**Author details**

14 Will-be-set-by-IN-TECH

*<sup>m</sup>*

*p* ∑ *l*=0

<sup>1</sup> <sup>−</sup> *<sup>π</sup>*2(*<sup>m</sup>* <sup>+</sup> <sup>1</sup>) 6*m*

region. In addition, *u*¯ represents the mean of the width distribution function *P*(*u*) involved in the transport of Cooper pairs through the sample. Eq. (39) reproduces the quasi-linear

Typical *Jc* measurements are performed using the four-probe technique with automatic control of the sample temperature, the applied magnetic field and the bias current [14]. Details of the technique and the experimental setup are in Ref. [14] and of the synthesis and sample

Figure (6) shows the experimental results of [38] for the critical current as a function of the applied field, together with the theoretical expression derived above for the critical field *H*∗

the figure and for *m* = 2 and 3. A very close fit to the experimental data is evident for *m* = 2

Theoretical models of the magnetic field dependence of the transport critical current density for a polycrystalline ceramic superconductor have been studied at last years [37, 39–41]. Here we have described a tunneling critical current between grains follows a Fraunhofer diffraction pattern or a modified pattern . It is important to emphasize that we followed the same approach as in [21] and extended the analytical results to all applied magnetic fields. A characteristic field (*H*∗) was identified and different regimes were considered, leading to analytical expressions for *Jc*(*H*): (i) analysis for low applied magnetic fields (*α* � 1) revealed quasi-linear behavior for *Jc* (*H*) vs. *H*∗; (ii) for high applied magnetic fields (*α* � 1), *Jc* (*H*) is

*fm*−2*H* + *R*(*H*)

1 (*α*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*2)(*m*−*p*−2)/2 <sup>×</sup>

> *Kp*−*<sup>l</sup>* (*<sup>p</sup>* − *<sup>l</sup>*)! *<sup>α</sup>l*+<sup>1</sup>

> > *H H*∗ *o*

*<sup>u</sup>*¯2 is a characteristic field that determines the behavior of *Jc* (*H*) in this

 *e* . (37)

<sup>−</sup>*kα*, (38)

, (39)

<sup>0</sup> in

 *<sup>H</sup>*<sup>∗</sup> 0 *H*

> *<sup>m</sup>*−<sup>2</sup> ∑ *p*=0

Now Eq. (16) can be expressed as [37]:

378 Superconductors – Materials, Properties and Applications

where *H*∗

*<sup>o</sup>* <sup>=</sup> *<sup>φ</sup>*<sup>0</sup> �*ηm*�

<sup>2</sup> = *<sup>φ</sup>*<sup>0</sup>

**5. Critical current measurement**

characterization were published elsewhere [13].

for many different applied magnetic fields.

proportional to *H*−0.5, as reported in [21].

behavior that was also reported by Gonzalez *et al.* [21].

*Jc*(*H*) = *Jc*0(1)*<sup>m</sup>*

*EK*+1(*α*) <sup>≤</sup> *<sup>e</sup>*(*<sup>m</sup>* <sup>−</sup> <sup>2</sup>)!

The series error estimated in *R*(*α*) [Eq. (34)] is [37]:

(*m* − 1)!

*π*

where *EK*+<sup>1</sup> is defined as the *K* + 1-order error of the series in Eq. (34).

*Jc* (*H*) ≈ *Jc* (0)

A low applied magnetic field implies that *α* � 1, and Eq. (37) is transformed to:

C. A. C. Passos, M. S. Bolzan, M. T. D. Orlando, H. Belich Jr, J. L. Passamai Jr. and J. A. Ferreira *Physics Department, University Federal of Espirito Santo - Brazil* 

E. V. L. de Mello, *Physics Department, University Federal Fluminense - Brazil* 

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**Path-Integral Description of Cooper Pairing**

**Chapter 16**

Before we start applying path integration to treat Cooper pairing and superfluidity, it is a good idea to quickly review the concepts behind path integration. There are many textbooks providing plentiful details, such as Feynman's seminal text [1] and Kleinert's comprehensive compendium [2], and other works listed in the bibliography [3–5]. We will assume that the reader is already familiar with the basics of path-integral theory, so if the following paragraphs

Quantum mechanics, according to the path-integral formalism, rests on two axioms. The first axiom, the superposition axiom, states that the amplitude of any process is a weighed sum of the amplitudes of all possible possibilities for the process to occur. These "possible possibilities" should be interpreted as the alternatives that cannot be distinguished by the experimental setup under consideration. For example, the amplitude for a particle to go from a starting point "A" to a final point "B" is a weighed sum of the amplitudes of all the paths that this particle can take to get to "B" from "A". The second axiom assigns to the weight the complex value exp{*i*S/¯*h*} where S is the action functional. In our example, each path *x*(*t*) that the particle can take to go from A to B gets a weight exp {*i*S[*x*(*t*)]/¯*h*} since the action is the time integral of the Lagrangian. There is a natural link with quantum-statistical mechanics: in the path-integral formalism, quantum statistical averages are expressed as the same weighed averages but now the weight is a real value exp{−S[*x*(*τ*)]/¯*h*} and the path is

In the example of the above paragraph, we considered a particle which could take many different paths from A to B. However, the same axioms can be applied to fields. As an example we take a complex scalar field *φ***x**,*t*, where **x** and *t* denote position and time respectively. Let us mentally discretize space-time, and to make things easy, we assume there are only five moments in time and five places to sit. In this simple universe, the field *φ***x**,*<sup>t</sup>* is represented by a set of 25 complex numbers, i.e. an element of **C**<sup>25</sup> if **C** is the set of complex numbers. Summing over all possible realizations of the fields corresponds to integrating over **C**25, a 25-fold integral over complex variables, or a 50-fold integral over real variables. Writing

> ©2012 Tempere and Devreese, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

©2012 Tempere and Devreese, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

are not merely reminders to you, it is probably better to first consult these textbooks.

Jacques Tempere and Jeroen P.A. Devreese

http://dx.doi.org/10.5772/48458

taken in imaginary time *τ* = *it*.

work is properly cited.

**1. Introduction**

Additional information is available at the end of the chapter
