4 Then the expression of J ( q ) is given by J ( q ) = J$_{1}$ exp[ i q · c + a (1) + a (2) 2 ] + J$_{1}$ exp[ i q · − c + a (1) + a (2) 2 ] + J$_{1}$ exp[ i q · c + a (2) + a (3) 2 ] + J$_{1}$ exp[ i q · − c + a (2) + a (3) 2 ] + J$_{1}$ exp[ i q · c + a (3) + a (4) 2 ] + J$_{1}$ exp[ i q · − c + a (3) + a (4) 2 ] + J$_{1}$ exp[ i q · c + a (4) + a (1) 2 ] + J$_{1}$ exp[ i q · − c + a (4) + a (1) 2 ] + J$_{2}$ exp[ i q · c ] + J$_{2}$ exp[ − i q · c ] (11) + J$_{3}$ exp[ i q · a (1)] + J$_{3}$ exp[ − i q · a (1)] + J$_{3}$ exp[ i q · a (2)] + J$_{3}$ exp[ − i q · a (2)] . Then J ( q ) can be rewritten as 2. Finding maximum 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$_{3}$ . where with h , k , and l being dimensionless numbers and a ∗ and c ∗ being reciprocal lattice constants. 3. ListVectorPlot3D program 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 . 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. 1. Contour plot program 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$_{3}$ . 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), 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 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 B. Mathematica programs used in the present work q$_{x}$ = ( 2 π a ) h = a $^{∗}$h, q$_{y}$ = ( 2 π a ) k = a $^{∗}$k, q$_{z}$ = ( 2 π c ) l = c $^{∗}$l J ( q ) = J ( h, k, l ) = 8 J$_{1}$ cos( πh ) cos( πk ) cos( πl ) + 2 J$_{2}$ cos(2 πl ) + 2 J$_{3}$ [cos(2 πh ) + cos(2 πk )] , (12) J ( h, k, l ) = 8 J$_{1}$ ( − 1) $^{l}$cos( πh ) cos( πk ) + 2 J$_{2}$ + 2 J$_{3}$ [cos(2 πh ) + cos(2 πk )] ,