content_image/338106.jpg content_image/338105.jpg content_image/338102.jpg content_image/338103.jpg
Let us chose the surface at which J = 0 and transform an integration over r$_{3}$ in a way d $^{3}$r$_{3}$ → d $^{2}$R dR$_{⊥}$ where R is the coordinate on the surface in question while R$_{⊥}$ is a coordinate along the normal to the surface where we assume that R$_{⊥}$ = 0 corresponds to J = 0. In the lowest order in R$_{⊥}$ we have J = ( dJ/dR$_{⊥}$ ) R$_{⊥}$ . As it is seen, the integration of the logarithm term over R$_{⊥}$ gives 

Finally, the integration of the factor 

ϕJ$_{1}$ dJ/dR$_{1}$ 

J$_{1}$ dJ/dR$_{⊥}$ 

over d $^{2}$R approximately gives a volume accessible for the site 3. Note that we have r$_{13}$ + r$_{23}$ ≤ r$_{12}$ + a , thus the projection of r 23 to the plane normal to r 12 should be less than ( r$_{23}$a ) 1 / $^{2}$. 

As a result, the integration over the spatial coordinate r 3 gives 

<img src='content_image/338106.jpg'>

In its turn, the area of the interference loop (entering the estimate of ϕ ) is 

<img src='content_image/338105.jpg'>

Note that these estimates actually hold for all accessible values of r$_{23}$ up to r$_{23}$ ∼ r$_{12}$/ 2. 

The final result depends on the behavior of g ( ε ). For g = const (Mott-type hopping) the integration over E$_{3}$ is naturally controlled by the lower limit which accordingly to Eq. 15 corresponds to the larger possible value of r$_{23}$ ∼ r$_{12}$/ 2. In this case the r.h.s. of Eq.13 is ∝ r 5 / 2 h where r$_{h}$ ∼ r$_{12}$ . 

In contrast, for the Coulomb gap hopping the integration over E$_{3}$ is controlled by the upper level, E$_{C}$ , corresponding to the edge of the Coulomb gap. In this case r.h.s. of Eq.13 is ∝ r$_{h}$ since the value of r$_{23}$ does not depend on r$_{h}$ . 

Now let us consider a combination of NMR with a positive magnetoresistance related to wave function shrinkage which can be estimated as 

where 

<img src='content_image/338102.jpg'>

<img src='content_image/338103.jpg'>

