We study the influence of spin exchange interaction of impurity scattering on critical temperature of anisotropic impure superconductors. The model of random nonmagnetic and magnetic impurity are revised to cover the effect of spin exchange interaction . The sign of magnitude of the second-order Born scattering have been changed after consideration the spin exchange interaction effect that also effect to form of Tc equation. We can get the general Tc equation that can be described anisotropic impure superconductors well and cover all model done before.
The purpose of our research is to study the nuclear spin lattice relaxation rate of impure d-wave superconductors. We use the Green function method to derive the approximation equation of density of states including the impurity scattering potential. We can get the analytic equation of the nuclear spin lattice relaxation rate that contained the impurity scattering potential in case of weak scattering potential and strong scattering potential in the simple form as the power series of order parameter and temperature . The numerical calculations show that there is coherence peak in the weak impur…
In the electron-phonon model, the influence of nonmagnetic impurities on the transition temperature of superconductors is revisited. Anderson's pairing condition between time-reversed eigenstate pairs is derived from the physical constraint of the Anomalous Green's function. After this pairing condition is incorporated into the self-consistency equation, one may find that the phonon-mediated interaction decreases exponentially by localization.
The electron-electron relaxation in impure two-dimensional superconductors is studied. All channels of the electron-electron interaction classified in the Nambu representation are taken into account. It is shown that the recombination relaxation rate originates from quasipartical processes associated with fluctuations of the electron density and the phase of the order parameter. At low temperatures the recombination relaxation rate has a double exponential temperature dependence. The scattering relaxation rate at low temperatures has a power law temperature dependence due to contributions from…
Recently a generic model has been proposed for the single-particle properties of gapless superconductors with simultaneously present pair-conserving and pair-breaking impurity scattering (the so-called Dynes superconductors). Here we calculate the optical conductivity of the Dynes superconductors. Our approach is applicable for all disorder strengths from the clean up to the dirty limit and for all relative ratios of the two types of scattering, nevertheless the complexity of our description is equivalent to that of the widely used Mattis-Bardeen theory. We identify two optical fingerprints of…
The calculations of the local and global properties of two band superconductors have been presented with particular attention to the role of the inter-orbital scattering of pairs. The properties of such superconductors are very different from a single band or typical two band systems with dominant intra-band pairing interactions. The role of Van Hove singularity in one of the bands on the properties of intra-band clean superconductor has been discussed. It leads to marked increase of superconducting transition temperature in the weak coupling limit. We study the inhomogeneous systems in which …
We reconsider the ordinary impurity effect on the transition temperature $T_{c}$ of superconductors using the Eliashberg formalism. It is shown that the correspondence principle, which relates strong-coupling and weak-coupling theories, works only when Anderson's pairing condition between the time-reversed scattered-states is used. For an Einstein phonon model, the change of the electron density of states caused by the impurity scattering leads to a $T_{c}$ decrease proportional to ${1/ E_{F}τ}$ in the dirty limit. It is pointed out that the phonon-mediated interaction decreases by the same we…
The Ginzburg-Landau coefficients, and the jump of the specific heat are calculated for a disordered two-band superconductor. We start with the analysis of a more general case arbitrary anisotropy. While the specific heat discontinuity at the critical temperature T_c decreases with increasing disorder, its ratio to the normal state specific heat at T_c increases and slowly converges to the isotropic value. For a strong disorder the deviation from the isotropic value is proportional to the elastic electron scattering time. In the case of a two-band superconductor we apply a simplified model of t…
We study the influence of anisotropy of the impurity potential on isotope coefficient of an impure superconductor in Born limit. From the isotope coefficient of non-magnetic and magnetic impurities including an angle-resolved Fermi surface density of state, anisotropy order parameter and impure potential, we make the assumption that the form of them are in the harmonic form and are investigated. Our numerical calculation of isotope coefficient can fit well with experimental data of high-Tc superconductors.
Based on the Kubo formula approach, the spin Hall conductivity (SHC) of a two-dimensional (2D) Rashba s-wave superconductor in the presence of nonmagnetic impurities is calculated. We will show that by increasing the superconducting gap, the SHC decreases monotonically to zero, while by decreasing the concentration of impurities at zero gap, the SHC closes to the clean limit universal value . As a function of the impurity relaxation rate at and (is the spin-orbit coupling in unit of), we will show that in the dirty limit () the SHC vanishes, and by increasing the relaxation time () the SHC dep…
The structural correlation functions of a weakly disordered Abrikosov lattice are calculated in a functional RG-expansion in $d=4-ε$ dimensions. It is shown, that in the asymptotic limit the Abrikosov lattice exhibits still quasi-long-range translational order described by a {\it nonuniversal} exponent $η_{\bf G}$ which depends on the ratio of the renormalized elastic constants $κ={c}_{66}/ {c}_{11}$ of the flux line (FL) lattice. Our calculations clearly demonstrate three distinct scaling regimes corresponding to the Larkin, the random manifold and the asymptotic Bragg-glass regime. On a wide…
The influence of various kinds of impurities on the isotope shift exponent αof high temperature superconductors has been studied. In these materials the dopant impurities, like Sr in La_{2-x}Sr_xCuO_4, play different role and usually occupy different sites than impurities like Zn, Fe, Ni {\it etc} intentionally introduced into the system to study its superconducting properties. In the paper the in-plane and out-of-plane impurities present in layered superconductors have been considered. They differently affect the superconducting transition temperature T_c. The relative change of isotope shift…
The combined effect of both nonmagnetic and magnetic impurities on the superconducting transition temperature is studied theoretically within the BCS model. An expression for the critical temperature as a function of potential and spin-flip scattering rates is derived for a two-dimensional superconductor with arbitrary in-plane anisotropy of the superconducting order parameter, ranging from isotropic s-wave to d-wave (or any pairing state with nonzero angular momentum) and including anisotropic s-wave and mixed (d+s)-wave as particular cases. This expression generalizes the well-known Abrikoso…
Superfluid 3He confined to high porosity silica aerogel is the paradigm system for understanding impurity effects in unconventional superconductors. However, a crucial first step has been elusive; exact identification of the microscopic states of the superfluid in the presence of quenched disorder. Using a new class of highly uniform aerogel materials, we report pulsed nuclear magnetic resonance experiments that demonstrate definitively that the two observed superfluid states in aerogel are impure versions of the isotropic and axial p-wave states. The theoretically predicted destruction of lon…
We review the recent progress in the theory of inhomogeneous superconductors. It was shown that Gor'kov's self-consistency equation needs a pairing constraint derived from the Anomalous Green's function. The Bogoliubov-de Gennes equations also need a pairing constraint in order to obtain a correct vacuum state by the corresponding unitary transformation. This new study opens up a reinvestigation of inhomogeneous superconductors. We discuss (i) problems of the conventional Green's function theory, (ii) reinvestigation of impure superconductors, and (iii) impurity doping effect in high $T_{c}$ s…
The effect of anisotropic order parameter to specific heat of anisotropic two-band superconductors in BCS weak-coupling limit are investigated. An analytical specific heat jump and the numerical specific heat are shown by using anisotropic order parameters, and the electron-phonon interaction and non-electron-phonon interaction. The two model of anisotropic order parameters are used for numerical calculation that we find little effect to the numerical results . The specific heat jump of, and superconductors can fit well with the both of them. By comparing to experimental data overall range of …
Lattice distortions in which the axes are locally rotated provide an intrinsic source of frustration in anisotropic superconductors. A general framework to study this effect is presented. The influence of lattice defects and phonons in $d$ and $s+d$ layered superconductors is studied.
Sharp increase of pinning force was observed at the 3D-2D phase transition in strongly anisotropic high temperature superconductors of the BiPbSrCaCuO system.
The upper critical field, the lower critical field and the critical magnetic field ratio of anisotropic magnetic superconductors are calculated by Ginzburg-Landau theory analytically. The effect of the Ginzburg-Landau parameter(κ_0), magnetic susceptibility(χ) and magnetic-to- anisotropic parameter ratio(θ) on the critical field ratio are considered. We find that the value of critical magnetic field ratio with χ< 0, higher κ_0,and higher θcase are shown the higher value. And, the diamagnetic superconductors with highly anisotropic and the ferromagnetic superconductors are shown the highest and…
The effect of a homogeneous magnetic field and nonmagnetic impurities on the critical temperature of an anisotropic superconductor has been investigated. The role of these pair-breakers in relation to the anisotropy of the order parameter is clarified.
We discuss the effect of an anisotropic impurity potential on the critical temperature, local density of states in the vicinity of a single impurity, and the quasiparticle density of states for a finite impurity concentration in a d-wave superconductor. Different scattering regimes are concerned.
In a type II superconductor in a moderate magnetic field, the superconductor to normal state transition may be described as a phase transition in which the vortex lattice melts into a liquid. In a biaxial superconductor, or even a uniaxial superconductor with magnetic field oriented perpendicular to the symmetry axis, the vortices acquire elongated cross sections and interactions. Systems of anisotropic, interacting constituents generally exhibit liquid crystalline phases. We examine the possibility of a two step melting in homogeneous type II superconductors with anisotropic superfluid stiffn…
An exact analytical solution is given for the critical state problem in long thin superconductor strips in a perpendicular magnetic field, when the critical current density j_c(B) depends on the local induction B according to a simple three-parameter model. This model describes both isotropic superconductors with this j_c(B) dependence, but also superconductors with anisotropic pinning described by a dependence j_c(theta) where theta is the tilt angle of the flux lines away from the normal to the specimen plane.
The specific-heat jump $ΔC$ at a critical temperature $T_c$ in an anisotropic superconductor containing both potential and spin-flip scatterers is calculated within a weak-coupling mean-field approximation. It is shown that the presence of even a small amount of spin-flip scatterers in the sample leads to a drastic change in the dependence of $ΔC$ on $T_c$ in a disordered $(d+s)$-wave or a strongly anisotropic s-wave superconductor. The implications for experimental tests for the presence of an s-wave admixture in the superconducting order parameter of high-$T_c$ superconductors are discussed.
We report measurements of the critical temperature of YBCO-Co doped YBCO Superconductor-Normal bilayer films. Depending on the morphology of the S-N interface, the coupling between S and N layers can be turned on to depress the critical temperature of S by tens of degrees, or turned down so the layers appear almost totally decoupled. This novel effect can be explained by the mechanism of quasiparticle transmission into an anisotropic superconductor.
A formula for the Josephson current between two superconductors with anisotropic pairing symmetries is derived based on the mean-field theory of superconductivity. Zero-energy states formed at the junction interfaces is one of basic phenomena in anisotropic superconductor junctions. In the obtained formula, effects of the zero-energy states on the Josephson current are taken into account through the Andreev reflection coefficients of a quasiparticle. In low temperature regimes, the formula can describe an anomaly in the Josephson current which is a direct consequence of the exsitence of zero-e…
Ginzburg-Landau expansion for superconductors with anisotropic $s-$ and $d-$wave pairing is derived in the presence of anisotropic normal impurity scattering which makes $d-$pairing state more stable under disordering. It is demonstrated that the slope of the upper critical field $|dH_{c2}/dT|_{T_{c}}$ in superconductors with $d-$wave pairing has nonlinear dependence on disorder, i.e. for the low anisotropic scattering rate it drops rather fast with concentration of normal impurities, but as anisotropy of scattering increases it features initial nonlinear growth, approaches a maximum and drops…
The upper and lower critical field, and the critical field ratio of an anisotropic two-band magnetic superconductors in Ginzburg-Landau(GL) scenery is derived analytically. The temperature-dependent on the upper critical field is investigated and applied to Fe-based superconductors. We find that the very high value of zero-temperature upper critical field in Fe-based superconductors can be found in the negative differential susceptibility region. The temperature-dependent upper critical field is presented in two formulas as in the empirical view and in the GL two-band view that agrees with the…
We derive a pair potential from tight binding further neighbours attraction that leads to superconducting gap symmetry similar to that of the phenomenological spin fluctuation theory of high temperature superconductors (Monthoux, Balatsky, Pines, Phys. Rev. Lett. {\bf 67}, 3448). We show that higher anisotropic d-wave than the simpliest d-wave symmetry is one of the important ingredients responsible for higher BCS characteristic ratio.
This is a Comment on "Vortex Liquid Crystal in Anisotropic Type II Superconductors" by E. W. Carlson et al. in PRL, vol.90, 087001 (2003) [cond-mat/0209175].
The penetration depth of anisotropic two-band s-wave superconductors have been studied by Ginzburg-Landau approach. The anisotropic spherical Fermi surfaces and order parameters with the spherical ellipse or pancake Fermi shape are used for derive penetration depth equation. The numerical calculation of pancake-pancake and pancake-ellipse Fermi sphere shape are fits well with the experiment data of MgB2 and CaAlSi superconductor, consequently. Our model should be the way to increase understanding in the microscopic mechanism of occurrence of the penetration depth in superconductivity state of …
We propose a model and derive analytical expressions for conductivity in heterogeneous fully anisotropic conductors with ellipsoid superconducting inclusions. This model and calculations are useful to analyze the observed temperature dependence of conductivity anisotropy in various anisotropic superconductors, where superconductivity onset happens inhomogeneously in the form of isolated superconducting islands. The results are applied to explain the experimental data on resistivity above the transition temperature $T_c$ in the high-temperature superconductor $\mathrm{YBa_2Cu_4O_8}$ and in the …
We discuss the nuclear magnetic relaxation rate and the superfluid density with the use of the effective five-band model by Kuroki et al. [Phys. Rev. Lett. 101, 087004 (2008)] in Fe-based superconductors. We show that a fully-gapped anisotropic \pm s-wave superconductivity consistently explains experimental observations. In our phenomenological model, the gaps are assumed to be anisotropic on the electron-like βFermi surfaces around the M point, where the maximum of the anisotropic gap is about four times larger than the minimum.
To understand the anisotropy of flux pinning and critical current density in technological superconductors, the scaling law for the anisotropy of single-vortex collective pinning in uniaxial superconductors is extended to flux-bundle collective pinning in biaxial superconductors. The scaling results show that in a system of random uncorrected point defects, the critical current density is described by a unified function with the magnetic field of the scaled isotropic superconductor. The obtained angular dependence of the critical current density depicts the main features of experimental observ…
We study a single impurity effect on the local density of states in a d-wave superconductor accounting for the momentum-dependent impurity potential. We show that the anisotropy of the scattering potential can alter significantly the spatial dependence of the quasiparticle density of states in the vicinity of the impurity.
We argue that in channels cut out of anisotropic single crystal superconductors and narrow on the scale of London penetration depth, the persistent current must cause the transverse phase difference provided the current does not point in any of the principal crystal directions. The difference is proportional to the current value and depends on the anisotropy parameter, on the current direction relative to the crystal, and on the transverse channel dimension. An experimental set up to measure the transverse phase is proposed.
The effect of weak anisotropic (momentum-dependent) impurity scattering in unconventional superconductors has been investigated. It is shown that the anisotropic scattering can lead either to a small reduction or a small enhancement of the isotropic pair-breaking effect. The influence of the anisotropy of the scattering potential becomes significant for the order parameters with large Fermi surface average values. In that case an unexpected enhancement (up to 10%) of the critical temperature over the critical temperature in the absence of impurities is predicted for a small impurity concentrat…
The specific heat jump at a normal-superconducting phase transition in an anisotropic superconductor with nonmagnetic impurities is calculated within a weak-coupling mean field approximation. It is shown that its dependence on the impurity concentration is remarkably different for d-wave and (d+s)-wave states. This effect may be used as a test for the presence of an s-wave admixture in the cuprates.
In this paper we solve numerically the two dimensional elliptic sine-Gordon equation with appropriate boundary conditions. These boundary conditions are chosen to correspond to the Josephson interaction between two adjacent pancakes belonging to the same flux-line in a highly anisotropic superconductor. An extrapolation is obtained between the regimes of low and high separation of the pancakes. The resulting formula is a better candidate for use in numerical simulations than previously derived formulas.
We carry out simulations of the anisotropic uniformly frustrated 3D XY model, as a model for vortex line fluctuations in high Tc superconductors. We compute the phase diagram as a function of temperature and anisotropy, for a fixed applied magnetic field. We find that superconducting coherence parallel to the field persists into the vortex line liquid state, and that this transition lies well below the "mean-field" cross-over from the vortex line liquid to the normal state.
We present a geometry-based discussion of possible vortex configurations in the mixed state of anisotropic type-II superconductors. It is shown that, if energy considerations assign six nearest neighbors to each vortex, two distinct modifications of the vortex lattice are possible. It is expected that certain conditions lead to a first order phase transition from one modification of the vortex lattice to the other upon varying the external magnetic field.
The upper critical field Hc2 is evaluated for weakly-coupled two-band superconductors. By modeling the actual bands and the gap distribution of MgB2 by two Fermi surface spheroids with average parameters of the real material, we show that H_{c2,ab}/H_{c2,c} increases with decreasing temperature in agreement with available data.
The response field of a half-space anisotropic superconductor is evaluated for an arbitrary weak external field source. Example sources of a point magnetic moment and a circular current are considered in detail. For the penetration depth $λ\ll L$ with $L$ being any other relevant distance (the source size, or the distance between the source and the superconductor), the major contribution to the response is the $λ$ independent field of the source image. It is shown that the absolute value of $λ$ cannot be extracted from the response field with a better accuracy than that for the source position…
We discuss a case of a strong anisotropic impurity scattering within the model introduced in our previous paper [Phys. Rev. B54, 15463 (1996)] and clarify our former statement about a possible enhancement of the critical temperature in this scattering regime. We show that for the anisotropy of the impurity potential determined by the functions from the non-identity irreducible representations of the crystal point group, the critical temperature decreases with the impurity scattering rate which is consistent with a generally understood role of disorder in superconductors.
The free energy, non-gradient terms of the Ginzburg-Landau expansion, and the jump of the specific heat of a multiband anisotropic-gap clean BCS superconductor are derived in the framework of a separable-kernel approximation. Results for a two-band superconductor, d-wave superconductor, and some recent models for MgB_2 are derived as special cases.
We present direct imaging of magnetic flux structures in the anisotropic, spin-triplet superconductor Sr$_{2}$RuO$_{4}$ using a scanning $μ$SQUID microscope. Individual quantized vortices were seen at low magnetic fields. Coalescing vortices forming flux domains were revealed at intermediate fields. Based on our observations we suggest that a mechanism intrinsic to the material stabilizes the flux domains against the repulsive vortex-vortex interaction. Topological defects like domain walls can provide this, implying proof for unconventional chiral superconductivity.
Here we use highly sensitive magnetisation measurements performed using a Hall probe sensor on single crystals of highly anisotropic high temperature superconductors $Bi_{2}Sr_{2}CaCu_{2}O_{8}$ to study the dynamic interactions between the two species of vortices that exist in such superconductors. We observe a remarkable and clearly delineated high temperature regime that mirrors the underlying vortex phase diagram. Our results map out the parameter space over which these dynamic interaction processes can be used to create vortex ratchets, pumps and other fluxonic devices.
We study the quantum interference between impurities in d-wave superconductors within a potential scattering formalism that easily applies to multiple impurities. The evolution of the low-energy local density of states for both magnetic and nonmagnetic short-ranged scatterers are studied as a function of the spatial configuration of the impurities. Further we discuss the influence of subdominant bulk superconducting order parameters on the interference pattern from multiple impurities.
It is shown theoretically that the Sommerfeld coefficient $γ(B)$ in the mixed state contains useful information on the gap topology in a superconductor. We accurately evaluate $γ(B)$ against magnetic induction $B$ by using microscopic quasi-classical theory. The linear portion in {$γ(\propto B)$} relative to $B_{\rm c2}$ and its slope in low field faithfully corresponds to the gap anisotropy, so that we can distinguish the nodal gap and the anisotropic full gap, and estimate the magnitude of the gap anisotropy.
We discuss the weak-coupling BCS theory of a superconductor with the impurities, accounting for their anisotropic momentum-dependent potential. The impurity scattering process is considered in the t-matrix approximation and its influence on the superconducting critical temperature is studied in the Born and unitary limit for a d- and (d+s)-wave superconductors. We observe a significant dependence of the pair-breaking strength on the symmetry of the scattering potential and classify the impurity potentials according to their ability to alter T_c. A good agreement with the experimental data for …
We theoretically determine the Ginzburg-Landau slopes of the anisotropic upper critical magnetic field in a quasi-one-dimensional superconductor and correct the previous works on this issue. By using the experimentally measured values of the Ginzburg-Landau slopes in the superconductor (TMTSF)$_2$ClO$_4$, we determine band parameters of its electron spectrum. Our main result is that the so-called quantum dimensional crossover has to happen in this material in magnetic fields, $H = 3-8 \ T$, which are much lower than the previously assumed. We discuss how this fact influences metallic and super…
The model for high frequency electrodynamics in anisotropic type-II superconductors in the vortex state is studied considering arbitrary orientations between the applied field, the applied current and the anisotropy axis. An anisotropic treatment is provided for the vortex dynamics, taking into account all the phenomena relevant at high frequency, which include flux flow, pinning and creep. The coupling between vortex motion and high frequency currents is included, providing an entirely tensor model of the electromagnetic response to high frequency fields. Examples of data analysis of angular …
A thin flat superconductor of arbitrary shape and with arbitrary in-plane and out-of-plane anisotropy of flux-line pinning is considered, in an external magnetic field normal to its plane. It is shown that the general three-dimensional critical state problem for this superconductor reduces to the two-dimensional problem of an infinitely thin sample of the same shape but with a modified induction dependence of the critical sheet current. The methods of solving the latter problem are well known. This finding thus enables one to study the critical states in realistic samples of high-Tc supercondu…
For a completely general anisotropic order parameter (including changes of sign), we show that weak coupling theory is incompatible with high values of the maximum Δ_{M} of the zero temperature gap as compared to the critical temperature T_{c}, such as those found experimentally in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} where 2 Δ_{M} / T_{c} \approx 7.5 . This gives evidence for strong coupling effects. In particular this comes as a major support for a spin fluctuation mechanism with strong coupling, if one assumes that only a repulsive pairing interaction is at work in high T_{c} superconductors.
The thermal conductivity of borocarbide superconductor LuNi_2B_2C was measured down to 70 mK (T_c/200) in a magnetic field perpendicular to the heat current from H = 0 to above H_c2 = 7 T. As soon as vortices enter the sample, the conduction at T -> 0 grows rapidly, showing unambiguously that delocalized quasiparticles are present at the lowest energies. The field dependence is very similar to that of UPt_3, a heavy-fermion superconductor with a line of nodes in the gap, and very different from the exponential dependence characteristic of s-wave superconductors. This is strong evidence for a h…
As shown in recent experiments [Auslaender et al., Nature Physics 5, 35 (2009)] magnetic force microscopy permits one not only to image but also to manipulate an individual vortex in type-II superconductors, and this manipulation provides a new powerful tool to study vortex dynamics and pinning. We derive equations that describe the deformation of an individual vortex in an anisotropic biaxial type-II superconductor under the action of the microscope's magnetic tip. These equations take into account the driving force generated by the tip, the elastic force caused by the vortex deformation, and…
This article develops a Fermi-liquid theory for superconductors with anisotropic Fermi surfaces, Fermi-liquid interactions, and energy gaps. For d-wave superconductors, the Fermi-liquid interaction effects are found to be classifiable into strong and negligible renormalizaton effects, for symmetric and antisymmetric combinations of the energies of $k\uparrow$ and $-k\downarrow$ quasiparticles, respectively. Furthermore, the leading clean-limit temperature-dependent correction to the superfluid density in a d-wave superconductor is found to be renormalized by a Fermi velocity (or mass) renormal…
The physical nature of the macroturbulence in the vortex matter in YBCO superconductors is verified by means of magnetooptic study of the instability in a single crystal prepared specially for this purpose. The instability develops near those sample edges where the oppositely directed flow of vortices and antivortices, guided by twin boundaries, is characterized by the discontinuity of the tangential component of the hydrodynamic velocity. This fact directly indicates that the macroturbulence is analogous to the instability of fluid flow at a surface of a tangential velocity discontinuity in c…
Physical properties of anisotropic superconductors like the critical temperature and others depend sensitively on the electron mean free path. The sensitivity to impurity scattering and the resulting anomalies are considered a characteristic feature of strongly anisotropic pairing. These anomalies are usually analyzed in terms of s-wave impurity scattering which leads to universal pair breaking effects depending on only two scattering parameters, the mean free path and the impurity cross section. We investigate here corrections coming from anisotropies in the scattering cross section, and find…
The vortex dynamics of uniaxial anisotropic superconductors in magnetic fields applied with arbitrary orientation is theoretically studied. Focus is on the model for electrical transport experiments in the linear regime. Relevant vortex parameters, like the viscous drag, the vortex mobility and pinning constant (with point pins), together with the flux flow and Campbell resistivities, are derived in tensor form, in the very different free flux flow and pinned Campbell regimes. The applicability to the various tensor quantities of the well-known scaling laws for the angular dependence on the fi…
