and Albedo, all obtained from spectral values of the refractive index, did not very strongly depend on wavelength.
3 Additional Problems
Risks arising during application of standard stability models have already been explained extensively in our previous papers [4 - 7]; the description shall not be repeated here. But thorough analysis of the physics behind superconductor stability imposes, even under non-transparency, at least three more problems: (i) Conflicts arising from relaxation time of the superconductor electron system in relation the integration time intervals in analytic and numerical simulations, (ii) Uncertainties resulting from strongly different propagation speed of heat transfer components, (iii) temporal localisabilty of events, like a quench In order to contribute to solutions of the three problems, the paper is organised as follows: (i) After a disturbance, how long does it take the electron system of the superconductor to arrive at a new thermodynamic equilibrium? When in a process a physical system reaches its equilibrium (if there is any), no more excursion with time of its degrees of freedom will be observed (apart from ubiquitous statistical fluctuations). The question to be answered is not lifetime of the equilibrium but how long it takes the superconductor electron system to actually reach this state. In the literature, there is no answer that would be suitable for solution of this problem. A general method to calculate lifetimes is provided by elements of perturbation theory. But perturbation theory breaks down near phase transitions. As another alternative, time-dependent Ginzburg-Landau theory of the order parameter would require an enormous amount of computational effort and can hardly be realised in case of transient temperatures (and, trivially, in complex superconductor geometry, like in multi-filament materials). It is the relaxation time that is needed, after a disturbance, to reorganise the whole number of normal conducting and superconducting electron states (single electrons, in the presence of residual electron pairs) to a new thermodynamic equilibrium. We are not primarily interested in detail which and how the corresponding new wave functions of the new equilibrium will be created, the temporal aspect is in the foreground. Completion of each single recombination event needs an elementary time interval, dt. Counting all the very large number of recombination events, i. e. summation over all intervals dt, taking into account all residual and, during cool-down, progressively restored electron pairs, offers an alternative to perturbation and Ginzburg-Landau theories to obtain the total relaxation time, τ El . The method is explained in [4]. The problem between relaxation time and length of time steps in numerical simulations, on the basis of a microscopic stability model [4] has repeatedly been discussed in our previous papers. In connection with items (ii) and (iii), it requests an update: (1) Finding a solution of the completeness problem of radiative transfer by means of an operator scheme, to separate fast from slow thermal transport mechanisms. We have (2) also to take into account a recently published paper on temporal localisability in quantum systems under relativity principles [10]. In the microscopic stability model [4], let under thermodynamic equilibrium the total wave function, ψ(t ), or a set of individual wave functions, υ i (t ), of which ψ(t ) is composed, describe the superconducting quantum state, at an original time, t . A thermal disturbance then requires the whole set of the υ i (t ), not only part of them, to progressively be re-arranged (re-ordered) to a new total wave function, ψ(t ). The wave function ψ(t ) subsequently has to be re-arranged and anti-symmetrized, under quantum-mechanical selection rules, to the new equilibrium wave function, ψ(t ). Time needed for physical realisation of the sequence ψ(t ) → ψ(t ) → ψ(t ), i. e. the difference, t – t , is the relaxation time, τ El , of the total electron system from the disturbed system (t ) to the final, new thermodynamic equilibrium state (t ). This model does not insist on a specific type of coupling in a BSC or any other superconductor. Instead of phonons that glue two electrons to a pair, coupling in the superconductor might as well be of excitonic or plasmonic or magnonic electron-electron type. In any case, we have to take into account the very small, but finite duration, dt, in time of a first step "selection" (according to selection rules) of the two electrons that preceeds "coupling" of the two, once their "selection" is accomplished. The time intervals dt are estimated from a formal analogue, the calculation of “coefficients of fractional parentage” in atomic and nuclear physics, and a “time of flight”-concept with a mediating Boson. In the Yukawa-model, as one of possible analogues, a pion, π, mediates binding of two nucleons. The large number of electron states, or of individual wave functions, υ i (t), is responsible for the divergence of τ El when T → T Crit . No convergence of the solutions apparently occurs at temperatures very close to T
Crit . When this result becomes available, two critical relations have to be considered: First, the length of numerical integration time steps, Δt, have to be reduced the more, the closer the simulation approaches critical temperature; this is because of the strong temperature dependence of the specific heat. It also becomes increasingly difficult to achieve convergence of the solutions of Fourier's differential equation. The strongly reduced Δt collide with the increasing relaxation time (Figure 8 of [11]. Second, the divergence of τ El to an apparently infinitely large value t ∞ near T Crit questions safe identification of superconductor (electron) temperature. Considering item (ii) of the above, standard radiative transfer theory is incomplete because it does not differentiate between particular transition times arising under multi-component heat transfer. Transition time of a thermal excitation, under solid conduction in parallel to radiative heat transfer, has recently been reported [7] for the YBaCuO 123 thin film superconductor. The calculated time lag between different heat transfer components (see below, Figure 7a) is substantial under stability aspects. The solution of the in-completeness problem very recently proposed in [7] divides length of a total time interval into appropriately chosen sub-intervals collected in a matrix wherein, line by line, the different heat transfer mechanisms, according to their propagation speed (and other criteria), are separated. Division into intervals not only separates "fast" from "slow" transport phenomena in multi-component heat transfer, it separates also transition times of scattering from those of absorption/remission. Their propagation speed is drastically different. The solution proposed in [7] is in the present paper extended to an operator, solution scheme, which simply is built up on the previous matrix of radiative transfer and energy equations but is more flexible and adds more clarity to the procedure. Description and an example are presented in Sect. 6. (iii) We have a three-fold problem with time scales: The first originates from divergence of the relaxation time, as mentioned in item (i). Also the second problem is trivial: It results from the time lag, item (ii). The third problem manifests itself in a cloud of "images" that arise from "events". Examples for events are local temperature variations that initiate emission of thermal radiation (mid-IR photons) that, when absorbed/remitted or scattered to other positions, induce corresponding, local temperature variations, the "images" (Figure 2). The point is: Any single, isolated event, by different heat transfer mechanisms (transport channels), and each transport channel individually running against a variety of obstacles produce a very large number of images. Obstacles result from e. g. contact resistances, materials inhomogeneities, voids, scattering centres, particle geometry, variations of the indices of refraction. For the same event, images are not identical. Unlike the time lag between different transport channels, item (ii), these uncertainties, manifested by the images, arise within each of the transport channels. While this uncertainty may seriously deteriorate temporal localisability of events, a yet more difficult question concerns existence of uniquely defined, physical time scales in general. This is a parallel to the already mentioned paper by Castro-Ruiz et al. [10]. Missing temporal localisability in both cases might be explained from a common origin, namely from a generalised non-transparency concept (Sect. 8). It appears that the uncertainties arising from items (i) to (iii) never have been investigated in connection with superconductivity, or they have simply been assumed as negligible. Before we in Sects. 6 to 8 in more detail discuss items (i) to (iii) and the completeness problem in radiative transfer within superconductors, a hypothesis shall be raised in Sect. 3.1: It is well known that electrical conductivity of superconductor materials at T > T Crit is larger than the expected, normal conduction value. Against traditional explanation of this phenomenon, the increased conductivity might be correlated with the superconductor order parameter. The extent of this contribution is unknown, but an attempt is made in Sect. 3.1 to quantify this contribution. If this hypothesis could be confirmed experimentally, validity of the microscopic stability model would strongly be supported, and the results of this model (Figure 1a) then also provide a key for the verification of items (i) to (iii). Crit
Increased electrical conductivity of homogeneous metallic superconductors at T > T
Crit , in relation to the normal conducting value, usually is understood as originating, as an intrinsic propery, from a "sort of fluctuating superconductivity", see Glover III [12]. This reference explains the increased conductivity as originating from superconducting carriers of finite lifetime when they are formed in the homogeneous superconductor material. Glover III claims evidence that rounding of the transition curve seen in thin samples with short electron mean free paths relies on this effect and cannot be explained from sample materials inhomogeneity. It is tempting to alternatively explain the increased conductivity, or at least estimate a contribution to this effect, from two observations: (a) non-uniformity of transient temperature distributions in superconductors and large temperature gradients, (b) finite relaxation time, both items as reported in [7] and the previous papers. During a disturbance (warm-up of the superconductor), excursion with temperature of the phonon system is ahead the electron system by a shift Δt
ElPh . The shift is a function of the (weak) coupling between both systems and of temperature. It has been calculated in [4] for the superconductors NbTi and YBaCuO 123, see Figures 13a,b to 14 of this reference. For T(t Ph ) > T Crit , the relation T(t El ) < T Crit can be fulfilled provided relaxation time of the electron system, τ El , is large enough. Excursion with time of superconductor transport properties in its two (normal conducting or superconducting) electronic phases then would be correlated over an "obstacle" that does not rely on materials inhomogeneity but is set by the critical temperature and temperature gradients. Large temperature gradients usually result from materials inhomogeneity but also induce materials interfaces simply because of temperature-dependent materials properties. Strong local temperature variations and large local temperature gradients, instead of a materials inhomogeneity, thus might constitute a mixture of regions of totally different electrical transport properties, like in a two-fluid model. Under this proviso, the contribution of superconducting charge carriers, after complete relaxation, to the overall, effective electrical conductivity could tentatively be estimated from a suitable cell model if the "porosity", π (describing the dominating, normal conducting contribution to the total electrical conductivity), or the part 1 - π (of the superconducting "inclusions"), is given. Since 1 - π is not known (but certainly is very small), we will provisionally treat it as a free variable. It is not clear that the porosity would be constant, independent of temperature. Not only zero resistance of the superconducting state but also the residual number of electron pairs has to be accounted for in the effective conductivity. The contribution 1 - π reflects the order parameter. Since it strongly depends on temperature, the fraction 1 - π cannot be constant. The Russell cell model is a standard means to calculate materials and transport properties of two-component systems. The model is easy to handle: In its original version, it just contains porosity and resistivity of both, solid and porous phases, either for electrical or thermal transport, and is flexible (the role of particles and voids without much effort can be interchanged). For its description and an application see [13]. The model can be used to calculate the effective electrical resistance, R eff , of the thermal fluctuations state by assuming it is not an i ntrinsic effect but the consequence of a "two fluid" mixture of normal conducting and superconducting states. To apply the model for calculations, the residual electron pair contribution has to be assigned a pseudo, finite non-zero resistance (trivially by many orders of magnitude smaller than the resistance of the normal conducting state). Predictions of the model may diverge at very small values of the porosity (this is a weak point of all cell models). We have applied the model to the superconductor Pb, with T Crit = 7.2 K. Assuming π below 10 -6 at temperature exceeding but very close to T Crit and π = 1 at T = 7.3 K (zero contribution to electrical conductivity by electron pairs), and with calculation of order parameter and relaxation time as done previously [4], the result for the effective, specific electrical resistivity, ρ El , is shown in Figure 1b. It is in the order of 10 -8 Ohm cm and decays during cool-down to T
Crit . Reversely, the specific electrical conductivity increases the closer the system approaches T
Crit from temperatures above. This is the result that is observed for the electrical conductivity of the thermal fluctuations state described in standard textbooks on superconductivity. The result shown in Figure 1b is at least not in conflict with standard theory; these are not questioned in general; Figure 1b might just contribute to the explanation of the thermal fluctuations phenomenon of electrical conductivity. Qualitatively, the course of the curve in Figure 1b appears acceptable but quantitative agreement still has to be found. If it exists, the method to calculate the order parameter in [4] in turn would be confirmed. This needs more investigations.
4 Superconductor Samples used in the Stability Calculations
We have numerically investigated the stability problem in two superconductors: In the Powder in Tube (PiT) manufacturing process, tapes are prepared as first generation (1G) BSCCO 2223 multi-filamentary superconductors. The tapes consist of a large number of flat filaments of superconductor material embedded in a metallic (Ag) matrix. Each filament consists of thin, flat plate-like superconductor grains, see e. g. Figure 1a,b of [5], or Figures 1a,b and 5 of [6] or the microscopic sections in Figure 6a,b of the same reference. The cross section of the other superconductor, the "second generation" (2G) coated, YBaCuO 123 thin film, is described in Figure 1 of [11]. It shows a flat coil of in total 100 turns (of which in this reference the outermost five windings are numerically modeled). In the radiative propagation, not in the proper materials sense, a microscopic, mid-IR optical grain structure is identified even in this thin superconductor film that induces obstacles against radiative transfer, see Figure 7a-c of [6]. A flux flow resistivity, continuum cell model has been suggested recently [13]. It improves standard approximations and provides a third key for successful analysis of the stability problem and the predictability of quench. The continuum cell model improves the analyses of the stability problem at situations close to T
Crit and the related questions, items (i) to (iii) in Sect. 3.
5 Results: Transient Temperature Distributions
Simulated temperature distributions within the "first generation", (1G) BSCCO 2223 multi-filamentary superconductor are highly non-uniform, not only in the overall Ag/superconductor matrix (this is trivial) but also within the filaments of the multi-filamentary superconductor. In the Long Island Superconductor Cable, filament thickness is about 20 μm, thickness of a tape consisting of 91 filaments imbedded in the Ag-matrix is 264 μm (the cross section is shown in Figure 2a of [8]). For results of the calculations see [5] and [6]. Non-uniformity of the temperature distribution is observed also in the cross section of the (2G) coated, YBaCuO 123 thin film, see Figure 5c in [11]. In the following calculations, film thickness is D = 2 μm (D taken provisionally, see Subsect. 6.3.2 and Caption to Figure 6b). In both superconductors, the simulations indicate enormous rates, dT(x,y,t)/dt, of local conductor temperature increase once a disturbance is switched on, here a sudden increase of transport current (a fault), within 2.5 ms, to a multiple of 20 of its nominal value. In the multi-filamentary BSCCO superconductor, the dT(x,y,t)/dt increase from about 3 10 K/s at the beginning to about 10 K/s when critical temperature of this superconductor (108 K) is exceeded. If in the thin film, YBaCuO superconductor the ratio, I
Transp /I Crit , of transport to critical current is limited to 0.95 and the statistical variation, d
JCrit of J
Crit (an uncertainty resulting e. g. from conductor manufacture) is within only one percent, no losses from current transport in the superconductor thin film will be observed. Conductor temperature, under this condition is very uniform, see Figure 4a,b of [11]). But the situation changes significantly when the ratio I Transp /I Crit closely approaches 1; see Figures 5c, 6 and 7 of the same reference. In both BSCCO and YBCO superconductors, quench does not start uniformly in the conductor volume even if the disturbance, in its first stages, might result from uniformly distributed flux flow losses. Also this finding is contrary to standard stability models. In both BSCCO and YBCO superconductors, non-uniform temperature distribution and, accordingly, local generation of quench is in strong contrast to the traditional assumptions. In both superconductors, zero loss, flux flow resistive and Ohmic resistive states therefore may coexist, within finite time intervals in the conductor cross section. There is no laminar flow-like current transport; instead, transport current under disturbances percolates through the conductor. If there are no, or no longer, zero resistance transport channels, the current selects those of lowest resistance, a process that in summa provides minimum losses. Like distribution of transport current, losses in the conductor cross section may be different in each time step and at each length-position of the conductor.
6 The Completeness Problem of Radiative Transfer
The completeness problem, from the rigorous radiative transfer aspect, has been explained extensively in Sect. 4.2 of [7]. It is in some aspects similar to public traffic (vehicles with strongly different speed), or when considering diffusion of different species, like with mixed gases or in soldering at solid/liquid interfaces. However, in all these situations, the diffusing species (cars, gas particles, alloy atoms) can be distinguished clearly which is not the case in radiative transfer; there is only one species (photons or beams thereof) that can be differentiated with respect to wavelength (after remission or inelastic scattering), polarisation, spatial distribution and intensity. Few attempts to solve the completeness problem in rigorous radiative transfer have been reported in the literature, like in Sect. 21.6 of Siegel and Howell [14] and citations therein. But only radiation propagation under absorption/remission (no other modes of heat transfer) is considered in the time dependent equation of radiative transfer. The theory of radiative transfer is explained in standard volumes like [14 - 15]; focus of [14] is rather on technical issues while the frequently cited [15] explains the classical theory of radiative transfer. Applications to single fibres, multi-filamentary superconductors and thin films has been described in our previous papers [4 - 9, 16]; details will not repeated here. In short, heat transfer including radiation requests the simultaneous solutions of the equation of radiative transfer (the ERT), the equation of conservation of energy (the "energy equation"). Both equations contain one and only one time variable, which raises the problem how to treat multi-component heat transfer in case of strongly different propagation speed of the contributing transport channels. After a single heat pulse generated at a position x , it is not clear at all that thermal (solid conductive or radiative) signals emitted from this pulse would arrive simultaneously at another co-ordinate. As a way out of the problem, we have in [7] suggested a matrix the elements of which are of the ERT and energy equations type. For this purpose, a series of ERTs and, correspondingly, a series of energy equations are applied, each of which defined in separate time intervals. This series can be re-arranged as operator equations, with a symmetric n x n matrix, M , and a column vector, V , of n lines. The matrix contains the ERT and the column vector the corresponding energy equations. The number of matrix M and vector V elements has to be chosen in appropriately dimensioned time intervals according to the physics of the transport process and the speed by which its components proceed through a sample. Propagation of radiation by scattering clearly is the fastest process (Figure 6a in Sect. 7). Specification of the length of time intervals and of the integration steps in the Finite Element procedure has to fulfil also convergence criteria. Consider Figure 3a for an example for application of the operator scheme: It describes a superconductor thin film and its co-ordinate system that specifies area elements and, by rotation against the symmetry axis (thick solid line), volume elements. The elements are used for both Finite Element and Monte Carlo simulations. The target, a circular surface, is subject to a sudden heat load. Origin of the load is arbitrary, it may be given by a conductor movement or absorption of high energy, particle radiation. In the Finite Element scheme, the load is thermalised in the conductor by solid conduction (described by the standard Fourier conduction law) and, in the Monte Carlo simulation, by radiation beams emitted from the target to the area or volume elements. All materials parameters depend on temperature. In this example, it is sufficient that the matrix for simplicity is of only the 4 x 4 type, and the column vector accordingly contains four rows. The results not only show that a difference exists between temperature distributions applying solid conduction plus radiation (the realistic scenario) or only solid conduction (both described as conductivities); the difference is clearly seen (later, Figure 4a-d against Figure 5a) when comparing temperature fields and maximum conductor temperature, Sect. 6.3.1,2. With these dimensions of M and V , a second difference is seen if, later, in Figure 6a,b the " standard procedure" (solid and radiative conductivity including the Monte Carlo simulation) is compared with results obtained with the operator concept. In all calculations, a check of the stagnation temperature obtained with the simple energy balance and the results from the operator concept yields almost perfect agreement. The matrix M operator reads and the column vector V is given by for the example given in Sect. 6.3. The symbol Ω denotes the Albedo for single scattering. From the results obtained in [6], Ω is nearly constant. Its weak temperature dependency has been integrated into the calculation of the extinction coefficients. But this is still an approximation only, because it is not clear that Ω constant is generally applicable to also multiple scattering in dispersed, radiative structures (Figure 7a-c of [6]) of YBaCuO 123 materials. For explanation of the other symbols and of structure of ERT and energy equations, see Sect. 4 of [7]. Dimension of all vectors q (like q elSC , q SolidCond+elSC , q SolidCond , q inelSC+AbsRem ) is W/m . The t i , t j (right to the column vector) define those time intervals within which elastic or inelastic scattering, absorption/remission and scattering, or only absorption/remission, or only solid conduction, or combinations of all, respectively, contribute to total heat transfer. All transport mechanisms occupy the time scale according to their individual transit times, Figure 7a, and by their individual strength. The matrix can be further extended to sets that correspond to different source functions or wavelengths. The operator product M × V (1) yields a series of solutions (a solution, column vector) for the transient temperature distribution, T(x,y,t), during each of the time intervals. By the symbol "×" in Eq. (1), lines and columns of M and V , respectively, are coupled to specify the solution problem within the given time intervals. In this scheme, boundary conditions (intensity, temperature) may replace modelling exponential decay of radiation incident into and absorbed/remitted in the superconductor sample. Application of the operator equations is laborious. But if the object under study is non-transparent, modelling of radiative transfer drastically simplifies to diffusion solutions. This allows to apply the Additive Approximation (provided the sample is really non-transparent to mid-IR radiation). Once a distribution of instantaneous, initial heat sources is provided by the Monte Carlo simulation of the absorption of a large number of bundles, this distribution is equivalent to an initial temperature distribution (Carslaw and Jaeger [17]). This theorem allows to treat the whole thermalisation problem (solid conduction in parallel to radiation) as a conduction process, a very strong reduction of the complexity of the method. Non-transparency is the ideal situation to explain incompleteness of Radiative Transfer and to justify separation of the total simulated period into time intervals by application of the operator scheme. It has to be shown that the scheme explained in the previous Subsection really works. This is not clear: The scheme involves repeated changes of the input variables, mostly radiative, during the running, numerical integration procedure, which means convergence might seriously be disturbed. In order to enhance the effect exerted by the anisotropic thermal conductivity in this example, with an anisotropy ratio χ (approximately χ = λ ab /λ c = 10), the crystallographic c-axis in Figures 4a-d, 5a,b and 6a,b (not in Figure 6c) has been oriented parallel to the sample x-axis (this is in this study for systematic purposes only; this orientation would not be suitable to obtain large, zero-loss current transport). For an over-all view of what is to be expected from the calculations, Figures 4a-d and 5a,b first show results applying the standard procedure, which means without dividing the time axis into intervals. Later, Figure 6a-c, the standard procedure is replaced by the operator concept. Figure 4a-d shows temperature distribution within the sample at t = 5 10 -8 s after start of the disturbance. In Figure 4a,b, emission of the beams is not from a sharply defined, single position but is variable within the target area: It is assumed, as a realistic case, that the positions are randomly distributed but concentrated around given cluster points (like elements of a mathematical series but with limited element number, here the individual emission points from which beams are emanated). A strong temperature increase would result (Figure 5a) if the excitation is only on the target surface, without generating heat sources within the sample. This is due to the small, solid thermal diffusivity of the material that induces a thermal "mirror" effect. In contrast, the Monte Carlo simulations distribute part of the total thermal load by emission to the interior of the sample and thus serves for strongly reduced front side temperature, by conservation of energy. Temperature of a considerable part of the conductor cross section exceeds T
Crit (lower part of Figure 5b).
Application of the operator concept to the same physical conditions using matrix M , column vector V and Eq. (1) yields the dark-blue diamonds in Figure 6a. Total length of the pulse is divided into eight single, short pulses to allow repeated application of the Carslaw and Jaeger theorem: Each single pulse (and its distribution by the heat transfer mechanisms prepares the initial temperature distribution for the next pulse. Near the final end of the pulse (t = 8 ns), the difference to the standard procedure, at T < T Crit , is very small, below 0.2 K (see the results within the black ellipse in Figure 6a). But this difference reflects only the temperature excursions at a s ingle position (x=0,y=0). A possible influence on the stability function can be found only from the integral over the total cross section and from mapping of the temperature field onto the field of critical current density. See a check of this expectation in a subsequent paper; we will in the present paper concentrate on primarily the applicability of the operator scheme. If t( nlast ) denotes the temperature obtained at the end of the load step n last , the division of the time axis is for all times t given by t( nlast - 1 ) + dtime,j. For all load steps (each of length 1 ns), the values of dtime,j are given as noted in the Caption to Figure 6a. In comparison to Figures 4a-d, 5a,b and 6a,b, the orientation of the c-axis in Figure 6c is parallel to the y-axis of the sample. Within t ≤ dtime,1, the Albedo is set to Ω = 1 (solely scattering) and for solely solid conduction heat transfer. During this period, a considerable part of the intensity emitted from the target thus is lost from the sample; scattering may direct the residual beams to e. g. neighbouring thin films, substrates or electrical insulation, which means it does not contribute to increase of temperature in the superconductor thin film. In all four intervals, Ω is set according to the specifications of the column vector, see above. While in Figure 6a, the deviations between standard procedure and operator concept are very small, this changes significantly when different weight is given to radiation by variations of the dtime,j (Figure 6b). In this Figure, decreasing values of dtime,3, from 0.9 to 0.5 and 0.1, are selected according to transit times shown in Figure 7a in order to put increasingly more weight to solid conduction. Reduction of dtime,3, because of the much larger solid conductivity, accordingly results in a continuous decrease of nodal temperature, compare the diamonds in the black ellipse in Figure 6b. The uncertainty of solid temperature seen around T
Crit , here in the early stages of temperature excursion, t = 10 ns after onset of the disturbance, Q, may question superconductor stability. At later times, data points almost coincide but then J Crit is zero, in any case, and accordingly does not have impacts on stability against quench . The same result would inevitably apply if film thickness is reduced to below D = 2 μm because of increasing contribution by radiation to total heat transfer. In magnetic coils, thin YBaCuO 123 films frequently are prepared with thickness below this value. Short load steps in Figure 6a,b (in units of 10 -9 s) shall account for e. g. incident, high energy particle radiation. Extended length of load steps (10 -5 s, Figure 6c, is applied to simulate the release and transformation of mechanical stress to thermal energy by conductor movement under magnetic (Lorentz) forces. While Figure 6a,b demonstrates that the operator concept is really applicable, Figure 6c is intended to confirm, within the same sequence in time of heat transfer mechanisms, but of increased length of the pulses, the concept works also if variations of materials, here the dominating (by strength and length of the intervals) conductive properties (this is the reason to expand length of the pulses and time intervals). After t = 8 ns (Figure 6a,b) or 8 10 -5 s (Figure 6c), the calculation is continued with solid conduction plus radiation, like in the standard procedure or conventionally by simple solid conduction. Pure radiative signals from the incident pulses have died out at the end of each load-step. In turn, if temperature distributions are given, variations of the dtime,j according to Figure 6a-c may be suitable to extract, from approximations to given temperature excursion, the temporal sequence of heat transfer mechanisms, by an n-dimensional χ -fits of calculated to experimental data (in the present case, n = 4, per load-step). Measurements have to be taken at least at n positions in the cross section. While this procedure again is laborious, this is an extension of traditional laser-flash experiments: The operator concept yields a time sequence of thermal diffusivity that resolves the contributions from different heat transfer mechanisms. Traditional laser flash experiments do not provide this information. The suggested operator scheme allows solution of the combined solid conduction parallel to radiative transfer also in increasingly complicated modelling problems, e. g. if a radiative disturbance is emitted from a source outside the proper superconductor, like a heat pulse from a normal conducting layer that, under current sharing, as a stabiliser contacts and mechanically stabilises the superconductor thin film. Explicit solution of the Equation of Radiative Transfer then would not be possible. A very similar problem arises in the well-known solution problem experienced in the Parker and Jenkins method to remotely determine the thermal diffusivity of thin films (see the explanations given in Sect. 4 of [7]).
7 Time Lag from Monte Carlo Simulations
The average radiative temporal length, the effective transition time, t
Trans , of beams under purely scattering and direct transmission, which means the temporal dimension of the corresponding cloud of events, is in the order of only 10 -14 s, with a standard deviation of between 1 to 2 x 10 -14 s, see Figure 10a,b of [7]. Direct transmission means: The scattered, residual intensity that remains after each interaction of the beam with an "obstacle". It is detected at the rear surface of the 2 μm, YBaCuO 123 thin film. The number, N, of radiative/solid interactions amounts to about 30 to 60 (black diamonds in Figure 8 of [7]), for a mean value obtained with 5 10 bundles. It substantially exceeds the critical (direct transmission) optical thickness, τ = 15, because only few of the radiation paths are strictly parallel to the surface normal; the large majority follows zig-zag paths. The large number, N, accordingly confirms applicability of the non-transparency approach. But the real situation (heat transfer, not only by solely radiation) involves combined solid conduction and radiation by scattering and absorption/remission. Each bundle after absorption at an obstacle would be split into an arbitrarily large number of remitted, "conductive" bundles that is too complicated to be simulated with a solely Monte Carlo method. We yet can treat this realistic but difficult, total thermal transport problem by application of an approximate diffusion solution of Fourier's differential equation, see standard textbooks on Heat Transfer, e. g. [18], Eq. 4.3-26, L = C (a Th t) of which an extension to also radiative transfer can be applied: With a Total = a SC + a Rad, the transition time is estimated from t
Trans = (D/C) /(a SC +a Rad ) (2) with C a constant; C is obtained from the standard, "1 percent" condition, C = 3.6 in a flat sample. But this is only an approximation to the present problem: Eq. (2) assumes that surface temperature of a flat sample suddenly jumps to a constant, stable value. Nevertheless, Eq. (2) has successfully been applied in the literature to also other disturbances (except for very short pulses that require full solution of Fourier's differential equation, like in the series expansion reported in [16]). Application of Eq. (2) works if the Albedo Ω < 1 and if the diffusivity a
Rad is restricted to solely the absorption/remission part of the radiation. The summation a
Total = a SC + a Rad is again justified by the Additive Approximation (the diffusivities contain the corresponding thermal conductivities for which the Additive Approximation has been confirmed). The other values in the diffusivity are approximately constant specific heat and density. The results (Figure 7a) indicate existence of an enormous time span between the maximum (purely absorbed/remitted radiation, a fictitious case) and the physically realistic minimum (solid conduction plus radiation; pure scattering again is a fictitious situation). Here we note that there is no superconductor the radiation extinction properties of which would rely solely on absorption/remission (Ω = 0, but situations may come up, like in the example in Sect. 6 the simulation of which may be reduced to this case). The transmission time for the combined solid conduction and radiative (absorption/remission) heat transfer estimated from Eq. (2) thus delivers an upper limit, t
Trans ≤ 5 10 -2 s, at T close to T Crit . This is rather independent of the Albedo, Ω. The uncertainty interval, dT(x,y,t), of the temperature, at x = D (D the thickness of the thin film), then is calculated using temperature variations, dT(x,y,t)/dt, multiplied by the transit time, t
Trans . The (large) variations dT/dt are obtained from the simulated conductor temperatures. The final result, dT(x,y,t) (Figure 7b) is explained by the transit time of 5 10 -2 s transformed to a temperature uncertainty arising within this period of about 0.7 K. While this is small, a dT(x,y,t) = 0.7 K is already larger than uncertainties observed in standard, critical temperature measurements. This result is well comprehensible from the time lag betwee n different heat transport channels and from statistics within a particular channel. But according to item (iii) in Sect. 3, the situation is still more complex. The results obtained so far accordingly have to be confronted with existence or non-existence of local physical time scales near a phase transition, here in non-transparent media. This question, whether or not local physical time scales exist, is not just academic.
8 Existence of uniquely defined Time Scales
First conclusions can be drawn: (i) A considerable part of electron pairs, during a disturbance and its relaxation, remains un-decayed and continuously serves for zero-loss current transport (during relaxation, electron pairs and electrons generated by the disturbance temporarily co-exist, like in classical, two-phase systems). This view is the background of the attempt reported in Subsect. 3.1 to contribute to the explanation of additional electrical conductivity at T > T
Crit in a cell-model. This observation might also explain the standard R(T) vs. temperature diagram when closely inspecting on the T-scale the steep increase near T
Crit from zero to non-zero resistivity. (ii) As long as relaxation is not completed, which means, as long as no thermodynamic equilibrium is obtained, time cannot uniquely be defined if time is understood in the strict thermodynamic sense. Here "time" means: Time t El defined as a thermodynamic variable defined for solely the electron system. There is also the time shift Δt Ph/El between t El and its phonon counterpart of lattice excitations, t Ph . (iii) The same conclusions apply to existence of critical temperature: T Crit is a property of strictly the superconductor electron state and has to be understood as a thermodynamic equilibrium variable. Any temperature, even if infinitely close to T
Crit , is not defined as long as no thermodynamic equilibrium of the electron system is obtained. In standard experiments, with variation of the time t Ph (and consequently of t El , since both are coupled by conservation of energy), the approach to T Crit , from temperatures below, too strongly increases relaxation time to obtain equilibrium within reasonable computation time (Figure 8 of [11]). The question then is whether T
Crit , can be interpreted as the limes of the series of (simulated) T, and as a consequence, whether a definite, sharply defined T
Crit , within simulations and in reality, exists at all (if it is understood that traditionally T
Crit is interpreted as a sharply defined temperature above which strictly no electron-electron pairs should exist). Relaxation proceeds fast, but it is a continuous process, not a sudden event. A way out of the problem is to understand T
Crit as a strongly non- equilibrium quantity but this view does not solve the problem of uniqueness. The following in some aspects appears to be parallel to a very recently published paper by Castro-Ruiz et al [10]: The authors investigate temporal localisability in quantum systems under relativity principles. The authors explain: When clocks interact gravitationally, the temporal localisability of events becomes relative that indicates a signature of an indefinite metric. Let events, 1 and 2, at different positions be given, under the observation of three experimenters, A, B and C. Then A might report the events occur simultaneously, while B reports 1 occurs before 2 and C recognises 2 before 1. The reports provided by A, B and C depend on their reference systems; a universal time scale for the three observers that uniquely books 1 and 2 in a common logical order on a time scale does not exist. From relativity, there is also no universally defined "instant". This is the classical result. But the paper by Castro-Ruiz et al [10] goes forward: A set of multiple clocks, with description of the evolution of a quantum system in which the space-time metric is indefinite due to gravitation in these systems by superposition of energy or position eigenstates. Obviously, the authors consider existence and inter-dependencies of multiple time scales in transparent space. But this is really the question: On the other hand, the Schrödinger equation is form-invariant under transformation from one time reference frame to another. But when when checked by items (a) to (e) and their extensions in Sect. 8.2, the situation would be better explained by a more general notion of non-transparency. The present paper generally questions localisability if a system is non-transparent, either in classicial or relativistic states. In terms of a logical order that could be observed by the three experimenters A, B and C, unique localisability gets lost not only in view of three time scales related to relativistic conditions (namely those of A, B and C) but also for, in principle, all time scales if these were constructed within a non-transparent medium. In another step forward and in a more general conception than only in the mid-IR optical sense or between gravitationally coupled clocks (or in case of entanglement), non-transparency could be the common background of these three and possibly other situations. The problem then is how to define a general non-transparency. Here it is helpful to recall how transparency of a solid medium like a thin film is defined and how it opposes non-transparency in the strict optical sense. Assume a thin film sample positioned between two parallel planes at x = 0 and x = D and let radiation beams be emitted, in direction normal to the planes, from a source (a small target area located at x = 0, parallel to the planes). The following items (a) to (e) apply also if emission from the source is isotropic, or if emission comes from a thermal source of infinitely small or of extended, non-zero cross section (the assumed geometry is not very decisive, one can for the following imagine quite a different setup). If a detector responds to an original beam or to an original distribution of beams, all emitted from the target, an observer at a position x ≥ D, if the medium is transparent, will be able to differentiate between (a) radiation emitted by the source at constant power or wavelength, with the radiation source at different (axial) positions of the target, or (b) radiation emitted at variable power or wavelength, but with the radiation source at a fixed, single position, (c) monochromatic radiation intensity emitted by the source at different power, (d) radiation intensity emitted at constant power but at different wavelengths, (e) single isolated pulses, or series thereof, and periodic radiation sources, all emitted from any (stationary) position or at any wavelength or at any time or frequency are received at x ≥ D or at any other co-ordinate, under conservation of the temporal order of emission and of detection. If any of the above listed items (a) to (e) is violated, the sample is non-transparent. Violation of item (e) would not get this conclusion into contradiction with the HOM experiment [19, Figure 2]: Let two photons emitted from the same source hit a beam splitter, under different directions. If they are scattered onto two separate detectors, the photons are distinguishable only if they do not arrive simultaneously at the beam splitter, a result that cannot be explained by classical theory. If in a transmission experiment the residual angular distribution of radiation leaving the sample on its rear surface is isotropic (i. e., if it follows the cosine-law), the sample again is non-transparent. This property provides an easily applicable check to decide whether the sample is non-transparent and the Additive Approximation is justified. Trivially,, in order to successfully realise items (a) to (e) in an experiment, the time interval between any variation of wavelength, duration, intensity or position of the source must be shorter than the characteristic time of the detector. The observer accordingly will only then “almost immediately” notice any variation of the emitted signals. Violation of items (a) to (e) and their extensions might also provide a tool to identify a possibly existing common background of generalised non-transparency in other physical situations: Gravitationally coupled clocks and entanglement. Previously, we have described transparency of the space between two stationary, flat planes at x = 0 and x = D by means of mapping functions, f[e( s ,ζ)], that create images, e( x ,t), of events, e( s ,ζ); compare Figure 3 in [6], the detailed description is not repeated here. In short, assume that at the position, s , a set consisting of an arbitrarily large number of (stationary) events, e( s ,ζ), can be defined, with ζ a provisional "scale" at this position that simply counts the events, like emission of a radiation beam from x = 0 (or from any other position, s , within the space) according to their temporal order of emission. Provisionally, we can say the events simply are "booked" by their order on a "heuristic scale", ζ. For transfer ( promotion ) of this scale to a genuine, uniquely and unambiguously defined "time scale", t, the time interval between any two successively booked events, e( s ,ζ ) and e( s ,ζ ) at the same position, s , potentially must be infinitely small, which means the distribution of the events on this scale, ζ, must be "dense". The property "dense" means: A distribution of elements, e( s ,ζ), and of their images, f[e( s ,ζ)] = e( s ,t), that reflects the distribution of elements of the set R of real numbers. Between any of these, there exists an unlimited number of other elements belonging to the same set. The number of elements on the (then genuine) time scale, t, is uncountably large, yet the elements are successively ordered (like the elements of the R are ordered. The set R can be expanded to R , if needed. A complete (perfect) correlation between time scales is possible only if the medium is transparent. If so, the mapping functions can be interpreted likewise as "bijective" correlations (injective and surjective), in the mathematical sense; this definition applies to also sets of an unlimited (uncountable) number of elements like R . Time scales, t, to be uniquely and unambiguously defined, cannot exist, in the physical (optical) sense without dense sets of events or images. Reversely, while conservation of the order is provided in case of transparent media, this is not fulfilled in a non-transparent medium, neither in space nor in time. The bijective correlation between events and their images gets lost in non-transparent objects. A single time scale to exist in transparent media implies existence of potentially an arbitrarily large number of other time scales, all correlated among each other. A proof of this conclusion can be attempted as described in Appendix 2. Time scales can be fabricated also in non-transparent media, but they are not uniquely defined and cannot be correlated to each other. Totally uncorrelated time scales means: In non-transparent media, no correlated time scales exist at all, as a logical conclusion; otherwise, if a set of uncorrelated time scale existed, it is not clear which of these should be preferred. The enormous fluctuation of t
Trans seen in Figures 7a and 8a,b excludes any individual, unique mathematical correlation between events (bundles) emitted from the target and , physically , macroscopic observables (the images ), like a temperature variation, when radiative beams emitted at x = 0 are absorbed at x' ≠ x . By the lemma in Appendix 2, the enormous fluctuations in Figures 7a and 8a,b in non-transparent media also exclude more generally any correlation of time scales at x with time scales at the other co-ordinate x '.
9 Summary and Open Questions
The results reported in the previous Sections are intended to improve present understanding of time, temperature, quench and, tightly related to these items, non-transparency. Observation of images uniquely correlated to their events is the basis for successful, perfect local stability analysis (temperature variations as the answer of the superconductor to events arising at other conductor positions). The correlation between temperature fields and the field of critical current density is not bijective. It is not the situation at the beginning of a local quench that is the most critical, but those instants, when the superconductor very closely has approached its critical value, deserve most attention. If the superconductor is already close to a phase transition (the exact identification of which itself is uncertain), it is this situation that in applications of superconductivity requests immediate action. Also, it is this situation that requires thorough understanding in order to justify the computational efforts associated with precise, numerical simulations. But if time cannot be specified exactly and uniquely, how can stability of superconductors be predicted and a quench, initially local, be avoided with safety from spreading over the total conductor cross section? Not only stability predictions become indefinite but also the meaning of critical temperature (and perhaps also of the other critical parameters) of a superconductor. This leads to the same question as raised in [10]: Gravitating quantum systems can lead to an indefinite space-time metric. Since the Schrödinger equation relies on a time parameter, too, what then plays the role of such time parameter in the absence of a definite metric? In addition to the conclusions listed in Sect. 8.1, a central question arising from applied methods and findings reported in this paper remains: Can a generalised "non-transparency" be specified for the four scenarios (a) radiation in multi-component heat transfer, (b) quantum mechanical entanglement (the EPR experiment), (c) localisability of images on time scales, (d) loss of individuality of photons if they simultaneously hit a beam splitter (the HOM experiment). in a way that it is relevant for superconductors? Interestingly, non-transparency is the property that makes a thorough and exact analysis of transient superconductor states possible at all. In the EPR experiment (the instant when particle 2 immediately "knows" that a measurement at particle 1 has been performed), there is no flow of energy running in parallel to a flow of entropy. This situation, and the simulations reported in the present paper apparently have parallels in exchange of information. Accordingly, is “non-transparency” perhaps an information problem? The author will try to find an answer. Final note by the author: All the above is not new physics but reports corollaries and consequence resulting from analysis of multi-component heat transfer within filament structures and thin films. It is an attempt to understand which uncertainties may arise when standard analytic and numerical tools are applied to situations near phase transitions. Conventional stability models cannot provide this information. Comments are welcome.
10 References
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15 Chandrasekhar S, Radiative Transfer, Dover Publ. Inc., New York (1960) 16 Reiss H, Troitsky, O Yu, Radiative transfer and its impact on thermal diffusivity determined in remote sensing, in: A. Reimer (Ed.), Horizons of World Physics (2012) Chapter 1, pp. 1 – 67 17 Carslaw H S, Jaeger J C Conduction of Heat in Solids, 2 nd Ed., Oxford Science Publ., Clarendon Press, Oxford (1959), reprinted (1988), 256 and 356 18 Withaker St, Fundamental Principles of Heat Transfer, Pergamon Press, Inc., New York (1977) 19 Hong C K, Ou Z Y, Mandel L, Measurement of subpicosecond time intervals between two photons by interference, Phys. Rev. Lett. 59 (1987) 2044 -2046 20 Phelan P E, Flik M I, Tien C L, Radiative properties of superconducting Y-Ba-Cu-O thin films, Journal of Heat Transfer, Transact. ASME 113 (1991) 487 - 493 Figures
Temperature (K) R e l a t i ve d e n s i t y n S ( T ) / n S ( T = K ) YBaCuO T Crit = 92 K
Microscopicstability model
Minimum n S0 (T)/n S (T) to support J Crit (T)
Analytical
Figure 1a Relative density (the order parameter), f S = n S (T)/n S (T=4K), in dependence of temperature, calculated for the thin film, YBaCuO 123 superconductor (dark-blue diamonds). Dark-yellow diamonds indicate the minimum relative density of electron pairs that would be necessary to generate a critical current density of 3 10 A/m (YBaCuO) at 77 K, in zero magnetic field. The diagram compares predictions of the microscopic stability model with analytical results (light-green) calculated from Eq. (8) in [20]. The Figure is copied with slight modifications from Figure 11b of [6]. Temperature T Ph (K) S p ec i f i c e l ec t r i ca l r es i s t a n ce ( O h m c m ) Figure 1b Effective specific electrical resistance, ρ El , of Pb in the thermal fluctuations state, T > T Crit (7.2 K). Results are calculated using the Russell cell model. The resistance decays if temperature, T Ph , approaches T Crit from values above. At T = 7.3 K, the porosity of the normal charge carriers (the part that really competes with the superconducting charge carriers to current transport) is set to π = 1 in this model (the contribution of the superconducting charge carriers then is zero). Figure 2 The Figure explains mapping functions, f[e( s ,ζ)], that under purely radiative transfer assign a series of images, f[e( s ,ζ )], of the single event, e( s ,ζ ), like a temperature variation. In non-transparent samples, the number of images on a time scale, t, generated of the same event is arbitrarily large. At a given position, x , transit time (small, solid black circles) of beams emitted by the target at its position, s , reflects the effective lengths of their radiative path through the sample. In case of transparent samples, correlations provided by the mapping functions are bijective (the black, solid horizontal line), for a single event with one and only one image, here at t . Images generated by conduction heat flow would be overlaid onto the radiative images. An infinitely (uncountably) large number of images, like in a dense set, is required for generating a time scale at position x or at other coordinates. Fig. 3a Coordinate system of a superconductor, thin film. Sample radius and thickness of the film are 5 and 2 μm, respectively. The Figure (schematic, not to scale) shows area elements, k(i,j), radiation bundles (thick solid lines) and the target (thick horizontal red line, target radius 1 μm, small against sample radius). Volume elements are generated by rotating the area elements against the symmetry axis (dashed-dotted line). Scattering angle is denoted by θ. The scheme is used for both Finite Element (FE) and Monte Carlo (MC) calculations. Bundles may escape from the sample (index "Escaped") after a series of absorption/remission or scattering interactions. The blue bundle, as an example originally emerging from a position x > 0 finally escapes from positions x < 0 to neighbouring materials or to coolant; it shows that FE and MC simulations cannot be restricted to only one of the half-sections (we do not have axial symmetry). Figure 3b The Figure explains incompleteness of standard Radiative Transfer in non-transparent materials. Regular materials element (dV , Albedo Ω < 1) and a foreign, strongly absorbing inclusion (dV , Ω << 1), respectively, are indicated by the light-blue rectangle and the dark-brown, solid circle, respectively. Scattered part of a beam (dashed line, emitted from the target) hits the inclusion at an arbitrary depth position, x , after about 10 -14 s (compare Figure 7a), which causes a local temperature increase within dV . Absorbed intensity of the same beam (dashed-dotted line), with its absorption coefficient, A = (1 - Ω) E, being much larger than the scattering coefficient, S = Ω E, causes a temperature increase of dV at position x , but at about 10 -2 s after emission. Temperature excursions at both positions originate from the same basic event (emission of a beam from the target). In the non-transparent material, both generated temperature excursions, dT/dt, when considering the integral radiative process, are strictly separated in time (the thick, horizontal red line) if the geometrical distance x - x is large against a multiple 15 of the mean free paths of photons (the optical thickness between both positions, τ ≥ 15). An observer at x does not know what happens at x as long as solid conduction, as a differential process, from x to x does not couple the dT/dt. As a consequence, there is no common time scale, t, in the "cells" C and C . But solid conduction is a considerably slow process (signals emitted from the target need about 10 -7 s to arrive at x ). Figure 4a,b Temperature fields within the superconductor sample (the 2 μm YBaCuO 123 thin film). In this Figure and in Figures 4c,d and 5a,b and 6a, in order to enhance the effect exerted by anisotropic thermal conductivity on calculated temperature fields, the crystallographic c-axis has been oriented parallel to the (horizontal) sample x-axis. Results are obtained for the case "Solid conduction plus radiation ", which means from Finite Element (FE) with integrated Monte Carlo (MC) simulations, here shown for t = 5 10 -8 s simulation time. The calculations apply a short, weak (rectangular) pulse (a "disturbance") of, in this Figure, in total Q = 5 10 -12 Ws that is incident onto the target area during 0 ≤ t ≤ 8 10 -9 s. A Monte Carlo simulation to generate conductive and radiative heat sources to apply the Carslaw and Jaeger theorem is overlaid onto the target excitation. All heat transfer mechanisms (solid conduction, radiation) in total are treated as a conduction process; this is specified as the standard procedure (FE, MC) as in our previous reports. Like in Figure 5a of [7], sample thickness is divided into 0.1 μm thick boundary layers and the 1.8 μm core (see Subsect. 4.3.2 of [7] for justification). The different temperature distributions in Figure 4a (above) and 4b (below) result from positions, within the target area, of arbitrarily assigned cluster points around which the emission of the Monte Carlo beams are distributed (right or left to sample symmetry axis, respectively). Angular distribution of the residual, directional intensity of the beams, when released at the sample rear side surface, in both cases is isotropic. Figure 4c,d Temperature fields within the superconductor sample (the 2 μm YBaCuO 123 thin film). Same calculation as in Figure 4a,b but with the increased load Q = 2 10 -11
Ws. Results are shown as before at t = 5 10 -8 s simulation time. Due to the higher load Q, the difference between both temperature distributions load has become smaller; this results from the increased number of beams that survive the absorption/remission processes. But compare the asymmetric position of the index "MX relative to the sample symmetry axis that denotes temperature maximum. Figure 5a Same calculation as in Figure 4c,d, again with the increased rectangular heat pulse of in total Q = 2 10 -11
Ws, applied to the target area during 0 ≤ t ≤ 8 10 -9 s, but for solely solid conduction (obtained from only the FE result). Simulation time as before is 5 10 -8 s. The significant temperature increase in comparison to Figure 4c,d results from the condition that part of the heat pulse is not distributed by a Monte Carlo simulation to the total thin film volume but is applied to only the target surface (y = 0). Figure 5b Same calculation as in Figure 5a but for solid conduction plus radiation (using the standard procedure , with FE, MC) but for a rectangular heat pulse of in total Q = 2 10 -9 Ws applied to the target during 0 ≤ t ≤ 8 10 -9 s. Simulation time is 5 10 -8 s. Results (above): full range of temperatures, and below: for all T > T Crit = 92 K. Temperature of a considerable part of the superconductor cross section exceeds T
Crit . Zero loss current transport would be possible only within the shaded (grey) area. Figure 6a Transient nodal temperature at x = 0, y = 0 (the target centre of the the D = 2 μm YBaCuO 123 thin film). A rectangular heat pulse (the "disturbance" Q) of in total 2 10 -11
Ws, is applied to the target area) during 0 ≤ t ≤ 8 10 -9 s. Orientation of the crystallographic c-axis like in the previous Figures is parallel to the sample x-axis. The two curves show (small) differences (high-lighted by the black ellipse) between (1) solid and radiative heat transfer (both in the non-transparent medium) simulated as solely conduction processes (red diamonds, the standard procedure, with no division of the time axis), and (2) the operator concept (light-blue diamonds) using the standard procedure but additionally with intervals time,j and dtime,j (1 ≤ j ≤ 4) as given in the text; the times time,j are applied to divide each load step (nlast) into intervals, with Ω = 0 in 0 ≤ t ≤ t ). In (1) and (2), as in the previous Figures, the original excitation is at the target surface and, using the Monte Carlo simulation, with radiative and conductive heat transfer from sources generated within the sample (this is done to apply the Carslaw and Jaeger theorem, compare [17]). Like in Figures 4a-d and 5a,b, the curves are calculated using the Additive Approximation. Within the load steps, length of the intervals is given by dtime,1 = 10 -12 , dtime,2 = 8 10 -10 , dtime,3 = 9 10 -10 and dtime,4 = 10 -9 s. Length of integration time steps within these intervals is 10 -13 , 10 -10 , 10 -11 and 10 -11 s, respectively. Figure 6b Same calculation as in Figure 6a for the YBaCuO 123 thin film (D = 2 μm) using the operator concept (dark-blue, light-green and dark-yellow diamonds); red diamonds for comparison show the result obtained with the standard procedure, curve (1) in Figure 6a. In this calculation, we have applied in intervals 1 to 3 only radiative conduction while in interval 4, solid conduction is overlaid onto radiation. The small radiative conductivity is responsible for strongly increased conductor temperature (it acts like a thermal mirror). Increasing weight is assigned to temporal contributions of solid conduction by decreasing value of dtime,3 (from 0.9 to 0.5 and 0.1, given by the dark-blue, light-green and dark-yellow diamonds, respectively). Reduction of dtime,3, because of the much larger solid conductivity, accordingly results in a continuous decrease of nodal temperature, compare the diamonds in the black ellipse. The uncertainty of solid temperature seen around T
Crit , here in the early stages of temperature excursion after the disturbance, Q, may question superconductor stability. At later times, data points almost coincide but then J
Crit is zero, in any case, and have no impacts on stability against quench. Figure 6c Transient nodal temperature at x = 0, y = 0 (solid, dark-blue diamonds). Same calculation as in Figure 6a using the operator concept but with increased length of the load steps (each 10 -5 s) and with orientation of the c-axis (anti-) parallel to the sample y-axis. Total incident energy onto the target here is Q = 2 10 -10 Ws. The dark-green and yellow diamonds indicate the effect of strong (by one magnitude) parameter variations (anisotropy ratio, χ, of the solid thermal conductivity in ab- and c-axis direction) made during the sequence of time intervals (but orientation of the c-axis remains fixed). Within the black circle, temperature variations are significant for stability analysis. The Figure again is calculated using the Additive Approximation. Values of dtime,j (1 ≤ j ≤ 3) in this calculation are identical, dtime,j = 10 -13 s, which results in strong contribution by solid conduction, and accordingly reduced conductor temperature in relation to Figure 6b. Length of integration time steps within these extended intervals is 10 -14 , 10 -14 , 10 -14 and 10 -6 s. Figure 7a Transit time of signals (events) proceeding by different heat transfer mechanisms through the 2 μm YBaCuO 123 thin film. The sequence in time of heat transfer mechanisms after an event is specified as (1) solely elastic scattering, (2), (3) and (4) solid conduction in parallel to absorbed/remitted radiation (still under scattering, but scattering does not contribute to temperature excursion) The possible case of in-elastically scattered radiation is not considered. Cases (2) to (4) are calculated from the diffusion approximation L = C (a Th t) using anisotropy factors, m S , indicated in the Figure, and the Albedo Ω (taken from Figure 14b of [6]). Time spans, s , denotes time lag between two, fundamentally different, separate heat transfer mechanisms (solely solid conduction, radiation), s is the maximum difference between two completely different, radiative transport processes. Time (ms) T e m p e r a t u r e va r i a t i on dT ( x , y , t) ( K ) Figure 7b Variation dT(x,y,t) of temperature of the "second generation" (2G) coated, YBaCuO 123 thin film superconductor during its transit time under solid conduction parallel to radiation (solely absorption/remission), in the YBaCuO 123 thin film (the coil [11], turn 96). Results are given for positions close to the axis of symmetry (red diamonds), and at the outermost left thin film position of the thin film (blue diamonds). Bundle Number T r a n s i t t i m e ( - s ) Figure 8a Monte Carlo simulation of transit time, t
Trans , of individual bundles, again in the 2 μm "second generation" (2G) coated, YBaCuO 123 thin film superconductor, here under pure elastic scattering. Results are shown for the first 2 10 (of in total M = 5 10 ) bundles. Separately for the thin, 0.1 μm boundary layers and the 1.8 μm core of the thin film (Figure 5a of [7]), the calculations apply extinction coefficients for dependent and independent scattering, m S -factors (for forward scattering) and Albedo, all from [6]). The yellow, light-blue, light-green and red diamonds and the dashed lines indicate 2- or 3- or 4- or 6-sigma, safety intervals, respectively. Multiples of Sigma N u m b e r o f I m a g es ( i n p e r ce n t) Figure 8b Number of images (bundles, given in per cent of the total M = 5 10 ) that do not fall into multiples between 1σ to 6σ of the data shown in Figure 10a,b of [7]. Figure 9 Flow chart (the "master" scheme) showing two iteration levels (i, j) and one time loop (t j ) of the numerical simulation: Light-green circles and indices, i: Sub-steps, the proper Finite Element (FE) calculations; Light-yellow indices, j: Load steps involving FE and, within the blue rectangles, critical current, magnetic field and resistance (flux flow, Ohmic) calculations; Dark-yellow indices, t: Time loop, lines of the matrix M (equation numbers shown in the rectangles refer to [7]). The blue rectangles with sub-step numbers i = 1, 2, 3,...N are defined as First FE step, j, with data input of start values of temperature distribution, specific resistances, critical parameters of J, B and of initial (uniform) transport current distribution or of single, isolated radiation heat pulses, respectively; Results obtained after the first FE step (i), if converged, for the same parameters in the sam e load-step, j; calculation of T
Crit , B
Crit , J
Crit ; Calculation of resistance network and of transport current distribution (if applicable), all to be used as data input into the next FE calculation (sub-step i + 1), within the same load step, j; Results like in ; Sub-steps
5, 6,...N:
Results like in or convergence yes or no ? If "no", return to (iteration i = 1, in the same load step, j). If "yes" go to next load step j + 1, continue with . The number N of FE calculations (green circles) might strongly increase computation time. Length of simulation time, t ≤ t max , within each of the individual intervals, with t max indicating the maximum time of a corresponding particular interval, is selected according to the different transit times, source functions, different radiation propagation mechanisms, different ratios of solid conduction and radiation and to different wavelengths. More description of the FE steps can be found in Sect. 2.3 of [11] and for the possible correlation between non-convergence and quench in Figure 1 of [9]. By the time-loop, t, the Figure is an extension of Figure 12 of [9]. For an example, see Figure 14 of [8].
Appendix 1: Iterative Numerical Procedure
In order to relieve the reader from repeatedly consulting details in our previous papers, the following explains the principle how the simulation of the stability problem have been performed. An iterative "master" scheme (Figure 9) has been applied that incorporates Finite Element (FE) simulation load-steps to calculate transient temperature, T(x,y,t), from Fourier's differential equation. When convergence has been achieved, the scheme calculates electrical and magnetic superconductor states, critical values of temperature, magnetic induction and of current density, and the distribution of transport current density in the conductor cross section. All these are calculated as local, transient values. The Meissner state is checked in each element of the FE mesh and in each load-step. Details of the Finite Element part of the scheme (selection of FE elements, meshing, solvers, time steps, convergence criteria) are described in our previous papers. The simulations assume the superconductor (e. g. a coil or a current limiter) is integrated in a standard, low or medium voltage system that includes Ohmic and induction resistances.
Appendix 2 (Supplement): Time scales
Physical time scales do not exist a priori , they need events for their definition. Time scales cannot exist without events, like space cannot exist without bodies. The following applies to non-relativistic situations. For a concept to describe properties of time scales in situations close to phase transitions, and to apply these properties in numerical simulations, we assume: Time scales, i. e. elements incorporated in time scales, can be defined as isomorph to the set R of real numbers. Here, we are interested in physical time (its counterpart, psychological time, is not considered). The set R cannot be replaced by the set of rational numbers. The set R contains an uncountably large number of elements, which means the distribution of elements within R is dense (the property "dense" is explained in the text). This is the pure mathematical content. Physically , the temporal "distance" between two events or between two images cannot be infinitesimally small; there is at least a lower limit given by resolution limits set to practical experiments. The numerical procedure selects only those events or images that can be assigned to the physics of a process under consideration (like a temperature variation, decay of an electron pair, or a local quench). But potentially the whole set of elements of R would be available for the simulations. The question whether this applies to only positive elements is left open. Correlation between an arbitrary physical situation (a system located at a position, s , within which an event takes place) and the set R is realised by mapping functions that project events, e( s ,ζ), in their natural order, ζ, onto a time scale. The projections of the events are their images, f[e( s , ζ)] = e( s ,t); these are correlated to elements of R where they within R are ordered if mapping is bijective; which replaces ζ by t. Correlations provided by the mapping functions and bijective mapping exist per se, if the system is transparent. Mapping does not require any action exerted by individuals or by mechanisms, it is a purely mathematical construct. In Figure 10, the decisive steps 1 and 3 correlate events with images on the time scale, t, that in turn are bijectively correlated with (mapped onto) the set R . The set R in this Figure is expanded to R to explain that the correlations between events and images can be considered at arbitrary positions x and x' in the superconductor cross section or volume. Figure 10 A cycle (schematic) to explain coupling between events, e and e , their images, time scales and the set of real numbers, R . Time scales are coupled, by bijective mapping, to an infinitely large number of sets R , wherein all sets are isomorph to each other (the dashed circles thus encloses elements the number of which is infinitely large and is yet identical to R ). The cycle consisting of steps (1) to (4) can be completed with bijective mapping functions and isomorphisms only in case the medium is transparent (which means, if also relation 4 is bijective). The complete cycle is the identity (if correlation 4 is bijective). Once bijective mapping between time scales and the set R is accepted, the set R takes the role as a "vehicle". In non-transparent media, relation (4) in Figure 10 is not uniquely defined. If distances of x ' from x are smaller than optical thickness of the medium, exceptions might be possible. These conclusions apply the following lemma: Lemma: A single time scale to exist implies existence of an arbitrarily large number of other time scales. A proof can be attempted by considering the construction of the set, R , of real numbers from four axioms: Trichotomy, transistivity, compatibility with addition and multiplication. No any other automorphism apart from identity is allowed in R . Time scales, by definition, shall be isomorph to R . Therefore, an arbitrarily number of isomorph sets R and, referring to the initial assumption (see above), an arbitrarily large number of time scales, can be generated, the latter only if the medium is transparent. Temporal localisability relies on transparency, temporal non-localisability of events accordingly results from non-transparency. A question remains: Is time itself transparent?transparent?