content_image/392383.jpg content_image/392400.jpg content_image/392399.jpg content_image/392401.jpg
Then the expression of J ( q ) is given by 

<img src='content_image/392383.jpg'>

Then J ( q ) can be rewritten as 

<img src='content_image/392400.jpg'>

where 

<img src='content_image/392399.jpg'>

with h , k , and l being dimensionless numbers and a ∗ and c ∗ being reciprocal lattice constants. 

## B. Mathematica programs used in the present work 

In order to determine the magnetic Bragg peaks in the reciprocal lattice plane, we use the following three Mathematica programs, ContourPlot for the equi-energy contour plot, FindMaximum for finding maximum, and ListVectorPlot3D for drawing the spin directions in the lattice points of the real space. 

This program is used to determine the overview on the positions of the magnetic Bragg peaks in the ( h, k, l ) reciprocal lattice plane, where one of h , k , and l are fixed. When the values of J$_{1}$ , J$_{2}$ , and J$_{3}$ are given, using the Mathematica program [ContourPlot], we can make a plot of the contours of J ( h, k, l ) = a (constant) in the ( h, k, l ) plane, where the parameter a is changed appropriately such that the maximum value of J ( h, k, l ) is obtained. This program is very useful to find the selection rule for the position of the magnetic Bragg peaks, as a function of J$_{1}$ , J$_{2}$ , and J$_{3}$ . 

1. Contour plot program 

This program is used to determine the exact position of the magnetic Bragg peak in the ( h, k, l ) reciprocal lattice plane, where one of h , k , and l are fixed. For convenience, here we assume that l is fixed such that l = 0. Using the Mathematica program [FindMaximum], we find the max- imum value of J ( h, k, l ) for given J$_{2}$ and J$_{3}$ , where J$_{1}$ (= -1) is fixed, and the regions of h and k are appropriately chosen. This program is very convenient when one wants to know how the position of the Bragg point changes as a function of J$_{2}$ and J$_{3}$ . 

This program is used to draw the spin directions at each lattice sites in the real space. The spin structure depends on the values of Q . 

## IV. SELECTION RULE FOR THE MAGNETIC BRAGG PEAKS FROM SIMPLE ANALYSIS 

## A. Distribution of magnetic Bragg peaks in the ( h , k , l = fixed integer) reciprocal lattice plane 

The magnetic Bragg peaks can be located at the wavevector q , where J ( h, k, l ) takes a maximum. When l = fixed integer, J ( h, k, l ) is given by 

<img src='content_image/392401.jpg'>

suggesting that the location of the magnetic Bragg peaks in the ( h , k , l = fixed integer) reciprocal lattice plane depends only on J$_{1}$ and J$_{3}$ ,and is independent of the J$_{2}$ . The selection rule can be derived as follows. (i) The distribution of the magnetic Bragg peaks is the same in the reciprocal lattice planes, ( h , k , 1), ( h , k , 3), 

2. Finding maximum program 

3. ListVectorPlot3D program 

