Understanding the field normalization int egrals

Hello,

I want to be sure that I understand how to read the fields of different modes and manipulate them (if needed), after computing bands in mpb. As a test I looked at field normalization. As I understand it, the integrals over the fields should be as written in

http://ab-initio.mit.edu/wiki/index.php/MPB_User_Reference#Field_normalization

I want to understand how to relate to that when computing a 2D structure with triangular lattice. Before reading the data, I used the mpb-data command to align the axis to cartesian coordinates. Then I used the following algorithm to compute the integral values of H:

1. extract the field components - Hx,Hy,Hz - I got a set of 10 by 11 grid points, a total of N=110 grid points

2. evaluate |H|^2 for each grid point

3. compute the volume element as dV = dx*dy / N, where dx=1 and dy=sqrt(3)/2 due to the triangular skew

3. sum over all points of |H|^2 and multiply dV

When I ran this I received a total of 0.9875

I repeated the steps for eps*|E|^2 and received 0.9442

so, please help me understand:

1. Is this the way to validate the normalization of fields?

2. Are the above values differ from a solid 1 due to the poor discretization of my grid? or am I always supposed to "compute the integrals" over the grid points and receive 1?

3. Is the sqrt(3)/2 factor a valid correction for dy? Would I not be needing it had I not used the mpb-data command to align the axis?

4. Are the integrate-fields commands in mpb work in the same manner?

thanks in advance,

Amnon

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