Cooper pair

Coolwater Coomassie Brilliant Blue Cooperite Cooper pairs Coordination Coordination catalysts... 

Nakamura, Y., Pashkin, Y.A. and Tsai, J.S. (1999) Coherent control of macroscopic quantum states in a single-Cooper-pair box. Nature, 398, 786-788. 

In a superconducting metal, some of the electrons are paired into the so-called cooper pairs which are all in the same zero entropy state and do not carry heat. Heat is carried only by unpaired electrons which are in energy states separated from cooper pairs by an energy gap AE(T). The number of unpaired electrons varies as exp(—AE/kBT). Hence ... 

The quantum of magnetic flux is only 2.07 x 10-15 Wb, which is approximately equal to the amount of the earth s magnetic field enclosed by a ring of 10p,m in diameter. The Josephson effect is observed when two superconductors are separated by a very thin insulating layer (about 20 nm). Single electrons and Cooper pairs can tunnel through such a layer. The characteristics of the Josephson junction are now used to define the volt and have enabled the uncertainty in the maintained standard to be reduced to 0.1 p,V. 

The interaction between charge carriers (electrons or holes) and phonons (lattice) may give origin to several phenomena in a crystal, e.g. the production of Cooper pairs. 

If transition temperatures other than those allowed by superconducting pure metals are required, two metallic layers can be deposited to form a bi-layer TES. In most cases, only one of the two metals is a superconductor. In this case, the Cooper pairs diffuse from... 

Further synergistic enhancement of amide resonance and H-bonding occurs when both monomers can participate in two complementary H-bonds, once as a Lewis base and once as a Lewis acid. Such concerted (cooperative) pairs of H-bonds occur in the cyclic formamide dimer, as illustrated in Fig. 5.20. In this case the strength of each H-bond is further enhanced (to 6.61 kcalmol-1, about 4.7 times that of the prototype (5.31c)), the bond orders ben and bco are further shifted (to 1.384 and 1.655, respectively), and the bond lengths undergo further shifts in the... 

While, in the BCS theory, such attractive force for electron Cooper pair is provided by phonons, for dense quark matter, where phonons are absent, the gluon exchange interaction provides the attraction, as one-gluon exchange interaction is attractive in the color anti-triplet channel1 One therefore expects that color anti-triplet Cooper pairs will form and quark matter is color superconducting, which is indeed shown more than 20 years ago [13, 14],... 

It is quite likely to find dense quark matter inside compact stars like neutron stars. However, when we study the quark matter in compact stars, we need to take into account not only the charge and color neutrality of compact stars and but also the mass of the strange quark, which is not negligible at the intermediate density. By the neutrality condition and the strange quark mass, the quarks with different quantum numbers in general have different chemical potentials and different Fermi momenta. When the difference in the chemical potential becomes too large the Cooper-pairs breaks or other exotic phases like kaon condensation or crystalline phase is more preferred to the BCS phase. 

The Fermi sea of up and strange quarks is shown in Fig. 9. Because of the strange quarks mass, they have different Fermi momenta. Note that the Cooperpairing occurs for quarks with same but opposite momenta. Therefore, at least one of the pairing quarks should be excited away from the Fermi surface, costing some energy. Let us suppose that the Cooper-pair gap opens at p p between two Fermi surfaces, psF < p < pf. 

Consider again non-relativistic fermions. Their BCS spectrum (for homogeneous systems) is isotropic when the polarizing field drives apart the Fermi surfaces of spin-up and down fermions the phase space overlap is lost, the pair correlations are suppressed, and eventually disappear at the Chandrasekhar-Clogston limit. The LOFF phase allows for a finite center-of-mass momentum of Cooper pairs Q and the quasiparticle spectrum is of the form... 

To motivate our next step recall that the LOFF spectrum can be view as a dipole [oc Pi (a )] perturbation of the spherically symmetrical BCS spectrum, where Pi(x) are the Legendre polynomials, and x is the cosine of the angle between the particle momentum and the total momentum of the Cooper pair. The l = 1 term in the expansion about the spherically symmetric form of Fermi surface corresponds to a translation of the whole system, therefore it preserves the spherical shapes of the Fermi surfaces. We now relax the assumption that the Fermi surfaces are spherical and describe their deformations by expanding the spectrum in spherical harmonics [17, 18]... 

In practice, the deformation parameters e/ (l > 2) are determined from the minimization of the free-energy of the system in full analogy to the total momentum Q of a Cooper pair. And they can be determined in a volume conserv-... 

For large enough asymmetries the homogeneous state becomes unstable towards formation of either the LOFF phase - a superconducting state with nonzero center-of-mass momentum of the Cooper pairs, or the DFS phase - a superconducting state which requires a quadrapole deformation of Fermi surfaces. A combined treatment of these phases in non-relativistic systems shows that while the LOFF phase corresponds to a local minimum, the DFS phase has energy lower that the LOFF phase. These phases break either the rotational, the translational or both symmetries. 

In a superconducting system, when one increases the temperature at a given chemical potential, thermal motion will eventually break up the quark Cooper pairs. In the weakly interacting Bardeen-Copper-Schrieffer (BCS) theory, the transition between the superconducting and normal phases is usually of second order. The ratio of the critical temperature TcBCS to the zero temperature value of the gap AbGS is a universal value [18]... 

Like there always exists a vapor under the water, there are excitations on the ground of any condensate. They appear due to quantum and thermal fluctuations. In classical systems and also at not too small temperatures in quantum systems, quantum fluctuations are suppressed compared to thermal fluctuations. Excitations are produced and dissolved with the time passage, although the mean number of them is fixed at given temperature. Pairing fluctuations are associated with formation and breaking of excitations of a particular type, Cooper pairs out of the condensate. Fluctuation theory of phase transitions is a well developed field. In particular, ten thousands of papers in condensed matter physics are devoted to the study of pairing fluctuations. At this instant we refer to an excellent review of Larkin and Varlamov [15]. 

For nonvanishing A in Eq. (1) the color symmetry is broken. Two of the three quark color degrees of freedom are coupled to bosonic Cooper pairs in the color antitriplet state which can form a Bose condensate.One can combine the chemical potentials Hu, Hd of u and d quarks by introducing Hq = (hu + Hd)/2 and hi = (hu — Hd)/% as the Lagrange multipliers related to, respectively, the quark number density nq and the isospin asymmetry n/. In thermal equilib-... 

From BCS theory it is known, that in order to form Cooper pairs at T = 0 in a dense Fermi system, the difference in the chemical potentials of the Fermions to be paired should not exceed the size of the gap. As previous calculations within this type of models have shown [24], there is a critical chemical potential for the occurrence of quark matter pf > 300 MeV and values of the gap in the region A < 150 MeV have been found. Therefore it is natural to consider the problem of the color superconducting (2SC) phase with the assumption, that quark matter is symmetric or very close to being symmetric (pu pd). 

We make the observation that, contrary to naive expectations of the mass defect as a result of the binding energy from Cooper pairing [16], AM c2... 

As the positive metal ions are heavier than the electron that caused the displacement, they return to their original positions fairly slowly. The relatively long time taken to re-establish the original positions allows the displaced metal ions to attract a second electron, which, given the proximity of the just passed first electron, leads to the formation of a Cooper pair. 

The opening of a band gap at in the superconducting state can also be interpreted within this framework. The charge carriers of a superconducting state are not individual electrons as in the normal metallic state but electrons coupled in pairs, the so-called Cooper pairs, having opposite wave vectors. Hence, Cooper pairs are described by product functions (p k)(p —k)). Following the discussion given above. 

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