BCS Results Research Articles

Thermal pairing treatment within the path integral formalism

A method for the treatment of pairing correlations at finite temperature is proposed within the path integral formalism, based on the square root extraction of the pairing term in the Hamiltonian of the system. Gap equations and expressions for the pairing gap parameter Δ, energy E, and heat capacity C are established. The formalism is first tested using the Richardson model, which enables comparison with an exact solution. The results obtained using this formalism are also compared with the finite temperature BCS (FTBCS) results. An improvement over the FTBCS model is noted, especially at low temperatures. Indeed, the agreement between the Δ values of this study and the exact values is good at low temperatures. This leads to better agreement between the values of E and C of this model and the exact values than with the FTBCS values. However, a critical value of temperature remains. Subsequently, realistic cases are considered using single-particle energies of a deformed Woods-Saxon mean-field for the nuclei Dy and Yb. In the framework of the current approach, pairing effects persist beyond the FTBCS critical temperature. Moreover, at low temperatures, a good agreement between the model and semiexperimental values of the heat capacity is observed, and a clear improvement compared to the FTBCS method is noted. This is no more the case at higher temperatures.

Coverage of the Subcutaneous Layer of the Breast and Chest Wall: Monte Carlo Calculation at Various Energies

Histological studies have determined that breast skin thickness is 3 mm. The subcutaneous layer is at risk of local recurrence for patients with breast cancer; because it lies within the buildup region for megavoltage radiation treatment, placement of bolus for patients receiving chest wall (CW) radiotherapy (RT) is routine at many centers. However, bolus is rarely used for breast (BCS) RT. The risk of Grade 3 acute toxicity is 9.6% with bolus and 1.2% without bolus. We hypothesized that the coverage of the subcutaneous layer was similar for the chest wall and breast through different beam energies.For each beam energy (6, 10 and 15 MV), one patient received CWRT and a second patient received BCSRT. Within the tangent fields, a shell was created using a high-resolution structure from 3 to 5 mm below the skin. Monte Carlo calculation (MC) plans were done for each patient using a 1 mm x 1 mm x 1 mm grid. For each energy, the V90 and V95 were calculated and compared between CW and BCS RESULTS: The V90 and V95 for the 3-5 mm shell was better for CWRT than for BCSRT at each energy. For 3 to 5mm, the V90 CW coverage was 14.0 % better than BCS for 6MV, 28.4% better than BCS for 10 MV and 36.3% better than BCS for 15MV. V95 CW coverage was also better for each energy.This comparison of V90 and V95 coverage of the subcutaneous layer for CWRT and BCSRT demonstrates that CWRT delivers a higher dose at clinically relevant energies. This raises the question of why bolus is deemed to be important for routine chest wall RT when it is not used for routine breast RT.

Competition of electron-phonon mediated superconductivity and Stoner magnetism on a flat band

The effective attractive interaction between electrons, mediated by electron-phonon coupling, is a well-established mechanism of conventional superconductivity. In metals exhibiting a Fermi surface, the critical temperature of superconductivity is exponentially smaller than the characteristic phonon energy. Therefore such superconductors are found only at temperatures below a few Kelvin. Systems with flat energy bands have been suggested to cure the problem and provide a route to room-temperature superconductivity, but previous studies are limited to only BCS models with an effective attractive interaction. Here we generalize Eliashberg's theory of strong-coupling superconductivity to systems with flat bands and relate the mean-field critical temperature to the microscopic parameters describing electron-phonon and electron-electron interaction. We also analyze the strong-coupling corrections to the BCS results, and construct the phase diagram exhibiting superconductivity and magnetic phases on an equal footing. Our results are especially relevant for novel quantum materials where electronic dispersion and interaction strength are controllable.

Calculating the thermal properties of 93,94,95Mo using the BCS model with an average value gap parameter

The gap parameter of the standard BCS model is replaced by the order parameter of the modified Ginzburg–Landau theory. Using this new form of the BCS model, the energy, entropy, and heat capacity of \(^{93,94,95}\)Mo nuclei are calculated. The results are compared with the experimental data and standard BCS results. Since the order parameter does not drop to zero at a critical temperature, our results for thermal properties are free of singularities. We have shown that the heat capacity as a function of temperature behaves smoothly and it is highly in agreement with the experimental heat capacity, while heat capacity behaves singularly and discontinuously in the standard BCS model. A smooth peak in the heat capacity is observed which is interpreted as a signature of the transition from the superfluid to the normal phase.

The role of the chemical potential in the BCS theory

We study the effect of the chemical potential on the results of the BCS theory of superconductivity. We assume that the pairing interaction is manifested between electrons of single-particle energies in an interval $[\mu - \hbar\omega_c, \mu + \hbar\omega_c]$, where $\mu$ and $\omega_c$ are parameters of the model--$\mu$ needs not be equal to the chemical potential of the system, denoted here by $\mu_R$. The BCS results are recovered if $\mu = \mu_R$. If $\mu \ne \mu_R$ the physical properties change significantly: the energy gap $\Delta$ is smaller than the BCS gap, a population imbalance appears, and the superconductor-normal metal phase transition is of the first order. The quasiparticle imbalance is an equilibrium property that appears due to the asymmetry with respect to $\mu_R$ of the single-particle energy interval in which the pairing potential is manifested. For $\mu_R - \mu$ taking values in some ranges, the equation for $\Delta$ may have more than one solution at the same temperature, forming branches of solutions when $\Delta$ is plotted vs $\mu_R-\mu$ at fixed $T$. The solution with the highest energy gap, which corresponds to the BCS solution when $\mu = \mu_R$, cease to exist if $|\mu-\mu_R| \ge 2\Delta_0$ ($\Delta_0$ is the BCS gap at zero temperature). Therefore the superconductivity is conditioned by the existence of the pairing interaction and also by the value of $\mu_R - \mu$.

Supersymmetric BCS: effects of an external magnetic field and spatial fluctuations of the gap

Recently an N=1 supersymmetric model of BCS superconductivity was proposed realizing spontaneous symmetry breaking of a U(1)_R symmetry. Due to scalar contributions the superconducting phase transition turned out to be first order rather than second order as in standard BCS theory. Here we consider the effects of an external magnetic field and spatial fluctuations of the gap in that model. This allows us to compute the magnetic penetration length and the coherence length, and also to distinguish between type I and type II superconductors. We compare the supersymmetric and standard relativistic BCS results, where the main differences come from the different orders of the phase transition.

High Superconducting T c and Suppressed Isotope Effect in the Instantonic Condensate State of the Fermi-System: Analytic Solution

A superconducting pairing instability and iso- tope effect on the superconducting transition temperature Tc in the electronic system with instantonic condensate are derived analytically using previously proposed theory (Mukhin, J. Supercond. Nov. Magn. 22:75-80, 2009; J. Su- percond. Nov. Magn. 24:1165-1171, 2011) of the instan- tonic nature of the "hidden order" in high-Tc cuprates. The self-organized instantonic condensate breaks Matsub- ara "time" invariance of the Fermi-system, and occurs due to strongly nonlinear interaction between antiferromagnetic or charge density fluctuations. The Cooper-channel suscepti- bility (polarization loop) is calculated using a single-particle Green's function that we have derived recently (Mukhin and Galimzyanov, Physica B 407:1882-1884, 2012 )f or the Fermi-system with instantonic condensate. We found that superconducting Tc could be strongly enhanced and an iso- tope effect strongly suppressed in the presence of the instan- tonic condensate. The usual weak-coupling BCS results are recovered in the limit of vanishing the instantons. Relevance of the theoretical predictions for the high-Tc compounds is discussed.

Quality Assurance At the Point of Care: PT/INR: What Is the Gold Standard for Patient Treatment?.

Abstract 3170 Background:The Prothrombin time (PT) can be performed in laboratories by instruments such as BCS (Siemens), and at the Point of Care using devices such as the Protime 3 (ITC) and the Coaguchek XS (Roche). Laboratory based instruments such as the BCS have been regarded as the Gold Standard for PT/INR assuming results can be obtained in under 24 hours from blood drawn in an uncentrifuged sample. A previous study performed in the Netherlands recommends that INR testing be completed within 6 hours of blood draw. This recommendation has not been universally adopted. In order to assess this effect, a study comparing the laboratory instrument (BCS) to POC instruments was performed while simultaneously comparing lab based results at or greater than 6 hours. Methods:The INR values of the BCS in two separate laboratories, Coaguchek XS, and Protime 3 were compared to each other. The INR values and turnaround times for each were recorded. The results were compared using paired observation T-tests and regression analysis including all data and within sub-therapeutic (<2), therapeutic (2.0–3.0), and supratherapeutic range (>3.0). Additionally, point of care results were compared to BCS test results before and after 6 hours. Results:60 patients had 3 INR test methods performed. All samples were tested in two different reference laboratories, both using BCS testing method. Reference lab A performed the tests under optimal conditions including a turnaround times of less than 2 hours, while Reference lab B performed the tests under usual workflow conditions. While the total Protime 3 results were significantly different from Reference lab A, the Coaguchek XS results were not. In the therapeutic range, the Coaguchek XS and Reference lab B had significant correlation (r = .556 and .634 respectively) to reference lab A while the Protime 3 did not (r = .281).Reference lab B results were significantly different from Reference lab A when testing was performed after 6 hours (P-value = .0252), but had similar results to Reference lab A when performed within 6 hours (P-value = .402). The Coaguchek XS showed no significant differences when compared to Reference lab A results (p-value = .1262). Another indicator of lack of concordance between Reference lab A and B, when testing is performed greater than 6 hours, was the change in directionality of the bias (p-value = .0396). Additionally, Protime 3 results in the therapeutic range correlated significantly better with Reference lab B results when they were resulted within 6 hours (r= .191 and r= .782). Conclusions:Based on these findings, the BCS method appears to lose reliability when the test is performed on uncentrifuged blood greater than 6 hours after blood draw. Interestingly, the Coaguchek XS correlated best with Reference lab A, BCS testing under optimal conditions. The Protime 3, on the other hand, was less well correlated with Reference lab A. The change in directionality of the bias also suggests that the BCS results were flawed when testing was performed after 6 hours. Although additional preanalytic factors (i.e. temperature) may have had an effect on these results, these data suggest that laboratory based INR testing should be performed within six hours of sample collection. Disclosures:No relevant conflicts of interest to declare.

Multipair approach to pairing in nuclei

The ground state of a general pairing Hamiltonian for a finite nuclear system is constructed as a product of collective, real, distinct pairs. These are determined sequentially via an iterative variational procedure that resorts to diagonalizations of the Hamiltonian in restricted model spaces. Different applications of the method are provided that include comparisons with exact and projected BCS results. The quantities that are examined are correlation energies, occupation numbers and pair transfer matrix elements. In a first application within the picket-fence model, the method is seen to generate the exact ground state for pairing strengths confined in a given range. Further applications of the method concern pairing in spherically symmetric mean fields and include simple exactly solvable models as well as some realistic calculations for middle-shell Sn isotopes. In the latter applications, two different ways of defining the pairs are examined: either with J=0 or with no well-defined angular momentum. The second choice reveals to be more effective leading, under some circumstances, to solutions that are basically exact.

PARTICLE-PARTICLE HOLE-HOLE TDA – AND BEYOND – FOR THE NUCLEAR PAIRING HAMILTONIAN

A comparison of different seniority zero solutions to the picket-fence model for the nuclear pairing hamiltonian problem is performed. These solutions are calculated, in the normal regime, within the self-consistent Random Phase Approximation (SCRPA) and various simplifications of this formalism, and also with the Tamm-Dancoff approach in the particle-particle-hole-hole channel (pphh-TDA). The latter formalism represents a first approximation to the earlier developped so-called P-Symmetric Many-Body method (PSY-MB). In the superfluid regime, the solutions are compared with the BCS results. By comparing the results with the exact ones, obtained by the Richardson method, it is shown that the PSY-MB method provides a powerful tool in solving the problem with good accuracy both in the normal and the superfluid regime, for single-particle space sizes adapted to typical nuclear structure calculations.

THERMODYNAMIC PROPERTIES OF A BCS SUPERCONDUCTOR NEAR ZERO TEMPERATURE

We theoretically study the thermodynamic properties of a BCS superconductor near zero temperature. Derivations of the temperature dependence of the order parameter and an exact formula for the free-energy difference between the superconducting and normal states are presented as functions of temperature and material parameters of the superconductor. Under the condition that the cut-off energy is much greater than the temperature, formulas for the critical field and specific heat in the superconducting state are presented. Our expressions for these thermodynamic quantities show new corrections to the BCS's results.

Role of intraband and interband pairings in two-band superconductors: deviation from BCS results

We study the stability of the intraband ( r=dd) and interband ( r=sd) couplings in the presence of the k -dependent hybridization in a two-band system. We calculate the critical temperature T c, r associated to both type of couplings, and we verified that there is a critical value of the hybridization V c, for which T c, r vanishes. Then we obtain the value of ε r=2Δ ∗ r(0)/k B T c,r and the isotope coefficients γ r showing that under some circumstances, one can obtain values for ε r and γ r which differ considerably from standard BCS results for both Cooper pairings.

Fixed number and quantum size effects in nanoscale superconductors

In recent experiments on nanoscale A1 particles, whose electron number was fixed by charging effects, a “negative gap” was observed in particles with an odd number of electrons. This observation has called into question the use of a grand canonical ensemble in describing superconductivity in such ultrasmall particles. We have studied the effects of fixed electron number and finite size in nanoscale superconductors, by applying the canonical BCS theory for the attractive Hubbard model. The ground state energy and the energy gap are compared with the conventional and parity-projected grand canonical BCS results, and in one dimension also with the exact solutions by the Bethe ansatz. The crossover from the bulk to quantum limit is studied for various regimes of electron density and coupling strength. A “negative gap” for odd electron number emerges most naturally in the canonical scheme. For even electron number, the gap is particularly large for “magic numbers” of electrons for a given system size or of atoms for a fixed electron density. These features are in accordance with the exact solutions, but are essentially missed in the grand canonical results.

Some properties of two-band superconductors: Role of k-dependent hybridization

We discuss the influence of a k-dependent hybridization on a superconducting material described by a two-band model, within an extended BCS picture. The effect of hybridization is to create in the normal state new bands with mixed features and the k-dependence arises from its non-local character. We consider a broad s-p conduction band hybridized with a narrow d-band, assuming that the pairing occurs at the d-band ( singlet d-d pairing). We calculate self-consistently the BCS type order parameter Δ and the critical temperature T c . We verify that T c is stable when hybridization is turned on until a critical value of V = V c and both the “phononic” and electronic parts of T c are renormalized by a parameter α, which gives the ratio of the effective band masses. We calculate the value of 2 Δ(0)/ k B T c and compare with standard BCS results. The isotope effect is almost the same in the whole range of 0 ≤ V < V c energy.

Effects of incomplete superconducting condensation inS-wave superconductors

We have reexamined quasiparticle excitations in s-wave superconductors by considering a finite range ${T}_{d}$ of pairing interaction energy. When ${T}_{d}$ is finite, low-energy states which are believed to be a hallmark of other pairing symmetries such as d-wave symmetry are also found in s-wave superconductors. We have numerically computed (i) the quasiparticle density of states, (ii) the order parameter, (iii) the superfluid density as a function of ${T}_{d}{/T}_{c},$ and (iv) the critical temperature ${T}_{c}$ as a function of effective coupling constant. These calculations show unusual features which arise as a result of these low-energy states when ${T}_{d}$ becomes comparable to ${T}_{c}.$ In conventional superconductors, where ${T}_{d}{/T}_{c}\ensuremath{\gg}1,$ the BCS results are recovered as the effect of the low-energy states becomes negligible.

Structure of the vacuum states in the presence of isovector and isoscalar pairing correlations

The long standing problem of proton-neutron pairing and, in particular, the limitations imposed on the solutions by the available symmetries, is revisited. We look for solutions with non-vanishing expectation values of the proton, the neutron and the isoscalar gaps. For an equal number of protons and neutrons we find two solutions where the absolute values of proton and neutrons gaps are equal but have the same or opposite sign. The behavior and structure of these solutions differ for spin saturated (single l-shell) and spin unsaturared systems (single j-shell). In the former case the BCS results are checked against an exact calculation.

Brueckner correlations following a boson mapping of the two-color delta model.

We study the application of boson mapping methods to a two-color delta model of interacting quarks, with the purpose of extracting information of relevance to quark-model descriptions of nuclei. In an earlier treatment, the boson Hamiltonian resulting from the mapping was approximately diagonalized at the level of Hartree-Bose approximation. The results differed from those of an exact solution of the model at first order in the density. Furthermore, the results were worse than those arising from a BCS treatment of pair correlations at the quark level. Here we extend the analysis to a Brueckner treatment, so as to properly take into account the short-range repulsion between bosons originating from the quark Pauli principle. The resulting energy per quark reproduces the exact results through first order in the density and is significantly better than the BCS results. At higher densities, deviations appear, reflecting the need for a full cluster expansion in the treatment of short-range correlation effects. The relevance of these results to more realistic three-color quark models of nuclei is discussed. {copyright} {ital 1996 The American Physical Society.}

Coherence lengths for three-dimensional superconductors in the BCS-Bose picture.

Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering {ital t}-matrix, the low-energy form of which is expressible in terms of the {ital s}-wave scattering length {ital a}{sub {ital s}}. The BCS gap equation for {ital s}-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap {Delta} as well as the chemical potential {mu} as functions of the dimensionless coupling variable {lambda}{equivalent_to}{ital k}{sub {ital F}}{ital a}{sub {ital s}}, where {ital k}{sub {ital F}} is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of {lambda} and compared with experimental values.

Pairing theory of high Tc and low Tc superconductors

New solutions for T c, the order parameter, and the density of states are obtained, based on the fact that pairs are formed within the pairing interaction range, T D (Debye), via the BCS pairing theory (not the BCS results). They are valiid for all ify = T D πT c , and are applicable to low T c (LTC) and high T c (HTS) superconductors. The order parameter variation with y is shown to account for all features of HTS. A new density of states via the zero order parameter outside the pairing interaction range is found to account for low energy states observed in HTS. For large y (LTS), the BCS results are reproduced.

Finite temperature correlated quasiparticle approach

A correlated quasiparticle approximation, based on the inclusion of diagonal two-body terms in the exponent of the statistical operator, is discussed. The scheme is seen to yield a substantial improvement over the conventional finite temperature mean-field approach for finite systems. The method is illustrated in two different pairing models and compared with exact and finite temperature BCS results.