Coherence lengths

The coherence length is an important parameter that characterizes the behavior of superconductors. This is the length scale over which the paired carriers interact, or more specifically the size of the Cooper pair. One can make a rough estimate of this length from the uncertainty principle. If the binding energy of the Cooper pair is AE, we know the energy to this precision. Since 

The imcertainty principle states that AxAp = h = AxHAk. Therefore 

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This is attributed to the different nature of the bonding of sulphur to silver as compared to gold and the slightly different packing density. The coherence length detennined with He atom diffraction was found to be 12 mn [162]. 

Neutron reflectivity measures the variation in concentration normal to the surface of the specimen. This concentration at any depth is averaged over the coherence length of the neutrons (on the order of 1 pm) parallel to the sur ce. Consequendy, no information can be obtained on concentration variadons parallel to the sample surface when measuring reflectivity under specular conditions. More imponantly, however, this mandates that the specimens be as smooth as possible to avoid smearing the concentration profiles. 

X-ray diffraction peaks were rather broad with coherence lengths as low as 20 nm and this was attributed to rapid quenching. It was proposed that the carbon atoms are arranged in polyyne chains (n = 4) along the c-axis. The density of Carbolite (1.46 g-cm ) is lower than values for other carbynes and for diamond and graphite - hence the name - and this was attributed to a rapid quenching process. 

Static defects scatter elastically the charge carriers. Electrons do not loose memory of the phase contained in their wave function and thus propagate through the sample in a coherent way. By contrast, electron-phonon or electron-electron collisions are inelastic and generally destroy the phase coherence. The resulting inelastic mean free path, Li , which is the distance that an electron travels between two inelastic collisions, is generally equal to the phase coherence length, the distance that an electron travels before its initial phase is destroyed ... 

At low temperatures, in a sample of very small dimensions, it may happen that the phase-coherence length in Eq.(3) becomes larger than the dimensions of the sample. In a perfect crystal, the electrons will propagate ballistically from one end of the sample and we are in a ballistic regime where the laws of conductivity discussed above no more apply. The propagation of an electron is then directly related to the quantum probability of transmission across the global potential of the sample. 

As for the coherent length in CNTs, a very interesting paper has been published from the group at the Georgia Institute of Technology about the conductance of individual MWCNTs [34], They have observed the quantisation of conductance by changing the distance between the two electrodes. This result indicates ballistic conduction in a CNT, which suggests the formation of stationary waves of electrons inside CNTs. 

Adaptive optics requires a reference source to measure the phase error distribution over the whole telescope pupil, in order to properly control DMs. The sampling of phase measurements depends on the coherence length tq of the wavefront and of its coherence time tq. Both vary with the wavelength A as A / (see Ch. 1). Of course the residual error in the correction of the incoming wavefront depends on the signal to noise ratio of the phase measurements, and in particular of the photon noise, i.e. of the flux from the reference. This residual error in the phase results in the Strehl ratio following S = exp —a ). 

Figure 4. Propiigiilicii IVom llic lasdr spol ai linile disiance and I rom the astmphysica] source ai iuliniiy. Beams (mm the IXiS and from (lie source to the edge of the pupil cross a uii luilem layer at distance each other larger than the coherence length (Court, M, Tallon).

In the hrst case, the degree of self coherence depends on the spectral characteristics of the source. The coherence time Tc represents the time scale over which a held remains correlated this hme is inversely proportional to the spectral bandwidth Au) of the detected light. A more quantitative dehnition of quasi-monochromatic conditions is based on the coherence time all relevant delays within the interferometer should be much shorter than the coherence length CTc. A practical way to measure temporal coherence is to use a Michel-son interferometer. As we shall see, in the second case the spatial coherence depends on the apparent extent of a source. 

The nematic phase of all the compounds CBn is characterized by a coherence length of about 1.4 times the elongated structure of the molecule. Based on this behaviour local associations in form of dimers with cyano-phenyl interactions were postulated. For the smectic A phase a partial bilayer arrangement of the molecules (SAd) is most likely. But there are also example for the smectic A phase with a monolayer (Sai) or a bilayer (Sa2) arrangement of the molecules as well as a commensurate structure A large number of X-ray measurements were carried out in the liquid crystalline state to clear up the structural richness and variability (see Chap. 2, this Vol. [52]). 

The transversal coherence length Tc trms is, however, limited. It is given by the expression ... 

Inwa (x/c)2(lw)nsin2(irz/2/c), where lQ is the coherence length (distance for accrual of a it phase mismatch). 

Figure 2. Depiction of the length dependence of second harmonic intensity. The lower curve shows the effect of decreasing the coherence length to 30% of the upper curve.

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